CHAPTER 3: BASIC FACTORS OF BIODYNAMICS AND JOINT STABILITY
CHAPTER 3:
BASIC FACTORS OF BIODYNAMICS AND JOINT STABILITY



This is Chapter 3 from R. C. Schafer, DC, PhD, FICC's best-selling book:
“Clinical Biomechanics: Musculoskeletal Actions and Reactions”

Second Edition ~ Wiliams & Wilkins


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Structural Motion

Kinematics

Summary of Major Static and Kinetic Factors Influencing Mechanical Efficiency of Muscular Effort

Biomechanical Stress

Biomechanical Aspects of Articular Cartilage 

Chapter 3: Basic Factors of Biodynamics and Joint Stability

The techniques used for analyzing static positions of the body are only approximate inasmuch as forces accompanying movement incorporate such dynamic factors as acceleration, momentum, friction, the changing positions of rotational axes, and the resistance and support offered by tissues other than muscles. This chapter discusses the basic concepts and terms of biodynamics, biomechanical stress, and the biomechanical aspects of articular cartilage pertinent to the clinical setting.


     Structural Motion

The study of dynamics is concerned with loads and the motions of bodies (kinematics) and the action of forces in producing or changing their motion (kinetics). Kinematics lets us describe the characteristics of motion position, acceleration, and velocity such as in gait or scoliotic displacements. Here we are concerned with the position of the center of mass of the body and its segments, the segmental range of motion, and the velocity and direction of their movements. In kinetics, we become concerned with the forces that cause or restrict motion such as muscle contraction, gravity, and friction. A complete biomechanical analysis of human motion or motion of a part would include both kinematic and kinetic data.

Motion can be defined as an object's relative change of place or position in space within a time frame and with respect to some other object in space. Thus, motion may be determined and illustrated by knowing and showing its position before and after an interval of time. While linear motion is readily demonstrated in the body as a whole as it moves in a straight line, most joint motions are combinations of translatory and angular movements that are more often than not diagonal rather than parallel to the cardinal planes. In addition to muscle force, joint motion is governed by factors of movement freedom, axes of movement, and range of motion.


Degrees of Freedom

      JOINT AXES

As previously discussed, the body is composed of numerous uniaxial, biaxial, and multiaxial joints. Joints with one axis have one degree of freedom to move in one plane such as pivot and hinge joints, joints with two axes have two degrees of freedom to move in two different planes, and joints with three axes have three degrees of freedom to move in all three planes, eg, the ball-and-socket joints. Thus, that motion in which an object may translate to and fro along a straight course or rotate one way or another about a particular axis equals one degree of freedom.

In Chapter 1, joint classification was given under the major divisions of synarthrodial, amphiarthrodial, and diarthrodial joints. This is the classic anatomic classification. However, from a purely biomechanical viewpoint, joint motion can be reduced to just two types: (1) ovoid, which permits motion in one plane, X; and (2) sellar, which permits motion in two planes, Y and Z (Fig. 3.1).


      JOINT FREEDOM

To know the actual degrees of freedom available to a part of the body, one must sum the degrees available of adjacent joints to appreciate the amount of free motion of one part relative to another part. The degrees of freedom of a fingertip relative to the trunk, for example, are the sum of the degrees of freedom of all the joints from the distal phalanges to the shoulder girdle. While the distal phalanges have only one degree of freedom, the entire extremity has 17 degrees in total. This summation process is an example of a living, open kinematic chain.


      COMBINED MOVEMENTS

Simple translatory motions of a body part involve movements of more than one joint. This requires reciprocating actions of three or more segments at two or more joints if parallel lines are to be followed. For example, a fingertip cannot be made to follow the straight edge of a ruler placed in front if the wrist and elbow joints are locked. If the wrist and elbow are fixed, the fingertip must follow an arc and not a straight line. Thus, human motion can be described as translatory motion that has major contributions from linear, angular, and curvilinear motions. The term general or three-dimensional motion implies that an object may move in any direction by combining multidirectional translation and multiaxial rotation.


Plane and Out-of-Plane Motion

      PLANE MOTION

Any motion in which all coordinates of a rigid body move parallel to a fixed point is referred to as plane motion. Such motion has three degrees of freedom; eg, sliding A-P, sliding lateral, and spinning. In other words, plane motion has two translatory degrees of motion along two mutually perpendicular axes and one rotatory degree of motion around an axis perpendicular to the translatory axes. For instance, if a person curves his trunk forward, the thoracic vertebrae flex and rotate in a single plane about an axis that is perpendicular to the sagittal plane (Fig. 3.2). In such plane motion, various points on a particular vertebra move in parallel planes.


      THE INSTANTANEOUS AXIS OF ROTATION

Plane motion is described by the position of its instantaneous axis of rotation and the motion's rotatory magnitude about this axis. In the above example of cervical flexion, for instance, as a vertebra moves in a plane, there is a point at every instant of motion somewhere within or without the body that does not move. If a line is drawn from that point so it perpendicularly meets the line of motion, the point of intersection is called the instantaneous axis of rotation for that motion at that particular point in time (Fig. 3.3). Most joint movement is to a great degree rotatory motion, but the axis of motion may change its location and/or its orientation during a complete range of motion.


      OUT-OF-PLANE MOTION

As contrasted to plane motion, out-of-plane motion is a version of general body motion with three degrees of freedom consisting of two rotations about mutually perpendicular axes and a translation perpendicular to the plane formed by the axes. Thus, in out-of-plane motion, the body moves more than in a single plane. For example, if a person bends laterally, a midthoracic vertebral body translates from the sagittal plane towards the horizontal plane (Fig. 3.4). This is not plane motion because various points on the vertebra do not move in parallel planes.


Range of Motion

A freely moving body such as a vertebra has six degrees of freedom as it moves in three-dimensional space: eg, translations along and rotations about each of the three cardinal axes. The range of motion (ROM) of a joint is the quantities that indicate two points at the extreme range of physiologic rotation and translation for each of the joint's six degrees of freedom. Thus, the range of motion is the difference between the extremes of physiologic motion. It is measured in linear units for translation and in degrees for rotation.

An articulation may have several degrees of freedom and a most limited range of motion, and a joint with one degree of freedom may have a large range of motion. Keep in mind that degrees of freedom refer to the ability to move in planes (ie, the number of axes), while range of movement is dependent upon soft-tissue restraints, the number of joint axes, the joint architecture, and the size and position of adjacent tissue which may affect motion of a part (eg, biceps hypertrophy restricting forearm flexion). The knee joint has one degree of freedom and a relatively large range of motion; L5 has six degrees of freedom and a most restricted range of motion. In a joint having one or more degrees of motion, a range of motion can be expressed for each degree.


Kinematics

Kinematics, without considering the forces involved, is the study of the motion of objects, with emphasis on displacement, acceleration, and velocity. Many of the concepts previously discussed such as coordinate systems, translation, rotation, coupling, degrees of freedom, range of motion, and axes of motion are major concerns within kinematics. The importance of this sometimes complex study is underscored in the analysis and management of structural problems of the body, in understanding trauma and clinical stability, in coping with spinal imbalance and other postural problems, and in evaluating orthopedic x-ray films.

It might be helpful here to review some major terms of kinematics. Motion infers a continuous change (displacement) of position. Translation refers to movement where various points in an object at a specific interval of time have the same direction of displacement relative to some fixed point. Rotation (spinning or angular motion) refers to movement where various points in a straight line in an object (or a hypothetical extension of such a line) at a specific interval of time have a velocity of zero relative to some fixed point. Rotation involves angular displacement of an object about some axis that is located within or exterior to the revolving object and where the points of the object describe concentric circles around an axis. Coupling is motion in which translation or rotation of an object along or about an axis is constantly related to simultaneous translation along or rotation about another axis.


Acceleration

The application of Newton's second law is an important facet of kinematic studies. That is, a net force acting on the body gives it an acceleration that is proportional to the force in both direction and magnitude and inversely proportional to the body's mass.


      LINEAR ACCELERATION

During human movement, the importance of body weight as a variable is explained in Newton's second law. Acceleration is the rate of change in velocity of an object. The greater a person's mass, the greater his or her equilibrium as measured by the quantity of force necessary to accelerate it positively or negatively. Thus, a heavy body has the advantage in maintaining a state of equilibrium, but it has a disadvantage in producing acceleration. This is exhibited in the weight need of a wrestler vs that of a sprinter. Also, the analysis and treatment of gait disorders and the application of braces and supports, for example, depend on the mechanical parameters of acceleration, displacement, and velocity.


      ANGULAR ACCELERATION

Angular acceleration is produced by a force when its line of action does not pass through the center of rotation. The further the distance is between the line of action and the center of rotation, the more effective is the force in producing angular motion (moment, torque). The same factors involved in kinematic and kinetic translational relationships apply to rotational motion if we substitute moment of inertia for mass, moment of force for force, angular change for displacement, and moment for force.

Angular acceleration is the rate of change of angular velocity. Neither a body in linear motion or a rotating body needs to have a uniform angular velocity. If a body's angular velocity changes to a new value in a time frame, the angular velocity is changed (Fig. 3.5). The unit of measurement of angular acceleration's magnitude is in degrees or radians per second per second. Mathematically, angular acceleration equals the new angular velocity minus the original angular velocity times the time involved.

In rotational movements, moment = mass moment of inertia X angular velocity. This physical property reflects how mass is distributed about an axis, and it is concerned with differences in motion by various rotating body segments. The moment of inertia of a body will differ about its various axes because the motion of any one segment for a specific angular rotation is determined by the distance of that part from the center of rotation. For example, when an automobile is struck from the rear, the trunk of a person in the car is accelerated forward in relation to the head. The head responds slower than the trunk because of its inertia, and it is angularly accelerated backward. Thus, the mechanism of cervical whiplash injury is determined by the angular acceleration of the head and its inertia.


Velocity

Kinematic analyses invariably incorporate the concepts of linear velocity, angular velocity, and instantaneous velocity. When velocity increases, acceleration is said to be positive; when velocity decreases, acceleration is negative.


      LINEAR VELOCITY

Linear velocity is an object's rate of linear position change. The term speed refers to velocity's magnitude. If acceleration is constant and an object starts moving from rest, the final velocity that will be required will be directly proportional to the duration of the event. As a position change is measured by changes in distance or length, linear velocity = distance/time. If a muscle shortens 3 inches in half a second during contraction, the contraction's linear velocity is 6 inches/second.


      ANGULAR VELOCITY

Angular velocity occurs when the motion is rotation. A thrown ball always has a linear velocity, and it may have an angular velocity if it spins. A spinning top always has an angular velocity, and it may have a linear velocity if it simultaneously moves across a surface.

When a bony lever is moved by a muscle, the length of the lever can be considered the radius of a circle. The difference between a starting position and a stopping position can be measured by the angle formed. Thus, angular velocity equals the angle turned through per unit of time. A radian is a unit of angular measurement which is commonly used to describe angular velocity that is defined as the ratio of arc length to the length of the radius.

All but a few body movements are rotations of bones about their joints, and these rotations are seldom confined to one simple arc. Rather, they tend to vary irregularly to compensate for the restrictions of other joints and for the transfer of power from one set of muscles to another. During such irregular rotations, the bones also twist about their own axes. For example, during walking, the legs not only move forward and backward, they also rotate laterally during the forward swing, during contact of the foot with the ground, and again during the recovery phase of the stride.

In addition, all human movement is governed by the positive and negative acceleration of joint actions. During locomotion, for instance, the thigh changes its linear and angular positions and its velocities during gait. Linear acceleration during straight-line locomotion is partially produced by the increased angular velocity (angular acceleration) of the lower extremities. In all joint movements, the principle that linear acceleration equals the product of the radius and the angular acceleration applies.


      INSTANTANEOUS VELOCITY

Both linear and angular velocity are vector quantities because they have magnitude and direction. The term instantaneous velocity is the average velocity when the time interval approaches zero. It is measured in linear units per second. An object moving toward a goal may not have consistent speed and direction; eg, an automobile driving up a winding and dipping road to reach the summit of a hill. At any instant in time, both acceleration and direction may be above or below the average for the journey. Thus, an accurate description of velocity may necessitate the full plotting of its instantaneous velocity vectors.


