1 Basic Concepts from Physics and Mechanics
Basic concepts like force, torque (also termed moment of force, or, for short, just moment), mechanical stress, energy, or work are unambiguously defined in physics and mechanics. These concepts have the same meaning in the field of biomechanics. A detailed treatise of the laws of mechanics and material properties can be found in any textbook on physics or mechanics (see for example Alonso et al 1 and Nelson et al 2). This chapter, therefore, reiterates only those items that are important for an understanding of later chapters in this book, illustrated by examples from the human body.
When discussing biomechanical problems in orthopedics or physiotherapy, the term mechanical load (or, briefly, load) is frequently used. In its strict sense (as used in this book), the term load designates a force or a moment. For example, it may designate the force transmitted by a joint or the bending moment exerted on a bone. When used in this way, loading by a force is measured in newtons [N], and loading by a moment in newton meters [Nm] (see Eq. 1.1 below). The term mechanical load is, however, occasionally used in the literature to designate just some mechanical effect on the biological tissues; it is then up to the reader to find out what is actually meant. When used in this fashion, mechanical loading of cartilage or skin might, for example, designate pressure, friction, or deformation. Obviously these are quite different effects which require their own specific description and evaluation. To avoid confusion, it is preferable to adhere strictly to the well-defined mechanical terms when discussing biomechanical problems; only in rare instances are new technical terms required.
1.1 Force
Forces can be neither measured nor observed directly; forces are merely a theoretical concept of physics. However, the effects of forces are measurable and observable. For example, as an effect of a force, the acceleration or deformation of a body can be observed. In this context, the term body designates some object composed of inert material or the human body (or a part of it). The term rigid designates a body that does not deform, or deforms only negligibly, under the action of a force.
Newton’s second law
describes the relation between a mass m, its acceleration a, and the force F which effected the acceleration ( Fig. 1.1 ). Mass is measured in units of kilograms [kg]. Acceleration is the change over time in velocity. Velocity is measured in units of meters per second [m/s], and the change in velocity over time in meters per second squared [m/s2]. Since the units (or dimensions) on the two sides of an equation must be identical, force is thus measured in units of kilograms times meters per second squared [kg · m/s2]. This unit has been given its own name and is called the newton [N].
One example: A body of 5 kg mass may have zero velocity at the beginning of the observation period and one second later it may have a velocity of 20 m/s. Thus, during the observation period the object has been accelerated by 20 m/s per second = 20 m/s2. On the assumption that the velocity increased linearly in the observation period, the force that effected the acceleration is calculated as F = 5 · 20 kg · m/s2 = 100 N.
In anticipation of the chapter on vector algebra (Chapter 4 ), it should be pointed out that Newton’s second law is really an equation between the vector of the force F and the vector of the acceleration a.
In this form, the law states that the magnitude of the acceleration is proportional to the magnitude of the force. The mass m is the proportionality factor. As an equation between two vectors, Newton’s law states that the acceleration is effected in the direction of the force.
The deformation of a body by a force is illustrated in Fig. 1.2 by the example of a coil spring. When loaded by a force, the spring shortens. For an elastic spring, the relation between the deformation and the magnitude of the force is given by Hooke’s law:
In this equation L designates the initial length of the spring and dL its change in length. L and dL are stated in units of meters [m]. The value of the constant c depends on the material properties and the form of the spring. c designates how much the spring deforms under a given force; that is, whether the spring is “hard” or “soft.” Any force acting on a body effects a deformation. Depending on the material properties of the body, the deformation can be elastic (reversible), viscoelastic (time-dependent), or plastic (irreversible). In the case of viscoelastic or plastic materials, the relation between force and deformation is no longer described by Hooke’s law, but by formulae adapted to the specific materials (see Chapter 2 ).
