Subjects: Classical Mechanics
Links: Oscillations and Hooke's Law, Vibrations, Second Order Linear Differential Equations, Main definitions for Fourier Analysis, Convergence of Fourier Series

There are several possibilities for the resistive force. Ordinary sliding friction is approximately constant in magnitud, but always directed opposite to the velocity. As we have seen when studying Air Resistance, it is a reasonable assumption that the resistive force is proportional to $v$ or $v^2$.

If we consider an object that is subject to a Hooke's law force $-kx$ and a resistive force $-b\dot{x}$. The net force on the object is $-b\dot{x}-kx$, and Newton's second law reads $$m\ddot x +b\dot x +kx=0.$$
We are going to consider two constants $$2\beta = \frac{b}{m}, \qquad \omega_0 = \sqrt{\frac km}.$$
We call ${\omega }_0$ the system's natural frequency, the frequency at which it would oscillate if there were no resistive force present. The equation of motion for the damped oscillator becomes $$\ddot x + 2\beta\dot x + \omega_0^2x = 0.$$
If $x(t)=e^{rt}$, we get the solutions of to be $r=-\beta \pm \sqrt{{\beta }^2-{\omega }_0^2}$, and we get the general solution as $$x(t) = e^{-\beta t}\left(C_1 e^{\sqrt{\beta^2-\omega_0^2}t}+ C_2 e^{-\sqrt{\beta^2-\omega_0^2}t}\right).$$

Undamped Oscillation

If there is no damping, meaning that $\beta =0$, then we get $$x(t) = C_1 e^{i\omega_0t}+ C_2 e^{-i\omega_0t}.$$

Weak Damping

Suppose that $\beta$ is small, we mean that $$\beta < \omega_0,$$a condition sometimes called underdamping. In this case we define another constant $$\omega_1 := \sqrt{\omega_0^2-\beta^2}.$$We see that ${\omega }_1$ is a frequency, which is less than the natural frequency ${\omega }_0$. In the important case of very weak damping $(\beta \ll {\omega }_0)$, ${\omega }_1$ is very close to ${\omega }_0$. With this in mind, we get the solution $$x(t) = e^{-\beta t}\left(C_1 e^{i\omega_1t}+ C_2 e^{-i\omega_1t}\right) = A e^{-\beta t} \cos(\omega_1 t-\delta).$$

We get that the decay parameter is $\beta$.

Strong Damping

Suppose instead that $\beta$ is large. Specifically suppose that $$\beta > \omega_0,$$a condition sometimes called overdamping. In this case, $r$ is real, and our solution is $$x(t) = C_1 e^{\left(-\beta +\sqrt{\beta^2-\omega_0^2}\right)t}+ C_2 e^{\left(-\beta -\sqrt{\beta^2-\omega_0^2}\right)t}.$$
Here we have that the decay parameter is $-\beta +\sqrt{{\beta }^2-{\omega }_0^2}$, then we get that as $\beta \to \mathrm{\infty }$, then $-\beta +\sqrt{{\beta }^2-{\omega }_0^2}\to 0$.

Critical Damping

The boundary between underdamping and overdamping is called critical damping and occurs when the damping constants equal to the natural frequency $$\beta = \omega_0.$$With this in mind, we get the solution $$x(t) = e^{-\beta t}(C_1 + C_2 t).$$
The decay parameter is $\beta$.

The decay parameter reaches its maximum, when $\beta ={\omega }_0$.