Kinematic Analyses of Human Motion

Frame-by-frame motion pictures of a body or cineroentgenology of a region in motion can be analyzed to gain a great appreciation of human movement. The problem is often to determine the levers involved in each joint action and compute the types of external load and internal stress involved. This analysis should incorporate at least the following data:

1.   The name of the movement under study, the time frame (frame number) at which the movement is initiated and concludes, and the joint or joints at which movement occurs.

2.   The segment or segments (lever) making up the kinematic chain being moved that are the effect of joint action should be described.

3.   The force or forces causing the movement and the point of force application should be described. These forces would include such factors as gravity, isotonic or concentric muscle shortening, or some other force contributing to the production of movement.

4.   The force or forces offering resistance to the movement and the point of force application should be described. These forces would include such factors as gravity, eccentric muscle lengthening while exerting tension, or some other force contributing to movement resistance.

5.   A description of the stabilized and/or relaxed joints in the lever and the forces which stabilize these articulations.

6.   A description of the adjacent stabilized and/or relaxed joints outside the lever and the forces which stabilize these articulations.

When more sophisticated data are desired, body markers (often illuminated), specialized photographic equipment, timing devices, and electromyographic and electrogoniometric appliances can be used to:

1.   Plot the trajectory (X vs Y axes) for each body marker, and Y coordinate for each marker in relation to time.

2.   Plot velocity and/or acceleration of each marker relative to time.

3.   Plot joint angles, angular velocity, and/or angular acceleration relative to time or instant of movement cycle.

4.   Plot average velocity in terms of direction, movement length, and range of movement.

5.   Plot the angle or one joint against another at various intervals of time.


Body Links

During human motion, several moving segments are involved in each study where one segment moves on one adjacent which moves on another. This is similar to engineering links that involve overlapping segments held together by pins (joints) which serve as axes of rotation. Such overlapping does not occur in the human body except in a few places such as at the ankle and the C1-odontoid articulation.

A link is a straight line (eg, a rod) of constant length that extends from axis to axis (Fig. 3.6). Such a system of links can serve as a geometric model to analyze motion. If power is to be transmitted, the links of a machine must form a closed system where each link has a particular relation to every other link in the system. The closed system guarantees that forces are transmitted in a positive and predictable manner.


      OPEN BODY SYSTEMS

The body can be thought of as a system of body links which form one or more kinematic chains. Most skeletal links are open chains rather than closed systems. This is because the distal ends of the extremities are free (unconnected to another link).

The dimensions of human links are determined by the length of bone from joint axis to joint axis. The joint axes are used because few articulations have overlapping bones. In kinematic analysis of human motion, the rotational axes are not located at the bony junctures. For example, the axis of the shoulder is within the humeral head; that of the hip, within the femoral head. In the knee and elbow, the axes are proximal to the articulating surfaces.

As mentioned earlier in the discussion of degrees of freedom, a proximal link in a segmental chain cannot be moved without causing displacement in one or more adjacent links. Thus, the "lever" in gross body movements is the kinematic chain involved, which is made up of a number of linked levers. Hip flexion and extension, for example, is not determined solely by the skeletal lever of the femur. When a person takes a step, the pelvis, hip, thigh, knee, leg, ankle, foot, and toes are involved.

In addition to the segmental levers involved in body movement and the forces producing the movement, consideration must be given to muscular and other softtissue forces which stabilize a segment involved in the kinematic chain. It is for this reason that a kinematic analysis must include a description of the stabilized joint both within and without the lever and the forces which stabilize the involved articulations.


      CLOSED BODY SYSTEMS

While most kinematic chains in the body are open systems, there are two excellent examples of closed systems which are of special interest in chiropractic. One is found in the thoracic cage where the dorsal spine, upper ten ribs, and sternum are united by the costovertebral and costosternal joints to form a closed kinematic chain. No movement of one segment can be made that does not affect the other in a predictable manner. The other example is found in the pelvic girdle where the ilia and sacrum are united by the sacroiliac and pubic joints to form a closed kinematic chain. Again, no movement of one segment can be made that does not affect the others.

Of course, the body as a whole or a part could be considered a closed system if the ends of the system are fixed. For instance, when a person tries to dislodge the overlapping bumpers of two cars, the person's feet are fixed to the ground by body weight and lifting compression and the hands are fixed by the weight of the car being lifted. The same principle applies when trying to displace any firmly fixed object (Fig. 3.7). One can appreciate that this concept has several implications in the analysis of fixated and ankylosed vertebrae during spinal movement.


Segmental Parameters

It is frequently important in kinematic and kinetic studies to know the center of mass for a particular body part as well as the body as a whole which is located slightly anterior to S2. Most researchers utilize the segmental centers of mass loci determined by Dempster with slight adaptations. See Table 3.1. The average weight percentages for body segments are shown in Table 3.2.



     Kinematics

Muscles are the primary source of force within and gravity the primary source without the body to produce motion of the body. As with kinematics, kinetic analyses are essentially based on Newton's laws of motion.

Several methods are used by the research scientist in analyzing kinetic problems. Briefly, these include the application of acceleration, impulsemomentum, and work-energy factors. An acceleration approach is used in kinetics to analyze either linear or rotational forces and instantaneous acceleration problems. An impulse-momentum approach is used to analyze force acting over a time duration, and it is essential in problems of falls or when two or more objects collide. A work-energy approach is used in kinetics to analyze forces known as a function of body position and the forces are acting over a distance. While detailed descriptions of these approaches are beyond the scope of this text, several basic concepts and concerns of kinetics pertinent to the physician are discussed in this section.


     Table 3.1.   Locations of Segmental Centers of Mass
Segment(s)           Center of Mass
Head, neck,          Slightly anterior to the vertebral body of T11.
and trunk

Head and neck        On the occipital bone's under surface or within from 3/4"
                     to 1-1/8" from the crest of the dorsum sellae.

Head                 Within the sphenoid sinus, about 3/16" past the
                     anteroinferior edge of the sella turcica.

-------------------------------------------------------------------------------

Upper limb           Slightly above the elbow joint.

Arm                  Within the medial head of the triceps adjacent to the
                     radial groove, about 3/16" proximal to the far end of
                     the deltoid insertion.

Forearm              About 1/2" proximal to the distal aspect of the insertion
                     of the pronator teres and 3/8" anterior to the
                     interosseous membrane.

Hand                 On the axis of the 3rd metacarpal, nearly centered
                     between the volar and palmar skin at the angle formed
                     between the radial longitudinal and proximal transverse
                     crease of the palm.

-------------------------------------------------------------------------------

Lower limb           Slightly above the knee joint.

Thigh                Within the adductor muscles, 1/2" medial to the linea
                     aspera, below the adductor canal, 1-3/16" beneath the
                     apex of the femoral triangle, and 3/4" proximal to the
                     farthest fibers of the adductor brevis.

Leg                  About 1-1/2" under the popliteus at the posterior aspect
                     of the posterior tibialis, 5/8" superior to the proximal
                     aspect of the Achilles tendon, and 3/8" posterior to the
                     interosseous membrane.

Foot                 On a line between the center of the ankle joint and the
                     ball of the foot in the plane of the 2nd metatarsal;
                     either within the plantar ligaments or just superficial
                     to the adjacent deep foot muscles, below the proximal
                     halves of the 2nd and 3rd cuneiform bones.


     Table 3.2.   Average Weight of Body Segments
                            Percentage
                            of Total
     Segment                Body Weight
     Head                       7.3
     Trunk                     50.7
     ---------------------------------
     Arm                        2.6
     Forearm                    1.6
     Hand                       0.7
     ---------------------------------
     Thigh                     10.3
     Leg                        4.3
     Foot                       1.5

Note: The above computations are based on a 150-lb male mesomorph.
Figures do not add to 100% because limbs are not computed bilaterally in the above tabulation.


Momentum

The quantity of a body's motion is its momentum. Linear momentum (L) is the product of an object's mass (M) and velocity (v): L = Mv. Its quantity unit of measure is in pound-feet (lb-ft) or kilogram-meters (kg-mps) per second. When a body is acted upon by a force, the momentum of that body is changed in the direction of the force. Also, the momentum is proportionate to the amount of the force applied and to the duration of the force acting upon the body (Fig. 3.8). Momentum is directly associated with inertia. The greater the inertia of the body, the greater the force that must be applied to change the momentum.


      MOMENTUM AND POWER

Momentum affects the amount of power that must be applied in moving, stopping, or changing the direction of a load. A greater amount of power must be applied to overcome the stationary object's inertia than to maintain the speed of the moving object. Likewise, a greater amount of power is necessary to stop a moving object quickly than to stop it by gradually reducing the speed. A greater momentum can be imparted to a movable object if the weight arm is lengthened. Less power is necessary to change the direction of a moving object if the object is kept moving than if the object is brought to a stop before it is moved in the new direction. Thus, the turn in swimming is a continuous motion through a short circle, not an abrupt reversal of the direction of movement.


      MOMENTUM TRANSFERENCE

When an object receives a force, the momentum lost by one object is gained by another. Thus, the momentum of a jumper's landing is transferred to the landing pad and then to the earth. The momentum of a fly ball is transferred to the outfielder's glove and body and then to the earth. The momentum of a dynamic adjustment is transferred to the patient's body, to the adjusting table, to the floor, and then to the ground.


      THE MOMENT OF INERTIA

As previously discussed, the moment of inertia is the measure of resistance of an object at rest to rotation or to change the state of rotation of a rotating object. This is the quantitative measure of inertia for change in angular velocity, measured in lb-ft524 or kg-m524, and this inertia is equal to the mass of the object (lb) times the square of its radius of gyration (ft). Torque (T) ex- erts a turning force that is equal to the product of the moment of inertia (I) and the angular acceleration (a): T = Ia.


      ANGULAR MOMENTUM

An object's angular momentum (A) is the product of its mass moment of inertia (I) and its angular velocity (w): A = Iw. Its unit of measure is lb-ft52/sec4 or kg-m524/sec. If there is no external torque acting on a closed system, the total of angular momentum remains unchanged even when the moment of inertia is changed.

This principle is often expressed in athletics. An acrobat, tumbler, high diver, or free-fall parachutist can regulate his or her speed as the body rotates about its center of gravity and by the postures assumed. If the limbs are tucked in, the radius of gyration is shorter to that when the limbs are abducted, the moment of inertia will be relatively small, and the body will spin rapidly about a transverse axis in the coronal plane. It is for this reason that the spinning figure skater can increase rotational speed by bringing the arms toward the center of the body. Conversely, the speed of rotation can be slowed by opening the extremities to decrease the radius of gyration and increase the moment of inertia.

The sum total of momentum also has many clinical implications. For example, the physician often sees signs showing evidence that a small force applied for a sustained period of time can cause momentum changes that are comparable to a much larger force applied for a shorter interval.


Friction


In the action of one object pressing against another, friction is the resistance to relative lateral motion between the two objects in contact (Fig. 3.9). The resistance to the force that develops at the contact surfaces has a magnitude called frictional force, and the quantity of force necessary to produce motion of one surface relative to the other is governed by the physical properties of the materials in contact (eg, roughness), how tightly the surfaces are compressed, lubrication, and the type of movement between the surfaces. When mechanical work is done by moving against frictional forces, one effect is heat between the articulating surfaces.


      THE COEFFICIENT OF FRICTION

The coefficient of friction is the ratio of tangential force to interbody pressure necessary to start a sliding motion between two objects. One can skate more easily on a hard, smooth surface than a soft, rough surface because the ratio of effort to body weight is smaller on the firm, flat surface. This action between the skater and the flat surface demonstrates a low coefficient of friction, where the action between the skater and the rough surface demonstrates a high coefficient of friction (Fig. 3.10).


