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WOLFRAM|DEMONSTRATIONS PROJECT

Spring-Mass-Damping System with Two Degrees of Freedom

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dynamic system
mass
m
1
110
mass
m
2
60
stiffness
k
1
1
stiffness
k
12
4000
stiffness
k
2
1
damping
c
1
1
damping
c
12
10
damping
c
2
500
force
F
1
-5
force
F
2
-10
initial condition
displacement
x
1
-0.5
displacement
x
2
0
velocity
x
1
'
0.8
velocity
x
2
'
0.8
time
This Demonstration shows the dynamics of a spring-mass-damping system with two degrees of freedom under external forces. The motion of the system is represented by the positions
x
1
(t)
and
x
2
(t)
of the masses
m
1
and
m
2
at time
t
. Both masses have a spring connected to a stationary base, with spring constants
k
1
and
k
2
; also
k
12
for the spring connecting the two masses. The motion of the masses is damped, with damping factors
c
1
and
c
2
; also
c
12
for damping between the two masses. The masses are acted upon by external forces
F
1
and
F
2
.
The Demonstration gives the solution of the differential equations from Newton's second law of motion, then shows free-body diagrams at the top of the graphic, with the natural frequencies of the system. The two plots at the bottom show the position and velocity of the two masses as functions of time.
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