By B. A. Auld

Quantity One starts with a scientific improvement of simple innovations (strain, tension, stiffness and compliance, viscous clamping) and coordinate ameliorations in either tensor and matrix notation. the elemental elastic box equations are then written in a sort analogous to Maxwell's equations. This analogy is then pursued while reading wave propagation in either isotropic and anisotropic solids. Piezoelectricity and bulk wave transducers are taken care of within the ultimate bankruptcy. Appendixes record slowness diagrams and fabric homes for numerous crystalline solids. quantity applies the cloth built in quantity One to a number of boundary worth difficulties (reflection and refraction at airplane surfaces, composite media, waveguides, and resonators). Pursuing the electromagnetic analogue, analytic innovations general in electromagnetism (for instance, common mode emissions), are utilized to elastic difficulties. ultimate chapters deal with perturbation and variational tools. An appendix lists homes of Rayleigh floor waves on unmarried crystal substrates.

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Goldstein, More on the prehistory of the Laplace-Runge-Lenz vector, Am. J. Phys. 44, 1123 (1976) 31 3. THE RIGID BODY Forward: Due to its particular features, the study of the motion of the rigid body has generated several interesting mathematical techniques and methods. In this chapter, we briefly present the basic rigid body concepts. 1 Definition A rigid body (RB) is defined as a system of particles whose relative distances are forced to stay constant during the motion. 2 Degrees of freedom In order to describe the general motion of a RB in the three-dimensional space one needs six variables, for example the three coordinates of the center of mass measured with respect to an inertial frame and three angles for labeling the orientation of the body in space (or of a fixed system within the body with the origin in the center of mass).

74) The equation (74) implies that ω3 = const. The equations (72) and (73) are rewritten as follows: ω˙ 1 = −Ωω2 where Ω = ω3 ω˙ 2 = −Ωω1 . 45 I3 − I1 I1 (75) (76) Multiplying (76) by i and summing it to (75), we have (ω˙ 1 + iω˙ 2) = −Ω(ω2 − iω1 ) (ω˙ 1 + iω˙ 2) = iΩ(ω1 + iω2 ). If we write η(t) = ω˙ 1 (t) + iω˙ 2(t), then η(t) ˙ − iΩη(t) = 0 . The solution is η(t) = A exp(iΩt) . This implies (ω1 + iω2 ) = A cos(Ωt) + i sin(Ωt) . Thus, ω1 = A cos(Ωt) (77) ω2 = A sin(Ωt). (78) The modulus of the vector ω does not change in time ω = ||ω|| = √ 2 ω1 + ω2 + ω3 = 2 A2 + ω32 = const .

EXAMPLE We give here an example of employing the previous equation. Determine the final amplitude of oscillations of a system acted by an extenal force F0 = const. during a limited time T . For this time interval we have ξ = exp(iwt) T ξ= F0 exp(iwt) m ξ= F0 [1 − exp(−iwt)] exp(iwt) . iwm 0 exp(−iwt)dt , Using |ξ|2 = a2 w2 we obtain a= 2F0 1 sin( wT ) . 3 DAMPED HARMONIC OSCILLATOR Until now we have studied oscillatory motions in free space (the vacuum), or when the effects of the medium through which the oscillator moves are negligeable.

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