Statics 7

(1) Active forces are external forces capable of doing virtual work during possible virtual displacements. In Fig. 7/7a forces P and F are active forces because they would do work as the links move.

(1) Active forces are external forces capable of doing virtual work during possible virtual displacements. In Fig. 7/7a forces P and F are active forces because they would do work as the links move.

(2) Reactive forces are forces which act at fixed support positions where no virtual displacement takes place in the direction of the force. Reactive forces do no work during a virtual displacement. In Fig. 7/7b the horizontal force FB exerted on the roller end of the member by the vertical guide can do no work because there can be no horizontal displacement of the roller. The reactive force FO exerted on the system by the fixed support at O also does no work because O cannot move.

(2) Reactive forces are forces which act at fixed support positions where no virtual displacement takes place in the direction of the force. Reactive forces do no work during a virtual displacement. In Fig. 7/7b the horizontal force FB exerted on the roller end of the member by the vertical guide can do no work because there can be no horizontal displacement of the roller. The reactive force FO exerted on the system by the fixed support at O also does no work because O cannot move.

(3) Internal forces are forces in the connections between members. During any possible movement of the system or its parts, the net work done by the internal forces at the connections is zero. This is so because the internal forces always exist in pairs of equal and opposite forces, as indicated for the internal forces FA and FA at joint A in Fig. 7/7c. The work of one force therefore necessarily cancels the work of the other force during their identical displacements

(3) Internal forces are forces in the connections between members. During any possible movement of the system or its parts, the net work done by the internal forces at the connections is zero. This is so because the internal forces always exist in pairs of equal and opposite forces, as indicated for the internal forces FA and FA at joint A in Fig. 7/7c. The work of one force therefore necessarily cancels the work of the other force during their identical displacements

A method based on the concept of the work done by a force is more direct. Also, the method provides a deeper insight into the behavior of mechanical systems and enables us to examine the stability of systems in equilibrium. This method is called the method of virtual work.

A method based on the concept of the work done by a force is more direct. Also, the method provides a deeper insight into the behavior of mechanical systems and enables us to examine the stability of systems in equilibrium. This method is called the method of virtual work.

A torsional spring, which resists the rotation of a shaft or another element, can also store and release potential energy. If the torsional stiffness, expressed as torque per radian of twist, is a constant K, and if is the angle of twist in radians, then the resisting torque is M K. The potential energy becomes Ve K d or

A torsional spring, which resists the rotation of a shaft or another element, can also store and release potential energy. If the torsional stiffness, expressed as torque per radian of twist, is a constant K, and if is the angle of twist in radians, then the resisting torque is M K. The potential energy becomes Ve K d or

As a second example, consider the screw jack described in Art. 6/5 and shown in Fig. 6/6. Equation 6/3 gives the moment M required to raise the load W, where the screw has a mean radius r and a helix angle , and where the friction angle is tan1 k. During a small rotation of the screw, the input work is M Wr tan ( ). The output work is that required to elevate the load, or Wr tan . Thus the efficiency of the jack can be expressed as

As a second example, consider the screw jack described in Art. 6/5 and shown in Fig. 6/6. Equation 6/3 gives the moment M required to raise the load W, where the screw has a mean radius r and a helix angle , and where the friction angle is tan1 k. During a small rotation of the screw, the input work is M Wr tan ( ). The output work is that required to elevate the load, or Wr tan . Thus the efficiency of the jack can be expressed as

As an example, consider the block being moved up the inclined plane in Fig. 7/10. For the virtual displacement s shown, the output work is that necessary to elevate the block, or mg s sin . The input work is T s (mg sin kmg cos ) s. The efficiency of the inclined plane is, therefore,

As an example, consider the block being moved up the inclined plane in Fig. 7/10. For the virtual displacement s shown, the output work is that necessary to elevate the block, or mg s sin . The input work is T s (mg sin kmg cos ) s. The efficiency of the inclined plane is, therefore,

Because of energy loss due to friction, the output work of a machine is always less than the input work. The ratio of the two amounts of work is the mechanical efficiency e. Thus,

Because of energy loss due to friction, the output work of a machine is always less than the input work. The ratio of the two amounts of work is the mechanical efficiency e. Thus,

Because the force acting on the movable end of a spring is the negative of the force exerted by the spring on the body to which its movable end is attached, the work done on the body is the negative of the potential energy change of the spring.

Because the force acting on the movable end of a spring is the negative of the force exerted by the spring on the body to which its movable end is attached, the work done on the body is the negative of the potential energy change of the spring.

The sum is zero, since ΣF 0, which gives ΣFx = 0, ΣFy = 0, and ΣFz 0. The equation U 0 is therefore an alternative statement of the equilibrium conditions for a particle. This condition of zero virtual work for equilibrium is both necessary and sufficient, since we may apply it to virtual displacements taken one at a time in each of the three mutually perpendicular directions, in which case it becomes equivalent to the three known scalar requirements for equilibrium.

The sum is zero, since ΣF 0, which gives ΣFx = 0, ΣFy = 0, and ΣFz 0. The equation U 0 is therefore an alternative statement of the equilibrium conditions for a particle. This condition of zero virtual work for equilibrium is both necessary and sufficient, since we may apply it to virtual displacements taken one at a time in each of the three mutually perpendicular directions, in which case it becomes equivalent to the three known scalar requirements for equilibrium.

