- An object that is attached to a spring that is slightly stretched or compressed by a displacement x will experience a restoring force given by Hooke's Law:
F_{r}(x)= - k x ,where k is a constant
The solution to this equation is a simple harmonic oscillation (we assume that the mass of the spring is negligible).
- Consider mass that is attached to a spring that is hung from the ceiling:
If the spring is stetched, or compressed, by an amount x, then the force exerted on the mass is equal to F(x) = m g - k x
The equilibrium position is x = m g / k
- Damping forces:
A viscous damping force can be modeled as F_{b}= - b v, where b is a constant and v is the velocity of the load.
- External, harmonic forces:
F_{ext} =
f_{o}sin(
cwt)
- w^{2} = w_{o}^{2} - (b/2m)^{2} where w_{o}^{2} = k/m, w_{o} is the nature frequency of the system
- if c=0. then f_{o} = 0.
- The net force acts on the mass is F = m g - k x - b v + f_{o}sin(cwt)