# Mass between Two Damped Springs

Mass between Two Damped Springs

Consider a mass between two springs attached to opposing walls. Let and be the respective spring constants. A displacement of the mass by a distance results in the first spring lengthening by , while the second spring is compressed by (and pushes in the same direction). Suppose that the equilibrium position is at , where the two springs have their force-free lengths. The equation of motion is then given by x++x=0, where is the viscous damping coefficient in and =+ is the natural frequency of the system. Evidently, the two springs are dynamically equivalent to a single spring with constant . The solution is . When (no damping), this reduces to . The motion is traced by the red curves in the right panel.

m

k

1

k

2

x

x

x

x=0

2

d

d

2

t

γ

m

dx

d t

2

ω

0

γ

Ns/m

ω

0

k

1

k

2

m

k=+

k

1

k

2

x(t)=Asin

-γt/2m

e

1

2m

4-

t2

m

2

ω

0

2

γ

γ=0

x(t)=Asin(t)

ω

0