      PRINCIPLES OF FRICTION

It is well to keep two points of friction in mind. The first point is that once an object is set in motion (eg, an adjusted vertebra), it takes less force to keep it moving. The friction effect of contacting surfaces at rest is constant, but this friction effect is greatly reduced once motion takes place. As it takes more force to get a heavy box sliding than to keep it sliding, it takes more force to free a fixated-subluxated articulation in its initial position than to adjust it to its normal site. This concept is demonstrated in the quick "dynamic" chiropractic adjustments as opposed to the relatively slow "leveraged" adjustments of osteopathy.

The second point is that contacting surfaces (eg, articular surfaces) produce a frictional effect that has no relation with the size of the contact area. Thus, a certain contacting force produces a given frictional effect whether the contact surfaces are large or small. Obviously, there are many other considerations which must be made, and these will be discussed further in the chapters concerning regional problems and their biomechanics.


      INTRINSIC FRICTION

Friction is also a factor between flexible and rigid objects such as between a tendon and a bony prominence or between the soft tissues within the intervertebral foramen and its bony borders. Forces of friction in the body are especially important between articulating surfaces, layers of tissue, and around structures which glide upon each other. At common sites of friction wear, bursae are genetically located to decrease the effects of friction. At uncommon sites, a bursa will form by physiological demand. In acute situations, traumatic effects of friction are seen both macroscopically and microscopically.


      CLINICAL APPLICATIONS

There are many instances in practice where the effects of friction are desirable and undesirable. Lubricants are used in deep tissue goading to reduce skin friction. A felt or foam rubber pad is used under an elastic bandage to enhance gripping the area firmly and to more evenly distribute the compressive forces of the bandage. The frictional forces within joints help greatly in stabilizing the body at rest. However, these forces must be assisted especially by the stabilizing forces of muscles and ligaments and the deformation of articular cartilage under load.


Damping


Damping is the property of material that offers resistance to speed. For example, it takes more force to move a bicycle pump's plunger quickly than to move the plunger slowly. This principle is also exhibited in an automobile's shock absorbers, in the body's ligaments and joints, and dramatically in the viscoelastic intervertebral discs. The blood within the vertebral capillaries also offers a damping effect in rapid spinal loading because the blood has difficulty in rapidly escaping through the foramina.

The damping coefficient (measure of damping) is the ratio of force exerted to the deformation speed when the damping resistance is proportional to the speed. For translatory motion, it is measured in poundforce sec/ft or newton sec/meter. For rotatory motion, it is measured in poundforce sec/degree or newton sec/radian. The term viscous damping refers to a situation where damping resistance is proportional to the speed.


Work and Power


Work is accomplished when the velocity of the body is increased as the result of an applied force. It can be measured in various ways, but the most common is to consider it the product of the force applied and the distance moved in the direction of force. For example, when a force (F) is applied to the body so that it moves a distance (d), the energy expended by the body or the work (W) done is expressed by the equation W = Fd. If a muscle lifts a 20-lb weight through a distance of 3 ft, the mechanical work accomplished is 60 ft-lb.

Note that work should not be confused with the effort expended as great force may be used against an immovable object that would result in zero work. Work, in the specific mechanical sense, is the result of a force overcoming a resistance and moving an object through a distance: work = force X distance.


      WORK CAPACITY

The work capacity of muscle has been variously estimated to be from 80-85 psi of cross-sectional area. Work capacity varies from muscle to muscle and in various conditions of fatigue, training, temperature, and state of nutrition. The computation of the force exerted by a muscle is a function of the work capacity, size, and shortening of the muscle. For example, if a sartorius with a work-capacity of 80 psi of cross-sectional area has a cross-sectional area of 2 inches, fibers 18 inches long, and contracts one-third of its length during a movement against maximal resistance, the work accomplished will be: cross section in inches X strength psi X distance muscle contacts in inches divided by 12 = force exerted in foot pounds.

Although torque is measured in similar units and also equals force times distance, it should not be confused with work. Work concerns displacement and lies in the same plane as the distance the object is displaced, while torque is perpendicular to a lever-arm distance. However, torque can produce work; ie, torque X angular displacement = work.


      TYPES OF WORK

The term energy means the capacity to do work. Positive work is accomplished when a muscle contracts and shortens (eg, in lifting a load). Negative work is achieved when an external force overcomes internal muscle contraction force to the extent that the muscle lengthens during contraction (eg, in lowering a load). That is, negative work occurs when a force acts parallel to the movement but in a direction that is opposite to the movement.

There is a close association in the same unit of time between the work accomplished by a weight lifter and that of a sprinter.


      POWER

Muscle contraction force is synonymous with strength; ie, that energy which tends to produce motion or work. The term power, as used in biomechanics, should not be confused with strength. Power relates to the time element of the work; ie, the rate at which energy is expended, the rate at which work is done. Power = force X distance/time, and it is expressed in footpounds, horsepower (1 hp = 55 ft-lb/sec), or joules/second. When force is measured in newtons and distance in meters, the unit of work is a joule (J): 1 J = 1 N continuously applied during 1 m of displacement.



     Summary of Major Static and Kinetic Factors Influencing Mechanical Efficiency of Muscular Effort

The different size, quantity, and length of muscle fiber structure govern the force which a muscle can produce and the distance which a muscle can contract. Most of these characteristics are genetically determined by usual body needs such as the strength and range of motion required.

When muscles act upon a bone segment, the result depends on:

(1) the length of the lever arms,
(2) the range and force developed by each muscle,
(3) the stretched condition of the contracting muscle,
(4) the individual angles of muscle pull,
(5) the site of the muscle relative to the joint axis, and
(6) the action of synergic and antagonistic muscles.


Lever Arm Length and Mechanical Advantage

To a great extent, the human body has been designed for speed rather than for overcoming great loads. In most of the levers of the body, the distance between the fulcrum and the point where force is applied (the power arm of the lever) is relatively shorter than the distance between the fulcrum and the center of the weight (load arm). In man-made mechanical levers, the power arm is relatively long and the load arm is short so that great loads can be overcome by the application of minimum force. Thus, the source of body power is placed at a mechanical disadvantage, and strong forces must be applied through short power arms that act through short distances. The muscles that are the source of power must be strong and efficient to work effectively at such a mechanical disadvantage.

A long power arm, in comparison with the length of the load arm, gives the lever a mechanical advantage which enables heavy loads to be lifted. A short power arm results in a mechanical disadvantage in lifting loads, but it imparts speed to the movement if the muscular force is sufficient. For example, the calcaneus presents a long power arm which enables a person to rise on the toes while carrying a heavy weight, and the olecranon is a short power arm which allows great speed in throwing.

A long load arm is a disadvantage in lifting heavy loads, but it is an advantage in movements of speed and in imparting momentum to light objects. A short load arm, in relation to the power arm, gives the lever a mechanical advantage in lifting loads. The further a light object is held from the fulcrum, the greater is the advantage for speed. Thus, a light weight is thrown with the arm extended, while a heavy weight (eg, shot put) is put near the shoulder.


Muscle Range of Movement and the Force Developed

A primary attribute of muscle tissue is its ability to develop tension against resistance. When a muscle contracts, it does so in the direction of the muscle fibers; ie, fiber arrangement determines the line of pull of an individual muscle.


      RANGE OF MOVEMENT

Muscles with long parallel fibers produce the greatest range of movement. Some muscle fibers extend the entire length of a muscle, while others terminate intramuscularly in connective tissue stroma. While fiber length helps govern the overall range of effective movement, other factors are important such as the shape of the articular surfaces, leverage, and the degree of stabilizing, antagonistic, and synergistic action. Because contraction may shorten a muscle's length by half, the long muscles where the fasciculi are fairly parallel produce the greatest range of movement.


      CONTRACTION FORCE AND ANATOMICAL FIBER LENGTH

Pennate fibers, which attach at an angle to the line of pull, have shorter fibers (Fig. 3.11). Thus, the distance which a muscle can shorten is not related to the length of the muscle as a whole but to the length of its fibers. However, the strength of contraction is determined by the size and number of contraction fibers (cross-sectional mass) and not on the distance over which a muscle contracts. Thus, in most instances, there are more fibers in a cross section of pennate muscle than in nonpennate muscle. The range of muscle motion is dependent on fiber length; strength on size and quantity. The range of movement efficiency loss of oblique fibers is usually compensated for by an increased quantity of fibers. This allows for powerful short-range movements. The inverse relationship between movement strength and range of movement follows the mechanics of levers where a bone serves as a rigid bar, the joint as a fulcrum, the muscle furnishes the force, and the load or resistance serves as the force which must be overcome to move the lever.


      CONTRACTION FORCE AND PHYSIOLOGIC FIBER LENGTH

As discussed previously, tension, a scalar quantity, and force, a vector quantity, are not synonymous. Tension has magnitude but not direction. Under load, the tension developed in the body is produced to the physiologic maximum in proportion to the number of links in the kinematic chain. Internal muscle force (stress) implies a tension magnitude acting in the direction of a muscle's action line under various conditions to resist a load. The elastic components within muscle shorten during contraction, but whether or not the muscle shortens in toto depends on the relation of the internal force developed as compared to the force of the external load.

In addition to a muscle's length at rest, its length at the instant of activation greatly affects its ability to develop tension. The maximum quantity of this tension can be judged as the greatest load that can be overcome. Many normal and abnormal mechanisms can be explained in that a slightly stretched muscle at the time of stimulation contracts with a great amount of force, whereas a shortened muscle contracts with much less force. This is true regardless of the type of contraction. It is for this reason of greater force under stretch that isotonic contraction, which increases tension during stretch, can produce more work with a lower energy cost than the other types of contraction. When fibers are parallel, maximum tension is produced at lengths that are only slightly longer than those at rest. Nonparallel fibers develop their maximum tension at a somewhat greater stretch.


      TWO-JOINT MUSCLES

Muscles of the thigh and upper arm that cross two joints are called two-joint muscles by Morehouse/Cooper. There are several mechanical advantages offered by this arrangement for the two-joint muscle can act as a towline. During motion in one joint, two-joint muscles exert a contrary action on the second joint so that the origin and insertion move in the same direction and energy can be transferred from one part to another (Fig. 3.12).

The duplication of effort saved when two-joint muscles act instead of two separate muscles has been calculated to be the algebraic sum of the rates of work of the one-joint muscles they replace. This relationship exists whether the work consists of that done by the tissue during shortening or during stretching. With a two-joint muscle, the change in length is less rapid. This slower rate of shortening puts the muscle in a more favorable state for the production of muscle tension.

The length at rest of a two-joint muscle is inadequate to allow full motion in the opposite direction at the same time in both joints over which it passes. Extension cannot take place naturally at any one of the articulations without also occurring at the other. Because of two-joint muscle tension, the articulations are moved to about the same extent. For example, the trunk and lower leg are usually parallel in movement. It is difficult to bend the knees and keep the trunk erect.

During action, the distance between the origin and insertion of these pluriarticular muscles may be changed. This change in length of the muscle involves a change by either shortening or lengthening to maintain constant tension. For instance, when the distance between origin and insertion of the biceps is shortened by raising the arm above the shoulder, the stretch of the biceps is reduced and the strength of its maximum contraction is diminished.


Muscle Speed and Velocity

Most muscles at rest shorten to about half their resting length when loaded to the maximum. The distance to which a particular muscle can shorten is essentially determined by the length and arrangements of its fibers, the resistance of antagonists, the joint architecture, and the amount of resisting load.

In addition to the biochemical reactions involved, the speed of muscle contraction varies inversely with the load; ie, the speed approaches maximum when the load decreases. Thus, the magnitude of the load determines the speed at which the lever can be moved. The heavier the load, the slower the movement. If a fast movement is required, the weight must be relatively light. A heavy weight can be moved faster if it is held closer to the fulcrum of the lever system. In the case of heavy loads, the quantity of load determines the distance through which it can be lifted.

The importance of the most effective application of muscular power and mechanical efficiency becomes greater as the load increases. A load which is difficult to lift by the use of inefficient movements becomes less difficult if efficient mechanical principles are applied. Some investigators have shown that even the chemical reactions associated with contraction are governed by the degree of load and force capabilities of the tissue involved.

