User:Prof McCarthy/virtualwork

Virtual work arises in the application of the principle of least action to the study of forces and movement of a mechanical system. Historically, virtual work, and the associated calculus of variations, was formulated to analyze systems of rigid bodies, but it has also been developed for the study of the mechanics of deformable bodies.

If a force acts on a particle as it moves from point A to point B, then, for each possible trajectory that the particle may take, it is possible to compute the total work done by the force along the path. The principle of virtual work, which is the form of the principle of least action applied to these systems, states that the path actually followed by the particle is the one for which the difference between the work along this path and other nearby paths is zero. The formal procedure for computing the difference of functions evaluated on nearby paths is a generalization of the derivative known from differential calculus, and is termed the calculus of variations.

Let the function x(t) define the path followed by a point, then a nearby path can be defined by adding the function δx(t) to the original path, so the new path is given by x(t)+δx(t). The function δx(t) is called the variation of the original path, and each of the components of δx=(δx, δy, δz) is called a virtual displacement. This can be generalized to an arbitrary mechanical system defined by the generalized coordinates q_i, i=1,..., n. In which case, the variation of the trajectory q_i(t) is defined by the virtual displacements δq_i, i=1,..., n.

Virtual work can now be described as the work done by the applied forces and the inertial forces of a mechanical system as it moves through a set of virtual displacements.

History

The formulation of virtual work implements the view that nature selects from a set of "tentative" realities in order to minimize a quantity. This is a version of the "simplicity hypothesis" that can be traced to Aristotle. Another form of this hypothesis is Ockham's "razor" which states that "it is futile to employ many principles when it is possible to employ fewer." These ideas illustrate a view of physics that nature optimizes in some way.

Leibnitz formulated Newton's laws of motion in terms of work and kinetic energy, or vis viva (living force), which are minimized as a system moves. Maupertuis adapted Liebnitz's ideas as the principle of least action that nature minimizes action. But it was Euler and Lagrange who provided the mathematical foundation of the calculus of variations and applied it to the study of the statics and dynamics of mechanical systems.

Hamilton's reformulation of the principle of least action and Lagrange's equations yielded a theory of dynamics that is the foundation for modern physics and quantum mechanics.

Introduction

In this introduction basic definitions are presented that will assist in understanding later sections.

Consider a particle P that moves along a trajectory r(t) from a point A to a point B, while a force F is applied to it, then the work done by the force is given by the integral

$W=\int _{\mathbf {r} (t_{0})=A}^{\mathbf {r} (t_{1})=B}\mathbf {F} \cdot d\mathbf {r} =\int _{t_{0}}^{t_{1}}\mathbf {F} \cdot \mathbf {v} dt,$

where dr is the differential element along the curve that is the trajectory of P, and v is its velocity. It is important to notice that the value of the work W depends on the trajectory r(t).

Now consider the work done by the same force on the same particle P again moving from point A to point B, but this time moving along the nearby trajectory that differs from r(t) by the variation δr(t)=εh(t), where ε is a scaling constant that can be made as small as desired and h(t) is an arbitrary function that satisfies h(t₀)=h(t₁)=0,

${\bar {W}}=\int _{A}^{B}\mathbf {F} \cdot d(\mathbf {r} +\epsilon \mathbf {h} )=\int _{t_{0}}^{t_{1}}\mathbf {F} \cdot (\mathbf {v} +\epsilon {\dot {\mathbf {h} }})dt.$

The variation of the work δW associated with this nearby path, known as the virtual work, can be computed to be

$\delta W={\bar {W}}-W=\int _{t_{0}}^{t_{1}}(\mathbf {F} \cdot \epsilon {\dot {\mathbf {h} }})dt.$

Now assume that r(t) and h(t) depend on the generalized coordinates q_i, i=1, ..., n, then the derivative of the variation δr=εh(t) is given by

