The Steady-State Vibration Response of Abaffled Plate

THE STEADY-STATE VIBRATION RESPONSE OF ABAFFLED PLATE

SIMPLY-SUPPORTED ON ALL SIDESSUBJECTED TO RANDOMPRESSUREPLANE WAVE EXCITATION AT OBLIQUE INCIDENCE

Revision A

ByTom Irvine

December 27, 2014

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The random excitation paper is very similar to Reference 1 which covered harmonic excitation.

Thebaffled, simply-supported plate in Figure 1 is subjected to an obliqueplane pressure wave on one side. Only a side view along the length is shown because the pressure is assumed to be uniform with width. This diagram and the corresponding pressure field equation are taken from Reference 1.

Figure 1.

is the acoustic wavelength. t is the trace wavelength.

(1)

The governing differential equation from Reference 2 is

(2)

The plate stiffness factor D is given by

(3)

where

The mass-normalized mode shapes are

(4)

(5)

(6)

(7)

(8)

The natural frequencies are

(9)

(10)

Let

(11)

Thus

(12)

(13)

The generalized force is

(14)

Let

(15)

(16)

(17)

(18)

(19)

(20)

(21)

(22)

(23)

For ,

(24)

Note that for all m values,

(25)

(26)

(27)

(28)

(29)

(30)

(31)

(32)

The phase angle ε will be regarded as arbitrary since only a steady-state solution is sought.

(33)

Note the following relationship for the modal wavelength .

(34)

(35)

(36)

(37)

(38)

For an arbitrary phase angle, the equation may be rewritten as

for ,

(39)

For ,

(40)

Again, only the steady-state solution is needed. So define a participation factor mn and represent the time varying term as a harmonic excitation function.

(41)

where

(42)

For ,

(43)

For ,

(44)

Define a joint acceptance function Jmn.

(45)

(46)

For ,

(47)

The equation of motion for the modal coordinates is

(48)

(49)

The temporal variable response to the applied force transposed to the frequency domain is

(50)

(51)

Recall

(52)

Transpose to the frequency domain.

(53)

(54)

The frequency response function relating the displacement to the oblique pressure field is

(55)

(56)

The bending moments are

(57)

(58)

The bending stresses from Reference 3 are

(59)

(60)

(61)

is the distance from the centerline in the vertical axis

An example is given in Appendix B.

References

1. T. Irvine, The Steady-State Vibration Response of a Baffled Plate Simply-Supported on All Sides Subjected to Harmonic Pressure Wave Excitation at Oblique Incidence, Revision E, Vibrationdata, 2014.
   
2. T. Irvine, The Mean Square Force Due to Random Forces on a Rigid Beam, Revision B, Vibrationdata, 2014.
3. J.S. Rao, Dynamics of Plates, Narosa, New Delhi, 1999.
4. T. Irvine, Steady-State Vibration Response of a Plate Simply-Supported on All Sides Subjected to a Uniform Pressure, Revision C, Vibrationdata, 2014.
5. E. Richards & D. Mead, Noise and Acoustic Fatigue in Aeronautics, Wiley, New York, 1968.

APPENDIX A

Magnitude of Complex Trigonometric Term

(A-1)

(A-2)

(A-3)

(A-4)

(A-5)

(A-6)

(A-7)

APPENDIX B

Example

Consider a rectangular plate with the following properties:

The normal modes and frequency response function analysis are performed via a Matlab script.

The normal modes results are:

The fundamental mode shape is shown in Figure B-1. The corresponding joint acceptance function is shown in Figure B-2.

Now apply the sound pressure level from Mil-Std-1540C in Figure B-3 at an angle of

The resulting displacement and stressat the center of the plate are shown in Figures B-4 and B-5, respectively.

Figure B-1.

Figure B-2.

Figure B-3.

Figure B-4. Center of Plate, 45 degrees Incidence

Figure B-5. Center of Plate, 45 degrees Incidence

APPENDIX C

Limiting Case, Uniform Pressure

Recall

For ,

(C-1)

(C-2)

The pressure field becomes a uniform and normal as .

In this case,

(C-3)

The following equation is equivalent to (C-3) for m as an integer and ignoring the polarity shift.

(C-4)

Equation (C-4) is essentially similar to the result for uniform, normal pressure in Reference 4, allowance for a difference in the way that the mass density was accounted for.

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