Simple Harmonic Motion
The definition of simple harmonic motion is simply that the acceleration causing the motion a of the particle or object is proportional and in opposition to its displacement x from its equilibrium position.
Where k is a constant of proportionality. This remembering that the acceleration is the second derivative of position, also leads us to the differential equation
x”(t) = – k x(t)
Simple Harmonic Motion is closely related to circular motion as can be seen if we take an object that moves in a circular path, like a ball stuck on a turntable. If we consider just the y-component of the motion the path with time we can see that it traces out a wave:
y(t) = A sin(ωt)(2)
We can also see that the period of the motion is equal to the time it takes for one rotation. Therefore, if we know the angular velocity ω = θ/t. For one rotation, ω = 2π/T therefore the period is also equal to
T = 2π/ω(3)
The particle can also at different speeds is conected with the period. The frequency is the number of oscilations per second.
The velocity of the particle can be calculated by differentiating the displacement. The result is also a wave but the maximum amplitude is delayed, so that when the displacement is at a maximum the velocity is at a minimum and when the displacement is zero the velocity has its greatest value. Simple Harmonic Motion is characterised by the acceleration a being oppositely proportional to the displacement, y.
We set this out mathematically, using a differential equation as in equation (4). We specify the equation in terms of the forces acting on the object. The acceleration is the second derivative of the position with respect to time and this is proportional to the position with respect to time. The minus sign indicates that the position is in the opposite direction to the acceleration.
m y”(t) = – k y(t)(4)
The derivation of the solution can be found here
For which the general solution is a wave like solution.
y(t) = c1 cos(ωt) + c2 sin(ωt)
Where, ω is the angular frequency. (ω=2πf) The values of c1 and c2 are determined by the initial conditions. Specifically, c1 = y0 and c2 = v0/ω These two initial conditions specify the starting position and the initial velocity.
The general solution can also be written more compactly as
y(t)= A cos(ωt – φ)(5)
Where φ = tan-1(ωy0/v0), A = (y02 + (v0/ω)2)1/2
Differentiating once with respect to time, we obtain the velocity. (The derivative of cos x = – sin x)
v = y‘(t)= – ωA sin(ωt – φ)(6)
Finally, the acceleration is the derivative of the velocity with respect to time. (The derivitive of -sin x = – cos x)
a = y”(t) = – ω2A cos(ωt – φ)(7)
Substituting equations (5) and (7) into equation (4) we verify that this does indeed satisfy the equation for simple harmonic motion. With the constant of proportionality k = ω2
Thus
a(t) = – ω2y(t)
The time for the maximum velocity and acceleration can be determined from these equations. From equation (6) the maximum magnitude of the velocity occurs when sin(ωt – φ) is 1 or -1. Therefore the maximum velocity is ±ωA. Intuitively, we can imagine that this velocity occurs when the oscillating system has reached the equilibrium position and is about to overshoot. The minus sign indicates the direction of travel is in the opposite direction.
The maximum acceleration occurs where the argument of cosine in eqn (7) is also -1 or 1. Thus the maximum acceleration is ±ω2A which occurs at the ends of the oscillations, as this is where the direction changes.