Dynamics of simple harmonic motion
For one-dimensional simple harmonic motion, the equation of motion, which is a second-order linear ordinary differential equation with constant coefficients, could be obtained by means of Newton's second law and Hooke's law.
where m is the inertial mass of the oscillating body, x is its displacement from the equilibrium (or mean) position, and k is the spring constant.
Solving the differential equation above, a solution which is a sinusoidal function is obtained.
In the solution, c1 and c2 are two constants determined by the initial conditions, and the origin is set to be the equilibrium position. Each of these constants carries a physical meaning of the motion: A is the amplitude (maximum displacement from the equilibrium position) , ω = 2πf is the angular frequency, and φ is the phase.
Position, velocity and acceleration of a harmonic oscillator
Using the techniques of differential calculus, the velocity and acceleration as a function of time can be found:
Position, velocity and acceleration of a SHM as phasors
Acceleration can also be expressed as a function of displacement:
Then since ω = 2πf,
and since T = 1/f where T is the time period,
These equations demonstrate that the simple harmonic motion is isochronous (the period and frequency are independent of the amplitude and the initial phase of the motion).