${\frac {d}{dt}}\delta \mathbf {r} =\epsilon {\dot {\mathbf {h} }}=\epsilon {\big (}{\frac {\partial \mathbf {h} }{\partial q_{1}}}{\dot {q}}_{1}+\ldots +{\frac {\partial \mathbf {h} }{\partial q_{n}}}{\dot {q}}_{n}{\big )},$

then we have

$\delta W=\int _{t_{0}}^{t_{1}}{\big (}\mathbf {F} \cdot {\frac {\partial \mathbf {h} }{\partial q_{1}}}\epsilon {\dot {q}}_{1}+\ldots +\mathbf {F} \cdot {\frac {\partial \mathbf {h} }{\partial q_{n}}}\epsilon {\dot {q}}_{n}{\big )}dt=\int _{t_{0}}^{t_{1}}{\big (}\mathbf {F} \cdot {\frac {\partial \mathbf {h} }{\partial q_{1}}}{\big )}\epsilon {\dot {q}}_{1}dt+\ldots +\int _{t_{0}}^{t_{1}}{\big (}\mathbf {F} \cdot {\frac {\partial \mathbf {h} }{\partial q_{n}}}{\big )}\epsilon {\dot {q}}_{n}dt.$

The requirement that the virtual work be zero for an arbitrary variation δr(t)=εh(t) is equivalent to the set of requirements

$F_{i}=\mathbf {F} \cdot {\frac {\partial \mathbf {h} }{\partial q_{i}}}=0,\quad i=1,\ldots ,n.$

The terms F_i are called the generalized forces associated with the virtual displacement δr.

Static equilibrium

Static equilibrium is the condition in which the applied forces and constraint forces on a mechanical system balance such that the system does not move. The principle of virtual work states that the virtual work of the applied forces is zero for all virtual movements of the system from static equilibrium, that is, δW=0 for any variation δr. This is equivalent to the requirement that the generalized forces for any virtual displacement are zero, that is F_i=0.

Let the forces on the system be F_j, j=1, ..., m and let the virtual displacement of each point of application of these forces be δr_j, j=1, ..., m, then the virtual work generated by a virtual displacement of these forces from the equilibrium position is given by

$\delta W=\sum _{j=1}^{m}\mathbf {F} _{j}\cdot \delta \mathbf {r} _{j}.$

Now assume that each δr_j depends on the generalized coordinates q_i, i=1, ..., n, then

$\delta \mathbf {r} _{j}={\frac {\partial \mathbf {r} _{j}}{\partial q_{1}}}\delta {q}_{1}+\ldots +{\frac {\partial \mathbf {r} _{j}}{\partial q_{n}}}\delta {q}_{n},$

and

$\delta W={\big (}\sum _{j=1}^{m}\mathbf {F} _{j}\cdot {\frac {\partial \mathbf {r} _{j}}{\partial q_{1}}}{\big )}\delta {q}_{1}+\ldots +{\big (}\sum _{j=1}^{m}\mathbf {F} _{j}\cdot {\frac {\partial \mathbf {r} _{j}}{\partial q_{n}}}{\big )}\delta {q}_{n}.$

The n terms

$F_{i}=\sum _{j=1}^{m}\mathbf {F} _{j}\cdot {\frac {\partial \mathbf {r} _{j}}{\partial q_{i}}},\quad i=1,\ldots ,n,$

are the generalized forces acting on the system. Kane shows that these generalize forces can also be formulated in terms of the ratio of time derivatives,

$F_{i}=\sum _{j=1}^{m}\mathbf {F} _{j}\cdot {\frac {\partial \mathbf {v} _{j}}{\partial {\dot {q}}_{i}}},\quad i=1,\ldots ,n,$

where v_j is the velocity of the point of application of the force F_j.