The virtual change in gravitational potential energy is simply where h is the upward virtual displacement of the mass center of the body. If the mass center has a downward virtual displacement, then Vg is negative. The units of gravitational potential energy are the same as those for work and elastic potential energy, joules (J) in SI units and foot-pounds (ft-lb) in U.S. customary units.

The virtual change in gravitational potential energy is simply where h is the upward virtual displacement of the mass center of the body. If the mass center has a downward virtual displacement, then Vg is negative. The units of gravitational potential energy are the same as those for work and elastic potential energy, joules (J) in SI units and foot-pounds (ft-lb) in U.S. customary units.

The virtual work done by all external active forces (other than the gravitational and spring forces accounted for in the potential energy terms) on a mechanical system in equilibrium equals the corresponding change in the total elastic and gravitational potential energy of the system for any and all virtual displacements consistent with the constraints.

The virtual work done by all external active forces (other than the gravitational and spring forces accounted for in the potential energy terms) on a mechanical system in equilibrium equals the corresponding change in the total elastic and gravitational potential energy of the system for any and all virtual displacements consistent with the constraints.

Consider a spring, Fig. 7/11, which is being compressed by a force F. We assume that the spring is elastic and linear, which means that the force F is directly proportional to the deflection x. We write this relation as F kx, where k is the spring constant or stiffness of the spring. The work done on the spring by F during a movement dx is dU F dx, so that the elastic potential energy of the spring for a compression x is the total work done on the spring

Consider a spring, Fig. 7/11, which is being compressed by a force F. We assume that the spring is elastic and linear, which means that the force F is directly proportional to the deflection x. We write this relation as F kx, where k is the spring constant or stiffness of the spring. The work done on the spring by F during a movement dx is dU F dx, so that the elastic potential energy of the spring for a compression x is the total work done on the spring

Consider now the case of a mechanical system where movement is accompanied by changes in gravitational and elastic potential energies and where no work is done on the system by nonpotential forces. The mechanism treated in Sample Problem 7/6 is an example of such a system. With U 0 the virtual-work relation, Eq. 7/6, becomes

Consider now the case of a mechanical system where movement is accompanied by changes in gravitational and elastic potential energies and where no work is done on the system by nonpotential forces. The mechanism treated in Sample Problem 7/6 is an example of such a system. With U 0 the virtual-work relation, Eq. 7/6, becomes

Consider the constant force F acting on the body shown in Fig. 7/1a, whose movement along the plane from A to A is represented by the vector s, called the displacement of the body. By definition the work U done by the force F on the body during this displacement is the component of the force in the direction of the displacement times the displacement, or

Consider the constant force F acting on the body shown in Fig. 7/1a, whose movement along the plane from A to A is represented by the vector s, called the displacement of the body. By definition the work U done by the force F on the body during this displacement is the component of the force in the direction of the displacement times the displacement, or

Consider the particle or small body in Fig. 7/5 which attains an equilibrium position as a result of the forces in the attached springs. If the mass of the particle were significant, then the weight mg would also be included as one of the forces. For an assumed virtual displacement r of the particle away from its equilibrium position, the total virtual work done on the particle is

Consider the particle or small body in Fig. 7/5 which attains an equilibrium position as a result of the forces in the attached springs. If the mass of the particle were significant, then the weight mg would also be included as one of the forces. For an assumed virtual displacement r of the particle away from its equilibrium position, the total virtual work done on the particle is

Equation 7/8 states that a mechanical system is in equilibrium when the derivative of its total potential energy is zero. For systems with several degrees of freedom the partial derivative of V with respect to each coordinate in turn must be zero for equilibrium.* There are three conditions under which E

Equation 7/8 states that a mechanical system is in equilibrium when the derivative of its total potential energy is zero. For systems with several degrees of freedom the partial derivative of V with respect to each coordinate in turn must be zero for equilibrium.* There are three conditions under which E

Figure 7/7a shows a simple example of an ideal system where relative motion between its two parts is possible and where the equilibrium position is determined by the applied external forces P and F. We can identify three types of forces which act in such an interconnected system. They are as follows:

Figure 7/7a shows a simple example of an ideal system where relative motion between its two parts is possible and where the equilibrium position is determined by the applied external forces P and F. We can identify three types of forces which act in such an interconnected system. They are as follows:

From Fig. 7/1b we see that the same result is obtained if we multiply the magnitude of the force by the component of the displacement in the direction of the force. This gives Because we obtain the same result regardless of the direction in which we resolve the vectors, we conclude that work U is a scalar quantity. Work is positive when the working component of the force is in the same direction as the displacement. When the working component is in the direction opposite to the displacement, Fig. 7/2, the work done is negative. Thus

From Fig. 7/1b we see that the same result is obtained if we multiply the magnitude of the force by the component of the displacement in the direction of the force. This gives Because we obtain the same result regardless of the direction in which we resolve the vectors, we conclude that work U is a scalar quantity. Work is positive when the working component of the force is in the same direction as the displacement. When the working component is in the direction opposite to the displacement, Fig. 7/2, the work done is negative. Thus

If F denotes the magnitude of the force F and ds denotes the magnitude of the differential displacement dr, we use the definition of the dot product to obtain We may again interpret this expression as the force component F cos in the direction of the displacement times the displacement, or as the displacement component ds cos in the direction of the force times the

If F denotes the magnitude of the force F and ds denotes the magnitude of the differential displacement dr, we use the definition of the dot product to obtain We may again interpret this expression as the force component F cos in the direction of the displacement times the displacement, or as the displacement component ds cos in the direction of the force times the

