Introduction to Structural Dynamics by Bruce K. Donaldson

By Bruce K. Donaldson

This can be a one semester engineering textbook on linear structural dynamics research for complex scholars.

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If a positive δθ were the only virtual displacement, the ruler would in the analyst’s imagination, rotate through a small angle, counterclockwise about the point located by the u and v generalized coordinates. 6(a). Of course, all three generalized coordinates, symbolized generally as q j , where j = 1, 2, 3, could be simultaneously augmented by virtual displacements, symbolized in general as δqj . The lowercase δ is the prefix that identifies all types of virtual quantities. Finally, an example of a prohibited displacement, imagined or real, would be any movement of the ruler off the z plane because such a motion would violate the constraints of the mathematical model.

Be sure to note that the above equation for generalized forces is not nearly as efficient a way of determining the values of generalized forces as that previously illustrated. Recall that the more efficient approach was simply taking, in turn, positive variations in the generalized coordinates and calculating the work done by the applied forces and moments with each such virtual displacement. When a generalized force of the summation is known to be energy conservative, such as an external generalized force resulting from a gravitational field, it is usually convenient to go one step further by introducing a potential (point) function that is associated with the energy conservative work done by such a force.

7 Virtual Quantities and the Variational Operator Before deriving the Lagrange equations, it is also necessary to briefly review the concept of virtual quantities and introduce the variational operator. A virtual displacement may be defined as any displacement of arbitrarily small magnitude that does not violate the constraints of the system. A virtual displacement may be a real displacement (a rare choice) or entirely a figment of the imagination of the analyst. An arbitrary, and therefore imaginary, small displacement is the usual case for a virtual displacement because generally it is a more useful choice than an actual displacement.

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