Thermal Effects on Transverse Vibrations

It will be assumed that the temperature field is uniform, that is, the sample is homogeneously cooled or heated, and for this reason there is no spatial temperature dependence. Also a constant dilatation coefficient and a sudden jump in the temperature will be assumed. This thermal history, though not very realistic, refers to the most unfavorable situation. 

To complete the solution of the problem, it is necessary to specify the viscoelastic properties of the sample. For simplification reason, the Poisson ratio will be assumed to be constant. According to the results of Problem 16.5, which presents a similar situation, the Laplace transform of the stress is given by 

This expression indicates that the viscoelastic response for this special case has the same formal structure as the elastic one, except for the shifting variable, = Uj-t. Therefore the longitudinal tension Q in the beam will 

Obviously, the sign of the longitudinal force will depend on the sign of the temperature jump. If the sample is clamped at room temperature or higher, then under cooling (AT 0), the sample will be stretched. On the other hand, if the sample is clamped at low temperature, a compression will appear in the sample after heating (AT 0). 

It should be pointed out that the analysis of the complete problem requires us to take into account the longitudinal stretching of the sample when it is transversely loaded by an alternating force. In this case, even in the absence of an external longitudinal stress, a temperature jump gives rise to a force along the rigidly clamped bar. This force is a consequence of the 

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