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Simple Harmonic Motion

Oscillations are a natural occurrence in nature. From sub-atomic particles and atoms to planets and stars, the universe works by forces and accelerations in repetitive motions.

A particular type of oscillation is one in which there is a fixed point of equilibrium, where there is no net force. A mass with a displacement from this equilibrium point experiences a force, and therefore acceleration, directly proportional to the size of the displacement. This type of motion is called SHM, simple harmonic motion.

SHM

In simple harmonic motion, the acceleration is proportional to the distance from, and at all times directed towards,a fixed equilibrium point.

Equations

Fundamental equation of simple harmonic motion:

$$a = -ω^2x$$

where a is the acceleration towards the equilibrium point, ω is the angular frequency, and x is the displacement from the equilibrium point.

Derived equations:

$v = v_0sinωt$, $v = v_0cosωt$, $v = ±ω√{(x_0^2 - x^2)}$, $x = x_0sinωt$, $x = x_0cosωt$

Two examples are the pendulum and a spring.

Angular frequency

The frequency of a full cycle is the number of times the oscillation returns to its initial point (careful: one full cycle means two swings for a pendulum) per second, measured in hertz (Hz). for circular motion, the angular frequency is therefore the angle, measured in radians, moved through per unit time: $ω = 2πf$, where f is the frequency of oscillation.

Displacement-time
A helical spring demonstrates simple harmonic motion in accordance with Hooke's Law, F = -kx.

Take the example of a helical spring, on which a mass is released. It enters SHM, whereby the acceleration due to the force exerted on the mass by the extended or compressed spring is always towards the equilibrium point.

Since the force and acceleration are varying with time, the displacement is not linear, but sinusoidal, with equation: $x = x_0cosωt$, where $x_0$ is the maximum displacement and ω is the angular frequency.

Velocity-time

The velocity is also sinusoidal. Zero velocity occurs at both extremities, and maximum velocity occurs at the equilibrium point.

$v = ±v_0sinωt$, where $v_0$ is the maximum velocity.

Acceleration-time

The acceleration is also sinusoidal. The acceleration is always towards the equilibrium point, where it is zero. It increases in magnitude directly proportional to the displacement from the equilibrium point.

$a = ±a_0sinωt$, where $a_0$ is the maximum acceleration.

Hooke's Law
Hooke's Law states that the force is linearly proportional to the distance of compression or extension of a helical spring

Damping

There are many cases in engineering when it is not desirable to allow a spring to oscillate too long. For example, a car's shock absorbers applying a counter-force to the force provided by the suspension springs. This causes the amplitude of oscillation after the car hits a bump to reduce rapidly to zero, so the passengers do not get motion sickness.

$β = b/{2m}$, where β is the damping constant, and b is damping coefficient.

$w_0 = √{k/m}$, where $w_0$ is the natural (frictionless) angular frequency.

Resonance

$ω_{res} = √{k/m}$, where $ω_{res}$ is the angular frequency in resonance.

$f_{res} = ω_{res}/{2π} $, where $f_{res}$ is the resonance frequency of the system.

Content © Andrew Bone. All rights reserved. Created : January 29, 2014 Last updated :April 19, 2016

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