Plataformas

Linear
Linear
response
Purdue University
Purdue University
CE571 -- Earthquake Engineering
CE571 Earthquake Engineering
Spring 2003
Spring 2003
Mete A. Sozen and Luis E. García
Mete A. Sozen and Luis E. García

Newton (1642-1727) 2nd Law
Newton (1642-1727) 2nd Law
"The force acting on a body and
"The force acting on a body and
causing its movement,
causing its movement,
isequal to the rate of change of
is equal to the rate of change of
momentum in the body."
momentum in the body."
Momentum Q,, is equal to the product of the body
Momentum Q is equal to the product of the body
mass by its velocity:
mass by its velocity:

dx
dx = mx
&&
Q = mv = m
Q = mv = m
= mx
dt
dt

Newton 2nd Law (cont.)
Newton 2nd Law (cont.)
Under the premise that the massremains
Under the premise that the mass remains
constant, the forces that act on the body are
constant, the forces that act on the body are
equal to the rate of change in momentum:
equal to the rate of change in momentum:

&&
dQ d
dQ = d ( mv ) = m dv = m dx = mx = ma
dv
dx
&&
F=
&&
F=
= ( mv ) = m
=m
= mx = ma
dt dt
dt
dt
dt dt
dt
dt

F = ma

D’Alambert PrincipleD’Alambert Principle
D’Alambert (1717-1783) suggested that
D’Alambert (1717-1783) suggested that
Newton's 2nd Law should be written in a
Newton's 2nd Law should be written in a
similar way that the principle of
similar way that the principle of
equilibrium in statics ((F=0):
equilibrium in statics F=0):
inertial force

F − ma = 0

“A simple device can yield
“A simple device can yield
perhaps80% of the truth
perhaps 80% of the truth
whereas the next 10% would
whereas the next 10% would
be difficult to obtain and the
be difficult to obtain and the
last 10% impossible …”
last 10% impossible …”
H.. M. Westergaard
H M. Westergaard

Response to base excitation
Response to base excitation
of a single degree of
of a single degree of
freedom (SDOF) oscillator
freedom (SDOF)oscillator
l Some analytical devices are efficient
l Some analytical devices are efficient

as design tools and some are good
as design tools and some are good
to organize experience, but very few
to organize experience, but very few
can serve both functions.
can serve both functions.
l The SDOF oscillator is better at
l The SDOF oscillator is better at
organizing and understandingorganizing and understanding
experience.
experience.

Base excitation
Base excitation
00

z
z

00

00

u
u

z
z

00

mass
mass
damper
damper
structural
structural
element
element

((a)
a)

((b)
b)

u
u

Position of
Position of
the base of
the base of
the structure
the structure
with respect
with respect
to the ground
to the ground

0
0

z

00

u

Absolute position of
Absolute position of
the mass with
the mass with
respect to a distant
respect to a distant
frame of reference
frame of reference

mass
mass
damper
damper
structural element
structural element

using equilibrium and D’Alambert principle
using equilibrium and D’Alambert principle
inertial force
inertial force
0
0

z

0
0

u
damping forcedamping force

&&
&&&
&
mu&
mu

c ((u − z ))
c && − &&
uz

k ((u − z ))
k u−z

Structural element force
Structural element force

&&
mu + c ((u − z ))+ k ((u − z )) = 0
&&
mu + c && − && + k u − z = 0
uz

We now do a simple transformation
We now do a simple transformation
We use the relative displacement
We use the relative displacement

x = ((u − z ))
x = u− zDeriving against time:
Deriving against time:

&& = && − &&
x = ((u − z ))
x uz
And deriving again:
And deriving again:

&& = ((&& − &&))
x = u −z
&& && &&
x uz

Now the equilibrium equation:
Now the equilibrium equation:
&&
m ((&& + &&)) + cx + kx = 0
&& &&
m x + z + cx + kx = 0
xz
m
m

&&
&&
x
&&&
&z
&
z
&&
&
c
c

= mass concentrated at its center of...