Oscillations

Period and Frequency of Oscillations

The frequency is defined as the inverse of the period. For example, a large period implies a small frequency.

$$f=\frac1T$$

Where \(f\) is the frequency in hertz, \(\mathrm{Hz}\), and \(T\) is the period in seconds, \(\mathrm{s}\).

The period is the time required to complete one oscillation cycle. The period of an oscillation cycle is related to the angular frequency of the object's motion. The expression for the angular frequency will depend on the type of object that is oscillating. The equation that relates the angular frequency denoted by \(\omega\) to the frequency denoted by \(f\) is

$$\omega=2\pi f.$$

Substituting \(\dfrac{1}{f}\) for \(T\) and rearranging for \(T\) we obtain

$$T=\frac{2\pi}\omega.$$

Where \(\omega\) is the angular frequency in radians per second, \(\frac{\mathrm{rad}}{\mathrm s}\). If we think about it, this expression makes sense, as an object with a large angular frequency will take a lot less to make one complete oscillation cycle.

Harmonic Oscillators

A harmonic oscillation is a type of oscillation in which the net force acting on the system is a restoring force. A restoring force is a force acting against the displacement in order to try and bring the system back to equilibrium. An example of this is Hooke's Law given by

$$F_s=ma_x=-k\Delta x,$$

where \(m\) is the mass of the object at the end of the spring in kilograms, \(\mathrm{kg}\), \(a_x\) is the acceleration of the object on the \(\text{x-axis}\) in meters per second squared, \(\frac{\mathrm m}{\mathrm s^2}\), \(k\) is the spring constant that measures the stiffness of the spring in newtons per meter, \(\frac{\mathrm{N}}{\mathrm m}\), and \(\Delta x\) is the displacement in meters, \(\mathrm{m}\).

If this is the only force acting on the system, the system is called a simple harmonic oscillator. This is one of the most simple cases, as the name suggests.

Most oscillations occur in the air or other mediums, where there is some type of force proportional to the system's velocity, such as air resistance or friction forces. These may act as damping forces. The equation for the damping force is

$$F_{damping}=-cv,$$

where \(c\) is a damping constant in kilograms per second, \(\frac{\mathrm{kg}}{\mathrm s}\), and \(v\) is the velocity in meters per second, \(\frac{\mathrm{m}}{\mathrm s}\).

As a consequence, part of the system's energy is dissipated in overcoming this damping force, so the amplitude of the oscillation will start to decrease as it reaches zero. These types of harmonic oscillators are called damped oscillators. We can write Newton's Second Law for the case where there is a restoring force and a damping force acting on the system,

$$ma=-cv-kx.$$

Writing the above expression as a differential equation, we obtain

$$m\frac{\operatorname d^2x}{\operatorname dt^2}+c\frac{\operatorname dx}{\operatorname dt}+kx=0.$$

The solution to the above equation is an exponential function. The damping term will exponentially dissipate the oscillations until the system decays to rest.

\[x=A_0e^{-\gamma t}\cos\left(wt+\phi\right),\] where \(\gamma=\frac c{2m}\)

$$x=A_0e^{-\frac c{2m}t}\cos\left(wt+\phi\right)$$

We can prove this is a solution by differentiating it and substituting it into the differential equation:

$$\begin{array}{rcl}\frac{\operatorname dx}{\operatorname dt}&=&-A_0\omega e^{-\frac c{2m}t}\sin(\omega t+\phi)\;-A_0\frac c{2m}e^{-\frac c{2m}t}\cos(\omega t+\phi)\\\frac{\mathrm d^2x}{\mathrm dt^2}&=&\begin{array}{c}-A_0\omega^2e^{-\frac c{2m}t}\cos(\omega t+\phi)\;+A_0\omega\frac cme^{-\frac c{2m}t}\sin(\omega t+\phi)\;+A_0\frac{c^2}{4m^2}e^{-\frac c{2m}t}\cos(\omega t+\phi)\end{array}\end{array}.$$

Now we can go back to the differential equation and prove that we found a solution for it.

