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Energy_05_Virtual_Work

# Energy_05_Virtual_Work - Section 5.5 5.5 Virtual Work...

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Section 5.5 Solid Mechanics Part I Kelly 200 5.5 Virtual Work Consider a mass attached to a spring and pulled by an applied force apl F , Fig. 5.5.1a. When the mass is in equilibrium, 0 = + apl spr F F , where kx F spr = is the spring force with x the distance from the spring reference position. Figure 5.5.1: a force extending an elastic spring; (a) block in equilibrium, (b) block not at its equilibrium position In order to develop a number of powerful techniques based on a concept known as virtual work , imagine that the mass is not in fact at its equilibrium position but at an (incorrect) non-equilibrium position x x δ + , Fig. 5.5.1b. The imaginary displacement x δ is called a virtual displacement . Define the virtual work W δ done by a force to be the equilibrium force times this small imaginary displacement x δ . It should be emphasized that virtual work is not real work – no work has been performed since x δ is not a real displacement which has taken place; this is more like a “thought experiment”. The virtual work of the spring force is then x kx x F W spr spr δ δ δ = = . The virtual work of the applied force is x F W apl apl δ δ = . The total virtual work is ( ) x F kx W W W apl apl spr δ δ δ δ + = + = (5.5.1) There are two ways of viewing this expression. First, if the system is in equilibrium ( 0 = + apl F kx ) then the virtual work is zero, 0 = W δ . Alternatively, if the virtual work is zero then, since x δ is arbitrary, the system must be in equilibrium. Thus the virtual work idea gives one an alternative means of determining whether a system is in equilibrium. The symbol δ is called a variation so that, for example, x δ is a variation in the displacement (from equilibrium). Virtual work is explored further in the following section. spr F apl F spr F apl F e x x δ ) a ( ) b (

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