In order for the virtual work to be zero for an arbitrary virtual displacement, each of the generalized forces must be zero, that is

$\delta W=0\quad \Rightarrow \quad F_{i}=0,i=1,\ldots ,n.$

Constraint forces

An important benefit of the principle of virtual work is that only forces that do work as the system moves through a virtual displacement are needed to determine the mechanics of the system. There are many forces in a mechanical system that do no work during a virtual displacement, which means that they need not be considered in this analysis. The two important examples are (i) the internal forces in a rigid body, and (ii) the constraint forces at an ideal joint.

Lanczos presents this as the postulate: "The virtual work of the forces of reaction is always zero for any virtual displacement which is in harmony with the given kinematic constraints." The argument is as follows. The principle of virtual work states in equilibrium the virtual work of the forces applied to a system is zero. Newton's laws state that at equilibrium the applied forces are equal and opposite to the reaction, or constraint, forces. This means the virtual work of the constraint forces must be zero as well.

Couples

A pair of forces action on a rigid body can form a couple defined by the moment vector M. The virtual work of a moment vector is obtained from the virtual rotation of the rigid body.

For planar movement, the moment acts perpendicular to the plane with magnitude M and the virtual work due to this moment is

$\delta W=M\delta \phi ,$

where δφ is the virtual rotation angle of the body.

In order to extend this to three dimensional rotations, use the angular velocity vector ω of the body to obtain the virtual work as

$\delta W={\big (}\mathbf {M} \cdot {\frac {\partial {\vec {\omega }}}{\partial {\dot {\phi }}}}{\big )}\delta \phi .$

Now consider the moments M_j acting on m rigid bodies in a mechanical system. Let the angular velocity vectors ω_j, j= 1,..., m of each body depend on the n generalized coordinates q_j, i=1,... , n. Then the virtual work obtained from these moments for a virtual displacement from equilibrium is given by

$\delta W=\sum _{j=1}^{m}\mathbf {M} _{j}\cdot {\big (}{\frac {\partial {\vec {\omega }}_{j}}{\partial {\dot {q}}_{1}}}\delta q_{1}+\ldots +{\frac {\partial {\vec {\omega }}_{j}}{\partial {\dot {q}}_{n}}}\delta q_{n}{\big )}.$

Collect the coefficients of the virtual displacements δq_i to obtain

$\delta W={\big (}\sum _{j=1}^{m}\mathbf {M} _{j}\cdot {\frac {\partial {\vec {\omega }}_{j}}{\partial {\dot {q}}_{1}}}{\big )}\delta q_{1}+\ldots +{\big (}\sum _{j=1}^{m}\mathbf {M} _{j}\cdot {\frac {\partial {\vec {\omega }}_{j}}{\partial {\dot {q}}_{n}}}{\big )}\delta q_{n}.$

Forces and moments

Combine the virtual work above for couples with the virtual work of forces in order to obtain the virtual work of a system of forces and moments acting on system of rigid bodies displaced from equilibrium as

$\delta W={\big (}\sum _{j=1}^{m}\mathbf {F} _{j}\cdot {\frac {\partial \mathbf {v} _{j}}{\partial {\dot {q}}_{1}}}+\mathbf {M} _{j}\cdot {\frac {\partial {\vec {\omega }}_{j}}{\partial {\dot {q}}_{1}}}{\big )}\delta q_{1}+\ldots +{\big (}\sum _{j=1}^{m}\mathbf {F} _{j}\cdot {\frac {\partial \mathbf {v} _{j}}{\partial {\dot {q}}_{n}}}+\mathbf {M} _{j}\cdot {\frac {\partial {\vec {\omega }}_{j}}{\partial {\dot {q}}_{n}}}{\big )}\delta q_{n},$

where the generalized forces are now defined to be

$F_{i}=\sum _{j=1}^{m}\mathbf {F} _{j}\cdot {\frac {\partial \mathbf {v} _{j}}{\partial {\dot {q}}_{i}}}+\mathbf {M} _{j}\cdot {\frac {\partial {\vec {\omega }}_{j}}{\partial {\dot {q}}_{i}}},\quad i=1,\ldots ,n.$

