Simple Harmonic Motion

The harmonic motion is one of the forms of periodic motion. The Simple Harmonic Motion is represented in terms of circular sine and cosine functions. All harmonic motions are periodic but all the periodic motions are not always harmonic. Simple harmonic motion is the term used to describe a body’s oscillations around a fixed point. SHM possesses the following characteristics:

The motion is periodic
A restoring force acts on the particle to tend to bring it back to the mean position when it deviates from the fixed point or mean position.
Restoring force on the particle is directly proportional to its displacement.

Consider a harmonic motion of type $x\:= \: A\: sin(\omega_n t+ \: \Phi)$

where
x is the displacement,
A is the Amplitude,
$\omega_n$ is the frequency
$\Phi$ is phase angle

The velocity and acceleration are,
$\frac{dx}{dt}$ = velocity = $\dot{x}\:= \: A\: \omega_n sin(\omega_n t+ \: \Phi)$
$\frac{d^2x}{dt^2}$ = acceleration = $\ddot{x}\:= \: A\: \omega^2_n sin(\omega_n t+ \: \Phi)$

For the maximum value of displacement, $sin(\omega_n t+ \: \Phi)$ = 1
$x_{max}=A$

to get the maximum value of velocity, $cos(\omega_n t+ \: \Phi)$ = 1
$\dot{x}_{max}\$ = A $\omega_n$

For the maximum value of acceleration, $sin(\omega_n t+ \: \Phi)$ = 1
$\ddot{x}\: = -A\omega^2_n = - \omega^2_n(x)$

Thus the acceleration in an SHM is always proportional to its displacement and directed toward a particular fixed point

The velocity of the vibrating particle is maximum at the mean position of rest and zero at the maximum position of vibration. However, the acceleration of the vibrating particle is zero at the mean position of rest and maximum at the maximum positions of vibration. The acceleration is directly proportional to the displacement of the vibrating particle and is always oriented toward the mean position of rest. This type of motion occurs where the acceleration is directed towards a fixed point and is proportional to the displacement of the vibrating particle.

A body’s total energy during SHM is made up of both kinetic and potential energy. At the mean location, both the velocity and the resulting kinetic energy reach their maximum. Potential energy is zero at the mean position and is maximum at the extreme position.

kinetic energy = $\frac{1}{2} mv^2$

Representation of harmonic motion in complex form

Consider a vector x= x+ry where x and y denote the real and imaginary components of x, and $i=\sqrt{-1}$ . If the vector makes an angle $\theta$ with X axis, it can be written as

$X= A \:cos\theta \:+\: iA\:sin\theta$
$=Ae^{i\theta }$

where A is the modulus or the absolute value of vector X.
$\theta = tan^{-1}\frac{y}{x}$
$X =Ae^{i\theta }$

Velocity can be determined by

$\dot{x}= \frac{dx}{dx}=iA\omega e^{i\omega t}$ since $\theta = \omega t$

= $i\omega x$
$\dot{x}= i\omega x$

This is known as the velocity vector. Again differentiation, we get
$\ddot{x} = \frac{d^2x}{dt^2} = i^2\omega^2Ae^{i\omega t}$

Acceleration = $-\omega^2 = Ae^{i\omega t}$
$\ddot{x} -\omega^2x$

This is known as the acceleration vector and its amplitude is $\omega^{2}x$ . In Figure 1.9, it is shown that the velocity vector leads the displacement by 90° and the acceleration vector leads the displacement by 180°. With constant angular velocity, every vector rotates in the same direction.

The displacement that results from two SHMs passing through the same spot in a medium at the same time is the total of the displacements caused by the two components of motion. This superposition of motion is called interference. The phenomenon of beat occurs as a result of interference between the two waves of slightly different frequencies moving along the same straight line in the same direction.

Suppose that the two wave movements are in phase at a given moment. The resulting vibrational amplitude will reach its maximum at this point. On the other hand, when two motions are not in phase with each other, they produce a minimum amplitude of vibration. After a while, the two motions are once more in phase, producing maximum and then minimum amplitude. The resulting amplitude, which has a frequency equal to the difference between the frequencies, continuously changes from maximum to minimum as this procedure is repeated. This phenomenon is known as beat.

The frequency of beats, i.e. (Aw) should be small to experience the phenomenon.

In Simple Harmonic Motion, the concept of phase is crucial. Phase definition was covered in an earlier post. The amount by which two Simple Harmonic Motion (SHM) are out of sync with one another, by how many angles, or by how much time, is indicated by the phase difference between them. When two sinusoidal motions are out of phase then phase difference $(\Delta\omega)=180^{\circ}$ . When two sinusoidal motions are in phase, then phase difference $(\Delta\omega)=0$ .

Simple Harmonic Motion

Also Read:- Types Of Vibration And Basic Definition