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 is called a virtual displacement . Define the virtual work W done by a force to be the equilibrium force times this small imaginary displacement x . It should be emphasized that virtual work is not real work – no work has been performed since x 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 = = . The virtual work of the applied force is x F W apl apl = . The total virtual work is ( ) x F kx W W W apl apl spr + = + = (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 . Alternatively, if the virtual work is zero then, since x 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 is called a variation so that, for example, x 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 ) a ( ) b (
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Section 5.5 Solid Mechanics Part I Kelly 201 5.5.1 Principle of Virtual Work: a single particle A particle of mass m is acted upon by a number of forces, N f f f , , , 2 1 K , Fig. 5.5.2. Suppose the particle undergoes a virtual displacement u δ ; to reiterate, these impressed forces i f do not cause the particle to move, one imagines it to be incorrectly positioned a little away from the true equilibrium position. Figure 5.5.2: a particle in equilibrium under the action of a number of forces If the particle is moving with an acceleration a , the quantity a m is treated as an inertial force. The total virtual work is then (each term here is the dot product of two vectors) u a f = = m W N i i 1 (5.5.2) Now if the particle is in equilibrium by the action of the effective (impressed plus inertial) force, then 0 = W (5.5.3) This can be expressed as follows: The principle of virtual work (or principle of virtual displacements ) I : if a particle is in equilibrium under the action of a number of forces (including the inertial force) the total work done by the forces for a virtual displacement is zero Alternatively, one can define the external virtual work = u f i W ext and the virtual kinetic energy u a = m K in which case the principle takes the form K W = ext (compare with the work-energy principle, Eqn. 5.1.10).
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This note was uploaded on 01/20/2012 for the course ENGINEERIN 1 taught by Professor Staff during the Fall '11 term at Auckland.

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Energy_05_Virtual_Work - Section 5.5 5.5 Virtual Work...

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