The velocity at which a muscle shortens (ie, the rate of shortening in a direction) is governed biomechanically by the force necessary to overcome a load, thus contraction velocity is maximum with zero load. Whenever more fibers are put into action than are necessary to overcome a load, the excess force is expressed into increasing velocity and the distance of movement. This is exhibited when one lifts a light object that was perceived to be much heavier.


Angle of Pull

Mechanical efficiency is more important than muscular efficiency in determining strength of body force. A pull at right angles to the lever gives maximum mechanical efficiency. The greater the deviation from the right angle, the less efficient is the angle of the pull. Thus, the effect of the angle of pull of a muscle upon the force imparted to the lever is generally a decrease in the efficiency of the pull because the angle made by the bones on either side of a joint is either decreased or increased from a right angle. The reason that efficiency of pull is greatest when the joint is at a right angle is that in this position the muscle is pulling directly against bone. In this position of direct pull, none of the force of the muscle is wasted in pulling the bone of its insertion either toward or away from the joint.

A muscle that is pulling at an angle less than 90* has a stabilizing effect upon the joint. The smaller the angle of pull, the greater will be the portion of the total force which will be devoted to the stabilizing effect. This stabilizing effect relieves the stress and strain on the body, but it represents only a loss of effort in the accomplishment of external work.

Consider the strongest pull in flexion of the forearm. The strongest pull by the forearm can be made when placed in 90* flexion, even if the flexor muscles are not stretched enough to give their most powerful contraction, for it is in its position of maximum mechanical efficiency. If the forearm, is extended to 180*, the muscle is in a position for a powerful pull, but the pull in this position is at such a mechanical disadvantage that only a small force can be applied to a load. In the flexion of the fully extended arm, considerable force is wasted by pulling the radius and ulna against the humerus.

This principle is commonly demonstrated in professional sports when we observe the position of the body joints during different physical activities that require strong movements. See Figure 3.13.


Composition of Forces


As previously discussed, the effect produced when a force is applied to a lever is that of moving the lever in the direction of the force. If several forces act upon a single lever, the movement that results depends upon

(1) the magnitude of the forces, and
(2) the direction of the forces.

Obviously, the design of the musculoskeletal system is such that the majority of muscles pull at an angle against the bones they move. The resultant of two forces acting at an angle to each other can be computed rather simply by constructing a parallelogram of forces. This method, shown in Figure 3.14, shows the anterior and posterior fibers of the deltoid acting through a common tendon inserted into the humerus to abduct the humerus. The diagonal of the parallelogram represents the resultant of the two forces. If the resultant force of 100 lb in Figure 3.14 is in the direction of the pull of the middle deltoid, and if it is exerting an additional force of 60 lb, then the total force of all three deltoid parts acting on the common tendon is 160 lb.


Attachment Site

The amount of a muscle's in toto maximum stretch and shortening is determined by the maximum degree of angular displacement of the body part at a particular joint. In joints where rotation is customary, the muscle fibers are coiled. During contraction, they unwind to produce bone rotation. All muscles crossing a single joint, but not those crossing two or more joints, are affected by the degree of angular displacement to the extent that they are normally capable of contracting sufficiently enough to move the attached segment through it's maximum angular displacement and extensible enough to allow the full range of opposite-directed motion. If a muscle crosses two or more joints, tension creates force movements at each joint crossed to an extent that depends on the amount of muscle force exerted and the instantaneous length of the moment arm at each joint.

How a muscle is attached to a bone influences the effect of muscle contraction because the rotational and stabilizing components of force is determined by the angle at which a muscle pulls the bony lever (Fig. 3.15). Quite frequently, this factor is determined by the passage of the muscle tendon over one or more bony prominences that moves the muscle's action line distal to the joint axis and increases the muscle's moment arm.

A muscle's anatomic position determines the action line of a muscle producing motion about a joint. The perpendicular distance from the action line of the musculotendinous unit to the axis of rotation of the joint constitutes the lever arm. Also, the torque may be determined by the muscle force that is perpendicular to the length of the bony rod times the distance from its site of attachment to the axis of the joint. Thus, the moment of force possible to produce is determined by the distance of the muscle attachment from the axis of the particular lever system. A muscle attached close to a joint sacrifices range of movement for strength. The farther a tendon lies from the joint's axis, the better will be its turning effect on the segment about the joint.


The Action of Synergic and Antagonistic Muscles

If the pull of a muscle is not directly away from its point of insertion, additional muscles must be called into action to hold the lever in the desired position during the movement. Most arm movements in swimming, for example, are angular and require the contraction of synergic muscles.

The reciprocal action of synergic and antagonistic muscles increases the steadiness and accuracy of a movement. The greater quantity of muscles engaged in a movement, the more accurate and graceful is the movement. The more complete the relaxation of the antagonistic muscles, the more rapid and powerful is the movement. The more angular the direction of pull, the greater is the importance of the action of the synergic muscles in controlling the direction of the movement.


Opposing Forces on Joints

Newton's third principle, that for every action there must be an equal and opposite reaction, is the basis upon which muscle moment is based. If the muscles acting on a joint produce movement, there must be forces acting to oppose this movement. These opposing forces essentially consist of gravity, exterior reaction forces, and inertia:

  1. Gravity.   Although the gravitational force acting at the center of mass of any body segment is a constant, the moment produced is governed by the segmental centers of rotation and the locations of the centers of the segmental masses. As the body in motion is constantly changing position of its segments, the gravitational moments are constantly changing.

  2. Reaction forces.   In simple walking, this is expressed as the reaction force exerted by the ground surface against the foot.

  3. Inertia.   The inertial force of a body segment is proportional to segmental acceleration, but it acts in the opposite direction of the segmental acceleration.

Opposing muscle mechanisms are readily seen during the walking cycle. Just prior to heelstrike, for example, gravity, surface reaction forces, and inertia are producing an extension (counterclockwise) moment about the knee. However, the knee flexors, principally the hamstring group and gastrocnemius, are essentially two-joint muscles so that when knee flexion is produced they also produce hip extension (hamstrings) and plantar flexion (gastrocnemius).


Genetic Influences

It is not unusual to find an individual endowed with bony levers that have exceptional mechanical arrangements. Their muscles are inserted further than normal from the joints about which the levers move (Fig. 3.16). When this occurs, it seems to be characteristic of all joints in that particular person. The result is that the mechanical advantage of the various levers is greater than normal, and the individual appears to have superior muscular power. In this instance, the superior power is not due to the integrity of the muscle fibers, but rather to an improved application of force.

An example of this principle is seen with a 12-inch forearm used to raise a 20-lb weight. If the insertion of the biceps is 2 inches from the center of the elbow rather than only 1| inches, the mechanical advantage of the lever is increased from 0.125 to 0.166. This small increase in mechanical advantage would reduce the effort required by the biceps to lift a 20-lb load from 160 lb to 120 lb, an advantage of 40 lb. The total effective strength of a muscle is thus enhanced when the inherent mechanical advantage is greater.

Mechanical advantage and disadvantage have a distinct relationship with performance. For example, pace varies with limb length, thus long limbs are an advantage in running, especially in long-distance events. But tall individuals possessing long body levers can use these levers to an advantage only against light loads unless the muscles moving them are extremely strong. The strength of the muscles moving these long levers becomes the limiting factor in their use. Weak muscles place the long levers at a disadvantage. A higher center of gravity is a disadvantage in that it takes extra postural effort to maintain balance such as in gymnastics or skating, but it has its advantages in sports such as basketball where increased height places one closer to the goal.



     Biomechanical Stress

Clinical stability refers to the body's ability under load to limit various patterns of potential displacement in order to prevent damage or irritation to its components. In the spine, for instance, this is the ability of the vertebral column and its associated tissues to avoid cord, nerve, vascular, lymphatic, ligament, disc, or muscle irritation, stretch, pressure, deformation, or pain as the result of structural changes from loads (Fig. 3.17). Thus, any disruption of a musculoskeletal unit decreases the unit's stability in proportion to the degree of structural and physiologic impairment.

The body functions in accordance with its basic design --a relatively unstable-jointed frame, precariously balanced above a small base of support from the purely biomechanical viewpoint. Even slight movement requires tension adjustments of the muscles surrounding a joint. However, this inherent instability affords considerable mobility and a means for rapid change of direction. Work is fairly restricted to anterior tasks due to the ventral placement of most sense organs and the design of the extremity joints and their associated muscles, tendons, and ligaments.


Joint Stability vs Mobility

Joint stability depends primarily upon its resistance to displacement. Biomechanically, it is the opposite of joint mobility. Joint flexibility is determined by osseous structures, soft tissue bulk, restraining ligaments, synovial fluid viscosity, muscle tone, nonelastic connective tissue of muscle, and the restraint action of the skin. Joint stability, on the other hand, is determined by mechanical and anatomic factors.


      MECHANICAL FACTORS

Joint stability is provided mechanically by the biomechanical principles (eg, Newton's laws of motion) that interact with structural factors to govern the degree of stability of a particular joint such as mass, the lateral distance of the common line of gravity to the joint's center, height of the joint's partial centers of gravity, the line of action, acceleration, size and position of the base of support, etc. In regard to the base of support, it is well to keep in mind that each segment of an articulated system has as its base of support the structure immediately underneath it, which in turn often has another base of support beneath it.


      ANATOMIC FACTORS

Joint stability is provided anatomically by the design of the articular surfaces, the joint capsule, the associated ligaments and fascia, joint fibrocartilage, atmospheric pressure, synovial fluid viscosity, and the site and function of muscles. Joints with poorly mated articular surfaces usually have strong collateral ligaments to resist forces transmitted across the joint (eg, the knee). Besides providing movement, muscles absorb energy during load transmission, provide smooth load transmission from one joint to another (reduce jar), and help to stabilize the joint.

While many of the mechanical and anatomic factors affecting joint stability and mobility have been discussed, such factors as stress, flexibility, viscosity, elasticity, plasticity, stiffness, strain, fatigue, and deformation characteristics of pertinent tissues require further definition and clarification.

Biomechanical Stress and Strain

The terms stress and strain have specific meanings in mechanics which differ from their popular interpretations.


      STRESS

Stress equals load divided by the original cross-sectional area. In mechanics, measurement is recorded by pounds/inch524 (psi), newtons/meter524, or pascals. The two types of stress are normal stress and shear stress. Any stress normal to a cardinal plane is referred to as a principal stress.

Normal and Shear Stress.   Normal stress is that which is perpendicular to a cross-sectional plane (eg, compression, tension), while shear stress is parallel to a cross-sectional plane. Thus, the intensity of any normal or shear stress on any site of the body at any particular time is relative to the stress direction through the cross-sectional plane. For example, the axial fibers of a bone are subjected to normal stress during compression (negative stress), tension (positive stress), or bending (concave negative and convex positive stress). See Figure 3.18.

The Angle and Area of Application.   The change in angle of the application of force has a distinct effect on stress. When the sacral angle changes, for instance, there must be changes in the shear and compression components of the force upon the sacrum (Fig. 3.19). In addition, The area over which stress is applied governs the magnitude of stress (force concentration). For example, a pisiform contact has a higher stress concentration than a broad contact. Joint stress concentration is determined by shape (articular congruity) and the articular tissue's plasticity. The better the articular mating and the tissue plasticity, the lower the stress concentration because these factors increase the bearing surface under load.

Stress and bending moments are related.   For instance, when a load is applied to the head of the femur, the bending effect causes the bone's inner cortex to be under compression and its lateral cortex to be under tension. This places considerable compression and tensile stress in the subtrochanteric area.


      STRAIN AND DEFORMATION

Normal and Shear Strain.   Strain equals a change in shape or length divided by the original length of the structure, thus it represents a change in angle (Fig. 3.20). This alteration of shape or length is called deformation. There is no unit of measurement for strain as there is with stress. As with stress, the two types of strain are normal strain and shear strain. Stress is usually plotted on the Y axis, while strain is plotted on the X axis.

Ultimate Loads and Stresses.   When a load is increased upon a structure to the point of fracture, that point of maximum load is called the ultimate load of the structure. The ultimate stress of a structure is computed by dividing the ultimate load by the original cross-sectional area.


Yield Factors

Such factors as flexibility, viscosity, elasticity, stiffness, plasticity, viscoelasticity, and structural fatigue are important considerations when dealing with mechanical reactions to structural load.


