Plate vibration equations

The equilibrium equations for a beam are derived to illustrate the derivation process and to serve as a review in preparation for addressing plates. Then, the plate equilibrium equations are derived for use in Chapter 5. Next, the plate buckling equations are discussed. Finally, the plate vibration equations are addressed. In each case, the pertinent boundary conditions are displayed. Nowhere in this appendix is reference needed to laminated beams or plates. All that is derived herein is applicable to any kind of beam or plate because only fundamental equilibrium, buckling, or vibration concepts are used. 

AT and BT plates are made to vibrate primarily in the thickness shear mode. However, it is important to realize that quartz crystals can be made to vibrate in any one or combination of these modes. The vibrational modes can be induced electrically, acoustically, thermally or by some combination of all of these factors. The thickness shear frequency response of an AT plate can be described in terms of the fundamental frequency. For a rectangular AT cut plate, the equation for calculating the approximate frequency of vibration is ... 

GOVERNING EQUATIONS FOR BEAM EQUILIBRIUM AND PLATE EQUILIBRIUM, BUCKLING, AND VIBRATION... 

Vibration isolation 237—250 critical damping 239 pneumatic systems 250 quality factor, Q 239 resonance excitation 241 stacked plate-elastomer system 249 transfer function 240 Virus 341 Viton 250, 270, 272 Voltage-dependent imaging 16, 17 Si(lOO) 17 Si(lll)-2X1 16 Volterra equation 310 Vortex 334 W... 

The kinetics of the laser-induced polymerization was followed either by measuring the thickness of the insoluble polymer film formed on the quartz plate after laser exposure and solvent development, using UV spectroscopy or by monitoring the decrease of the IR absorption of the coating at 810 cm-1 which corresponds to the twisting vibration of the acrylate CH2=CH double bond. This last method permits accurate evaluation of the rate of polymerization (Rp) by observing the variation of the 810 cm-1 band, and using Equation 2 ... 

Fig. 22 shows the results of photometry of plates similar to that illustrated in Fig. 21. The relative intensities of suitable transitions were determined from the asymptotic limit at long time delays when the system attains equilibrium. (These resemble, but are not identical to, the relative/ values because of the usual instrumental effects which depend on line width.) The time variation of the relative concentrations is shown in Fig. 23 the upper four levels attain Boltzmann equilibrium amongst themselves after 100 /isec, to form a coupled (by collision) system overpopulated with respect to the 5DA state. The equilibration of the upper four levels causes the initial rise (Fig. 22) in the population of Fe(a5D3). Thus relaxation amongst the sub-levels is formally similar to vibrational relaxation in most polyatomic molecules, in which excitation to the first vibrational level is the rate determining step. In both cases, this result is due to the translational overlap term, for example, in the simple form of equation (14) of Section 3.

An interest represents the comparison of (po with AV-potential, i.e. the total potential difference at the solution/air interface. Fig. 3.24 plots q>o(C) and AV(Q dependences for a solution of non-ionic surfactant. The measurement of AV-potential is performed by the method of vibrating plate over the solution surface [203]. (po and AV change simultaneously and reach a maximum value at the same surfactant concentration. Surely, their absolute values are different, as expected from the following equation [204]... 

Another method recently developed for manipulating small particles uses the forces created by a two- or three-dimensional sound field that is excited by a vibrating plate, the surfaces of which move sinusoidally and emit an acoustic wave into a layer of fluid. Such a wave is reflected by a rigid surface and generates a standing sound field in the fluid, the forces of which act on particles by displacing them in one, two or three dimensions. In this way, particles of sizes between one and several hundred microns can be simultaneously manipulated in a contactless manner. Equations describing this behaviour have been reported [63]. 

In the first part of this chapter we studied the radial vibrations of a solid or hollow sphere. This problem was considered an extension to the dynamic situation of the quasi-static problem of the response of a viscoelastic sphere under a step input in pressure. Let us consider now the simple case of a transverse harmonic excitation in which separation of variables can be used to solve the motion equation. Let us assume a slab of a viscoelastic material between two parallel rigid plates separated by a distance h, in which a sinusoidal motion is imposed on the lower plate. In this case we deal with a transverse wave, and the viscoelastic modulus to be used is, of course, the shear modulus. As shown in Figure 16.7, let us consider a Cartesian coordinate system associated with the material, with its X2 axis perpendicular to the shearing plane, its xx axis parallel to the direction of the shearing displacement, and its origin in the center of the lower plate. Under steady-state conditions, each part of the viscoelastic slab will undergo an oscillatory motion with a displacement i(x2, t) in the direction of the Xx axis whose amplitude depends on the distance from the origin X2-... 

Here also the proper value parameter A depends on the nature of the plate and in effect represents the square of the frequency. The differential equation can be easily solved in polar co-ordinates (Appendix XVII, p. 297). In this case again we obtain possible forms of vibration... 

The derivation of a relationship between the phases of the different frequencies at the receiving transducer tn and the phases of the force components at the bonded interface is more compHcated. The continuity of stresses and displacements at the aluminum plate/couphng medium and couphng medium/re-ceiver probe interfaces have to be taken into account, which introduces the material parameters of both the couphng medium and the transducer into the calibration equations. The procedure described so far yielded the phases relative to the excitation at the transmitting transducer. To obtain the phases related to the interface vibration, similar considerations exploiting the continuity of stress and displacement at the bonded interface have to be carried out. Details have been presented elsewhere [10] and are omitted here due to lack of space. 

The cubic equation (4.20) allows determination of the adhesive modulus of elasticity E2 in a plate made of three layers. For this purpose it is necessary to find the frequency of the natural vibrations of the three-layer plate under investigation with one edge fixed and the other free. 

Summary Non-stationary random vibrations of polvcfonally shaped slightly damped Kirchhoff-plates are presented. The frequency response function of the undamped structure is calculated by an advanced bound-ary-integral equation method with Green s functions of finite domains. Subsequently, light hysteretic damping is built in by applying the quadrature type of elastic-viscoelastic correspondence. ... 

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