Continuum Electromechanics Exam Help

Continuum Electromechanics Exam Help

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Problem 1 Prob. 8.18.2 (Melcher, Continuum Electromechanics) Corrections: Part (b) Part (c)

Problem 2 Prob. 8.18.3 (Melcher) Correction: Prob. 8.18.1 (continued)

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where, consistent with the usage in Section 8.14, Eo is the equilibrium electric field evaluated at the interface between layers. (b) Show that the dispersion equation for the layer model, based on the results of Section 8.14, takes the normalized form

(c) Using k = 1, Dpm = 0, VoJVoJ = i1, Dq /jDqel = 1 and S = 1, compare the prediction of the first eigenfrequency to the first resonance frequency predicted in the weak-gradient approximation and to the "exact" result shown in Fig. 8.18.2a. Compare the analytical expression to that for the weak-gradient imposed field approximation in the long-wave limit. Should it be expected that the layer approximation would agree with numerical results for very short wavelengths? (d) How should the model be refined to include the second mode in the prediction?

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Prob. 8.18.2 A layer of magnetizable liquid is in static equilibrium, with mass density and permeability having vertical distributions ps(x) and Is(x) (Fig. P18.8.2). The equilibrium magnetic field Hs(x) is assumed to also have a weak gradient in the x direction, even though such a field is not irrotational. (For example, this gradient represents fields in the cylindrical annulus between concentric pole faces, where the poles have radii large compared to the annulus depth k. The gradient in H is a quasi-one-dimensional model for the circular geometry.) Assume that the fluid is perfectly insulating and inviscid. (a) Show that the perturbation equations can be reduced to

(b) As an example, assume that the profiles are constant. Show that solutions are a linear combination of expyx, where

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(c) Assume that boundary conditions are

and that eigenfrequencies are

(e) Discuss the analogous electric coupling with the analogous physical configuration.

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and describe

Prob. 8.18.3 As a continuation of Problem 8.18.2, prove that the principle of exchange of stabilities holds, and specifically that the eigenfrequencies are given by

where

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Solutions Problem 8.18.2 The basic equations for the magnetizable but insulating inhomogeneous fluid are

where H = Hs(x)iz + h.

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With the definitions µ = µs(x) + µ’ and ρ = ρs(x) + ρ’ , Eqs. 5 and 6 link the perturbations in properties to the fluid displacement

Thus, with the use of Eq. 8a and Eqs. 7, the linearized version of Eq. 3 is

and this represents the magnetic field, given the mechanical deformation. To represent the mechanics, Eq. 2 is written in terms of complex amplitudes:

and, with the use of Eq. 8b, the x component of Eq. 1 is written in the linearized form

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Similarly, the y and z components of Eq. 1 become

With the objective of making vˆx a scalar function representing the mechanics, these last two expressions are solved for vˆy and vˆz and substituted into Eq. 10:

This expression is then solved for pˆ, and the derivative taken with respect to x. This derivative can then be used to eliminate the pressure from Eq. 11:

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Equations 9 and 15 comprise the desired relations.

In an imposed field approximation where Hs = H0 = constant and the properties have the profiles ρs = ρm exp βx and µs = µm exp βx, Eqs. 9 and 15 become

where L ≡ D2 + βD − k2.

For these constant coefficient equations, solutions take the form exp γx and L → γ2 + βγ − k2. From Eqs. 16 and 17 it follows that

Solution for L results in

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From the definition of L, the γ’s representing the x dependence follow as

In terms of these γ’s,

The corresponding hˆz is written in terms of these same coefficients with the help of Eq. 17:

Thus, the four boundary conditions require that

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This determinant is easily reduced by first subtracting the second and fourth columns from the first and third respectively and then expanding by minors

Thus, eigenmodes are c+l = jnπ and c−l= jnπ. The eigenfrequencies follow from Eqs. 19 and 20:

For perturbations with peaks and valleys running perpendicular to the imposed fields, the magnetic field stiffens the fluid. Internal electromechanical waves propagate along the lines of magnetic field intensity. If the fluid were confined between parallel plates in the x − z planes, so that the fluid were indeed forced to undergo only two dimensional motions, the field could be used to balance a heavy fluid on top of a light one.... to prevent the gravitational form of Rayleigh-Taylor instability. However, for perturbations with hills and valleys running parallel to the imposed field, the magnetic field remains undisturbed, and there is no magnetic restoring force to prevent the instability. The role of the magnetic field, here in the context of an internal coupling, is similar to that for the hydromagnetic system described in Sec. 8.12 where interchange modes of instability for a surface coupled system were found. liveexamhelper.com

The electric polarization analogue to this configuration might be as shown in Fig. 8.11.1, but with a smooth distribution of in the x direction Problem 8.18.3 Starting with Eqs. 9 and 15 from Prob. 8.18.2, multiply the first by hˆ∗ z and integrate from 0 to l.

Integration of the first term by parts and use of the boundary conditions on hˆz gives integrals on the left that are positive definite:

Now, multiply Eq. 15 from Prob. 8.18.2 by v ˆx ∗ and integrate:

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Integration of the first term by parts and the boundary conditions on vˆx gives

and this expression takes the form

Multiplication of Eq. 28 by Eq. 31 results in yet another positive definite quantity

and this expression can be solved for the frequency

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Because the terms on the right are real, it follows that either the eigenfrequencies are real or they represent modes that grow and decay without oscillation. Thus, the search for eigenfrequencies in the general case can be restricted to the real and imaginary axes of the s plane.

Note that a sufficient condition for stability is N > 0, because that insures that I3 is positive definite.

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