# Forced Mechanical Vibrations

#### The model and solution

We consider the damped, mass-spring system with a sinusoidal external force:

where $F_0,\gamma >0$.

By using the method of undetermined coefficients, one can shown that

where complementary solution, $x_c(t)$, is the solution of the corresponding homogeneous equation.

In real-world damped systems, we must have $x_c(t) \to 0$ as $t \to \infty$, and therefore, $x_c(t)$ is called the transient part of the solution. The transient part depends only on the parameters of the system ($m$, $b$, and $k$) and the initial conditions. The other term of the solution is called the steady-state part or the steady-periodic part of the solution.

The expression

is called the gain factor. Notice that the gain factor depends on the system parameters and the frequency of the external force.

#### Practical resonance

Let's find the maximum value of $G(\gamma)$.

Upon differentiating, we find that

and $G'(\gamma)=0$ if and only if

We use $\gamma_r$ to denote the frequency on the right.

##### Case 1

If the system is over-damped ($b^2>4mk$) or critically damped ($b^2=4mk$), then $\gamma_r$ is imaginary, and we must have the maximum $G(\gamma)$ when $\gamma=0$. In this case, the maximum gain is $1/k$.

##### Case 2

If the system is under-damped ($b^2<4mk$), we have the maximum $G(\gamma)$ when $\gamma=\gamma_r$. In this case, the maximum gain is

and $\gamma_r$ is called the resonance frequency. For undamped systems, maximum gain occurs when the frequency of the external force is equal to the resonance frequency. Notice that as $b \to 0$, the gain increases without bound and $\gamma_r \to \sqrt{k/m}$ ($\sqrt{k/m}=\omega$ is the natural frequency of the system).

#### Example

Solve: $10x''+3x'+49x=20 \cos 4t, \quad x(0)=5, \, x'(0)=0$

Here is the Sage code.

t=var("t")x=function("x")(t)de=10*diff(x,t,2)+3*diff(x,t)+49*x==20*cos(4*t)desolve(de,x,[0,5,0])

The solution is

which can be written in the approximate form

The first term is the transient part. Notice that it decays with time. The second term is the steady-state part. It dominates the solution for large $t$.

The graph of $x(t)$ is shown here.

In this example, the resonance frequency is $\sqrt{971/200} \approx 2.203$, which is not particularly close to the frequency of the external force ($\gamma=4$). The maximum gain is rather small.