Electromechanical Dynamics, Part I: Discrete Systems

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The dependence of the longitudinal and shear stresses on the transverse coordinate z, is shown in Fig. Note that the x,-dependences of Tu. The situation shown in Fig. Depth w into paper T driven by a force f t. Output load resistor. Input signal. Input force plate UAlu -. Newell, "Ultrasonics in Integrated Electronics," Proc. The elastic beam provides an inher- ently stable resonant element of extremely small proportions see Fig. Courtesy of Westinghouse Electric Corp. Of course, to find the driven response, we shall also find the natural frequencies of the beam. In an experiment such as that shown in Fig.

Hence the dependence of the lowest eigenfrequency on the length I will also be found. The equation of motion is The evanescent waves are required in addition to the ordinary waves to satisfy the four boundary conditions imposed on the beam.

This is in contrast to the situation in Section Here, all four waves are present simultaneously. As we have seen, a boundary value problem of this kind is more conveniently solved in terms of trigonometric and hyperbolic functions, rather than complex exponentials traveling waves. This force is held in equilibrium by the shear stress T The constants D and C follow from 1 and m to complete the solution for given by e. For now we assume that the forcing function is independent of , that is, that P is a given complex constant. Then b provides x, t.

When the denominator of q is zero, the response to the forcing function F is infinite. Hence the first four modes have frequencies such that oa is as shown in Table Given the value of al, the resonance frequency follows from Eq. The numbers indicate the solutions for the eigenvalues of the lowest three natural modes given in Table Note that the resonance frequency of any given mode varies inversely as the square of the beam length 1,a fact that is easily verified by the experiment in Fig. The numerator of q is plotted in Fig. The role played by the evanescent wave portion of the solution is clear from these deflections.

In the lowest mode the deflection appears to have an "exponential" character, which indicates that the evanescent solutions dominate. The first four natural modes are shown, with cl as given in Table The amplitude is exaggerated, with a different normalization for each mode. By contrast the higher modes are dominated by the sinusoidal deflections of the ordinary wave solutions, with the evanescent solutions becoming apparent near the ends. This trend is also seen in Fig. These results are consistent with the notion that the evanescent waves are excited by the boundary conditions and affect only that region in the vicinity of the boundary.

The longitudinal and transverse modes considered in this section have been described in terms of quasi-one-dimensional models. As the frequency is increased, the longitudinal wavelengths take on the same magnitude as the transverse dimensions of the elastic structure. Under this condition the effect of higher order transverse modes cannot be ignored, as is illustrated in Section As a result, the higher order modes, which become significant as the frequency is raised, are often mathematically complicated. We can, however, illustrate the basic physical effects by considering a particular class of modes composed of a purely shearing and rotational motion.

We x3. Shearing motions of the material in the x 3-direction are considered as they propagate in the x1 -direction. Mason, PhysicalAcoustics, loc. These assumptions are justified if we can find solutions that satisfy Ta Solutions to Given the frequency o, the wave- number k follows from At a given frequency each of the modes has a different wavenumber and a different dependence on the transverse x2 -dimension. The relationship between frequency and wavenumber is shown graphically in Fig.

By contrast with the membrane, however, a principal mode now propagates without dispersion, even as the frequency approaches zero. The spatial dependence of the first two modes is illustrated in Fig. The evanescent modes arise because of the "stiffness" introduced by the walls. The principal mode is not affected by the transverse boundary conditions, hence does not possess a cutoff frequency. This condition illustrates the general relationship between the principal modes discussed in Section As long as the wavelength is long compared with the thickness, only the principal modes propagate and need be considered far from the point of excitation.

As we saw in Section Modes of the kind described here are often used in delay lines.

Electromechanical Dynamics Part I: Discrete Systems

The higher modes are dispersive, hence lead to a distortion of the transmitted signal. For this reason the cutoff frequency often represents an upper limit on the frequency spectrum that can be transmitted without distortion. Many electromechanical interactions with elastic media can be modeled in terms of terminal pairs. This was illustrated in Chapter 9, where, even though portions of the mechanical system required continuum descriptions, the effect of electrical forces could be accounted for by means of boundary conditions.