Experience shows that when a body 1 exerts a force F 12 on a body 2, the body 2 exerts a force F 21 of equal magnitude but opposite direction on body 1 (Newton’s third law):
In the static case (that is, when no accelerated movements occur), the force and counteracting force add up to zero. In the example shown in Fig. 1.3 , the downward-directed gravitational force of the mass is of equal magnitude and opposite in direction to the upward-directed force of the spring.
Both effects of a force, acceleration and deformation, can be observed in the course of the gait cycle when the heel strikes the floor ( Fig. 1.4 ). After a heel strike, the velocity of the body decreases; in addition, the heel deforms. Deceleration (negative acceleration) of the body and deformation of the heel are caused by a force exerted by the floor on the foot. There is an equal, opposite force exerted by the foot on the floor. This force deforms the floor. The deformation of a concrete floor is small, but still measurable. The deformation of an elastic floor in a gymnasium can be seen with the naked eye.
Forces exerted in specified directions with respect to a given surface receive additional designations ( Fig. 1.5 ). A force directed perpendicularly onto a surface is called a compressive force. A force directed perpendicularly away from the surface is called a tensile force. A force acting in or parallel to the plane of the surface is called a shear force. Occasionally, forces are named according to their origin. Gravitational force (weight) originates from the gravitational attraction between a body and the earth. Muscular force originates from biochemical processes in the muscle tissue. Frictional force originates from surface irregularities and electrochemical properties of the materials in contact.
In the dynamic case (that is, when accelerated movements are encountered), inertial forces come into play. If the acceleration of a mass m equals a, the inertial force F amounts to
The inertial force has the magnitude m · a and its direction is opposite to the direction of the acceleration. Fig. 1.6 illustrates the effect of the inertial force (or inertia) when a mass is lifted from the floor. During lifting, the acceleration is directed upwards. The inertial force is directed downwards and is of magnitude m · a. The object thus appears to be “heavier” in the initial phase of lifting than it does when it is simply being held.
Inertial forces become apparent when the human body is decelerated (that is, negatively accelerated), for example when hitting the floor after jumping from a wall or from gymnastic apparatus. The magnitude of the force between the body and the floor depends on the acceleration a. The magnitude of the acceleration can be calculated approximately by dividing the change in velocity Δv by the time interval Δt required to decelerate the body
This approximation rests on the assumption that the magnitude of the acceleration is constant during the time interval Δt. For a given change in velocity, a shorter time interval for deceleration will result in a larger inertial force; a longer time interval will result in a decrease in inertial force. Decreasing an inertial force by prolonging the time interval available for deceleration is called damping. In practical situations, the deformation must be large enough if an appreciable amount of damping is to be achieved. Two examples: If a person jumps off a wall and hits the floor with straight legs, the body’s deceleration time will be short and the force between floor and foot will be correspondingly high. If, however, the time interval available for decelerating the body is lengthened, either by bending the knees (deformation of the body) or by a deformable (sprung) gym floor, the magnitude of the inertial force will be smaller. Or, if a shoe has a low heel, the heel can in the nature of things only be slightly deformed upon heel strike. There is therefore little scope for damping the force between shoe and floor, whatever material is used for the heel.
In many biomechanical problems gravitational forces play an important role. The force of gravity is always directed perpendicularly downwards; its magnitude is given by
In this equation, m designates the mass, given in kilograms [kg]. g is a constant, determined from free fall or pendulum experiments; it is called gravitational acceleration. The constant g has the numerical value of 9.81 [m/s2]. It follows that a mass of 1 kg exerts a gravitational force equal to 1 · 9.81 kg · m/s2 = 9.81 N. If errors in the order of 2% are deemed to be negligible (as is usually the case in orthopedic biomechanics), the numerical value of g can be rounded off to 10.0 [m/s2]. Thus, the gravitational force (or weight) of a mass of 1 kg amounts (when rounded) to 10.0 N.
