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Biomechanical Principles

IP A R T

1

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

Fd

s

Fs

rs

rd

d

MAd = rdsin(d) = (20 cm)sin(5) 2 cmMAs = rssin(s) = (2 cm)sin(80) 2 cm

IP A R T

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

Introduction to Biomechanical AnalysisA N D R E W R . K A R D U N A , P H . 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

Newtons 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 wasnt 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

1C H A P T E R

3

4

completely useless. If a patient was told to perform a seriesof exercises for two, the patient would have no idea if thatmeant two days, weeks, months, or even years.

The units used in biomechanics can be divided into twocategories. First, there are the four fundamental units oflength, mass, time, and temperature, which are defined onthe basis of universally accepted standards. Every other unitis considered a derived unit and can be defined in terms ofthese fundamental units. For example, velocity is equal tolength divided by time and force is equal to mass multipliedby length divided by time squared. A list of the units neededfor biomechanics is found in Table 1.1.

TrigonometrySince angles are so important in the analysis of the muscu-loskeletal system, trigonometry is a very useful biomechanicstool. The accepted unit for measuring angles in the clinic is

MATHEMATICAL OVERVIEW

This section is intended as a review of some of the basic math-ematical concepts used in biomechanics. Although it can beskipped if the reader is familiar with this material, it wouldbe helpful to at least review this section.

Units of MeasurementThe importance of including units with measurements cannotbe emphasized enough. Measurements must be accompaniedby a unit for them to have any physical meaning. Sometimes,there are situations when certain units are assumed. If aclinician asks for a patients height and the reply is 5-6, itcan reasonably be assumed that the patient is 5 feet, 6 inchestall. However, that interpretation would be inaccurate if thepatient was in Europe, where the metric system is used. Thereare also situations where the lack of a unit makes a number

Part I | BIOMECHANICAL PRINCIPLES

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) 1 psi 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

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 26. The material in this chapter is an important reference for the force analysis

chapters throughout the text.

5Chapter 1 | INTRODUCTION TO BIOMECHANICAL ANALYSIS

b

a

c

b

a

c

A

B

Trigonometric functions:

Pythagorean theorem:

a2 + b2 = c2

Law of cosines:

a2 + b2 2abcos() = c2

Law of sines:

sin () = bccos () = ac

atan () = b

b = a = csin() sin() sin()

Figure 1.1: Basic trigonometric relationships. These are someof the basic trigonometric relationships that are useful forbiomechanics. A. A right triangle. B. A general triangle.

Vector AnalysisBiomechanical parameters can be represented as eitherscalar or vector quantities. A scalar is simply represented byits magnitude. Mass, time, and length are examples of scalarquantities. A vector is generally described as having bothmagnitude and orientation. Additionally, a complete de-scription of a vector also includes its direction (or sense) andpoint of application. Forces and moments are examples ofvector quantities. Consider the situation of a 160-lb man sit-ting in a chair for 10 seconds. The force that his weight is ex-erting on the chair is represented by a vector with magnitude(160 lb), orientation (vertical), direction (downward), andpoint of application (the chair seat). However, the time spentin the chair is a scalar quantity and can be represented by itsmagnitude (10 seconds).

To avoid confusion, throughout this text, bolded notationis used to distinguish vectors (A) from scalars (B). Alternativenotations for vectors found in the literature (and in class-rooms, where it is difficult to bold letters) include putting aline 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)is represented by the same letter, but not bolded (A).

By far, the most common use of vectors in biomechanicsis to represent forces, such as muscle and joint reaction andresistance forces. These vectors can be represented graphi-cally with the use of a line with an arrow at one end(Fig. 1.2A). The length of the line represents its magnitude,

the degree. There are 360 in a circle. If only a portion of acircle is considered, then the angle formed is some fractionof 360. For example, a quarter of a circle subtends an angleof 90. Although in general, the unit degree is adopted forthis text, angles also can be described in terms of radians.Since there are 2 radians in a circle, there are 57.3 per ra-dian. When using a calculator, it is important to determine ifit is set to use degrees or radians. Additionally, some com-puter programs, such as Microsoft Excel, use radians to per-form trigonometric calculations.

Trigonometric functions are very useful in biomechanicsfor resolving forces into their components by relating anglesto distances in a right triangle (a triangle containing a 90 an-gle). The most basic of these relationships (sine, cosine, andtangent) are illustrated in Figure 1.1A. A simple mnemonicto help remember these equations is sohcahtoasine is theopposite side divided by the hypotenuse, cosine is the adja-cent side divided by the hypotenuse, and tangent is the op-posite side divided by the adjacent side. Although mostcalculators can be used to evaluate these functions, someimportant values worth remembering are

sin (0) 0, sin (90) 1 (Equation 2.1)

cos (0) 1, cos (90) 0 (Equation 2.2)

tan (45) 1 (Equation 2.3)

Additionally, the Pythagorean theorem states that for a righttriangle, the sum of the squares of the sides forming the rightangle equals the square of the hypotenuse (Fig. 1.1A). Al-though less commonly used, there are also equations thatrelate angles and side lengths for triangles that do not containa right angle (Fig. 1.1B).

Figure 1.2: Vectors. A. In general, a vector has a magnitude,orientation, point of application, and direction. Sometimes thepoint of application is not specifically indicated in the figure. B. A polar coordinate representation. C. A componentrepresentation.

Direction

Magnitude

Orientation

Point ofapplication

A A = 5 N = 37

Ax = 4 NAy = 3 N

Ax

Ay

A. Graphical

B. Polar coordinates

C. Components

6 Part I | BIOMECHANICAL PRINCIPLES

the angular position of the line represents its orientation, thelocation of the arrowhead represents its direction, and thelocation of the line in space represents its point of applica-tion. Alternatively, this same vector can be representedmathematically with the use of either polar coordinates orcomponent resolution. Polar coordinates represent themagnitude and orientation of the vector directly. In polar co-ordinates, the same vector would be 5 N at 37 from hori-zontal (Fig. 1.2B). With components, the vector is resolvedinto its relative contributions from both axes. In this exam-ple, vector A is resolved into its components: AX 4 N andAY 3 N (Fig. 1.2C). It is often useful to break down vectorsinto components that are aligned with anatomical directions.For instance, the x and y axes may correspond to superiorand anterior directions, respectively. Although graphical rep-resentations of vectors are useful for visualization purposes,analytical representations are more convenient when addingand multiplying vectors.

Note that the directional information (up and to the right)of the vector is also embedded in this information. A vectorwith the same magnitude and orientation as the vectorrepresented in Figure 1.2C, but with the opposite direction(down and to the left) is represented by AX 4 N andAY 3 N, or 5 N at 217. The description of the point-of-application information is discussed later in this chapter.

VECTOR ADDITION

When studying musculoskeletal biomechanics, it is commonto have more than one force to consider. Therefore, it isimportant to understand how to work with more than onevector. When adding or subtracting two vectors, there aresome important properties to consider. Vector addition iscommutative:

A B B A (Equation 2.4)

A B A (B) (Equation 2.5)

Vector addition is associative:

A (B C) (A B) C (Equation 2.6)

Unlike scalars, which can just be added together, both themagnitude and orientation of a vector mu...