# Energy of driven damped oscillator

• Marin
In summary: This is because energy is being dissipated by the damping force. Therefore, the LHS is not equal to zero. It represents the rate of change of the total energy, which is not conserved in this system due to the presence of an external force and damping. In summary, the equation for the driven damped oscillator involves the rate of change of the total energy, which is not conserved in this system due to the presence of an external force and damping. This means that the LHS cannot be set to zero, as energy is being dissipated and exchanged with other systems. This leads to a different solution compared to the standard solution for a damped oscillator.

#### Marin

Hi all!

I was considering the Energy of a driven damped oscillator and came upon the following equation:

given the equation of motion:

$$m\ddot x+Dx=-b\dot x+F(t)$$

take the equation multiplied by $$\dot x$$

$$m\ddot x\dot x+Dx\dot x=-b\dot x^2+F(t)\dot x$$

and we rewrite it:

$$\frac{d}{dt}(\frac{m\dot x^2}{2}+\frac{Dx^2}{2})=-b\dot x^2+F(t)\dot x$$

now the LHS appears to be the total energy of the undamped harmonic oscillator, by energy conservation a constant, so it´s rate of change is 0:

$$0=-b\dot x^2+F(t)\dot x$$

Ok, so far so good :) But here lies my problem. Now we´ve obtained an equation for \dot x which we could solve for x(t). But the result appears to be different from the standart solution :(

$$0=\dot x(-b\dot x+F(t))$$
$$\dot x=0$$
$$-b\dot x+F(t)=0$$

=> $$\dot x_1=const:=C$$
$$x(t)=\frac{1}{b}\displaystyle{\int_{t_0}^{t}}F(t')dt'$$

so the solution would read: (would it? Do we have superposition here, since the DE is not linear any more?)

$$x(t)=C+\frac{1}{b}\displaystyle{\int_{t_0}^{t}}F(t')dt'$$

Marin said:
now the LHS appears to be the total energy of the undamped harmonic oscillator, by energy conservation a constant, so it´s rate of change is 0:

This is where you went wrong. The energy here is not conserved within the system, so you can't put the rate of change to zero. The SHM is a closed system, this is not. Here the energy keeps changing, the external force F(t) is pumping in energy while the damping is siphoning it out. Your equation tells you how this happens :
$$\frac{d}{dt}(\frac{m\dot x^2}{2}+\frac{Dx^2}{2})=-b\dot x^2+F(t)\dot x$$

But you can't put the LHS to zero, because energy is not being conserved. Suppose there was no external force and so the term F(t) was absent. The equation shows you how your energy would then decrease due to the damping term. And you know it must, because eventually it loses all the energy and stops.

So the moral of the story is : when your system is not closed, but interacting and exchanging energy with other systems (in this case whatever is doing the driving and the damping), you cannot use energy conservation for your system.

So it´s not like all the energy pumped in gets lost in damping, because this would imply a vanishing LHS?

the LHS isn't energy, but rate of change of it. If there is no driving term, eventually all the energy gets lost - and the thing comes to rest. Then as energy is changing no longer, the LHS is also zero. At this pint you can see that the rhs is also zero. Theoretically, this happens after infinite time.

Point is, the function of time on the LHS is not always zero, but becomes zero at t= infinity.

As already pointed out, the LHS is zero for an undamped oscillator. Let's take a look at the bracketed terms:

$$\frac{m\dot{x}^2}{2} = \frac{1}{2}mv^2 = T$$

$$\frac{Dx^2}{2} = V$$

The first term represents the kinetic energy, whilst the second term represents the potential energy. So the LHS represents the rate of change of the sum of the potential and kinetic energies, in other words the total energy. In this case of an undamped oscillator the total energy remains constant, however in the case of damped motion the total energy is not conserved.

## 1. What is a driven damped oscillator?

A driven damped oscillator is a physical system that experiences periodic oscillations due to the application of an external force, while also experiencing damping, which causes the oscillations to decrease in amplitude over time.

## 2. What is the role of energy in a driven damped oscillator?

The energy of a driven damped oscillator plays a crucial role in determining the behavior of the system. It is the energy that is transferred back and forth between kinetic and potential forms during the oscillations, and it is also responsible for the damping effect on the oscillations.

## 3. How is the energy of a driven damped oscillator affected by the damping coefficient?

The damping coefficient, which is a measure of the strength of the damping force, directly affects the amount of energy lost by the system during each oscillation. A higher damping coefficient leads to a faster decrease in energy and a shorter period of oscillation.

## 4. Can the energy of a driven damped oscillator be increased through external means?

Yes, the energy of a driven damped oscillator can be increased through the application of an external driving force. This can be seen in systems such as a child's swing, where the parent pushes the swing to increase its amplitude of oscillation.

## 5. How does the frequency of the driving force affect the energy of a driven damped oscillator?

The frequency of the driving force has a direct relationship with the energy of a driven damped oscillator. The closer the driving frequency is to the natural frequency of the system, the more energy will be transferred to the system and the higher the amplitude of oscillation will be.