2.3
2.4
2.2.2
Nonlineax Equations of Motion . . . . . . . . . . . . . . . . . . .
32
2.2.3
Linear Equations of Motion Derived Rom Linear Strain . . . . .
33
2.2.4
Linear Equations of Motion Derived From Nonlinear Strain . . . .
34
Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
2.3.1
Linear Transverse Vibrations . . . . . . . . . . . . . . . . . . . . .
38
2.3.2
Linear In-plane Vibrations . . . . . . . . . . . . . . . . . . . . . .
39
2.3.3
Nonlinear Boundary Conditions . . . . . . . . . . . . . . . . . . .
40
Discussion and Compazison Between the New and Existing Models . . .
42
2.4.1
Nonlinear Equat ions . . . . . . . . . . . . . . . . . . . . . . . . .
42
2.4.2
Lagragian vs Eulerian Variables . . . . . . . . . . . . . . . . . .
43
2.4.3
Decoupled In-plane and Transverse Plate Problems . . . . . . . .
-1-1
2-44 The Membrane Problem . . . . . . . . . . . . . . . . . . . . . . .
45
Presence of Additionai Terms . . . . . . . . . . . . . . . . . . . .
46
2.4.6
3 Linear Transverse Vibrations 3.1 Introduction
..................................
3.2 Equations of Motion
50 O;
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.3 Frequency .4n alysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
3.3.1
Solid Plate with Free Boundary . . . . . . . . . . . . . . . . . . .
57
3.3.2
Frequency Equation . . . . . . . . . . . . . . . . . . . . . . . . .
59
3.3.3
Mode Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60