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Forced Oscillation

We now explore the case of a forced oscillation. There are a few types of forced oscillations; the first is a damped forced oscillation, which is a forced oscillation with a damping force. The second is a driven forced oscillation, which is a forced oscillation with a driving force. Critical damping is a special case of a damped forced oscillation, where the damping force is equal to the spring force. The effect of this is that the system will damp out to zero in the shortest time possible. We will explore these cases (and more) in this section.

Table of Contents​

Damped Oscillation​

Suppose we have a spring-mass system submerged in a viscous fluid (like water). The spring-mass system will experience a damping force, which is proportional to the velocity of the mass.

Before we derive the mathematical model, let's first consider a qualitative picture of the system. If the spring-mass system is submerged in a viscous fluid, the oscillation will slow down over time. If we plot the position of the mass as a function of time, we will see that the amplitude of the oscillation decreases over time. This is because the damping force is always acting in the opposite direction of the velocity of the mass. The mass will eventually come to a stop, and the system will reach an equilibrium position.

There are three cases of damping: underdamped, critically damped, and overdamped. In the underdamped case, the system will oscillate with a decreasing amplitude. In the critically damped case, the system will return to equilibrium in the shortest time possible without oscillating. In the overdamped case, the system will return to equilibrium without oscillating, but it will take longer than in the critically damped case.

Now we can analyze the system mathematically. This damping force is given by

where is a damping constant that depends on the properties of the fluid. Combining this with the spring force, we have

or

We have a second-order linear ordinary differential equation (ODE) with constant coefficients. In the previous section, we proved that any linear and time-translation invariant system has a solution that is a linear combination of irreducible exponentials. As such, we can justifiably create the ansatz

and substitute this into the ODE. This gives us

which is a quadratic equation in . The power of exponentials is that they are their own derivatives. As such, linear differential equations simply become polynomials. The solutions to this quadratic equation are given by the quadratic formula;

The solutions to this equation depend on the value of . We can classify the solutions into three cases based on the discriminant .

Overdamped Oscillation​

In the overdamped case, we have , or equivalently . This means that we have two real, distinct solutions to the quadratic equation.

The general solution is just

where

Underdamped Oscillation​

In the underdamped case, we have , or equivalently . This means that we have two complex solutions to the quadratic equation. We know that polynomials have complex roots that come in conjugate pairs, so we expect the solutions to be complex conjugates.

We can write the solutions as

We can rewrite this as

where is the damped frequency. Then the general solution is

We can factor out the term to get

and we can use Euler's formula to rewrite this as

from which we can rearrange to get