文档介绍：Source: Formulas for Structural Dynamics: Tables, Graphs and Solutions
CHAPTER 1
TRANSVERSE VIBRATION
EQUATIONS
The different assumptions and corresponding theories of transverse vibrations of beams are
presented. The dispersive equation, its corresponding curve `propagation constant±
frequency' and parison with the exact dispersive curve are presented for each
theory and discussed.
The exact dispersive curve corresponds to the ®rst and second antisymmetrical Lamb's
wave.
NOTATION
p
cb Velocity of longitudinal wave,pcb E=r
ct Velocity of shear wave, ct G=r
4
D0 Stiffness parameter, D0 EIz=
2rH
E, n, r Young's modulus, Poisson's ratio and density of the beam material
2
E1, G Longitudinal and shear modulus of elasticity, E1 E=
1 À n , G E=2
1 n
Fy Shear force
H Height of the plate
Iz Moment of inertia of a cross-section
k Propagation constant
kb Longitudinal propagation constant, kb o=cb
kt Shear propagation constant, kt o=ct
4 2 4
k0 Bending wave number for Bernoulli±Euler rod, k0 o =D0
M Bending moment
p, q Correct multipliers
ux, uy Longitudinal and transversal displacements
w, c Average displacement and average slope
x, y, z Cartesian coordinates
sxx, sxy Longitudinal and shear stress
mt, l Dimensionless parameters, mt ktH, l kH
o Natural frequency
d
0 Differentiation with respect to space coordinate
dx
d
Á Differentiation with respect to time
dt
1
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TRANSVERSE VIBRATION EQUATIONS
2 FORMULAS FOR STRUCTURAL DYNAMICS
AVERAGE VALUES AND RESOLVING
EQUATIONS
The different theories of dynamic behaviours of beams may be obtained from the equations
of the theory of elasticity, which are presented with respect to average values. The object
under