The principle of virtual work requires that a system of rigid bodies acted on by the forces and moments F_j and M_j is in equilibrium if the generalized forces F_j are zero, that is

$\delta W=0\quad \Rightarrow \quad F_{i}=0,i=1,\ldots ,n.$

One degree-of-freedom mechanisms

In this section, the principle of virtual work is used for the static analysis of one degree-of-freedom mechanical devices. Specifically, we analyze the lever, a pulley system, a gear train, and a four-bar linkage. Each of these devices moves in the plane, therefore a force F=(f_x, f_y) has two components and acts on a point with coordinates r= (r_x, r_y) and velocity v=(v_x, v_y). A moment, also called a torque, T acting on a body that moves in the plane has one component as does the angular velocity ω of the body.

Assume the bodies in the mechanism are rigid and the joints are ideal so that the only change in virtual work is associated with the movement of the input and output forces and torques.

Applied Forces

Consider a mechanism such as a lever that operates so that an input force generates an output force. Let A be the point where the input force F_A is applied, and let B be the point where the output force F_B is exerted. Define the position and velocity of A and B by the vectors r_A, v_A and r_B, v_B, respectively.

Because the mechanism has one degree-of-freedom, there is a single generalized coordinate q that defines the position vectors r_A(q) and r_B(q) of the input and output points in the system. The principle of virtual work requires that the generalized force associated with this coordinate be zero, thus

$F_{q}=\mathbf {F} _{A}\cdot {\frac {\partial \mathbf {v} _{A}}{\partial {\dot {q}}}}-\mathbf {F} _{B}\cdot {\frac {\partial \mathbf {v} _{B}}{\partial {\dot {q}}}}=0.$

The negative sign on the output force F_B arises because the convention of virtual work assumes the forces are applied to the device.

Applied Torque

Consider a mechanism such as a gear train that operates so that an input torque generates an output torque. Let body E_A have the input moment T_A applied to it, and let body E_B exert the output torque T_B. Define the angular position and velocity of E_A and E_B by θ_A, ω_A and θ_B, ω_B, respectively.

Because the mechanism has one degree-of-freedom, there is a single generalized coordinate q that defines the angles θ_A(q) and θ_B(q) of the input and output of the system. The principle of virtual work requires that the generalized force associated with this coordinate be zero, thus

$F_{q}=T_{A}{\frac {\partial \mathbf {\omega } _{A}}{\partial {\dot {q}}}}-T_{B}{\frac {\partial \mathbf {\omega } _{B}}{\partial {\dot {q}}}}=0.$

The negative sign on the output torque T_B arises because the convention of virtual work assumes the torques are applied to the device.

Law of the Lever

A lever is modeled as a rigid bar connected a ground frame by a hinged joint called a fulcrum. The lever is operated by applying an input force F_A at a point A located by the coordinate vector r_A on the bar. The lever then exerts an output force F_B at the point B located by r_B. The rotation of the lever about the fulcrum P is defined by the rotation angle θ.

This is an engraving from Mechanics Magazine published in London in 1824.

Let the coordinate vector of the point P that defines the fulcrum be r_P, and introduce the lengths

$a=|\mathbf {r} _{A}-\mathbf {r} _{P}|,\quad b=|\mathbf {r} _{B}-\mathbf {r} _{P}|,$

which are the distances from the fulcrum to the input point A and to the output point B, respectively.

Now introduce the unit vectors e_A and e_B from the fulcrum to the point A and B, so

$\mathbf {r} _{A}-\mathbf {r} _{P}=a\mathbf {e} _{A},\quad \mathbf {r} _{B}-\mathbf {r} _{P}=b\mathbf {e} _{B}.$

This notation allows us to define the velocity of the points A and B as

$\mathbf {v} _{A}={\dot {\theta }}a\mathbf {e} _{A}^{\perp },\quad \mathbf {v} _{B}={\dot {\theta }}b\mathbf {e} _{B}^{\perp },$

where e_A^⊥ and e_B^⊥ are unit vectors perpendicular to e_A and e_B, respectively.

