Kinematics is the study of the geometric and time-dependent aspects of motion without analyzing the forces causing the motion.

Kinetics is the study of the forces that cause motion of a rigid body. These forces can be classified as either external forces or internal forces.

The general unconstrained movement in three-dimensional space requires the description of three translations and three rotations to fully describe joint motion.

External forces represent the action of objects contacting the body, gravitational forces, or force due to inertia of the body.

Internal forces are the body’s responses to external forces, which consist of muscle, ligament, and joint contact forces.

Because of relatively smaller mechanical advantages, large muscle and tendon forces, and thus internal joint forces, are expected when performing any activities.

The anatomic structure responsible for joint constraint can be divided into passive and active elements. Passive elements, consisting of the capsulo-ligamentous structures and bony articulating surfaces, provide the static constraints of the joint. The active elements include muscle-tendon units, which provide dynamic joint constraints.

## Introduction

Biomechanics combines the field of engineering mechanics with the fields of biology and physiology. Biomechanics applies mechanical principles to the human body to understand the mechanical influences on bone and joint health. Forces that load the joints are generated by muscles and transmitted by tendons. Bones must withstand these forces. Developments in the field of biomechanics have improved our understanding of normal and pathologic gait, mechanics of neuromuscular control, and mechanics of growth and form. This knowledge contributed to the development of medical diagnostic and treatment procedures; it provided the basis for the design and manufacture of medical implants and orthotic devices and enhanced rehabilitation therapy practices. Biomechanics is also used to improve human performance in the workplace and in athletic competition.

Mechanics is a branch of physics that is concerned with the motion and deformation of bodies that are acted on by mechanical forces. Mechanics is one of the oldest physical sciences, dating back to Aristotle (384–322 bc ), who conducted an organized analytical analysis of animal movement. Leonardo da Vinci (1452–1519) investigated the mechanics of the human body; his detailed anatomic sketches represent the birth of anatomy, as a discipline, and of mechanics as the science governing human motion. Although da Vinci wrote extensively on body mechanics, the man generally credited to be the father of modern biomechanics is Giovanni Alphonso Borelli (1608–1679). His book *De Motu Animalian* provided a quantitative graphical solution to a musculoskeletal biomechanics problem ( Fig. 6.1 ).

Engineering mechanics is the discipline devoted to the solution of mechanical problems through the integrated application of mathematical, scientific, and engineering principles. With roots in physics and mathematics, engineering mechanics is the basis of all engineering mechanical sciences. Engineering mechanics is an applied mechanics branch of the physical sciences. The broad field of applied mechanics can be further divided into three main parts: rigid body mechanics, deformable body mechanics, and fluid mechanics. In general, a material can be characterized as either a solid or fluid. Solid materials can be considered to be rigid or deformable. A rigid body is one that cannot be deformed. In reality, every object undergoes deformation to some extent when acted upon by external forces. However, this is a definition of convenience that is used to simplify complex problems. For example, during the study of movement in gait analysis, the bones are considered to be rigid bodies when compared with the soft tissues joining the bones. External loads applied to a rigid body result in internal loads, stresses, and deformations. The mechanics of deformable bodies deal with the relationships between externally applied loads and their internal effects. The mechanics of deformable bodies have strong ties with the field of materials science and are more complex when compared with the analyses required in rigid body mechanics. The purpose of this chapter is to introduce the reader to the field of biomechanics. We will focus on rigid body mechanics.

Basic biomechanics relies heavily on Newtonian mechanics. These laws were introduced by Sir Isaac Newton and form the basis for analyses in statics and dynamics. Statics analyzes the forces that occur in rigid bodies that are in static equilibrium. Dynamics is the study of bodies in motion. The general field of dynamics consists of two major areas: kinematics and kinetics. Ultimately, proper joint constraint and stability enables limb function in characteristic ways.

## Kinematics

Kinematics is the study of the geometric and time-dependent aspects of motion without analyzing the forces causing the motion. Kinematic analysis is used to relate displacement, velocity, acceleration, and time. To study kinematics in an organized manner, it is common to classify the motion as translational, rotational, or general. Translational motion occurs when a straight line drawn between two points on the body remains in the same direction during the motion. Translational motion can be either rectilinear motion (if the paths are straight lines) or curvilinear (if the paths are curved lines). Rotational motion occurs when the points on the body move in a circular path around an axis of rotation. The angular motion occurs about a central line known as the *axis of rotation* , which lies perpendicular to the plane of motion. The third class of motion is called *general motion* or *displacement* ; this type of motion occurs if a body undergoes both translational and rotational motion simultaneously.

