Lecture 8-2018 同济大学研究生结构动力学课件

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
u x, t x z t


Selection of shape function
The 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
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
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 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
1 1 2 2 2 & mu max mn u0 2 2 1 2 1 2 ES0 ku max ku0 2 2
EK 0
k n m


Natural frequency by Rayleigh’s method

Mass-spring system
Distributed-mass system
EOM

Natural frequency


Natural frequency by Rayleigh’s method

Mass-spring system
Distributed-mass system

Lumped-mass system
According to the principle of conservation of energy:
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.
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


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
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
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
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.
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

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.
u x, t x z t

Systems with distributed mass and elasticity

The virtual-work principle requires that the external virtual work performed by the external loadings acting through their corresponding virtual displacement should be equated to the internal virtual work


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.
Systems with distributed mass and elasticity
wk.baidu.com
The variable-separating method:
u x, t x z t u j x, t j z t


Generalized SDOF systems
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

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.

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
Structural Dynamics
Lecture 8 Generalized SDOF Systems
Contents


Generalized SDOF systems
Rigid-body assemblages


Systems with distributed mass and elasticity