Damped Oscillator Equation

[Paddyod1509]

the damped oscillator equation:

(m)y''(t) + (v)y'(t) +(k)y(t)=0

Show that the energy of the system given by

E=(1/2)mx'² + (1/2)kx²

satisfies:

dE/dt = -mvx'


i have gone through this several time simply differentiating the expression for E wrt and i end up with

dE/dt = x'(-vx')

im at a brick wall. Am i doing something wrong? Any help is much appreciated! Thanks

 

[Paddyod1509]

I have also assumed that y and x are interchangeable variables here, as no other information has been provided

 

[Mute]

I don't see anything wrong with your calculation. In fact, double check the units for your expressions - I don't think the suggested answer "$m\gamma \dot{x}$" even has the same units as $dE/dt$.

 

[pasmith]

the damped oscillator equation:

(m)y''(t) + (v)y'(t) +(k)y(t)=0

Show that the energy of the system given by

E=(1/2)mx'² + (1/2)kx²

satisfies:

dE/dt = -mvx'

That must be wrong: it requires that $E = C - mvx$ for some constant C, which is not the case.

i have gone through this several time simply differentiating the expression for E wrt and i end up with

dE/dt = x'(-vx')

That is the right expression for dE/dt.

 

[blue_raver22]

The damped oscillator equation is a fundamental equation used in the study of oscillating systems, particularly in physics and engineering. It describes the motion of a damped harmonic oscillator, where the motion is characterized by a restoring force (k) and a damping force (v).

In this equation, m represents the mass of the oscillator, y(t) is the displacement from equilibrium at time t, and the prime notation (') represents differentiation with respect to time.

The energy of the system, E, is a key quantity in understanding the behavior of the damped oscillator. It is given by the sum of the kinetic energy (1/2)mx'² and the potential energy (1/2)kx², where x is the displacement from equilibrium.

To show that dE/dt = -mvx', we can differentiate the expression for E with respect to time:

dE/dt = (1/2)m(2x'x'') + (1/2)k(2xx')

= mx'x'' + kxx'

We can then use the damped oscillator equation to substitute for x'' and x':

dE/dt = mx'(-vx') + kx(-vy')
= -mvx'x' - kvx'x'
= -vx'(mvx' + kvx')
= -vx'(m + v)x'
= -mvx'

Therefore, we have shown that dE/dt = -mvx', which satisfies the given condition. This result shows that the energy of the damped oscillator decreases over time, due to the damping force, as expressed by the negative sign in the equation.

I hope this explanation helps you understand the relationship between the damped oscillator equation and the energy of the system. Keep exploring and experimenting with this equation to deepen your understanding of oscillating systems.