Kinematics can be analyzed in two-dimensional (2-D) or three-dimensional (3-D) space. When all points of a rigid body move parallel to a plane, the motion is referred to as *planar motion* , which can be thought of as 2-D motion. The more general type of rigid body motion is 3-D motion. Six independent parameters are required to describe 3-D motion. These parameters are called *degrees of freedom* , or the number of independent coordinates in a coordinate system that is required to completely specify the position of an object in space. A rigid body in space has a maximum of six degrees of freedom, consisting of three translations (expressed by linear coordinates) and three rotations (expressed by angular coordinates). The general movement of an object is defined by a vector quantity that is a combination of both linear and angular displacement. Velocity is the time-related change of displacement. Linear velocity is expressed in units of length per time (e.g., m/sec). Angular velocity is expressed in units of angular measure per time (e.g., rad/sec). Because velocities reflect vector quantities, both magnitude and direction must be specified. Acceleration is the time-rate of change of velocity. Linear acceleration is expressed in units of length per time squared (e.g., m/sec2). Angular acceleration is the time rate of change in angular velocity (e.g., rad/sec2). Accelerations are also vector quantities, and both magnitude and direction must be specified.

Kinematic techniques have been used to study body movements in both 2-D and 3-D space. The human body is typically modeled as a number of interconnected rigid body segments ( Fig. 6.2 ). A coordinate system is affixed to each rigid body segment to establish an anatomic coordinate system. External markers are used to define orthogonal coordinate systems whose axes define the position of these body segments. Joint motion is then described as the relative motion of the distal body segment with respect to the proximal body segment. Limb segments are assumed to undergo angular displacement during human movement. However, more sophisticated analyses will also quantify the linear displacement that limb segments may undergo. These measures of relative segmental angles have been used to describe human walking and other activities of daily living.

For example, knee motion in the sagittal plane can be characterized throughout the gait cycle ( Fig. 6.3 ). At heel strike, the knee is nearly fully extended (with knee flexion of only five degrees). During midstance, the knee flexes to about 15 degrees, which occurs at 15% of the gait cycle. The knee joint is brought back into extension by midstance. At 50% of the gait cycle, opposite foot contact occurs. The weight is shifted to the opposite limb and the knee begins flexing. Toe-off occurs at 60% of the gait cycle. Peak knee flexion of 60 degrees occurs during the early portion of the swing phase. The knee motion can be described as two flexion waves, each starting in relative extension, progressing into flexion, and then returning again to extension. The first flexion wave, or stance phase knee flexion, acts as a shock absorber to aid weight acceptance. This curve peaks in early stance at opposite foot-off. The mechanical source for this shock absorber is the eccentrically contracting quadriceps muscles. The second flexion wave is necessary to clear the foot in the early swing phase. The knee is rapidly flexed beginning just after heel rise to a maximum in the swing phase just as the swinging foot passes the opposite limb.

The complexity of kinematic analysis increases substantially when one moves from planar analysis to 3-D analysis. The complexity of the analysis arises from a technical difficulty—that is, large rigid body rotations cannot be treated as vectors and hence do not obey the vectorial principles of transformation, independence, and interchangeability of operations. For finite spatial rotation, the sequence of rotations is extremely important and must be specified for a unique description of joint motion. For the same amount of rotation, different final orientations will result from different sequences of rotation ( Fig. 6.4 ). However, with proper selection and definition of the axes of rotation between two bony segments, it is possible to make finite rotation sequence independent or communitive. Investigators in orthopedic biomechanics have adopted the concept of Eulerian angles to unify the definition of finite spatial rotation. In the selection of reference axes, one axis is fixed to the stationary segment and another axis is fixed to the moving segment ( Fig. 6.5 ). In the knee joint, for example, the flexion/extension angle Φ occurs about a medial-laterally directed axis defined by a line connecting the medial and lateral femoral condyles. The axial rotation angle, Ψ, is measured about an axis defined by the line along the shaft of the tibia. The third axis, also defined as the *floating axis* , is orthogonal to the other two axes and defines abduction/adduction (Θ). These rotations match the Eulerian angle description and are thought to be performed in such a way as to bring the moving segment from the reference orientation into the current orientation. The advantage of using this system for description of the spatial rotation of anatomic joints is that the angular rotations do not have to be referred back to the neutral position of the joint because the rotation sequence can be independent, and thus, the measurement can be easily obtained and related to anatomic structures.