The work done by any force F acting on the particle during the virtual displacement r is called virtual work and is where is the angle between F and r, and s is the magnitude of r. The difference between dr and r is that dr refers to an actual infinitesimal change in position and can be integrated, whereas r refers to an infinitesimal virtual or assumed movement and cannot be integrated. Mathematically both quantities are first-order differentials. A virtual displacement may also be a rotation of a body. According to Eq. 7/2 the virtual work done by a couple M during a virtual angular displacement is U M

The work done by any force F acting on the particle during the virtual displacement r is called virtual work and is where is the angle between F and r, and s is the magnitude of r. The difference between dr and r is that dr refers to an actual infinitesimal change in position and can be integrated, whereas r refers to an infinitesimal virtual or assumed movement and cannot be integrated. Mathematically both quantities are first-order differentials. A virtual displacement may also be a rotation of a body. According to Eq. 7/2 the virtual work done by a couple M during a virtual angular displacement is U M

The work done on an elastic member is stored in the member in the form of elastic potential energy Ve. This energy is potentially available to do work on some other body during the relief of its compression or extension

The work done on an elastic member is stored in the member in the form of elastic potential energy Ve. This energy is potentially available to do work on some other body during the relief of its compression or extension

The work of the couple is positive if M has the same sense as d (clockwise in this illustration), and negative if M has a sense opposite to that of the rotation. The total work of a couple during a finite rotation in its plane becomes

The work of the couple is positive if M has the same sense as d (clockwise in this illustration), and negative if M has a sense opposite to that of the rotation. The total work of a couple during a finite rotation in its plane becomes

There is a separate class of problems in which bodies are composed of interconnected members which can move relative to each other. Thus various equilibrium configurations are possible and must be examined. For problems of this type, the force- and moment-equilibrium equations, although valid and adequate, are often not the most direct and convenient approach.

There is a separate class of problems in which bodies are composed of interconnected members which can move relative to each other. Thus various equilibrium configurations are possible and must be examined. For problems of this type, the force- and moment-equilibrium equations, although valid and adequate, are often not the most direct and convenient approach.

This condition is illustrated in Fig. 7/6, where we want to determine the reaction R under the roller for the hinged plate of negligible weight under the action of a given force P. A small assumed rotation of the plate about O is consistent with the hinge constraint at O and is taken as the virtual displacement. The work done by P is Pa , and the work done by R is Rb . Therefore, the principle U 0 gives

This condition is illustrated in Fig. 7/6, where we want to determine the reaction R under the roller for the hinged plate of negligible weight under the action of a given force P. A small assumed rotation of the plate about O is consistent with the hinge constraint at O and is taken as the virtual displacement. The work done by P is Pa , and the work done by R is Rb . Therefore, the principle U 0 gives

We consider now a particle whose static equilibrium position is determined by the forces which act on it. Any assumed and arbitrary small displacement r away from this natural position and consistent with the system constraints is called a virtual displacement. The term virtual is used to indicate that the displacement does not really exist but only is assumed to exist so that we may compare various possible equilibrium positions to determine the correct one.

We consider now a particle whose static equilibrium position is determined by the forces which act on it. Any assumed and arbitrary small displacement r away from this natural position and consistent with the system constraints is called a virtual displacement. The term virtual is used to indicate that the displacement does not really exist but only is assumed to exist so that we may compare various possible equilibrium positions to determine the correct one.

We may also characterize the stability of a mechanical system by noting that a small displacement away from the stable position results in an increase in potential energy and a tendency to return to the position of lower energy. On the other hand, a small displacement away from the unstable position results in a decrease in potential energy and

We may also characterize the stability of a mechanical system by noting that a small displacement away from the stable position results in an increase in potential energy and a tendency to return to the position of lower energy. On the other hand, a small displacement away from the unstable position results in a decrease in potential energy and

We may regard the force F or couple M as remaining constant during any infinitesimal virtual displacement. If we account for any U F r or U F s cos N m N m U M d 400 Chapter 7 Virtual Work change in F or M during the infinitesimal motion, higher-order terms will result which disappear in the limit. This consideration is the same mathematically as that which permits us to neglect the product dx dy when writing dA y dx for the element of area under the curve y ƒ(x).

We may regard the force F or couple M as remaining constant during any infinitesimal virtual displacement. If we account for any U F r or U F s cos N m N m U M d 400 Chapter 7 Virtual Work change in F or M during the infinitesimal motion, higher-order terms will result which disappear in the limit. This consideration is the same mathematically as that which permits us to neglect the product dx dy when writing dA y dx for the element of area under the curve y ƒ(x).

We now express ΣF in terms of its scalar sums and r in terms of its component virtual displacements in the coordinate directions, as follows:

We now express ΣF in terms of its scalar sums and r in terms of its component virtual displacements in the coordinate directions, as follows:

We now extend the principle of virtual work to the equilibrium of an interconnected system of rigid bodies. Our treatment here will be limited to so-called ideal systems. These are systems composed of two or more rigid members linked together by mechanical connections which are incapable of absorbing energy through elongation or compression, and in which friction is small enough to be neglected.

We now extend the principle of virtual work to the equilibrium of an interconnected system of rigid bodies. Our treatment here will be limited to so-called ideal systems. These are systems composed of two or more rigid members linked together by mechanical connections which are incapable of absorbing energy through elongation or compression, and in which friction is small enough to be neglected.

