# Simple Harmonic Motion Calculator

Frequency Hz
Time s

## Simple Harmonic Motion Calculator Information

Using this Calculator, users are able to input known amplitudes and frequencies of the system, from which the amplitudes of the other aspects of motion can then be calculated. Analysis can then go a step further and have the user input a specific time to and get the time specific values for displacement, velocity, and acceleration.

Example:
Trevor needs to know the characteristics of a vibrating surface, and is able to measure the acceleration using a data recorder (like a Slam Stick X), and then analyzes the data with an analysis tool (such as Slam Stick Lab) and determines the frequency and amplitude of the vibration to be 100Hz and 25g.

Using the calculator, Trevor enters the 100Hz in the Frequency box, and the 25g amplitude in the Acceleration box, before hitting the CALCULATE button.

The derived results for Displacement and Velocity appear in the shaded blue boxes.

To determine the Displacement, Velocity and Acceleration at a specified time, Trevor can enter a time value in the Time box, such as 0.00625, and then hit enter/return. The calculated values will appear below the plot, as well as appearing as red dots in the plot.

Trevor now knows the characteristics of his vibrating surface.

### Simple Harmonic Motion Equations

The motion of a vibrational system results in velocity and acceleration that is not constant, but is in fact modeled by a sinusoidal wave. A sinusoid, similar to a sine wave, is a smooth, repetitive wave, but may be shifted in phase, period, or amplitude. The important factors associated with this oscillatory motion are the amplitude and frequency of the motion. The general equation for motion that follows a sine wave is

where is the amplitude of displacement, is the frequency, is the time, and could be the displacement of at time in this instance. Angular velocity can be represented by the following equation.

The can then be reduced to

Committing to being the displacement of the system, it is known that we can learn more about the system’s motion through differentiation. This can be seen as

where is the velocity and is the acceleration of the system, with the first derivative of displacement being velocity and the second derivative being the acceleration. This knowledge can be applied to our sinusoidal equation, resulting in

as equations to relate the velocity and acceleration to displacement. If the amplitude and displacement of frequency are known, then functions for velocity and acceleration can be determined from them.

The same concept can be used if, along with the frequency, the amplitude of velocity or acceleration is known. For velocity, the displacement can be found through integration and the acceleration through differentiation, as seen in the equations

where is the amplitude of velocity. If going from known amplitude and frequency for acceleration, the velocity can be found by integrating once and the displacement by integrating twice, which mathematically looks like

where is the amplitude of acceleration.