A complete analysis of total joint movement (i.e., six degrees of freedom) can be obtained using markers embedded in the bone or dual fluoroscopic imaging techniques. This general unconstrained movement in 3-D space requires the description of three translations and three rotations to fully describe joint motion. The most commonly used analytic method for description of six-degrees-of-freedom displacement of a rigid body is the screw displacement axis or helical axis.

## Kinetics

Kinetics is the study of the forces that cause motion of a rigid body. When unbalanced forces or moments are acting on a rigid body, it is under a nonequilibrium, or dynamic, condition, resulting in motion. Understanding the kinetics of human movement provides a fundamental understanding of the musculoskeletal system. Before one can begin to analyze the forces during human movement, some basic definitions are necessary, and some basic assumptions must be made.

The key quantities in kinetics are *force, moment,* and *torque.* A force represents an interaction between two bodies. According to Newton’s Second Law, force is any action that tends to change the state of rest or state of motion of a body to which it is applied. Forces can be contact forces (as when bodies touch each other) or field forces (as when bodies are separated by a distance, such as gravitational, electric, or magnetic forces). Forces are represented by *vectors* , which are composed of four components: magnitude, direction, sense, and position (also called *point of application* ). The vector may be resolved into several component forces, usually along specified, mutually perpendicular coordinate axes. Conversely, forces can be summed using vectorial addition.

A moment represents the turning, twisting, or rotational effect of a force. A moment is a vector. A moment is defined as the product of the force and the perpendicular distance between the line of action of the force and the axis of rotation of the motion that the force produces ( Fig. 6.6 ). Its magnitude is the force times the perpendicular distance to the axis of rotation. The direction of the moment is along the axis of rotation (or potential rotation) and, thus, perpendicular to the plane in which the twisting force is applied. The moment arm—that is, the distance used to calculate the moment—is the shortest distance from the force action line to the actual or potential pivot point of the system, regardless of the state of motion. Skeletal motions are the result of moments applied by muscles that cross the joints on which they act. The moment of a force about an axis measures the tendency of the force to impart to the body a motion of rotation about a fixed axis.

A torque is a special type of moment that results when a pair of forces that have equal magnitude, parallel lines of action, and opposite senses act on a body ( Fig. 6.7 ). The magnitude of the torque is Fd, where *d* is the perpendicular distance between the two forces. The resultant force is zero because the two forces are equal and oppose each other.

Kinetics can be used to analyze forces affecting the musculoskeletal system. These forces can be classified as either external or internal. External forces represent the action of objects contacting the body, gravitational forces, or force due to inertia of the body. Internal forces are the body’s responses to external forces. Internal forces consist of muscle, ligament, and joint contact forces. In general, the limb segments are assumed to be rigid, which simplifies the analysis because it is assumed that the structure does not deform under load. Further, joints are assumed to be frictionless hinges.

Statics is the study of forces acting on a body at rest. When performing a force analysis, the body or part of a body at equilibrium may be isolated from the environment, and the environment is replaced by forces acting on the system, resulting in what is called a *free-body diagram.* Because both forces and moments are vectors, they must sum to zero in each of the three perpendicular directions (reference system). Consider a person standing quietly ( Fig. 6.8 ). The person’s weight (force of gravity) tends to pull the person toward the ground ( Fig. 6.8A ). The person does not move downward because the ground is pushing up with a total force equal in magnitude to the individual’s weight (Newton’s Third Law). If the person has his or her weight distributed symmetrically, then his or her weight is evenly shared by each lower extremity ( Fig. 6.8A ). The load under each foot can be represented by resultant ground reaction force (GRF) vectors, with one-half of the body weight supported by each foot. With one-half of body weight supported by each foot, the point of application of the resultant GRF passes approximately midway between the subject’s two feet ( Fig. 6.8B ). No motion occurs because the external forces (i.e., body weight and the GRF) are balanced—they are equal and opposite in magnitude. When the person leans to one side ( Fig. 6.8C ), the GRF shifts in the direction that the person leans. Thus, overall body posture can affect the GRF location. If the person leans more, he or she will become unstable (i.e., the downward body weight vector will fall outside the base of support). To remain in static equilibrium, the subject requires additional support ( Fig. 6.8D ). The horizontal force applied to the upper body by the wall is balanced with an equal and opposite horizontal GRF component. Also, the two equal and opposite vertical forces are no longer aligned (i.e., collinear). They form a force couple that would tend to rotate the body in a counterclockwise direction. A second clockwise force couple formed by the two equal and opposite horizontal forces balances the counterclockwise force couple. As a result, the subject remains in both translational and rotational static equilibrium.