$$m\frac{\operatorname d^2x}{\operatorname dt^2}+c\frac{\operatorname dx}{\operatorname dt}+kx=0$$

$$\begin{array}{rcl}\frac{A_0c^2e^{\displaystyle\frac{-bt}{2m}}\cos\left(\omega t+\phi\right)}{4m}+\cancel{A_0c\omega e^\frac{-bt}{2m}\sin\left(\omega t+\phi\right)}\;-A_0\omega^2me^\frac{-bt}{2m}\cos\left(\omega t+\phi\right)\;-\frac{A_0c^2e^{\displaystyle\frac{-bt}{2m}}\cos\left(\omega t+\phi\right)}{2m}&-\cancel{A_0c\omega e^\frac{-bt}{2m}\sin\left(\omega t+\phi\right)}+A_0ke^\frac{-bt}{2m}\cos\left(\omega t+\phi\right)=&0\end{array}$$

$$\begin{array}{rcl}-\frac{\cancel{A_0}c^2\cancel{e^{\displaystyle\frac{-bt}{2m}}\cos\left(\omega t+\phi\right)}}{4m}-\cancel{A_0}\omega^2m\cancel{e^\frac{-bt}{2m}\cos\left(\omega t+\phi\right)}\;+\;\cancel{A_0}k\cancel{e^\frac{-bt}{2m}\cos\left(\omega t+\phi\right)}&=&0\end{array}$$

$$-\frac{c^2}{4m^2}-\omega^2+\frac km=0$$

$$\omega=\sqrt{\frac km-\frac{c^2}{4m^2}}.$$

The damped oscillators with oscillations and an amplitude that decreases with time are called underdamped oscillators. While the ones that do not oscillate and immediately decay to equilibrium position are called overdamped oscillators. The boundary limit between under-damping and over-damping is called critical damping. To confirm the damped oscillator is undergoing critical damping we verify that the damping coefficient \(\gamma\) is equal to the system's natural angular frequency \(\omega_0\). The damping coefficient \(\gamma\) can be determined with the following equation:

$$\gamma=\frac c{2m},$$

where \(c\) is a damping constant measured in units of kilograms per second, \(\frac{\mathrm{kg}}{\mathrm s}\), and \(m\) is the system's mass in kilograms, \(\mathrm{m}\).

The angular frequency for the damped oscillator can be defined in terms of the damping coefficient and the natural angular frequency.

$$\begin{array}{rcl}\omega&=&\sqrt{\frac km-\frac{c^2}{2m}}\\\omega&=&\sqrt{\omega_0-\gamma}\end{array}$$

These 3 cases can be summarized as follows:

• Underdamping: \(\omega_0>\gamma\)
• Critical damping: \(\omega_0=\gamma\)
• Overdamping: \(\omega_0<\gamma\)

There is also another type of oscillator called forced oscillators. In these, the oscillations are caused by an external force that is a periodic force. If the frequency of this force is equal to the system's natural frequency, this causes a peak in the amplitude of oscillation. The natural frequency is the frequency at which an object will oscillate when it is displaced out of equilibrium.

Oscillations in a Spring-Mass System

We will consider the simplest case of Simple Harmonic Motion to understand oscillations in a spring-mass system. For a spring, we already know the equation for Newton's second law:

$$F_s=ma_x=-k\Delta x.$$

Rearranging for the acceleration we obtain

$$a_x=-\frac km\Delta x.$$

So, comparing the equation for a spring with the general equation for harmonic motion \(a=-\omega_0^2x\), we can derive the angular frequency \(\omega\) for a spring, which is given by the equation

$$\omega_0^2=\frac km,$$

expressed more explicitly as

$$\omega_0=\sqrt{\frac km}.$$

Where \(m\) is the mass of the object at the end of the spring in kilograms, \(\mathrm{kg}\), and \(k\) is the spring constant that measures the stiffness of the spring in newtons per meter, \(\frac{\mathrm N}{\mathrm m}\).

The formula for the time period of an oscillating spring-mass system is

$$T_s=2\pi\sqrt{\frac mk}.$$

Graphing oscillations

If we plot the displacement as a function of time for an object undergoing simple harmonic motion, we would identify the period as the time between two consecutive peaks or any two analogous points on two waves with the same phase. To locate the amplitude, we look at the highest peak in distance.

We can also graph the displacement as a function of time for damped oscillators, to visually understand and compare their characteristics. Critical damping provides the quickest way for the amplitude to reach zero. Overdamping takes you faster to the zero position, but decaying oscillations still occur. Underdamped oscillations take more time to reach an amplitude of zero.