To describe the effect of a gravitational force on a body, a specific point, the center of mass, can be defined. The mass of the body is thought of as concentrated at this point and the force is thought of as acting on this point. Determination of the location of the center of mass is explained in Chapter 11 .
Two remarks on gravitational force:
Colloquially, the terms mass and weight are used synonymously. Strictly speaking, this usage is not correct. The mass of a body is given in kilograms [kg], and its weight in newtons [N]. In everyday life, the imprecise use of the technical terms does not give rise to any misunderstanding. In biomechanical calculations, however, the weight of a mass must always be expressed in newtons (as the product m · g). Otherwise both the dimension and the numerical value of the result will be wrong.
g has the dimension of an acceleration. If a body is resting on a table, gravitational force is acting on it. Why don’t we see any acceleration? The answer is, of course, that the table is exerting a force on the body in the opposite direction, and the accelerations of the gravitational force and the opposing force cancel each other out to zero (see also Fig. 1.3 ). If the body is allowed to fall freely, by contrast, no opposing force will be acting on it and the acceleration of the body will be clearly visible.
Forces, like all vectors, are described by magnitude and direction. This description does not include the point at which the force is applied to the body: in other words, the point of application is not a property of the force vector. For this reason, force vectors may be shifted at will along or parallel to their direction in graphic representations, as this changes neither the individual forces nor the vector sum. The effect of a force on a body depends of course on the point of application. If, for example, the force is applied such that its line of action (a line drawn to extend the force vector) passes through the center of mass, the body will be linearly accelerated in accordance with Newton’s second law. If the line does not pass through the center of mass, the force also effects a moment. In this case the body will also show an accelerated rotation.
1.2 Moment
The moment effected by a force can be defined in different ways, as a number or as a vector (see Chapter 4 ). If calculations are limited to two-dimensional problems where the axis of rotation is perpendicular to the plane of interest, the moment can be defined for reasons of simplicity as a (positive or negative) number.
In this equation L is the magnitude of the perpendicular distance of the line of action of the force from a reference point. The line of action is an imagined line in the direction of the force and running through the point of force exertion. F is the magnitude of the force. If one is interested in the moment with respect to a joint, the position of the axis of rotation is chosen as reference point. If one is interested in the bending of a bone or an implant, a point in the bone or the implant is chosen as reference point. In the context of orthopedic biomechanics, the distance L is often called the moment arm of the force F—for example, the moment arm of a muscular force with respect to the axis of rotation of a joint, or the moment arm of the ground reaction force with respect to a point in the tibia where fracture may occur.
In mathematical notation, where the bars indicate the magnitude (positive value) of distance and force, we can write
Alternatively, the moment can be given as a product of the distance L1 of the point of the force exertion from the fulcrum, the sine of the angle φ between the direction of the force and the line connecting the point of force exertion and the fulcrum, and the magnitude of the force:
In this equation, φ is the smaller of the two angles between the direction of the force and the distance line. φ is ≤ 90° and the numerical value of sinφ lies between 0 and +1. In the case of a distance line perpendicular to the direction of the force, φ equals 90° and sinφ equals +1; the two defi nitions of M above are thus identical.
Fig. 1.7 illustrates the definition of moment. The gravitational force F = m · g of the mass m of the body held in the hand exerts a moment M with respect to the axis of rotation in the center of the humeral condyle. Using both definitions stated above
L and L1 form the side and the base of a right-angled triangle; thus L = L1 sinφ.
If the moment is defined as a product of the magnitudes of distance and force (and not as a product of vectors; see Chapter 4 on vector algebra), an additional rule has to be followed to obtain the positive or negative sign of the moment. If the force effects a rotation in a clockwise direction, by convention the moment is counted as positive; if the rotation effected is counterclockwise, the moment is given a negative sign. In the example given in Fig. 1.7 , the force effects a clockwise rotation; this moment is positive. In graphic illustrations, the moments defined as numbers (and not as vectors) are depicted by circular curved arrows with the arrowhead pointing in the direction of the rotation. It should be pointed out that, for historical reasons, the direction of rotation where moments are counted positive (that is, clockwise) does not match the definition of positive angles (angles are defined as positive when rotation is in a counterclockwise direction).