Driven Damped Oscillations

Any natural oscillator eventually comes to a rest, as the damping forces drain its energy. If we want the oscillations to continue we need an external driving force. If we denote the external driving force by $F(t)$, and if we assume as before the damping force is of the form $-bv$, then the net force on the oscillator is $-b\dot{x}-kx+F(t)$, and the equation can be written as $$m \ddot x + b\dot x + kx = F(t).$$
Just as before we can consider $$f(t) = \frac{F(t)}{m},$$the force per unit mass. With this notation we get the equation $$\ddot x + 2\beta \dot x+ \omega_0^2 x = f(t).$$
We shall see the special case for the driving force $f(t)$ as a sinusoidal function on time, $$f(t) = f_0 \cos(\omega t),$$where $f_0$ denotes the amplitud of the driving force, and $\omega$ is the angular frequency of the driving force, also called the driving frequency. Then we get the following equation $$\ddot x + 2\beta\dot x+\omega_0^2 x= f_0 \cos(\omega t).$$
We can consider a similar equation $$\ddot y + 2\beta\dot y+\omega_0^2 y= f_0 \sin(\omega t),$$with these equations in mind, we can make a substitution. Let $z=x+iy$, then $$\ddot z + 2\beta\dot z+\omega_0^2 z= f_0 e^{i\omega t}.$$
If we look for the particular solution to this equation, we get that $$z = \frac{f_0}{(\omega_0^2-\omega^2) + 2i \beta\omega} e^{i\omega t}.$$This while correct needs a bit of tidying up. We get a polar form on the constant, then we see that $$ A = \frac{f_0}{\sqrt{(\omega_0^2-\omega^2)^2+4\beta^2\omega^2}}, \qquad \text{and}\qquad \delta = \arctan\left(\frac{2\beta\omega}{\omega_0^2-\omega^2}\right). $$Thus we get that $$z(t) = Ae^{i(\omega t -\delta)},$$and the solutions to the other differential equations $$x(t) = A \cos(\omega t -\delta), \qquad y(t) = A \sin(\omega t-\delta).$$
We get the general solution of $$x(t) = A \cos(\omega t-\delta) + C_1 e^{r_1 t} + C_2 e^{r_2 t}.$$
We see that if $\beta >0$, then the terms in the general solution will decay exponentially, and just leave the motion of the particular solution irrespective of the initial conditions. For this reason, the motion $x(t)=A\cos (\omega t-\delta )$ is called an attractor.

Lastly, we get the solution of the underdamped version $$
\begin{align*}
x(t) &= A \cos(\omega t -\delta) + A_\text{tr} \cos(\omega_1 t-\delta_\text{tr})\ &= A \cos(\omega t -\delta) +e^{-\beta t}(B_1\cos(\omega_1 t)+ B_2\sin(\omega_1 t)),
\end{align*} $$where we get that $A_{\text{tr}}$ and ${\delta }_{\text{tr}}$ are arbitrary constants, and have the subscript $\text{tr}$ because it is they are transient terms.

We can be even more explicit in this case, and let $x(0)=x_0$, and $\dot{x}(0)=v_0$, $$\begin{align*}
B_1 &= x_0-A\cos\delta\
B_2 &= \frac{1}{\omega_1}(v_0-\omega A \sin\delta + \beta B_1)
\end{align*}$$

Resonance

We We say that apart from transient motions that die out quickly, the system's response is to oscillate sinusoidally at the same frequency as the driving force $$x(t) = A \cos(\omega t-\delta),$$where the amplitude $A$ is given by $$ A^2 = \frac{f_0^2}{(\omega_0^2-\omega^2)^2+4\beta^2\omega^2}.$$We see that $A\propto f_0$. We also need to examine how does the $A$ depend on the ${\omega }_0$, $\omega$ and $\beta .$ The most interesting case is when $\beta$ is small, so the second term of the denominator is small. If ${\omega }_0$ and $\omega$ are very different, then the first term is large, and the amplitude of the driven oscillations are small. On the other hand, if ${\omega }_0$ and $\omega$ are really closes, then both terms are very small, and the amplitude $A$ is very large.

This means that the oscillator would vibrate at its natural frequency ${\omega }_0$. If we try to force it to vibrate at a frequency $\omega$, then the for values of $\omega$ close to ${\omega }_0$ the oscillator responds very well, but if $\omega$ is far from ${\omega }_0$, it hardly responds at all. We call resonance as the phenomenon of the greater response of an oscillator when driven at the right frequency.