      FLEXIBILITY

Flexibility refers to a substance's ability to be flexed and yield without fracture; thus, its pliability and nonrigidity. The ratio of the quantity of displacement produced to the load applied is the flexibility coefficient (responsiveness) of the structure. A relatively high flexibility coefficient is seen in a readily yielding physiologic scoliosis where only small forces result in large deformities.


      VISCOSITY

Viscosity refers to that property of a substance whereby, when flow occurs within it, forces arise in such a direction as to oppose the flow. For example, when the viscosity of synovial fluid becomes higher than normal, friction increases and movement is inhibited. Mathematically, viscosity is expressed as the ratio of the shearing stress to the shearing strain rate for viscous solids or the shearing stress to the velocity gradient for fluids.


      ELASTICITY

In the mechanical sense, elasticity is that springiness or resilience property of a substance that causes it to resist deformation by storing energy and thereby recovering its original shape and size without permanent deformation by releasing energy when the deforming forces are removed.

The Range of Elasticity.   All substances are elastic to some degree, but the elastic range under load to the point of permanent deformation and fracture varies widely. For most practical purposes, the vertebrae are considered rigid bodies as compared to the easily deformed intervertebral discs even though both bone and fibrocartilage have elastic characteristics under load.

The Modulus of Elasticity.   The degree of stiffness is the ratio of the normal stress to the normal strain of a substance. The lower the value, the less stiff is the substance.


      STIFFNESS

Stiffness is that property of a substance that resists deformation when the substance is under load. Although similar, stiffness and elasticity are not opposites. Elasticity is a property of the substance, while stiffness represents the mechanical behavior, size, and shape of a substance under load. The brittleness of bone is due to its high mineral content.

Coefficient of Stiffness.   The coefficient of stiffness varies with the magnitude of the load; ie, a stiff (eg, ankylosed) adult spine has a higher stiffness coefficient than that of an infant. The stiffness coefficient equals the applied load divided by the displacement produced (Fig. 3.21).

Torsional Rigidity.   The term torsional rigidity refers to a substance's rotational stiffness; ie, the torque per unit of angular deformation. All joints present this characteristic.


      PLASTICITY

Plasticity refers to the ability of a substance to retain a permanent shape attained by pressure deformation beyond its elastic range in any direction without fracture.

The Range of Plasticity.   Any range beyond the elastic range to the point of rupture is referred to as the plastic range of deformation.

Yield Stress.   A substance's yield stress is that point where appreciable deformation occurs without an appreciable load increase; ie, that point where plastic deformation begins. After the load is removed, the deformation which occurs after the point of yield stress is relatively permanent (Fig. 3.22).

Allowable Stress.   The term allowable stress refers to a value of stress that is above that of normal loads but lower than that of the yield stress; ie, that margin of safety necessary to withstand accidental overloads during exceptional activities. For example, an athlete's body is commonly subjected to stresses far above that necessary for average activity but which are within the range of allowable stress for that particular individual.

Ductility.   That property of a substance to absorb relatively large amounts of plastic deformity energy prior to fracture, either in elongated length or decreased cross section, is called ductility. Substances with low ductility (less than 5%) are referred to as being brittle (eg, cortical bone). Substances with high ductility have a high energy absorption capacity.


      VISCOELASTICITY

All biologic materials have a loading rate where, for example, a slow gradual pull will produce considerable deformation before fracture as contrasted to a fast pull that produces fracture with little deformation. This loading rate characteristic concerns the viscoelastic nature of the substance in question such as bone, fibrocartilage, ligaments, muscles, and tendons. Thus, any material whose mechanical properties vary depending upon the rate which load is applied is a viscoelastic material.

Relaxation.   The combined components of viscosity and elasticity allow for relaxation and creep. To measure relaxation, a load is applied to produce a deformation which is then fixed. Relaxation, sometimes popularly called "give", is a steady deformation that occurs with less force over a period of time (Fig. 3.23). This is demonstrated in a tissue being stressed at a constant magnitude where the force necessary to maintain the deformation decreases with time.

Creep.   To measure creep, a load is applied suddenly and sustained at a constant magnitude. Creep is a steady deformation that occurs over a period of time. This is exhibited in the loss in an individual's height from many hours in the upright position due to the creep phenomena occurring in the intervertebral discs where a constant weight has been borne over a period of time.

Viscoelastic Stability.   As mentioned, the critical factor in elastic stability is the magnitude of the load. However, the critical factor in viscoelastic stability involves both load and a time element. That is, a viscoelastic material may withstand a certain load for a certain period of time and then fail without the load being altered. Thus, all musculoskeletal structures have a time-dependent stability factor, but this is often structurally adapted to in living tissue if the time element is prolonged (eg, redesign of trabeculae).

Hysteresis.   During cyclic loading and unloading, a viscoelastic substance shows a loss of energy in the form of heat. This phenomenon is called hysteresis (Fig. 3.24). For example, when an intervertebral disc is subjected to repetitive cycles of load and unload (eg, hopping), the shock waves directed from the feet to the head are substantially dissipated by disc hysteresis. Hysteresis decreases when the load-unload cycle is prolonged (eg, constant bumping) and during old age when viscoelasticity diminishes.


      BIOMECHANICAL FATIGUE AND ENDURANCE

The process of developing structure cracks when subjected to cyclic loading is called fatigue. The magnitude of the load is usually far below that of the ultimate load of the particular structure, and thus well within the elastic range. The result is a summation effect, in which a fatigue crack reaches a size that causes the remainder of the structure to become so stressed that the entire structure fails. This factor is popularly called the time or aging factor of a body structure, and the time of failure decreases as the magnitude of the load increases. The term endurance limit refers to the least load that produces a failure from structural fatigue.

If healing processes are inhibited or do not have adequate time to repair structural cracks in bone, for example, a fatigue fracture occurs. From a biomechanical standpoint, the term stress fracture is a misnomer because all fractures are the result of excessive stress.



     Biomechanical Aspects of Articular Cartilage

The principal mechanical functions of articular cartilage are exhibited in its ability to control joint motion by its shape, sustain a load (absorb energy), reduce friction during movement, transmit load from adjacent limb segments, and maintain joint stability by the specific shape of the bearing surfaces.


Load Carriage

All articulating bony surfaces possess a variety of incongruities and small projections where dynamic stress would be concentrated if not for the smoothing effect of articular cartilage. These small projections are the minute bony asperities that produce surface roughness. This characteristic of articular cartilage allows deformation during loading to increase the contact area and significant recovery during the unloaded stage. Load carriage is particularly enhanced by the cartilage's hydrophilic proteoglycans to retain matrix water as a result of Donnan osmotic pressure and by cartilage collagen to resist matrix tensile forces.


      JOINT REACTIVE FORCE

When any joint in the body is subjected to loading, certain internal reactive forces act at the surfaces in contact. Clinically, we see this exhibited in most low-back strains from lifting (Fig. 3.25).

Normal and Shear Forces.   At a finger, for example, one major reaction (normal) is perpendicular and compressive to the contact surfaces, and the other much smaller secondary tensile reaction (tangential) is parallel to the contact surfaces (Fig. 3.26). This tangential force is opposite to the sliding movement, and the low coefficient of friction at the contact surfaces is the ratio between normal and tangential components.

Direction of Force.   Articular cartilage is predominantly loaded perpendicular to the articulating surfaces. Tensile forces in all directions within the matrix are secondary to the perpendicular compressive forces. The magnitude and direction of these forces are governed by varieties of load application and the area of application that are dependent on the joint's point in its range of movement.

Fiber Arrangement.   When articular cartilage is tensile loaded in planes parallel to the articulate surface, the stiffness of the tissue (far below that of bone) is closely related to the quantity and arrangement of collagen fibers lying parallel to the direction of the tensile force. Fibers in the midzone are randomly oriented to accommodate the tensile stresses occurring in different directions. In the deepest zones, the collagen fibers have a distinct tendency to be perpendicular to the direction of tensile force and help tether the matrix to subchondral bone.


      DEFORMATION FROM LOADING

When articular cartilage is subjected to loading, deformation develops instantaneously according to the tissue's stiffness. This initial rapid deformation stage has a negligible matrix fluid flow, and tissue contour changes but not its volume. This stage is followed by a slower time-dependent creep that is related to the flow of water through the matrix according to the magnitude of the load, the fiber elasticity, the quantity of surface area loaded, the uniformity of force distribution, the matrix permeability which is low even when unloaded, the osmotic pressure of the matrix colloid, and the length of the flow path.

When the load is removed during rest, the stressed cartilage begins to return to its original thickness quickly at first (90%) because of the elastic recoil of the collagen fibers and then slowly further from the absorption of water that is governed by the Donnan osmotic pressure of the proteoglycans in the matrix gel. This recovery by absorption is enhanced by oscillation of the unloaded joint and limited by the collagen fiber's stiffness and strength that are subjected to increasing tensile forces as the swelling develops.


      RESPONSES TO CYCLIC LOADING

A dynamic load, as contrasted with that of a static load, is any load that varies with time. If a dynamic load has a repetitive pattern of variation, it is referred to as a cyclic load. For example, all weight-bearing structures are subjected to dynamic load as body weight is shifted from one limb to the other during locomotion. Because this weight has a repetitive pattern, it is also a cyclic load.

Cyclic Loading Magnitude and Duration.   Almost all weight-bearing joints are continually subjected to high loads of short duration that are rapidly applied. During a normal walking cycle, for instance, the weight-bearing surfaces of the sacrum are subjected to loads up to five times body weight at an approximate rate of one time per second. The typical cycle time between heelstrike and toeoff is about half a second. These and greater load forces are sustained by mechanical forces and liquid pressures within the matrix and bone. Although shock-absorbing articular cartilage is quite thin, peak forces, often twice the compressible strength of bone, would undoubtedly produce bone failure if not for the damping and attenuating viscoelastic properties of articular cartilage.

Recovery During Cyclic Loading.   Because fluid flow within the matrix is time dependent, cartilage response to compression depends upon the magnitude of the load, the length of time the load is applied, and if the load is applied statically or cyclically. A small amount of water is expressed through the matrix even during a briefly applied load, and its absorption is time dependent. If a second load is applied before the matrix is fully reimbibed, as during cyclic loading, the result is incomplete recovery which summates as the cyclic loading continues. In addition, all cartilage can be considered fatigue prone.


      EFFECTS OF CYCLIC LOADING ON THE SPINE

There appears to be strong, yet not definitive, evidence that spines exposed to dynamic loads may have an increased risk to low-back disorders. Current evidence suggests that chronic low-back pain relates to disc degeneration and that this, in turn, may be accelerated by mechanical failure in the region of the end-plate and subchrondral bone. Two hypotheses are proposed by Sandover that relate fatigue-induced failure of vertebral tissues to disc degeneration.5(169)4 After investigating the extent of dynamic stresses on the vertebral column motion segment arising from transport, the results indicated that fatigue-induced failure is a distinct possibility. Liu and associates conducted a study of low-cycle fatigue on 11 lumbar intervertebral joints placed under axial compression in a range of 37%-80% of failure load.5(170)4 The maximum deformation, in regard to the number of cycles applied, showed two distinct results: one group showed a gradual-stable increase, and the other group exhibited an abrupt-unstable increase in deformation. Before-and-after radiographs showed a 1:1 correspondence between unstable specimens and generalized bony failure. Following roentgenography of end-plate slices showed crack propagation from the periphery of the subchondral bone centrally. When the organic matrix from the cracked specimens were removed, it disintegrated into small particles. The normal controls and stable specimens in the experiment, however, retained their normal size and shape. Adams and Hutton flexed and fatigue loaded 41 cadaveric intervertebral joints from 18 spines to stimulate a vigorous day's activity. (167) Posttest examination showed that 23 out of the 41 discs exhibited distortions in the lamellae of the anulus fibrosus and, in a few of these, complete radial fissures in the posterior anulus were found.