The angle θ is the generalized coordinate that defines the configuration of the lever, therefore using the formula above for forces applied to a one degree-of-freedom mechanism, the generalized force is given by

$F_{\theta }=\mathbf {F} _{A}\cdot {\frac {\partial \mathbf {v} _{A}}{\partial {\dot {\theta }}}}-\mathbf {F} _{B}\cdot {\frac {\partial \mathbf {v} _{B}}{\partial {\dot {\theta }}}}=a(\mathbf {F} _{A}\cdot \mathbf {e} _{A}^{\perp })-b(\mathbf {F} _{B}\cdot \mathbf {e} _{B}^{\perp }).$

Now, denote the components of the forces that are perpendicular to the radial segments PA and PB as F_A and F_B. These forces are given by

$F_{A}=\mathbf {F} _{A}\cdot \mathbf {e} _{A}^{\perp },\quad F_{B}=\mathbf {F} _{B}\cdot \mathbf {e} _{B}^{\perp }.$

Finally, the principle of virtual work states the lever is in static equilibrium, when the generalize force is zero, that is

$F_{\theta }=aF_{A}-bF_{B}=0.\,\!$

This equation allows the calculation of the mechanical advantage of the lever, which is the ratio of the output force F_B to the input force F_A, that is

$MA={\frac {F_{B}}{F_{A}}}={\frac {a}{b}}.$

This equation shows that if the distance a from the fulcrum to the point A where the input force is applied is greater than the distance b from fulcrum to the point B where the output force is applied, then the lever amplifies the input force. If the opposite is true that the distance from the fulcrum to the input point A is less than from the fulcrum to the output point B, then the lever reduces the magnitude of the input force.

This is the law of the lever, which was proven by Archimedes using geometric reasoning.

Gear train

A gear train is formed by mounting gears on a frame so that the teeth of the gears engage. Gear teeth are designed to ensure the pitch circles of engaging gears roll on each other without slipping, this provides a smooth transmission of rotation from one gear to the next. For this analysis, we consider a gear train that has one degree-of-freedom, which means the angular rotation of all the gears in the gear train are defined by the angle of the input gear.

The size of the gears in the gear train and the sequence in which they engage define the ratio of the angular velocity ω_A of the input gear G_A to the angular velocity ω_B of the output gear G_B, known as the speed ratio, or gear ratio, of the gear train. Let R be the speed ratio of the gear train, then

${\frac {\omega _{A}}{\omega _{B}}}=R.$

The input torque T_A acting on the input gear G_A is transformed by the gear train into the output torque T_B exerted by the output gear G_B. If we assume, that the gears are rigid and that there are no losses in the engagement of the gear teeth, then the principle of virtual work can be used to analyze the static equilibrium of the gear train.

Let the angle θ of the input gear be the generalized coordinate of the gear train, then the speed ratio R of the gear train defines the angular velocity of the output gear in terms of the input gear, that is

$\omega _{A}=\omega ,\quad \omega _{B}=\omega /R.\!$

The formula above for the principle of virtual work with applied torques yields the generalize force

$F_{\theta }=T_{A}{\frac {\partial \omega _{A}}{\partial \omega }}-T_{B}{\frac {\partial \omega _{B}}{\partial \omega }}=T_{A}-T_{B}/R=0.$

The mechanical advantage of the gear train is the ratio of the output torque T_B to the input torque T_A, and the above equation yields

$MA={\frac {T_{B}}{T_{A}}}=R.$

Thus, the speed ratio of a gear train also defines its mechanical advantage. This shows that if the input gear rotates faster than the output gear, then the gear train amplifies the input torque. And, if the input gear rotates slower than the output gear, then the gear train reduces the input torque.