Like forces, moments cannot be observed directly; only their effects can be observed. These effects are the acceleration of a rotational motion and the deformation of a body in torsion or bending. Fig. 1.8 illustrates these effects: angular acceleration and torsional deformation. If a mass m is attached to the rope around a wheel, the wheel is loaded by a moment of magnitude
and the wheel performs an accelerated rotation. If the wheel were fixed (nonrevolving) on the axis, application of the moment would result in a torsional deformation of the axis. For a thin axis of elastic material, the relation between the moment M and the angle of torsion φ is given by
c is a constant which depends on the material properties, the cross section, and the length of the axis. (The equation is analogous to the equation describing the deformation of a spring under a force.)
Fig. 1.9 illustrates a situation where a torsional deformation of the lower extremity occurs. When the ski hits an obstacle, a sideways-directed force of magnitude F applied at a distance L from the long axis of the leg exerts a moment L · F on the leg. The range of rotational motion about the long axis of the leg amounts to approx. 15° in the hip joint while the range in the ankle and knee joint is very small, so once the endpoint of the range of motion is reached, torsional deformation of the leg follows. If the ski is long and the force is high, the resulting moment may be great enough to effect a torsional fracture of the tibia or to rupture the ligaments of the knee.
The relation between moment and accelerated rotation is given by
α designates the angular acceleration. Angles are dimensionless quantities; they are measured in degrees (°) or radians (rad). A full rotation of 360°) corresponds to an angle of 2π rad (π = 3.14159…); accordingly 1 rad corresponds to an angle of 57.2958°. Angular velocity ω is the rotational angle covered per second, measured in radians or degrees per second (1/s). Angular acceleration α is the change over time in angular velocity, measured in radians or degrees per second squared (1/s2). By analogy to the definition of the inertial force, the quantity
characterizes resistance of a body against accelerated rotation. Mi opposes the moment that causes the acceleration. I denotes the moment of inertia of the body in relation to the axis of rotation. I depends on the distribution of the body’s mass with respect to the axis of rotation. A mass m concentrated at a point located at a distance L from the axis has a moment of inertia equal to m · L2. For the calculation of moments of inertia, see Chapter 11 .
In general, a force acting on a body effects a rotation as well as a linear acceleration. According to Newton’s second law, the acceleration depends on the force F and the mass m of the body. If, by contrast, two forces of equal magnitude but opposing direction act on a body, the forces effect a moment (provided the forces are not applied at the identical point), but the sum of the forces equals zero. In this case, the two forces exert a pure moment. Fig. 1.10 illustrates this with the example of a pivoted beam loaded by a pair of forces (force couple). Related to the fulcrum, the moment is the sum of the moments exerted by the two forces and amounts to
The reference point can, however, be chosen arbitrarily. Related to the left end of the beam, the moment would be
Choosing any other reference point would result in the identical moment. Because the moment is independent of the choice of the reference point, it is called a free moment. A single force F which loads a body at a distance L from the fulcrum ( Fig. 1.11 ) can be replaced by a pure moment 2 · (L · 1/2 F) = L · F and a force F exerted at the fulcrum. This replacement is advantageous if the linear acceleration of the whole setup and the rotational acceleration are to be investigated separately.
