Virtual Work - Introduction

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

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,

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

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

then we have

\delta W = \int_{t_0}^{t_1}\left(\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\right)dt = \int_{t_0}^{t_1}\left(\mathbf{F}\cdot \frac{\partial \mathbf{h}}{\partial q_1}\right) \epsilon\dot{q}_1 dt + \ldots + \int_{t_0}^{t_1}\left(\mathbf{F}\cdot \frac{\partial \mathbf{h}}{\partial q_n}\right) \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

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

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