We now generalize the definition of work to account for conditions under which the direction of the displacement and the magnitude and direction of the force are variable. Figure 7/3a shows a force F acting on a body at a point A which moves along the path shown from A1 to A2. Point A is located by its position vector r measured from some arbitrary but convenient origin O. The infinitesimal displacement in the motion from A to A is given by the differential change dr of the position vector. The work done by the force F during the displacement dr is defined as

We now generalize the definition of work to account for conditions under which the direction of the displacement and the magnitude and direction of the force are variable. Figure 7/3a shows a force F acting on a body at a point A which moves along the path shown from A1 to A2. Point A is located by its position vector r measured from some arbitrary but convenient origin O. The infinitesimal displacement in the motion from A to A is given by the differential change dr of the position vector. The work done by the force F during the displacement dr is defined as

When sliding friction is present to any appreciable degree in a mechanical system, the system is said to be "real." In real systems some of the positive work done on the system by external active forces (input work) is dissipated in the form of heat generated by the kinetic friction forces during movement of the system. When there is sliding between contacting surfaces, the friction force does negative work because its direction is always opposite to the movement of the body on which it acts. This negative work cannot be regained.

When sliding friction is present to any appreciable degree in a mechanical system, the system is said to be "real." In real systems some of the positive work done on the system by external active forces (input work) is dissipated in the form of heat generated by the kinetic friction forces during movement of the system. When there is sliding between contacting surfaces, the friction force does negative work because its direction is always opposite to the movement of the body on which it acts. This negative work cannot be regained.

Thus, for a mechanical system with elastic members and members which undergo changes in position, we may restate the principle of virtual work as follows:

Thus, for a mechanical system with elastic members and members which undergo changes in position, we may restate the principle of virtual work as follows:

Thus, the kinetic friction force kN acting on the sliding block in Fig. 7/9a does work on the block during the displacement x in the amount of kNx. During a virtual displacement x, the friction force does work equal to kN x. The static friction force acting on the

Thus, the kinetic friction force kN acting on the sliding block in Fig. 7/9a does work on the block during the displacement x in the amount of kNx. During a virtual displacement x, the friction force does work equal to kN x. The static friction force acting on the

Thus, the potential energy of the spring equals the triangular area in the diagram of F versus x from 0 to x. During an increase in the compression of the spring from x1 to x2, the work done on the spring equals its change in elastic potential energy or which equals the trapezoidal area from x1 to x2. During a virtWhen we have a spring in tension rather than compression, the work and energy relations are the same as those for compression, where x now represents the elongation of the spring rather than its compression. While the spring is being stretched, the force again acts in the direction of the displacement, doing positive work on the spring and increasing its potential energual displacement x of the spring, the virtual work done on the spring is the virtual change in elastic potential energy During a decrease in the compression of the spring as it is relaxed from x x2 to x x1, the change (final minus initial) in the potential energy of the spring is negative. Consequently, if x is negative, Ve is also negative

Thus, the potential energy of the spring equals the triangular area in the diagram of F versus x from 0 to x. During an increase in the compression of the spring from x1 to x2, the work done on the spring equals its change in elastic potential energy or which equals the trapezoidal area from x1 to x2. During a virtual displacement x of the spring, the virtual work done on the spring is the virtual change in elastic potential energy During a decrease in the compression of the spring as it is relaxed from x x2 to x x1, the change (final minus initial) in the potential energy of the spring is negative. Consequently, if x is negative, Ve is also negativeWhen we have a spring in tension rather than compression, the work and energy relations are the same as those for compression, where x now represents the elongation of the spring rather than its compression. While the spring is being stretched, the force again acts in the direction of the displacement, doing positive work on the spring and increasing its potential energ

When using the method of virtual work, you should draw a diagram which isolates the system under consideration. Unlike the free-body diagram, where all forces are shown, the diagram for the method of virtual work need show only the active forces, since the reactive forces do not enter into the application of U 0. Such a drawing will be termed an active-force diagram. Figure 7/7a is an active-force diagram for the system shown.

When using the method of virtual work, you should draw a diagram which isolates the system under consideration. Unlike the free-body diagram, where all forces are shown, the diagram for the method of virtual work need show only the active forces, since the reactive forces do not enter into the application of U 0. Such a drawing will be termed an active-force diagram. Figure 7/7a is an active-force diagram for the system shown.

Work has the dimensions of (force) (distance). In SI units the unit of work is the joule (J), which is the work done by a force of one newton moving through a distance of one meter in the direction of the force (J ). In the U.S. customary system the unit of work is the footpound (ft-lb), which is the work done by a one-pound force moving through a distance of one foot in the direction of the force

Work has the dimensions of (force) (distance). In SI units the unit of work is the joule (J), which is the work done by a force of one newton moving through a distance of one meter in the direction of the force (J ). In the U.S. customary system the unit of work is the footpound (ft-lb), which is the work done by a one-pound force moving through a distance of one foot in the direction of the force

a tendency to move farther away from the equilibrium position to one of still lower energy. For the neutral position a small displacement one way or the other results in no change in potential energy and no tendency to move either way. When a function and its derivatives are continuous, the second derivative is positive at a point of minimum value of the function and negative at a point of maximum value of the function. Thus, the mathematical conditions for equilibrium and stability of a system with a single degree of freedom x are:

a tendency to move farther away from the equilibrium position to one of still lower energy. For the neutral position a small displacement one way or the other results in no change in potential energy and no tendency to move either way. When a function and its derivatives are continuous, the second derivative is positive at a point of minimum value of the function and negative at a point of maximum value of the function. Thus, the mathematical conditions for equilibrium and stability of a system with a single degree of freedom x are:

