System mass-spring-damper and Energy

[Jhenrique]

Hellow!

A general system mass-damper-spring is describied by differential equation:
$$m \frac{d^2x}{dt^2}+\gamma \frac{dx}{dt}+kx=F$$
For calculate the energy in this system just integrate the equation wrt x. However, I know the 3th parcel integrated is the elastic potential energy of spring, but what are it the 1st and 2nd parcels integrated?

PS: the 1st should be the kinetic energy, I think...

 

[tiny-tim]

Hello Jhenrique!

… I know the 3th parcel integrated is the elastic potential energy of spring, but what are it the 1st and 2nd parcels integrated?

PS: the 1st should be the kinetic energy, I think...

Yes, the 1st term is mv dv/dx, = d/dx (1/2 mv²), = d/dx (KE).

(and the 3rd term is obviously d/dx (1/2 kx²), = d/dx (PE))

The 2nd term is v, and ∫ v dx isn't anything particular (other than the energy lost to friction) …

nor would you expect it to be …

this is a damping term, so non-conservative, and you only expect a potential energy from a conservative force!