When dealing with objects that are very long in comparison with their cross section (beams or long bones, for example), moments that load the objects in certain directions have specific names. A moment that twists a beam about its long axis is called torsional moment or torque. A moment that bends the beam is called a bending moment. In Fig. 1.12 , the force F on the free end of the angled beam exerts a bending moment on part a and both a bending and a torsional moment on part b of the beam. According to the definition of the moment, the bending moment in part a increases with the distance from the point at which the force is applied. The same is true for part b: the highest bending moment occurs at the location where the beam is fixed to the wall. In contrast to this, the torsional moment is constant throughout the length of part b, because the distance between the point at which the force is applied and the long axis of part b does not change along the length of part b. On the end face A, and on all cross sections that are parallel to A, a shear force acts that is equal to F. The beam and wall undergo downward acceleration and accelerated clockwise rotation exerted by force F. That the beam and wall remain where they are instead of accelerating away is due to the opposing force and moment exerted by the ground on the wall.
If a force acts on a body, the moment of this force can be stated with respect to any chosen point inside or outside of the body. Which is the best point to choose for reference depends on the problem to be solved. Here are four examples:
To obtain information about muscle forces and joint loads, the reference point chosen is the center of rotation. Fig. 1.13 illustrates this using the example of the elbow joint. In relation to the center of rotation in the humeral condyle, the moment of the load F amounts to L1 · F, and the moment of muscular force B amounts to −L2 · B (the negative sign is dictated by the sign convention for moments). The moment of joint load G is zero, because this force passes through the center of rotation and thus its moment arm is equal to zero. If the precise location of a joint’s center of rotation is unknown, a point at the center of the surface of the joint can be chosen as an approximation.
To investigate mechanical stress in the bones of the lower arm ( Fig. 1.14 ), a reference point is chosen within the bone, for example at a distance L3 from the force application point. At this point the bones are acted on by a moment L3 · F. This moment bends the humerus and ulna downwards. Equilibrium is established by compressive and tensile stresses in the bones.
To work out how to help transfemoral amputees stop the prosthetic knee from buckling at the moment of heel strike, one focuses on the moment of the ground reaction force F with respect to the axis of rotation ( Fig. 1.15 ). With respect to the axis of the healthy, anatomical knee (filled circle), force F exerts a flexion moment. To prevent flexion (that is, to keep the leg straight), healthy subjects activate their knee extensors (quadriceps) at heel strike. In an amputee without knee extensors, however, another solution must be found. The answer is to employ, instead of a simple hinge joint, a so-called polycentric knee that deliberately shifts the joint’s axis of rotation (open circle in Fig. 1.15 ) to a position posterior to the anatomical axis. The effect of this is that the ground reaction force F generates an extension moment at heel strike, thus allowing the amputee to maintain stability. Alternatively, an electronically controlled prosthesis may be employed, which identifies the heel strike phase (extended knee, loaded) and blocks joint flexion during this phase.
Mechanical equilibrium exists when the sum of all forces and moments equals zero (see Chapter 12 ). In the state of equilibrium (and only in this state), any point can be chosen as the reference in order calculate the sum of moments. This theorem can be used in any given setup to check whether the equilibrium of moments is fulfilled. As an example, Fig. 1.16 analyzes an exercise designed for isometric activation of the quadriceps muscle. The lower leg is acted on by the force F 2 of the patellar tendon (activated by the quadriceps) and the tibiofemoral force F 1. An external, posteriorly directed force F 3 acting on the lower leg prevents extension of the knee. If the application point of F 3 is chosen such that its line of action runs through point P (the intersection of the lines of action of forces F 1 and F 2), it follows that the sum of moments equals zero. This is because the moment arms of all three forces in relation to point P equal zero and thus the sum of the moments equals zero. This holds in relation to any other point of reference, including the center of rotation of the knee joint. If, by contrast, the line of action of F 3 did not pass through P, mechanical equilibrium would not exist.
Related posts:
2 Mechanical Properties of Solid Materials
3 Deformation and Strength of Structures
6 Translation and Rotation in a Plane
11 Dimensions, Mass, Location of the Center of Mass, and Moment of Inertia of the Segments of the Human Body
12 Determination of Joint Load in a Model Calculation
4 Vector Algebra
Stay updated, free articles. Join our Telegram channel
Full access? Get Clinical Tree