body, since all internal forces occur in pairs of equal, opposite, and collinear forces, and the net work done by these forces during any movement is zero. As in the case of a particle, we again find that the principle of virtual work offers no particular advantage to the solution for a single rigid body in equilibrium. Any assumed virtual displacement defined by a linear or angular movement will appear in each term in U 0 and when canceled will leave us with the same expression we would have obtained by using one of the force or moment equations of equilibrium directly

body, since all internal forces occur in pairs of equal, opposite, and collinear forces, and the net work done by these forces during any movement is zero. As in the case of a particle, we again find that the principle of virtual work offers no particular advantage to the solution for a single rigid body in equilibrium. Any assumed virtual displacement defined by a linear or angular movement will appear in each term in U 0 and when canceled will leave us with the same expression we would have obtained by using one of the force or moment equations of equilibrium directly

hanical systems composed of individual members which we assumed to be perfectly rigid. We now extend our method to account for mechanical systems which include elastic elements in the form of springs. We introduce the concept of potential energy, which is useful for determining the stability of equilibrium.

hanical systems composed of individual members which we assumed to be perfectly rigid. We now extend our method to account for mechanical systems which include elastic elements in the form of springs. We introduce the concept of potential energy, which is useful for determining the stability of equilibrium.

ights by the negative of the respective potential energy changes. We can use these substitutions to write the total virtual work U in Eq. 7/3 as the sum of the work U done by all active forces, other than spring forces and weight forces, and the work (Ve Vg) done by the spring and weight forces. Equation 7/3 then becomes or (7/6) where V Ve Vg stands for the total potential energy of the system. With this formulation a spring becomes internal to the system, and the work of spring and gravitational forces is accounted for in the V term.

ights by the negative of the respective potential energy changes. We can use these substitutions to write the total virtual work U in Eq. 7/3 as the sum of the work U done by all active forces, other than spring forces and weight forces, and the work (Ve Vg) done by the spring and weight forces. Equation 7/3 then becomes or (7/6) where V Ve Vg stands for the total potential energy of the system. With this formulation a spring becomes internal to the system, and the work of spring and gravitational forces is accounted for in the V term.

To carry out this integration, we must know the relation between the force components and their respective coordinates, or the relations between F and s and between cos and s. In the case of concurrent forces which are applied at any particular point on a body, the work done by their resultant equals the total work done by the several forces. This is because the component of the resultant in the direction of the displacement equals the sum of the components of the several forces in the same direction.

To carry out this integration, we must know the relation between the force components and their respective coordinates, or the relations between F and s and between cos and s. In the case of concurrent forces which are applied at any particular point on a body, the work done by their resultant equals the total work done by the several forces. This is because the component of the resultant in the direction of the displacement equals the sum of the components of the several forces in the same direction.

We can easily extend the principle of virtual work for a single particle to a rigid body treated as a system of small elements or particles rigidly attached to one another. Because the virtual work done on each particle of the body in equilibrium is zero, it follows that the virtual work done on the entire rigid body is zero. Only the virtual work done by external forces appears in the evaluation of U 0 for the entire

We can easily extend the principle of virtual work for a single particle to a rigid body treated as a system of small elements or particles rigidly attached to one another. Because the virtual work done on each particle of the body in equilibrium is zero, it follows that the virtual work done on the entire rigid body is zero. Only the virtual work done by external forces appears in the evaluation of U 0 for the entire

namely, when the total potential energy is a minimum (stable equilibrium), a maximum (unstable equilibrium), or a constant (neutral equilibrium). Figure 7/15 shows a simple example of these three conditions. The potential energy of the roller is clearly a minimum in the stable position, a maximum in the unstable position, and a constant in the neutral position

namely, when the total potential energy is a minimum (stable equilibrium), a maximum (unstable equilibrium), or a constant (neutral equilibrium). Figure 7/15 shows a simple example of these three conditions. The potential energy of the roller is clearly a minimum in the stable position, a maximum in the unstable position, and a constant in the neutral position

nergy Equation We saw that the work done by a linear spring on the body to which its movable end is attached is the negative of the change in the elastic potential energy of the spring. Also, the work done by the gravitational force or weight mg is the negative of the change in gravitational potential energy. Therefore, when we apply the virtual-work equation to a system with springs and with changes in the vertical position of its members, we may replace the work of the springs and the work of the weights by the negative of the respective potential energy changes.

nergy Equation We saw that the work done by a linear spring on the body to which its movable end is attached is the negative of the change in the elastic potential energy of the spring. Also, the work done by the gravitational force or weight mg is the negative of the change in gravitational potential energy. Therefore, when we apply the virtual-work equation to a system with springs and with changes in the vertical position of its members, we may replace the work of the springs and the work of the weights by the negative of the respective potential energy changes.

reaction force. In Fig 7/14b, where the particle alone is isolated, U includes the virtual work of all forces shown on the active-force diagram of the particle. (The normal reaction exerted on the particle by the smooth guide does no work and is omitted.) In Fig. 7/14c the spring is included in the system, and U is the virtual work of only F1 and F2, which are the only external forces whose work is not accounted for in the potentialenergy terms. The work of the weight mg is accounted for in the Vg term, and the work of the spring force is included in the Ve term.

reaction force. In Fig 7/14b, where the particle alone is isolated, U includes the virtual work of all forces shown on the active-force diagram of the particle. (The normal reaction exerted on the particle by the smooth guide does no work and is omitted.) In Fig. 7/14c the spring is included in the system, and U is the virtual work of only F1 and F2, which are the only external forces whose work is not accounted for in the potentialenergy terms. The work of the weight mg is accounted for in the Vg term, and the work of the spring force is included in the Ve term.