Joint Lubrication

During articulation, movement is restricted by frictional forces at the contact surfaces, and its coefficient is the ratio of frictional force to load weight. The coefficient of friction is governed by the the extent of the area in contact and the shear strength of the contacting surfaces. According to Newton's first law, a greater force is necessary to start a joint's movement (static frictional force) than is necessary to keep it moving (sliding frictional force) at a constant velocity.


      SURFACE IRREGULARITIES

As the load is increased and the area of contact increases, friction increases. Any apposing surface irregularities of articular cartilage, or bone in its absence, has the highest points sheared off as the articulating surfaces slide on one another.

The more articulating surfaces are congruous (well mated), the more stress force per unit of surface area can be distributed uniformly. It is for this reason that incongruous (trauma, disease) weight-bearing joints (hip, knee, ankle) are more apt to exhibit degenerative changes.


      BODY BEARINGS

Every machine with moving parts, including the body, requires lubrication to prevent wear of articulating parts, reduce friction, and minimize heat production. If the synovial membrane is injured, hemarthrosis and a thick, boggy, blood-stained synovium result. Later, adhesions may develop to produce stiffness and restrict movement, but these effects can be reduced or eliminated by early mobilization.

The complex lubrication system of human joints far exceeds that of similarly designed man-made bearings. Much of this is due to:

(1) the renewable coating of glycoprotein molecules that blanket the surface of articular cartilage, (2) the ingress and egress of fluid from the cartilage's matrix, (3) the porosity and elasticity of cartilage which also affords fluid imbibement and expulsion during load compression and relief, and
(4) the unique folding and sliding action of synovial folds during movement.


      FACTORS OF SELF-LUBRICATION

An important factor in joint lubrication is the fact that the curves of the articulating surfaces are not exactly congruous (ie, do not fit reciprocally). This is necessary to afford better lubrication of the joint. Self-lubrication depends on a thin wedge-shaped space (provided by the slight incongruity), a viscous fluid, and a degree of rapid movement. During loading, the hydrostatic pressure developed in the matrix counteracts the pressure of the lubricant trapped within the area of surface deformation.

Joint lubrication is greatly diminished in prolonged immobilization such as in posttraumatic support or paralysis. The effect is atrophy of muscle, cartilage, and subcutaneous fat, along with chronic circulation disturbances, capsular contraction, and bone decalcification. This syndrome is referred to as "fracture disease" by Gozna/Harrington.

      THE VISCOSITY FACTOR IN JOINT LUBRICATION

There are several possible lubrication mechanisms for living joints. One such mechanism, the squeeze film compliant bearing, has been indirectly studied by through the use of a computational model. (175) It was found that changes in viscosity of the lubricant had a greater influence than changes in the cartilage stiffness, and the lubrication qualities of the compliant system were superior to those of the system with rigid surfaces. It was also found, however, that a compliant system with a low viscosity has poorer lubrication properties than a rigid system with a higher viscosity.


Factors Disturbing Normal Function

      AGING

Aging decreases articular cartilage permeability up to the age of about 40; after which, an increase in permeability begins to develop. However, aging has little or no influence on the elastic properties of articular cartilage. The effect of aging on tensile strength is unknown.


      TRAUMA AND INFLAMMATION

Both trauma and inflammation increase permeability of synovial vessels which allows large molecules of plasmin to enter the joint and act on articular cartilage to adversely alter its mechanical properties. Sustained joint blood, as that associated with prolonged trauma, also tends to alter the physical and chemical properties of articular cartilage. Any joint malalignment (eg, subluxation, genu distortions) produces abnormal unilateral loads which tend to break down articular cartilage.


      TENSION

Normal cartilage may fail mechanically when the collagen network is overly tensed. This usually requires a single impact of great magnitude sufficient to burst the aqueous gel. High loads of continual pressure produce cartilage degeneration and chondrocyte necrosis that contribute to cartilage failure. In cyclic loading, tension failure results from fatigue.


      TISSUE NUTRITION

The nutrition of articular cartilage is provided by the ingress and egress of matrix fluid during cycles of loading and relief. Thus, either sustained compression or sustained relief invites degeneration from lack of nutrition.

Even in a well-lubricated joint, continuous cartilage-against-cartilage pressure results in a irreparable pressure necrosis in a few days due to the prevention of synovial nutritive elements from entering surface pores of the cartilage. When this occurs, such as in immobilization in a forced position, chondrocytes die and degenerative arthritis is encouraged and its extent, from superficial to deep involvement, is proportional to the duration of the compression. In addition, the protein-polysaccharide complexes become depleted. The cartilage becomes unable to withstand stress as movement occurs, and degenerative changes develop. In osteoarthritis, articular cartilage loses its stiffness, especially at the surface.

The role of exercise on the nutrition of intervertebral discs was studied by Holm and Nachemson. (179) Experiments on dogs under a supervised exercise program utilizing moderate, violent, or specific regimens showed a distinct increase in aerobic metabolism in the outer part of the IVD anulus and the central portion of the nucleus pulposus, which resulted in a reduction of lactate concentration. The conclusion of their findings was that prolonged spinal movements effect positive nutritional variations and that these changes might be significant for human disc problems inasmuch as several studies have demonstrated nutritional similarities between the discs of humans and canines.


      DRUG EFFECTS

Certain drugs and biologic deficiencies have been shown to have a harmful effect on articular cartilage. Injections of corticosteroids especially tend to soften and fibrillate articular cartilage. Mid-zone areas of cell degeneration, failure of chondrocyte proliferation, abnormal patterns of subchondral ossification, suppression of intracytoplasmic metabolic processes, and matrix weakening have been described by some researchers. Proteoglycan-deficient cartilage is more deformable, and the matrix is more permeable than usual under loading. It also fatigues earlier than normal under cyclic loading.

**************************************************************

Chapter 4 References:

---1.	Hakim NS, King AI: A three dimensional finite element dynamic response
      analysis of a vertebra with experimental verification. Journal of Bio-
      mechanics, 12:277-292, 1979.

---2. Steindler A: Kinesiology of the Human Body. Springfield, IL, Charles C.
      Thomas, 1973.

---3. White AA, Panjabi MM: Spinal kinematics. In Goldstein M (ed): The Re-
      search Status of Spinal Manipulative Therapy. NINCDS Monograph No. 15,
      DHEW Publication No. (NIH) 76-998, Stock No. 017-049-00060-7, Washington,
      DC, U.S. Government Printing Office, 1975, pp 93-94.

---4. Williams JGP, Sperryn PN: Sports Medicine, ed 2. Baltimore, Williams &
      Wilkins, 1976, pp 113-114.

---5. Cochran GVB: A Primer of Biomechanics. New York, Churchill Livingstone,
      1982, pp 9-10.

---6.	Cailliet R: Soft Tissue Pain and Disability. Philadelphia, F.A. Davis, p
      10.

---7. Gowitzke BA, Milner M: Understanding Scientific Basis of Human Movement,
      ed 2. Baltimore, Williams & Wilkins, 1980, p 4.

-- 8.	Gonzna ER, Harrington IJ: Biomechanics of Musculoskeletal Injury. Balti=-
      more, Williams & Wilkins, 1982, pp 31-32.

---9. LeVeau B: Williams and Lissner: Biomechanics of Human Motion, ed 2. Phila-
      delphia, W.B. Saunders, 1977, pp 223, 159, 160.

--10.	Dumbleton JH, Black J: Principles of mechanics. In Black J, Dumbleton JH
      (eds): Clinical Biomechanics: Case History Approach.  New York, Churchill
      Livingstone, 1981, pp 374-376.

--11. White AA, Panjabi MM: Clinical Biomechanics of the Spine. Philadelphia,
      J.B. Lippincott, 1978, pp 479, 489.

--12. White AA, Panjabi MM: Spinal kinematics. In Goldstein M (ed): The Re-
      search Status of Spinal Manipulative Therapy. NINCDS Monograph No. 15,
      DHEW Publication No. (NIH) 76-998, Stock No. 017-049-00060-7, Washington,
      DC, U.S. Government Printing Office, 1975, pp 93-101.

--13. White AA, Panjabi MM: Clinical Biomechanics of the Spine. Philadelphia,
      J.B. Lippincott, 1978, p 63.

--14. Tyson HN: Kinematics. New York, John Wiley & Sons, 1966.

--15.	Frankel VH, Nordin M: Basic Biomechanics of the Skeletal System. Philadel-
      phia, Lea & Febiger, 1980.

--16. White AA, Panjabi MM: Spinal kinematics. In Goldstein M (ed): The Re-
      search Status of Spinal Manipulative Therapy. NINCDS Monograph No. 15,
      DHEW Publication No. (NIH) 76-998, Stock No. 017-049-00060-7, Washington,
      DC, U.S. Government Printing Office, 1975, pp 93-101.

--17. LeVeau B: Williams and Lissner: Biomechanics of Human Motion, ed 2. Phila-
      delphia, W.B. Saunders, 1977, pp 4, 159, 223.

--18. LeVeau B: Williams and Lissner: Biomechanics of Human Motion, ed 2. Phila-
      delphia, W.B. Saunders, 1977, pp 159-160.

--19. LeVeau B: Williams and Lissner: Biomechanics of Human Motion, ed 2. Phila-
      delphia, W.B. Saunders, 1977, pp 10, 162.

--20. Morehouse LE, Cooper JM: Kinesiology. St. Louis, C.V. Mosby, 1950, pp
      122-123.

--21. Barham JN, Wooten EP: Structural Kinesiology. New York, Macmillan, 1973,
      pp 98, 131.

--22. Gowitzke BA, Milner M: Understanding Scientific Basis of Human Movement,
      ed 2. Baltimore, Williams & Wilkins, 1980, p 46.

--23. White AA, Panjabi MM: Clinical Biomechanics of the Spine. Philadelphia,
      J.B. Lippincott, 1978, pp 455-456.

--24. Grieve DW: The assessment of gait. Physiotherapy, 55:452-460, 1975.

--25. Barham JN, Wooten EP: Structural Kinesiology. New York, Macmillan, 1973,
      p 99.

--26. LeVeau B: Williams and Lissner: Biomechanics of Human Motion, ed 2. Phila-
      delphia, W.B. Saunders, 1977, pp 165, 167.

--27. White AA, Panjabi MM: Clinical Biomechanics of the Spine. Philadelphia,
      J.B. Lippincott, 1978, pp 456-457.

--28. Barham JN, Wooten EP: Structural Kinesiology. New York, Macmillan, 1973,
      p 96.

--29. LeVeau B: Williams and Lissner: Biomechanics of Human Motion, ed 2. Phila-
      delphia, W.B. Saunders, 1977, pp 164, 165.

--30. Barham JN, Wooten EP: Structural Kinesiology. New York, Macmillan, 1973,
      pp 95, 96-97.

--31. Gowitzke BA, Milner M: Understanding Scientific Basis of Human Movement,
      ed 2. Baltimore, Williams & Wilkins, 1980, p 45.

--32. White AA, Panjabi MM: Clinical Biomechanics of the Spine. Philadelphia,
      J.B. Lippincott, 1978, pp 479-480.

--33. Plagenhoef SC: Computer programs for obtaining kinetic data on human
      movement. Journal of Biomechanics, 1:221-234, 1968.

--34. Plagenhoef SC: Patterns of Human Motion: A Cinematographical Analaysis.
      Englewood Cliffs, NJ, Prentice-Hall, 1971.

--35. Dilman CJ: A kinetic analysis of the recovery leg during sprint running.
      In Cooper JM (ed): Selected Topics on Biomechanics. Chicago, The Athletic
      Institute, 1971.

--36. Carlsoo S: How Man Moves. London, William Heinemann, 1972.

--37. Gowitzke BA, Milner M: Understanding Scientific Basis of Human Movement,
      ed 2. Baltimore, Williams & Wilkins, 1980, p 152.

--38. LeVeau B: Williams and Lissner: Biomechanics of Human Motion, ed 2. Phila-
      delphia, W.B. Saunders, 1977, p 206.

--39. Gowitzke BA, Milner M: Understanding Scientific Basis of Human Movement,
      ed 2. Baltimore, Williams & Wilkins, 1980, p 3.