We can now readily imagine using electromechanical transducers to excite or detect the waves discussed in Section At least in simple situations a discussion in this regard would parallel that given in Section 9. In Example In a similar manner we could use a transducer to excite or detect shear waves propagating through the slab of elastic material shown in Fig. By contrast, in this section we highlight a few illustrative situations in which continuum coupling with elastic media is important, but even in these cases the terminal pair concept is useful. A case in point is the design of large rotating machines, such as in Chapter 4.

Here the energy conversion process depends on a large magnetic torque being transmitted between the rotor and stator. Because action equals reaction, the rotor and stator materials are necessarily under significant stress due to the magnetic forces; for example, this is the primary reason that conductors are placed in slots. With the conductor imbedded in a highly permeable material, the bulk of the magnetic force is on the magnetic material rather than on the conductor. If this were not the case, it would be difficult to hold the con- ductors down in many machines.

In fact, a significant number of machine failures have been traced to fatigue ofconductors and their support structures stressed by magnetic forces. In a less obvious class of situations in which electromagnetic stresses are a major design consideration the objective is not to convert energy electro- mechanically.

Algebraic systems examples

Rather the forces of electrical origin are a necessary evil. Examples in which this is the case are transformers and magnets. In an ordinary transformer, electromechanical effects come into play in at least three mechanisms, two of which involve magnetization forces on the laminated magnetic core of the transformer. These forces arise because of inhomogeneities of the core introduced with the laminations and because of changes in the volume of the magnetic material magnetostriction. These forces were discussed in Section 8. A third mechanism for electromechanical effects is simply the J x B force density on the individual conductors in a transformer.

Transformers must be designed to withstand 25 or more times their rated currents in power applications to prevent mechanical damage under short circuit conditions. This is a step-down transformer with large. Note how reaction forces on the inner secondary coil have buckled it inward on the long sides of the rectangle. Also note that forces on the outer secondary coil have rounded it outward on the long sides.

Original shape of the coils on the long sides was flat. Courtesy of the General Electric Co. Courtesy of General Electric Company. Used with permission. With the secondary short-circuited, the ampere turns in the secondary are essentially equal to those in the primary.

Find courses by:

The arrangement of the core and windings is sketched in Fig. The secondary windings are constructed of sheets of aluminum which were originally wound in an essentially rectangular shape. As shown in Fig. The copper secondary turns bulge inward on the inside and outward on the outside.

Although, in this case, the result is not a gross mechanical failure of the structure, significant deformation of the insulation causes local damage that can lead to electrical breakdown.

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Also, the deformation increases the leakage reactance of the transformer. Increased leakage reactance increases regulation voltage drops as load current increases and this decreases the transformer efficiency, a crucial factor in distribution transformers.

So far in this chapter we have emphasized the elastic behavior of solid materials. Our main objective in this section is to draw attention to the fact that in many situations it is the inelasticbehavior of a solid that is of interest. If we wish to use solids to synthesize transducers, we must be careful to ensure that stresses are not so large that permanent or inelastic deformations will occur.

Even more, in many situations like the one shown in Fig. We are then faced with the problem of defining meaningful limits on the stress that can be supported by the material. Because the inelastic behavior is an upper bound on the elastic deformation of the material, we can use the elastic theory developed in earlier sections as a starting point for computing limiting stresses. A typical stress-strain relation for a polycrystalline metal is shown in Fig.

For small values of the stress and strain the relationship is essentially linear. As the stress is raised, however, a point is reached at which the resulting material strain increases more rapidly. Above this point, if the material is unloaded, it is likely that it will retain a permanent deforma- tion. An index of the degree of this permanent set is the yield strength of the material, which is defined in Fig. After the material has been loaded to the yield strength the limiting stress it is assumed that if it were unloaded it would return to the zero stress condition along a straight line parallel to the loading curve in the elastic range.

To fix the yield strength of a material we must define the hypothetical permanent set the strain taken by the material when the stress is returned to zero.

In practice this might be 0. Crandall and N.

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