which is analogous to the expression for the linear extension spring. The units of elastic potential energy are the same as those of work and are expressed in joules (J) in SI units and in foot-pounds (ft-lb) in U.S. customary units.

which is analogous to the expression for the linear extension spring. The units of elastic potential energy are the same as those of work and are expressed in joules (J) in SI units and in foot-pounds (ft-lb) in U.S. customary units.

which is simply the equation of moment equilibrium about O. Therefore, nothing is gained by using the virtual-work principle for a single rigid body. The principle is, however, decidedly advantageous for interconnected bodies, as discussed next.which is simply the equation of moment equilibrium about O. Therefore, nothing is gained by using the virtual-work principle for a single rigid body. The principle is, however, decidedly advantageous for interconnected bodies, as discussed next.

which is simply the equation of moment equilibrium about O. Therefore, nothing is gained by using the virtual-work principle for a single rigid body. The principle is, however, decidedly advantageous for interconnected bodies, as discussed next.

Active-Force Diagrams With the method of virtual work it is useful to construct the activeforce diagram of the system you are analyzing. The boundary of the system must clearly distinguish those members which are part of the system from other bodies which are not part of the system. When we include an elastic member within the boundary of our system, the forces of interaction between it and the movable members to which it is attached are internal to the system. Thus these forces need not be shown because their effects are accounted for in the Ve term. Similarly, weight forces are not shown because their work is accounted for in the Vg term. Figure 7/14 illustrates the difference between the use of Eqs. 7/3 and

Active-Force Diagrams With the method of virtual work it is useful to construct the activeforce diagram of the system you are analyzing. The boundary of the system must clearly distinguish those members which are part of the system from other bodies which are not part of the system. When we include an elastic member within the boundary of our system, the forces of interaction between it and the movable members to which it is attached are internal to the system. Thus these forces need not be shown because their effects are accounted for in the Ve term. Similarly, weight forces are not shown because their work is accounted for in the Vg term. Figure 7/14 illustrates the difference between the use of Eqs. 7/3 and

An alternative to the foregoing treatment expresses the work done by gravity in terms of a change in potential energy of the body. This alternative treatment is a useful representation when we describe a mechanical system in terms of its total energy. The gravitational potential energy Vg of a body is defined as the work done on the body by a force equal and opposite to the weight in bringing the body to the position under consideration from some arbitrary datum plane where the potential energy is defined to be zero. The potential energy, then, is the negative of the work done by the weight. When the body is raised, for example, the work done is converted into energy which is potentially available, since the body can do work on some other body as it returns to its original lower position. If we take Vg to be zero at h 0, Fig. 7/12, then at a height h above the datum plane, the gravitational potential energy of the body is (7/5) If the body is a distance h below the datum plane, its gravitational potential energy is mgh.

An alternative to the foregoing treatment expresses the work done by gravity in terms of a change in potential energy of the body. This alternative treatment is a useful representation when we describe a mechanical system in terms of its total energy. The gravitational potential energy Vg of a body is defined as the work done on the body by a force equal and opposite to the weight in bringing the body to the position under consideration from some arbitrary datum plane where the potential energy is defined to be zero. The potential energy, then, is the negative of the work done by the weight. When the body is raised, for example, the work done is converted into energy which is potentially available, since the body can do work on some other body as it returns to its original lower position. If we take Vg to be zero at h 0, Fig. 7/12, then at a height h above the datum plane, the gravitational potential energy of the body is (7/5) If the body is a distance h below the datum plane, its gravitational potential energy is mgh.

By constraint we mean restriction of the motion by the supports. We state the principle mathematically by the equation (7/3) where U stands for the total virtual work done on the system by all active forces during a virtual displacement. Only now can we see the real advantages of the method of virtual work. There are essentially two. First, it is not necessary for us to dismember ideal systems in order to establish the relations between the active forces, as is generally the case with the equilibrium method based on force and moment summations. Second, we may determine the relations between the active forces directly without reference to the reactive forces. These advantages make the method of virtual work particularly useful in determining the position of equilibrium of a system under known loads. This type of problem is in contrast to the problem of determining the forces acting on a body whose equilibrium position is known

By constraint we mean restriction of the motion by the supports. We state the principle mathematically by the equation (7/3) where U stands for the total virtual work done on the system by all active forces during a virtual displacement. Only now can we see the real advantages of the method of virtual work. There are essentially two. First, it is not necessary for us to dismember ideal systems in order to establish the relations between the active forces, as is generally the case with the equilibrium method based on force and moment summations. Second, we may determine the relations between the active forces directly without reference to the reactive forces. These advantages make the method of virtual work particularly useful in determining the position of equilibrium of a system under known loads. This type of problem is in contrast to the problem of determining the forces acting on a body whose equilibrium position is known

Equation 7/7 expresses the requirement that the equilibrium configuration of a mechanical system is one for which the total potential energy V of the system has a stationary value. For a system of one degree of freedom where the potential energy and its derivatives are continuous functions of the single variable, say, x, which describes the configuration, the equilibrium condition V 0 is equivalent mathematically to the requirement

Equation 7/7 expresses the requirement that the equilibrium configuration of a mechanical system is one for which the total potential energy V of the system has a stationary value. For a system of one degree of freedom where the potential energy and its derivatives are continuous functions of the single variable, say, x, which describes the configuration, the equilibrium condition V 0 is equivalent mathematically to the requirement