--40. Cochran GVB: A Primer of Biomechanics. New York, Churchill Livingstone,
      1982, p 6.

--41.	Dumbleton JH, Black J: Principles of mechanics. In Black J, Dumbleton JH
      (eds): Clinical Biomechanics: Case History Approach. New York, Churchill
      Livingstone, 1981, pp 369-372.

--42.	Hatze H: A complete set of control equations for the human musculoskeletal
      system. Journal of Biomechanics, 10:799-805, 1977.

--43.	Dempster WT: The anthropometry of body action. Annuals of the New York 
      Academy of Science, 63:559-585, 1955.

--44. Dempster WT: Free-body diagrams as an approach to the mechanics of human
      posture and locomotion. In Evans FG (ed): Biomechanical Studies of the
      Musculoskeletal System. Springfield, IL, Charles C. Thomas, 1961.

--45. Clauser CE, McConville JT, Young JW: Weight, volume, and center of mass of
      segments of the human body. AMRL-TR-69-70, Wright-Patterson Air Force
      Base, Ohio, 1969.

--46. Drillis R, Contini R: Body segment parameters. Technical Report 1166.03,
      New York, New York University, 1966.

--47.	Stish EE: Anthropokinetics. Journal of Health, Physical Education, and 
      Recreation, 35:33, November-December 1964.

--48. Barham JN, Wooten EP: Structural Kinesiology. New York, Macmillan, 1973,
      pp 101-102.

--49.	Eberhart HD, Inman VT, Bresler B: The principal elements in human locomo-
      tion. In Klopsteg PE, Wilson PD (eds): Human Limbs and Their Substitutes.
      New York, McGraw-Hill, 1954.

--50. Wartenweiler J, Wettstein A: Basic kinetic rules for simple human move-
      ments. In Biomechanics II, Medicine and Sports Series, Baltimore, Univer=-
      sity Park Press, 1971, pp 134-145.

--51. LeVeau B: Williams and Lissner: Biomechanics of Human Motion, ed 2. Phila-
      delphia, W.B. Saunders, 1977, p 172.

--52. Morehouse LE, Cooper JM: Kinesiology. St. Louis, C.V. Mosby, 1950, p 120.

--53. Gowitzke BA, Milner M: Understanding Scientific Basis of Human Movement,
      ed 2. Baltimore, Williams & Wilkins, 1980, pp 65-66

--54. White AA, Panjabi MM: Clinical Biomechanics of the Spine. Philadelphia,
      J.B. Lippincott, 1978, pp 483-486.

--55. Elftman H: The force exerted by the ground in walking. American Journal of 
      Physiology, 10:485-491, 1939.

--56. Morehouse LE, Miller AT Jr: Physiology of Exercise. St. Louis, C.V. Mosby,
      1948, p 225.

--57. Paul JP: Bio-engineering studies of the forces transmitted by joints. In
      Kenedi RM (ed): Biomechanics and Related Bio-engineering Topics. Oxford,
      England, Pergamon, 1965, pp 351-357.

--58.	Gonzna ER, Harrington IJ: Biomechanics of Musculoskeletal Injury. Balti=-
      more, Williams & Wilkins, 1982, pp 13, 120.

--59. LeVeau B: Williams and Lissner: Biomechanics of Human Motion, ed 2. Phila-
      delphia, W.B. Saunders, 1977, p 176.

--60. Gowitzke BA, Milner M: Understanding Scientific Basis of Human Movement,
      ed 2. Baltimore, Williams & Wilkins, 1980, p 78.

--61.	Dapena J: A method to determine the angular momentum of a human body about
      three orthogonal axes passing through its center of gravity. Journal of 
      Biomechanics, 11:251-256, 1978.

--62. Gowitzke BA, Milner M: Understanding Scientific Basis of Human Movement,
      ed 2. Baltimore, Williams & Wilkins, 1980, pp 78-80.

--63. LeVeau B: Williams and Lissner: Biomechanics of Human Motion, ed 2. Phila-
      delphia, W.B. Saunders, 1977, p 182.

--64.	Dumbleton JH, Black J: Principles of mechanics. In Black J, Dumbleton JH
      (eds): Clinical Biomechanics: Case History Approach.  New York, Churchill
      Livingstone, 1981, pp 396-399.

--65. LeVeau B: Williams and Lissner: Biomechanics of Human Motion, ed 2. Phila-
      delphia, W.B. Saunders, 1977, pp 143-144.

--66. Cochran GVB: A Primer of Biomechanics. New York, Churchill Livingstone,
      1982, pp 128-131.

--67. Gowitzke BA, Milner M: Understanding Scientific Basis of Human Movement,
      ed 2. Baltimore, Williams & Wilkins, 1980, pp 80-81.

--68. White AA, Panjabi MM: Clinical Biomechanics of the Spine. Philadelphia,
      J.B. Lippincott, 1978, p 462.

--69. Barnett CH, Cobbold AF: Lubrication within living joints. Journal of Bone
      and Joint Surgery, 44B:662-784, 1962.

--70. Linn FC: Lubrication of animal joints. Journal of Biomechanics,
      1:193-205, 1968.

--71.	Jackson R: The Cervical Syndrome, ed 2. Springfield, IL, Charles C. Thom-
      as, 1958.

--72. Knocke FJ, Knocke LS: Orthopaedic Nursing. Philadelphia, F.A. Davis, 1945.

--73. Markolf KL: Stiffness and damping characteristics of the thoracic-lumbar
      spine. Proceedings: Workshop on Bioengineering Approaches to the Problems
      of the Spine. National Institutes of Health, September 1970.

--74. Travers PR, Campbell WR: The organism and speed and power. In Larson LA
      (ed): Fitness, Health, and Work Capacity. New York, Macmillan, 1974, pp
      100-110.

--75. Barham JN, Wooten EP: Structural Kinesiology. New York, Macmillan, 1973,
      pp 107-108.

--76. Morehouse LE, Cooper JM: Kinesiology. St. Louis, C.V. Mosby, 1950, pp
      196-204.

--77.	Richards JG, Cooper J: Implementation of an on-line isokinetic analysis
      system. Journal of Orthopedics and Sports, 4:36-38, 1982.

--78. Barham JN, Wooten EP: Structural Kinesiology. New York, Macmillan, 1973,
      pp 82, 85.

--79. LeVeau B: Williams and Lissner: Biomechanics of Human Motion, ed 2. Phila-
      delphia, W.B. Saunders, 1977, pp 184-185.

--80. Morehouse LE, Cooper JM: Kinesiology. St. Louis, C.V. Mosby, 1950, p 115.

--81. Barham JN, Wooten EP: Structural Kinesiology. New York, Macmillan, 1973,
      p 86.

--82. LeVeau B: Williams and Lissner: Biomechanics of Human Motion, ed 2. Phila-
      delphia, W.B. Saunders, 1977, p 193.

--83. Gowitzke BA, Milner M: Understanding Scientific Basis of Human Movement,
      ed 2. Baltimore, Williams & Wilkins, 1980, pp 69-70.

--84. Morehouse LE, Cooper JM: Kinesiology. St. Louis, C.V. Mosby, 1950, pp
      101, 103.

--85. Barham JN, Wooten EP: Structural Kinesiology. New York, Macmillan, 1973,
      pp 78-81.

--86. LeVeau B: Williams and Lissner: Biomechanics of Human Motion, ed 2. Phila-
      delphia, W.B. Saunders, 1977, pp 48, 50, 64.

--87. Gowitzke BA, Milner M: Understanding Scientific Basis of Human Movement,
      ed 2. Baltimore, Williams & Wilkins, 1980, pp 123-130.

--88. Morehouse LE, Miller AT Jr: Physiology of Exercise. St. Louis, C.V. Mosby,
      1948, pp 223-224.

--89. Lipovetz FJ: Medical Physical Education, rev ed. Minneapolis, Burgess,
      1946.

--90. McCloy CH: Some notes of differential actions of partite muscles. Research
      Quarterly, American Association of Health, Physical Education, and
      Recreation, 17:254, 1946.

--91. Travers PR, Campbell WR: The organism and speed and power. In Larson LA
      (ed): Fitness, Health, and Work Capacity. New York, Macmillan, 1974, pp
      99-100.

--92. Elftman H: Biomechanics of muscle. Journal of Bone and Joint Surgery,
      48A:363-377, 1966.

--93. Morehouse LE, Miller AT Jr: Physiology of Exercise. St. Louis, C.V. Mosby,
      1948, pp 217-218.

--94. Morehouse LE, Cooper JM: Kinesiology. St. Louis, C.V. Mosby, 1950, pp
      189-191.

--95. Barham JN, Wooten EP: Structural Kinesiology. New York, Macmillan, 1973,
      pp 108-111.

--96. Gowitzke BA, Milner M: Understanding Scientific Basis of Human Movement,
      ed 2. Baltimore, Williams & Wilkins, 1980, p 95.

--97. Huxley AF, Niedergerke R: Structural changes in muscle during contraction;
      interference microscopy of living muscle fibers. Nature (British),
      173:971, 1954.

--98. LeVeau B: Williams and Lissner: Biomechanics of Human Motion, ed 2. Phila-
      delphia, W.B. Saunders, 1977, pp 23-24.

--99.	Hollingshead WH, Jenkins DB: Functional Anatomy of the Limbs and Back.
      Philadelphia, W.B. Saunders, 1981, pp 32-34.

-100.	Rosse C, Clawson DK: The Musculoskeletal System in Health & Disease.
      Hagerstown, PA, Harper & Row, 1980, pp 39-47, 51-52.

-101. Henneman E, Olson CB: Relations between structure and function in the
      design of skeletal muscle. Journal of Neurophysiology, 28:581, 1965.

-102. Huxley HE: The mechanism of muscular contraction. Science, 164:1356, 1969.

-103. Morehouse LE, Cooper JM: Kinesiology. St. Louis, C.V. Mosby, 1950, pp
      111-113, 188-189.

-104. Gowitzke BA, Milner M: Understanding Scientific Basis of Human Movement,
      ed 2. Baltimore, Williams & Wilkins, 1980, p 115.

-105. Walker SM, Schrodt GR: Contraction of skeletal muscle. American Journal of
      Physical Medicine, 46:151, 1967.

-106. Jordan HE: The structural changes in striped muscle during contraction.
      Physiology Review, 13:301, 1933.

-107. Lupton H: An analysis of the effects of speed on the mechanical efficiency
      of human muscular movement. Journal of Physiology, 57:337, 1923.

-108. Gowitzke BA, Milner M: Understanding Scientific Basis of Human Movement,
      ed 2. Baltimore, Williams & Wilkins, 1980, pp 97?-102.

-109. Barham JN, Wooten EP: Structural Kinesiology. New York, Macmillan, 1973,
      pp 90-91.

-110. Morehouse LE, Miller AT Jr: Physiology of Exercise. St. Louis, C.V. Mosby,
      1948, p 225.

-111. Gowitzke BA, Milner M: Understanding Scientific Basis of Human Movement,
      ed 2. Baltimore, Williams & Wilkins, 1980, pp 61-65, 117-118.

-112. LeVeau B: Williams and Lissner: Biomechanics of Human Motion, ed 2. Phila-
      delphia, W.B. Saunders, 1977, pp 22, 23.

-113. Morehouse LE, Cooper JM: Kinesiology. St. Louis, C.V. Mosby, 1950, pp
      113-114.

-114.	Rosse C, Clawson DK: The Musculoskeletal System in Health & Disease.
      Hagerstown, PA, Harper & Row, 1980, p 57.

-115. Gowitzke BA, Milner M: Understanding Scientific Basis of Human Movement,
      ed 2. Baltimore, Williams & Wilkins, 1980, pp 109-112.

-116. Morehouse LE, Cooper JM: Kinesiology. St. Louis, C.V. Mosby, 1950, pp
      114-115.

-117. LeVeau B: Williams and Lissner: Biomechanics of Human Motion, ed 2. Phila-
      delphia, W.B. Saunders, 1977, pp 22, 23.

-118.	Rosse C, Clawson DK: The Musculoskeletal System in Health & Disease.
      Hagerstown, PA, Harper & Row, 1980, pp 57-58.

