PART I Biomechanical Principles Fd Fs θ s rs rd θ d θ ° ≈ MAd = rdsin( d) = (20 cm)sin(5 ) 2 cm θ ° ≈ MAs = rssin( s) = (2 cm)sin(80 ) 2 cm Chapter 1: Introduction to Biomechanical Analysis Chapter 2: Mechanical Properties of Materials Chapter 3: Biomechanics of Bone Chapter 4: Biomechanics of Skeletal Muscle Chapter 5: Biomechanics of Cartilage Chapter 6: Biomechanics of Tendons and Ligaments Chapter 7: Biomechanics of Joints 1 PART I This part introduces the reader to the basic principles used throughout this book to understand the structure and function of the musculoskeletal system. Biomechanics is the study of biological systems by the application of the laws of physics. The pur- poses of this part are to review the principles and tools of mechanical analysis and to describe the mechanical behavior of the tissues and structural units that compose the musculoskeletal system. The specific aims of this part are to ■ Review the principles that form the foundation of biomechanical analysis of rigid bodies ■ Review the mathematical approaches used to perform biomechanical analysis of rigid bodies ■ Examine the concepts used to evaluate the material properties of deformable bodies ■ Describe the material properties of the primary biological tissues constituting the musculoskeletal system: bone, muscle, cartilage, and dense connective tissue ■ Review the components and behavior of joint complexes By having an understanding of the principles of analysis in biomechanics and the bio- mechanical properties of the primary tissues of the musculoskeletal system, the reader will be prepared to apply these principles to each region of the body to understand the mechanics of normal movement at each region and to appreciate the effects of impairments on the pathomechanics of movement. 2 CHAPTER 1 Introduction to Biomechanical Analysis ANDREW R. KARDUNA, PH.D. MATHEMATICAL OVERVIEW . .4 Units of Measurement . .4 Trigonometry . .4 Vector Analysis . .5 Coordinate Systems . .7 FORCES AND MOMENTS . .8 Forces . .8 Moments . .9 Muscle Forces . .10 STATICS . .11 Newton’s Laws . .11 Solving Problems . .12 Simple Musculoskeletal Problems . .12 Advanced Musculoskeletal Problems . .14 KINEMATICS . .17 Rotational and Translational Motion . .17 Displacement, Velocity, and Acceleration . .18 KINETICS . .18 Inertial Forces . .18 Work, Energy, and Power . .18 Friction . .19 SUMMARY . .20 Although the human body is an incredibly complex biological system composed of trillions of cells, it is subject to the same fundamental laws of mechanics that govern simple metal or plastic structures. The study of the response of biological systems to mechanical forces is referred to as biomechanics. Although it wasn’t recognized as a formal discipline until the 20th century, biomechanics has been studied by the likes of Leonardo da Vinci, Galileo Galilei, and Aristotle. The application of biomechanics to the musculoskeletal system has led to a better understanding of both joint function and dysfunction, resulting in design improve- ments in devices such as joint arthroplasty systems and orthotic devices. Additionally, basic musculoskeletal biomechanics concepts are important for clinicians such as orthopaedic sur- geons and physical and occupational therapists. Biomechanics is often referred to as the link between structure and function. While a ther- apist typically evaluates a patient from a kinesiologic perspective, it is often not practical or necessary to perform a complete biomechanical analysis. However, a comprehensive 3 4 Part I | BIOMECHANICAL PRINCIPLES knowledge of both biomechanics and anatomy is needed to understand how the muscu- loskeletal system functions. Biomechanics can also be useful in a critical evaluation of current or newly proposed patient evaluations and treatments. Finally, a fundamental understanding of biomechanics is necessary to understand some of the terminology associated with kinesiology (e.g., torque, moment, moment arms). The purposes of this chapter are to ■ Review some of the basic mathematical principles used in biomechanics ■ Describe forces and moments ■ Discuss principles of static analysis ■ Present the basic concepts in kinematics and kinetics The analysis is restricted to the study of rigid bodies. Deformable bodies are discussed in Chapters 2–6. The material in this chapter is an important reference for the force analysis chapters throughout the text. MATHEMATICAL OVERVIEW completely useless. If a patient was told to perform a series of exercises for two, the patient would have no idea if that This section is intended as a review of some of the basic math- meant two days, weeks, months, or even years. ematical concepts used in biomechanics. Although it can be The units used in biomechanics can be divided into two skipped if the reader is familiar with this material, it would categories. First, there are the four fundamental units of be helpful to at