文档介绍：Review:
Set up partial differential equation of motion
Set solution form u(x,t)=U(x)ejωt or u(x,t)=U(x)est
get mode solution
and
(3) Boundary condition
(4) Determine coefficient A or B by one boundary condition
(5) Determine natural frequency by another boundary condition
(6) Mode equation and shape
(7) Solution of equation
General method for continuous system
Vibrating String
Lagrange’s equation
x
y
o
(1)ic energy
Potential energy
Lagrange integration
Vibrating String
(2)From Lagrange’ formula
Get equation of motion
Vibrating String
Let
(3)Solution of equation
for
One of methods
Vibration mode
depend space x
SHU
Vibrating String
Solution form substitute in motion equation
obtain
Solution is
SHU
Vibrating String
(4) Boundary condition
y(0,t)=0; y( l, t)=0
x
y
o
SHU
Vibrating String
y(0, t)=0; B=0
Because of
From y( l, t )=0
or
n=1,2,3…
(5)Natural frequency
(6) Mode function
while A=1
SHU
Vibrating String
Vibration mode shape
SHU
Vibrating String
(7) Solution of vibration
SHU
Vibrating String
Vibration equation for moving string
x
y
o
ic energy
Potential energy
Lagrange integration
y(x, t) is a function of x and t
v