The amplitude $A$ is a maximum when the denominator $$(\omega_0^2-\omega^2)^2+4\beta^2\omega^2$$is minimum. If we vary ${\omega }_0$ with $\omega$ fixed, this minimum occurs when ${\omega }_0=\omega$. On the other hand if we vary $\omega$ with ${\omega }_0$ fixed, then we see that the maximum occurs at $$\omega_2 := \sqrt{\omega_0^2-2\beta^2}.$$
However, when $\beta \ll {\omega }_0$, we see that ${\omega }_2$ is really close to ${\omega }_0$. As a recap of all the frequencies we have seen $$\begin{align*}
\omega_0 := \sqrt{\frac km} &\quad \text{natural frequency of undamped oscillator} \
\omega_1 := \sqrt{\omega_0^2-\beta^2} &\quad \text{frequency of damped oscillator} \
\omega &\quad \text{frequency of driving force} \
\omega_2 := \sqrt{\omega_0^2-2\beta^2} &\quad \text{value of $\omega$ at which response is maximum} \
\end

Width of the Resonance; the $Q$ factor

If we make the damping constant $\beta$ smaller, the resonance peak not only higher, but also gets narrower. We can make this idea more precises by defining full width at half maximum or FWHM as the interval between where $A^2$ is equal to its half maximum height.

We see that the points of half maximum are $$\omega = \sqrt{\omega_2^2 \pm \sqrt{12}\beta\omega_1}\approx \omega_0 \pm \beta. $$Thus the gull width at half maximum is $$\text{FWHM} = \sqrt{\omega_2^2 + \sqrt{12}\beta\omega_1}- \sqrt{\omega_2^2 - \sqrt{12}\beta\omega_1} \approx 2\beta,$$
or, equivalently, the half width at half maximum is $$\text{HWHM} = \frac{\sqrt{\omega_2^2 + \sqrt{12}\beta\omega_1}- \sqrt{\omega_2^2 - \sqrt{12}\beta\omega_1} }{2} \approx \beta.$$
The sharpness of the resonance peak is indicated by the ratio of its width $2\beta$ to its position ${\omega }_2$. Sometimes we want a very sharp resonance, so it is common to practice to define a quality factor $Q$ as $$Q = \frac{\omega_0}{2\beta} = \pi \frac{1/\beta}{2\pi /\omega_0} = \pi \frac{\text{decay time}}{\text{period}}.$$

Periodic Driving Force

Let $f$ be a periodic driving force, with period $\tau$. Then, we can define a $\omega :=1/\tau$. With this in mind, we can assume that $f$ is representable as Fourier series $$f(t) = \frac{a_0}{2} + \sum_{n = 1}^\infty a_n\cos(n\omega t) + b_n\sin(n\omega t),$$where $$a_n := \frac{2}{\tau}\int_{0}^\tau f(t)\cos(n\omega t), dt, \qquad b_n := \frac{2}{\tau}\int_{0}^\tau f(t)\sin(n\omega t), dt.$$
Fortunately, we can assume that $f$ is an even function since we only need to know what happens for $t>0$, then $b_n=0$ for all $n\in {\mathbb{N}}^{+}$. Then we get that $$f(t) = \sum_{n = 0}^\infty f_n\cos(n\omega t).$$
We can get a solution for the differential equation $$\ddot x + 2\beta \dot x + \omega_0^2 x = f(t),$$as $$x(t) = \sum_{n = 0}^\infty A_n \cos(n\omega t - \delta_n)$$where $$A_n := \frac{f_n}{\sqrt{(\omega_0^2- n^2\omega^2)+ 4 \beta^2n^2 \omega^2}}, \qquad \delta_n := \arctan\left(\frac{2\beta n \omega}{\omega_0^2-n^2\omega^2}\right).$$

The Root Mean Squared Displacement

It would be nice to have a single number to measure the oscillator's response and then just plot this number against the driving frequency, The most convenient quanitity to use is the mean sqare displacement $⟨x^2⟩$, and to give a quantity with the dimensions of length we usually discuss the root mean square of RMS displacement $$x_\text{rms} := \sqrt{\langle x^2\rangle}.$$We need a good definition for the average. It is usually defined as $$\langle x^2\rangle :=\frac1\tau \int_{-\tau/2}^{\tau/2} x^2, dt.$$
If $x$ has the following form $$x(t) = \sum_{n = 0}^\infty A_n \cos(n\omega t-\delta_n),$$then $$\langle x^2\rangle = A_0^2+\frac12 \sum_{n= 1}^\infty A_n^2.$$