least review this section. length, mass, time, and temperature, which are defined on the basis of universally accepted standards. Every other unit Units of Measurement is considered a derived unit and can be defined in terms of these fundamental units. For example, velocity is equal to The importance of including units with measurements cannot length divided by time and force is equal to mass multiplied be emphasized enough. Measurements must be accompanied by length divided by time squared. A list of the units needed by a unit for them to have any physical meaning. Sometimes, for biomechanics is found in Table 1.1. there are situations when certain units are assumed. If a clinician asks for a patient’s height and the reply is “5-6,” it Trigonometry can reasonably be assumed that the patient is 5 feet, 6 inches tall. However, that interpretation would be inaccurate if the Since angles are so important in the analysis of the muscu- patient was in Europe, where the metric system is used. There loskeletal system, trigonometry is a very useful biomechanics are also situations where the lack of a unit makes a number tool. The accepted unit for measuring angles in the clinic is TABLE 1.1 Units Used in Biomechanics Quantity Metric British Conversion Length meter (m) foot (ft) 1 ft 0.3048 m Mass kilogram (kg) slug 1 slug 14.59 kg Time second (s) second (s) 1 s 1 s Temperature Celsius (°C) Fahrenheit (°F) °F (9/5) °C 32° Force newton (N kg m/s2) pound (lb slug ft/s2)1 lb 4.448 N Pressure pascal (Pa N/m2 ) pounds per square inch (psi lb/in2)1psi 6895 Pa Energy joule (J N m) foot pounds (ft-lb) 1 ft-lb 1.356 J Power watt (W J/s) horsepower (hp) 1 hp 7457 W Chapter 1 | INTRODUCTION TO BIOMECHANICAL ANALYSIS 5 the degree. There are 360° in a circle. If only a portion of a Vector Analysis circle is considered, then the angle formed is some fraction of 360°. For example, a quarter of a circle subtends an angle Biomechanical parameters can be represented as either of 90°. Although in general, the unit degree is adopted for scalar or vector quantities. A scalar is simply represented by this text, angles also can be described in terms of radians. its magnitude. Mass, time, and length are examples of scalar Since there are 2π radians in a circle, there are 57.3° per ra- quantities. A vector is generally described as having both dian. When using a calculator, it is important to determine if magnitude and orientation. Additionally, a complete de- it is set to use degrees or radians. Additionally, some com- scription of a vector also includes its direction (or sense) and puter programs, such as Microsoft Excel, use radians to per- point of application. Forces and moments are examples of form trigonometric calculations. vector quantities. Consider the situation of a 160-lb man sit- Trigonometric functions are very useful in biomechanics ting in a chair for 10 seconds. The force that his weight is ex- for resolving forces into their components by relating angles erting on the chair is represented by a vector with magnitude to distances in a right triangle (a triangle containing a 90° an- (160 lb), orientation (vertical), direction (downward), and gle). The most basic of these relationships (sine, cosine, and point of application (the chair seat). However, the time spent tangent) are illustrated in Figure 1.1A. A simple mnemonic in the chair is a scalar quantity and can be represented by its to help remember these equations is sohcahtoa—sine is the magnitude (10 seconds). opposite side divided by the hypotenuse, cosine is the adja- To avoid confusion, throughout this text, bolded notation cent side divided by the hypotenuse, and tangent is the op- is used to distinguish vectors (A) from scalars (B). Alternative posite side divided by the adjacent side. Although most notations for vectors found in the literature (and in class- calculators can be used to evaluate these functions, some rooms, where it is difficult to bold letters) include putting a ෆ important values worth remembering are line under the letter (A), a line over the letter (A), or an ar- row over the letter (A៬). The magnitude of a given vector (A) sin (0°) 0, sin (90°) 1 (Equation 2.1) is represented by the same letter, but not bolded (A). By far, the most common use of vectors in biomechanics cos (0°) 1, cos (90°) 0 (Equation 2.2) is to represent forces, such as muscle and joint reaction and tan (45°) 1 (Equation 2.3) resistance forces. These vectors can be represented graphi- cally with the use of a line with an arrow at one end Additionally, the Pythagorean theorem states that for a right (Fig. 1.2A). The length of the line represents its magnitude, triangle, the sum of the squares of the sides forming the right angle equals the square of the hypotenuse (Fig. 1.1A). Al- though less commonly used, there are also equations that relate angles and side lengths for triangles that do not contain a right angle (Fig.