-119. Morehouse LE, Miller AT Jr: Physiology of Exercise. St. Louis, C.V. Mosby,
      1948, pp 225-226.

-120. Morehouse LE, Cooper JM: Kinesiology. St. Louis, C.V. Mosby, 1950, p 193.

-121. Williams JGP, Sperryn PN: Sports Medicine, ed 2. Baltimore, Williams &
      Wilkins, 1976, pp 109-110.

-122. Tanner JM: The Physique of the Olympic Athlete. London, George Allen and
      Unwin, 1964.

-123. Williams JGP, Sperryn PN: Sports Medicine, ed 2. Baltimore, Williams &
      Wilkins, 1976, pp 9, 23, 145.

-124. De Garay A, Carter L: Genetic and Anthropological Studies of Olympic Ath=-
      letes. New York, Academic Press, 1974.

-125.	Hollingshead WH, Jenkins DB: Functional Anatomy of the Limbs and Back.
      Philadelphia, W.B. Saunders, 1981, p 35.

-126. Schafer RC: Chiropractic Management of Sports and Recreational Injuries.
      Baltimore, Williams & Wilkins, 1982, p 101.

-127.	Salter RB: Textbook of Disorders and Injuries of the Musculoskeletal 
      System. Baltimore, Williams & Wilkins, 1981, p 396.

-128.	Rosse C, Clawson DK: The Musculoskeletal System in Health & Disease.
      Hagerstown, PA, Harper & Row, 1980, p 88.

-129. Barham JN, Wooten EP: Structural Kinesiology. New York, Macmillan, 1973,
      pp 119-121.

-130.	Gonzna ER, Harrington IJ: Biomechanics of Musculoskeletal Injury. Balti=-
      more, Williams & Wilkins, 1982, p 32.

-131.	Dumbleton JH, Black J: Principles of mechanics. In Black J, Dumbleton JH
      (eds): Clinical Biomechanics: Case History Approach.  New York, Churchill
      Livingstone, 1981, pp 382-385.

-132.	Evans FG: Stress and Strain in Bones. Springfield, IL, Charles C. Thomas,
      1957.

-133. White AA, Panjabi MM: Clinical Biomechanics of the Spine. Philadelphia,
      J.B. Lippincott, 1978, pp 496-497, 500-501.

-134.	Gonzna ER, Harrington IJ: Biomechanics of Musculoskeletal Injury. Balti=-
      more, Williams & Wilkins, 1982, pp 41, 120, 171.

-135. LeVeau B: Williams and Lissner: Biomechanics of Human Motion, ed 2. Phila-
      delphia, W.B. Saunders, 1977, pp 4, 223.

-136. Cochran GVB: A Primer of Biomechanics. New York, Churchill Livingstone,
      1982, pp 30-38.

-137.	Gonzna ER, Harrington IJ: Biomechanics of Musculoskeletal Injury. Balti=-
      more, Williams & Wilkins, 1982, p 120.

-138. Cochran GVB: A Primer of Biomechanics. New York, Churchill Livingstone,
      1982, pp 27-30.

-139.	Frankel VH, Nordin M: Basic Biomechanics of the Skeletal System. Philadel-
      phia, Lea & Febiger, 1980.

-140. White AA, Panjabi MM: Clinical Biomechanics of the Spine. Philadelphia,
      J.B. Lippincott, 1978, pp 470-472, 475, 484, 498-499, 507, 513.

-141.	Panjabi MM, Krag MH, Goel VK: A technique for measurement and description
      of three-dimensional six degree-of-freedom motion of a body joint with an
      application to the human spine. Journal of Biomechanics, 14:447-460, 1981.

-142. Cochran GVB: A Primer of Biomechanics. New York, Churchill Livingstone,
      1982, pp 76-78, 131-132.

-143.	Black J, Dumbleton JH (eds): Clinical Biomechanics: Case History Approach.
      New York, Churchill Livingstone, 1981, pp 18-19, 21, 387, 388.

-144. Cochran GVB: A Primer of Biomechanics. New York, Churchill Livingstone,
      1982, pp 73-75, 84-85.

-145.	Dumbleton JH, Black J: Principles of mechanics. In Black J, Dumbleton JH
      (eds): Clinical Biomechanics: Case History Approach.  New York, Churchill
      Livingstone, 1981, p 393.

-146. Cochran GVB: A Primer of Biomechanics. New York, Churchill Livingstone,
      1982, pp 84, 147

-147. White AA, Panjabi MM: Clinical Biomechanics of the Spine. Philadelphia,
      J.B. Lippincott, 1978, pp 212, 456, 469-470, 490, 514, 515.

-148. Cochran GVB: A Primer of Biomechanics. New York, Churchill Livingstone,
      1982, pp 75-76, 158-159, 192.

-149.	Burstein AH, et al: The ultimate properties of bone tissue: The effects of
      yielding. Journal of Biomechanics, 5:35, 1972.

-150. Cochran GVB: A Primer of Biomechanics. New York, Churchill Livingstone,
      1982, pp 87, 91-96.

-151. White AA, Panjabi MM: Clinical Biomechanics of the Spine. Philadelphia,
      J.B. Lippincott, 1978, pp 347-348, 352, 478, 511-513.

-152.	Twomey L, Taylor J: Flexion creep deformation and hysteresis in the lumbar
      vertebral column. Spine, 7:116-122, 1982.

-153. Cochran GVB: A Primer of Biomechanics. New York, Churchill Livingstone,
      1982, pp 89-90, 170-172.

-154. White AA, Panjabi MM: Clinical Biomechanics of the Spine. Philadelphia,
      J.B. Lippincott, 1978, pp 474-475.

-155.	Rosse C, Clawson DK: The Musculoskeletal System in Health & Disease.
      Hagerstown, PA, Harper & Row, 1980, pp 80-81, 82.

-156.	Salter RB: Textbook of Disorders and Injuries of the Musculoskeletal 
      System. Baltimore, Williams & Wilkins, 1981, pp 22-23, 194.

-157.	Gonzna ER, Harrington IJ: Biomechanics of Musculoskeletal Injury. Balti=-
      more, Williams & Wilkins, 1982, pp 31-32.

-158.	Goldie IF, Dumbleton JH: In Black J, Dumbleton JH (eds): Clinical Bio-
      mechanics: Case History Approach. New York, Churchill Livingstone, 1981,
      pp 83, 84, 87.

-159.	Goldie IF, Dumbleton JH: Intertrochanteric osteotomy of the femur. In
      Black J, Dumbleton JH (eds): Clinical Biomechanics. New York, Churchill
      Livingstone, 1981, p 83.

-160.	Edwards CC, Chrisman OD: Articular cartilage. In Albright JA, Brand RA
      (eds): The Scientific Basis of Orthopaedics. New York, Appleton-Century-
      Crofts, 1979, pp 329-330.

-161. King AI, Vulcan P: Elastic deformation characteristics of the spine. Pro-
      ceedings: Workshop on Bioengineering Approaches to Problems of the Spine.
      National Institutes of Health, September 1970, p 15.

-162. Cochran GVB: A Primer of Biomechanics. New York, Churchill Livingstone,
      1982, p 231.

-163.	Dumbleton JH, Black J: Principles of mechanics. In Black J, Dumbleton JH
      (eds): Clinical Biomechanics: Case History Approach. New York, Churchill
      Livingstone, 1981, pp 377-382, 388-389.

-164. White AA, Panjabi MM: Clinical Biomechanics of the Spine. Philadelphia,
      J.B. Lippincott, 1978, p 468.

-165. Cochran GVB: A Primer of Biomechanics. New York, Churchill Livingstone,
      1982, pp 23, 26-27, 82, 190-193.

-166.	McElhaney JH: Dynamic response of bone and muscle tissue. Journal of Ap-
      plied Physiology, 21:231-236, 1966.

-167.	Gonzna ER, Harrington IJ: Biomechanics of Musculoskeletal Injury. Balti=-
      more, Williams & Wilkins, 1982, pp 10, 32.

-168. White AA, Panjabi MM: Clinical Biomechanics of the Spine. Philadelphia,
      J.B. Lippincott, 1978, p 470.

-169.	Sandover J: Dynamic loading as a possible source of low-back disorders.
      Spine, 8:652-658, 1983.

-170.	Liu YK, Njus G, Buckwalter J, Wakano K: Fatigue response of lumbar inter=-
      vertebral joints under axial cyclic loading. Spine, 8:857-865, 1983.

-171.	Adams MA, Hutton WC: The effect of fatigue on the lumbar intervertebral
      disc. Journal of Bone and Joint Surgery, 65B:199-203, 1983.

-172.	Brand RA: Joint lubrication. In Albright JA, Brand RA (eds): The Scien=-
      tific Basis of Orthopaedics. New York, Appleton-Century-Crofts, 1979, pp
      349-363.

-173.	Dumbleton JH, Black J: Principles of mechanics. In Black J, Dumbleton JH
      (eds): Clinical Biomechanics: Case History Approach. New York, Churchill
      Livingstone, 1981, pp 396-399.

-174. Cochran GVB: A Primer of Biomechanics. New York, Churchill Livingstone,
      1982, pp 125, 133-138.

-175.	Rosse C, Clawson DK: The Musculoskeletal System in Health & Disease.
      Hagerstown, PA, Harper & Row, 1980, pp 87-88.

-176.	Wright V: Lubrication and Wear in Joints. Philadelphia, J.B. Lippincott,
      1969.

-177.	Edwards CC, Chrisman OD: Articular cartilage. In Albright JA, Brand RA
      (eds): The Scientific Basis of Orthopaedics. New York, Appleton-Century-
      Crofts, 1979, pp 330-331.

-178. Jones ED: Joint lubrication. Lancet, 1:1426, 1934.

-179.	Gonzna ER, Harrington IJ: Biomechanics of Musculoskeletal Injury. Balti=-
      more, Williams & Wilkins, 1982, pp 32-33.

-180.	Rybicki EF, Glaeser WA, Strenkowski JS, Tamm MA: Effects of cartilage
      stiffness and viscosity on a nonporous compliant bearing lubrication model
      for living joints. Journal of Biomechanics, 12:403-409, 1979.

-181.	Edwards CC, Chrisman OD: Articular cartilage. In Albright JA, Brand RA
      (eds): The Scientific Basis of Orthopaedics. New York, Appleton-Century-
      Crofts, 1979, pp 319, 331, 334.

-182.	Turek SL: Orthopaedics: Principles and Their Application, ed 3. Philadel-
      phia, J.B. Lippincott, 1977, pp 17-19, 331-332, 347.

-183. Cochran GVB: A Primer of Biomechanics. New York, Churchill Livingstone,
      1982, pp 124-125.

-184.	Holm S, Nachemson A: Variations in the nutrition of the canine interverte-
      bral disc induced. Spine, 8:866-874, 1983.


                               Uncited References  

Benedek GB, Villars FMH: Physics. Reading, MA, Addison-Wesley, vol 1: Mechanics,
  1973.

Halliday D, Resnick R: Fundamentals of Physics. New York, John Wiley & Sons,
  1974.

Kramer DJ: Kinematics for the Handicapped. New York, Exposition Press, 1973.

Metheny E: Body Dynamics. New York, McGraw-Hill, 1952.

Miller DI, Nelson RC: Biomechanics of Sport. Philadelphia, Lea & Febiger, 1973.

Miller F Jr: College Physics, ed 3. New York, Harcourt, Brace, Jovanovich, 1974.

Ralston HJ, Lukin L: Energy levels of human body segments. Ergonomics, 12:39,
  1969.

Ripley JA: The Elements and Structure of the Physical Sciences. New York, John
  Wiley & Sons, 1964.

Ruch TC, Patton HD: Physiology and Biophysics. Philadelphia, W.B. Saunders,
  1974.

Schenck JM, Cordova FD: Introductory Biomechanics, ed 2. Philadelphia. F.A.
  Davis, 1980.

Straley JW: Basic Physics. Englewood Cliffs, NJ, Prentice-Hall, 1974.

Wells KF: Kinesiology, ed 5. Philadelphia, W.B. Saunders, 1971.