In addition to the work done by forces, couples also can do work. In Fig. 7/4a the couple M acts on the body and changes its angular position by an amount d. The work done by the couple is easily determined from the combined work of the two forces which constitute the couple. In part b of the figure we represent the couple by two equal and opposite forces F and F acting at two arbitrary points A and B such that F M/b. During the infinitesimal movement in the plane of the figure, line AB moves to AB. We now take the displacement of A in two steps, first, a displacement drB equal to that of B and, second, a displacement drA/B (read as the displacement of A with respect to B) due to the rotation about B. Thus the work done by F during the displacement from A to A is equal and opposite in sign to that due to F acting through the equal displacement from B to B. We therefore conclude that no work is done by a couple during a translation (movement without rotation). During the rotation, however, F does work equal to rA/B Fb d, where drA/B b d and where d is the infinitesimal angle of rotation in radians. Since M Fb, we have

In addition to the work done by forces, couples also can do work. In Fig. 7/4a the couple M acts on the body and changes its angular position by an amount d. The work done by the couple is easily determined from the combined work of the two forces which constitute the couple. In part b of the figure we represent the couple by two equal and opposite forces F and F acting at two arbitrary points A and B such that F M/b. During the infinitesimal movement in the plane of the figure, line AB moves to AB. We now take the displacement of A in two steps, first, a displacement drB equal to that of B and, second, a displacement drA/B (read as the displacement of A with respect to B) due to the rotation about B. Thus the work done by F during the displacement from A to A is equal and opposite in sign to that due to F acting through the equal displacement from B to B. We therefore conclude that no work is done by a couple during a translation (movement without rotation). During the rotation, however, F does work equal to rA/B Fb d, where drA/B b d and where d is the infinitesimal angle of rotation in radians. Since M Fb, we have

In the previous article we treated the work of a gravitational force or weight acting on a body in the same way as the work of any other active force. Thus, for an upward displacement h of the body in Fig. 7/12 the weight W mg does negative work U mg h. If, on the other hand, the body has a downward displacement h, with h measured positive downward, the weight does positive work U mg h.

In the previous article we treated the work of a gravitational force or weight acting on a body in the same way as the work of any other active force. Thus, for an upward displacement h of the body in Fig. 7/12 the weight W mg does negative work U mg h. If, on the other hand, the body has a downward displacement h, with h measured positive downward, the weight does positive work U mg h.

In the previous chapters we have analyzed the equilibrium of a body by isolating it with a free-body diagram and writing the zeroforce and zero-moment summation equations. This approach is usually employed for a body whose equilibrium position is known or specified and where one or more of the external forces is an unknown to be determined.

In the previous chapters we have analyzed the equilibrium of a body by isolating it with a free-body diagram and writing the zeroforce and zero-moment summation equations. This approach is usually employed for a body whose equilibrium position is known or specified and where one or more of the external forces is an unknown to be determined.

Note that the datum plane for zero potential energy is arbitrary because only the change in potential energy matters, and this change is the same no matter where we place the datum plane. Note also that the gravitational potential energy is independent of the path followed in arriving at a particular level h. Thus, the body of mass m in Fig. 7/13 has Vg mgh Ve 1 2K2 0 418 Chapter 7 Virtual Work Vg = +Wh U = -W h Vg = 0 Vg = -Wh δh δ δ V h g δ = +Wδ W W G G +h +h alternative Datum plane or Figure 7/12 Reference datum Datum 2 Datum 1 m G G G h + Δh ΔVg = mg Δh h Figure 7/13 the same potential-energy change no matter which path it follows in going from datum plane 1 to datum plane 2 because h is the same for all three paths.

Note that the datum plane for zero potential energy is arbitrary because only the change in potential energy matters, and this change is the same no matter where we place the datum plane. Note also that the gravitational potential energy is independent of the path followed in arriving at a particular level h. Thus, the body of mass m in Fig. 7/13 has Vg mgh Ve 1 2K2 0 418 Chapter 7 Virtual Work Vg = +Wh U = -W h Vg = 0 Vg = -Wh δh δ δ V h g δ = +Wδ W W G G +h +h alternative Datum plane or Figure 7/12 Reference datum Datum 2 Datum 1 m G G G h + Δh ΔVg = mg Δh h Figure 7/13 the same potential-energy change no matter which path it follows in going from datum plane 1 to datum plane 2 because h is the same for all three paths.

The dimensions of the work of a force and the moment of a force are the same although they are entirely different physical quantities. Note that work is a scalar given by the dot product and thus involves the product of a force and a distance, both measured along the same line. Moment, on the other hand, is a vector given by the cross product and involves the product of force and distance measured at right angles to the force. To distinguish between these two quantities when we write their units, in SI units we use the joule (J) for work and reserve the combined units newton-meter ( ) for moment. In the U.S. customary system we normally use the sequence foot-pound (ft-lb) for work and pound-foot (lb-ft) for moment.

The dimensions of the work of a force and the moment of a force are the same although they are entirely different physical quantities. Note that work is a scalar given by the dot product and thus involves the product of a force and a distance, both measured along the same line. Moment, on the other hand, is a vector given by the cross product and involves the product of force and distance measured at right angles to the force. To distinguish between these two quantities when we write their units, in SI units we use the joule (J) for work and reserve the combined units newton-meter ( ) for moment. In the U.S. customary system we normally use the sequence foot-pound (ft-lb) for work and pound-foot (lb-ft) for moment.

