Lecture 8-2018 同济大学研究生结构动力学课件

u x, t x z t
The equation of motion
u j x, t j z t

The analysis method for normal SDOF can be readily applied to the above equation. The key step here is to determine the generalized mass, damping, stiffness and excitation for a given system.
u j x, t j z t

Systems with distributed mass and elasticity
WE =WI

External virtual work
u j x, t j z t

Systems with distributed mass and elasticity
u x, t x z t


Selection of sha来自e functionThe accuracy of Rayleigh method depends entirely on the shape function which is assumed to represent the vibration mode shape. In principle, any shape that satisfies the geometric boundary conditions can be selected. Better shape functions give lower estimates. The true natural frequency being a lower bound of all estimates

Rigid-body assemblages
u x, t x z t x t u x, t x z t x t

Natural frequency and damping ratio

Solution with c=0

where can be any reasonable approximation of the exact mode shape.

Selection of shape function


One common assumption is that the inertial loading p(x) is merely the weight, that is p(x)=m(x)*g, can be any reasonable approximation of the exact mode shape. The vibration frequency then is evaluated on the basis of the deflected shape resulting from the dead-weight load.
WE =WI
u j x, t j z t
WI = k j u j u j 1 u j u j 1
j 1
N

EOM


Systems with distributed mass and elasticity
u j x, t j z t
WE WI

Systems with distributed mass and elasticity
WE =WI

External virtual work
u x, t x z t

Systems with distributed mass and elasticity
Distributed-mass system

Lumped-mass system
u j x, t j z t
According to the principle of conservation of energy:

Natural frequency by Rayleigh’s method
WE =WI

Internal virtual work
u x, t x z t

Systems with distributed mass and elasticity
WE =WI
u x, t x z t


Systems with distributed mass and elasticity

Rigid-body assemblages
the critical or buckling axial load


Systems with distributed mass and elasticity
Assumed shape function: The assumed shape function must satisfy the displacement boundary conditions.

Lumped-mass system
u x, t x z t
According to the principle of conservation of energy:


Natural frequency by Rayleigh’s method

Mass-spring system


Rigid-body assemblages
u x, t x z t x t u x, t x z t x t


Rigid-body assemblages
EOM
u x, t x z t x t u x, t x z t x t
Systems with distributed mass and elasticity
The variable-separating method:
u x, t x z t u j x, t j z t


Generalized SDOF systems
Structural Dynamics
Lecture 8 Generalized SDOF Systems
Contents


Generalized SDOF systems
Rigid-body assemblages


Systems with distributed mass and elasticity


Selection of shape function

The properties of exact mode shape
If ψ(x) were the exact mode shape, static application of these inertia forces at each time instant will produce deflections u x, t ' An approximate shape function may be determined as the deflected shape due to the following static forces.
WE =WI

Internal virtual work
u j x, t j z t
WI = k j u j u j 1 u j u j 1
j 1
N

Systems with distributed mass and elasticity
EOM
u x, t x z t

Natural frequency


Systems with distributed mass and elasticity
EOM
u x, t x z t

Lumped-mass system: Shear building
Lumped-mass system: Shear building Natural frequency by Rayleigh’s method Selection of shape function

Generalized SDOF systems
Rigid-body assemblages Lumped-mass system
k n m Properties of Rayleigh’s Quotient: The approximate frequency obtained from an assumed shape function is never smaller than the exact value. Rayleigh’s quotient provides excellent estimates of the fundamental frequency, even with a mediocre shape function.
g
2 n
m x u x dx
0 2
L
u x m x
0
L
dx

Selection of shape function

In general, the selection of trial shapes goes through two steps 1. considers the flexibilities of different parts of the structure and the presence of symmetries to devise an approximate shape 2. the structure is loaded with constant loads directed as the assumed displacements, the displacements are computed and used as the shape function