1. Rectangular & Circular Plate Shock & Vibration
Unit 31
Rectangular & Circular Plate
Shock & Vibration
By Tom Irvine
Dynamic Concepts, Inc.

2. Introduction This unit will continue plate bending shock & vibration
Plates modeled as continuous systems
(Finite element analysis for plates will be covered in a future unit)
First perform normal modes analysis
Next Plates will be subjected to base excitation (enforced acceleration)
Also consider stress-velocity relationship for plates

3. Rayleigh’s Method
Exact natural frequencies and mode shape solutions can be found for certain boundary condition cases, such as rectangular plate simply-supported on all sides
The Rayleigh method is used in other cases to determine the natural frequency and mode shape of a structure
Rayleigh method is
where
KE = kinetic energy
PE = potential energy

4. Rayleigh’s Method (cont)
Rayleigh's method requires an assumed displacement function
The method thus reduces the dynamic system to a single-degree-of-freedom system.
The assumed displacement function introduces additional constraints which increase the stiffness of the system
Rayleigh's method yields an upper limit of the true fundamental frequency
So form a number of candidate trial displacement functions each of which satisfies the boundary conditions
Select the winner as the function which yields the lowest natural frequency

5. Read Input Arrays vibrationdata > Import Data to Matlab
Read in Library Arrays: NAVMAT PSD Specification & SRS 1000G Acceleration Time History

6. Rectangular Plate 12 x 8 x 0.125 inch
vibrationdata > structural dynamics > Plates, Rectangular & Circular >
Rectangular Fixed-Free-Fixed-Free

7. Fundamental Mode Shape

8. Enter Damping Q=10

9. PSD Base Input

10. Acceleration PSD

11. von Mises Stress

12. Shock Input

13. Base Excitation

14. Acceleration

15. von Mises Stress

16. Circular Bulkhead & Avionics Example in Sounding Rockets
Penn State
USC

17. Circular Bulkhead & Avionics Example in Sounding Rockets (cont)
Armadillo Aerospace

18. Circular Homogeneous Plate, Simply-Supported
vibrationdata > Structural Dynamics > Plates, Rectangular & Circular > Circular Plate > Homogeneous

19. Natural Frequency Results
(Hz) n k C D root PF EMM
n = nodal diameters
k = nodal circles
PF = participation factor
EMM = effective modal mass ratio

20. Mode 1

21. Mode 4

22. Enter Uniform Damping Q=10

23. PSD Base Input

24. Acceleration

25. Relative Velocity

26. Relative Displacement

27. von Mises Stress

28. Arbitrary Input

29. Base Input

30. Acceleration

31. Relative Velocity

32. Relative Displacement

33. von Mises Stress

34. Shock Summary Peak Response Values at Center Acceleration = 423.2 G
Relative Velocity = in/sec
Relative Displacement = in
von Mises Stress = psi
Peak Response Values at Half-Radius
Acceleration = G
Relative Velocity = in/sec
Relative Displacement = in
von Mises Stress = psi