The mechanical efficiency of simple machines which have a single degree of freedom and which operate in a uniform manner may be determined by the method of work by evaluating the numerator and denominator of the expression for e during a virtual displacement.

The mechanical efficiency of simple machines which have a single degree of freedom and which operate in a uniform manner may be determined by the method of work by evaluating the numerator and denominator of the expression for e during a virtual displacement.

The method of virtual work is especially useful for the purposes mentioned but requires that the internal friction forces do negligible work during any virtual displacement. Consequently, if internal friction in a mechanical system is appreciable, the method of virtual work cannot be used for the system as a whole unless the work done by internal friction is included.

The method of virtual work is especially useful for the purposes mentioned but requires that the internal friction forces do negligible work during any virtual displacement. Consequently, if internal friction in a mechanical system is appreciable, the method of virtual work cannot be used for the system as a whole unless the work done by internal friction is included.

The number of degrees of freedom of a mechanical system is the number of independent coordinates needed to specify completely the configuration of the system. Figure 7/8a shows three examples of onedegree-of-freedom systems. Only one coordinate is needed to establish the position of every part of the system. The coordinate can be a distance or an angle. Figure 7/8b shows three examples of two-degree-offreedom systems where two independent coordinates are needed to determine the configuration of the system. By the addition of more links to the mechanism in the right-hand figure, there is no limit to the number of degrees of freedom which can be introduced

The number of degrees of freedom of a mechanical system is the number of independent coordinates needed to specify completely the configuration of the system. Figure 7/8a shows three examples of onedegree-of-freedom systems. Only one coordinate is needed to establish the position of every part of the system. The coordinate can be a distance or an angle. Figure 7/8b shows three examples of two-degree-offreedom systems where two independent coordinates are needed to determine the configuration of the system. By the addition of more links to the mechanism in the right-hand figure, there is no limit to the number of degrees of freedom which can be introduced

The principle of virtual work U 0 may be applied as many times as there are degrees of freedom. With each application, we allow only one independent coordinate to change at a time while holding the others constant. In our treatment of virtual work in this chapter, we consider only one-degree-of-freedom systems.*

The principle of virtual work U 0 may be applied as many times as there are degrees of freedom. With each application, we allow only one independent coordinate to change at a time while holding the others constant. In our treatment of virtual work in this chapter, we consider only one-degree-of-freedom systems.*

The principle of zero virtual work for the equilibrium of a single particle usually does not simplify this already simple problem because U 0 and ΣF 0 provide the same information. However, we introduce the concept of virtual work for a particle so that we can later apply it to systems of particles.

The principle of zero virtual work for the equilibrium of a single particle usually does not simplify this already simple problem because U 0 and ΣF 0 provide the same information. However, we introduce the concept of virtual work for a particle so that we can later apply it to systems of particles.

The second derivative of V may also be zero at the equilibrium position, in which case we must examine the sign of a higher derivative to ascertain the type of equilibrium. When the order of the lowest remaining nonzero derivative is even, the equilibrium will be stable or unstable according to whether the sign of this derivative is positive or negative. If the order of the derivative is odd, the equilibrium is classified as unstable, and the plot of V versus x for this case appears as an inflection point in the curve with zero slope at the equilibrium value. Stability criteria for multiple degrees of freedom require more advanced treatment. For two degrees of freedom, for example, we use a Taylor-series expansion for two variables.

The second derivative of V may also be zero at the equilibrium position, in which case we must examine the sign of a higher derivative to ascertain the type of equilibrium. When the order of the lowest remaining nonzero derivative is even, the equilibrium will be stable or unstable according to whether the sign of this derivative is positive or negative. If the order of the derivative is odd, the equilibrium is classified as unstable, and the plot of V versus x for this case appears as an inflection point in the curve with zero slope at the equilibrium value. Stability criteria for multiple degrees of freedom require more advanced treatment. For two degrees of freedom, for example, we use a Taylor-series expansion for two variables.

rolling wheel in Fig. 7/9b, on the other hand, does no work if the wheel does not slip as it rolls. In Fig. 7/9c the moment Mƒ about the center of the pinned joint due to the friction forces which act at the contacting surfaces does negative work during any relative angular movement between the two parts. Thus, for a virtual displacement between the two parts, which have the separate virtual displacements 1 and 2 as shown, the negative work done is Mƒ 1 Mƒ 2 Mƒ(1 2), or simply Mƒ . For each part, Mƒ is in the sense to oppose the relative motion of rotation. It was noted earlier in the article that a major advantage of the method of virtual work is in the analysis of an entire system of connected members without taking them apart. If there is appreciable kinetic friction internal to the system, it becomes necessary to dismember the system to determine the friction forces. In such cases the method of virtual work finds only limited use.

rolling wheel in Fig. 7/9b, on the other hand, does no work if the wheel does not slip as it rolls. In Fig. 7/9c the moment Mƒ about the center of the pinned joint due to the friction forces which act at the contacting surfaces does negative work during any relative angular movement between the two parts. Thus, for a virtual displacement between the two parts, which have the separate virtual displacements 1 and 2 as shown, the negative work done is Mƒ 1 Mƒ 2 Mƒ(1 2), or simply Mƒ . For each part, Mƒ is in the sense to oppose the relative motion of rotation. It was noted earlier in the article that a major advantage of the method of virtual work is in the analysis of an entire system of connected members without taking them apart. If there is appreciable kinetic friction internal to the system, it becomes necessary to dismember the system to determine the friction forces. In such cases the method of virtual work finds only limited use.