# Simple Harmonic Motion Calculator

This tool calculates the variables of simple harmonic motion (displacement amplitude, velocity amplitude, acceleration amplitude, and frequency) given any two of the four variables. Click here to see How it works & for Governing Equations of Motion.

Values automatically update when you enter a value (Press F5 to refresh).

 Displacement: mm in Velocity: mm/s in/s Acceleration: g Frequency: Hz Number of Cycles for Plot:

## Simple Harmonic Motion Calculator - How it Works

### Displacement, Velocity, Acceleration, Frequency Calculations

This tool calculates the variables of simple harmonic motion (displacement amplitude, velocity amplitude, acceleration amplitude, and frequency) given any two of the four variables.

• When changing values for displacement, velocity or acceleration the calculator assumes the frequency stays constant to calculate the other two unknowns.
• When changing the frequency value, the calculator assumes acceleration to be constant and calculates velocity and displacement using this new value for frequency.

These governing equations of motion are explained in more depth below in the Simple Harmonic Motion Equations section.

All these values automatically update when changing any variable. An important note is that they represent amplitude (zero-to-peak) not peak-to-peak values. Refer to the plots for more information on how these motion waves change over time.

### Displacement, Velocity, Acceleration, Frequency Plots

The frequency, displacement, velocity, acceleration relationship plot above displays two lines that illustrate how velocity and frequency change if acceleration (blue line) or displacement (orange line) is kept constant. The intersection of these two lines occur at the variable values defined by the user in the above number fields.

The other plot displays a few cycles, as defined by the above dialog box, for displacement, velocity, and acceleration. These waveforms represent the displacement, velocity, and acceleration as a function of time given the provided or calculated amplitudes and frequency. These governing equations of motion are explained in more depth below in the Simple Harmonic Motion Equations section.

#### Aliasing

Please note that there are only 100 data points plotted in the simple harmonic motion plots.  This was done on purpose to help illustrate the importance of sampling rate. For example, the waveforms look "clean" at 10 or fewer cycles (a sample rate that is 10x the frequency of interest).  When more cycles are plotted with only those 100 data points, the plots become increasingly jagged.  This same phenomenon occurs in your vibration testing - if you are not sampling fast enough you will miss important characteristics of your vibration profile.

As the number of cycles increases to 1/2 the "sample rate" (100 data points), more and more information is missed in the waveform. At 1/2 the "sample rate" or Nyquist frequency the signal "folds" in on itself. At 99 cycles, due to the ineffectiveness of our sample rate, the data acquisition system would believe it cleanly measured 1 cycle. We explain more about Nyquist frequency and aliasing in a blog post.

### Slam Stick Accelerometer Logger Selection

The tool also suggests different Slam Stick accelerometer (sometimes called vibration data) loggers to consider. The tool is looking at the frequency and amplitude range to make this determination. Slam Sticks can be used for modest vibration/shock levels (amplitudes greater than a 0.2g but less than 2,000g & maximum frequency below 2,000 Hz); for levels outside these ranges the tool points to a blog on accelerometer selection that includes external accelerometer supplier companies who may have more appropriate product solutions.

## 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

$\inline&space;\dpi{120}&space;\bg_white&space;\large&space;x(t)=A\,&space;sin(2\pi&space;t)$

where $\inline&space;\dpi{120}&space;\bg_white&space;\large&space;A$ is the amplitude of displacement, $\inline&space;\dpi{120}&space;\bg_white&space;\large&space;f$ is the frequency, $\inline&space;\dpi{120}&space;\bg_white&space;\large&space;t$ is the time, and $\inline&space;\dpi{120}&space;\bg_white&space;\large&space;x(t)$ could be the displacement of $\inline&space;\dpi{120}&space;\bg_white&space;\large&space;x$ at time $\inline&space;\dpi{120}&space;\bg_white&space;\large&space;t$ in this instance. Angular velocity $\inline&space;\dpi{120}&space;\bg_white&space;\large&space;w$ can be represented by the following equation.

$\inline&space;\dpi{120}&space;\bg_white&space;\large&space;w=2\pi&space;f$

The $\inline&space;\dpi{120}&space;\bg_white&space;\large&space;x(t)$ can then be reduced to

$\inline&space;\dpi{120}&space;\bg_white&space;\large&space;x(t)=A\,&space;sin(wt)$

Committing to $\inline&space;\dpi{120}&space;\bg_white&space;\large&space;x(t)$ 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

$\inline&space;\dpi{120}&space;\bg_white&space;\large&space;v=\frac{dx}{dt},\;&space;a=\frac{d^{2}x}{dt^{2}}$

where $\inline&space;\dpi{120}&space;\bg_white&space;\large&space;v$ is the velocity and $\inline&space;\dpi{120}&space;\bg_white&space;\large&space;a$ 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

$\inline&space;\dpi{120}&space;\bg_white&space;\large&space;v(t)=\frac{dx}{dt}=A\,&space;w\,&space;cos(wt)$

$\inline&space;\dpi{120}&space;\bg_white&space;\large&space;a(t)=\frac{dv}{dt}=-A\,&space;w^{2}\,&space;sin(wt)$

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

$\inline&space;\dpi{120}&space;\bg_white&space;\large&space;v(t)=B\,&space;sin(wt)$
$\inline&space;\dpi{120}&space;\bg_white&space;\large&space;x(t)=-\frac{B}{w}\,&space;cos(wt)$
$\inline&space;\dpi{120}&space;\bg_white&space;\large&space;a(t)=B\,&space;w\,&space;cos(wt)$

where $\inline&space;\dpi{120}&space;\bg_white&space;\large&space;B$ is the amplitude of velocity. If going from known acceleration amplitude, $\inline&space;\dpi{120}&space;\bg_white&space;\large&space;C$, and frequency the velocity can be found by integrating once and the displacement by integrating twice, which mathematically looks like

$\inline&space;\dpi{120}&space;\bg_white&space;\large&space;a(t)=C\,&space;sin(wt)$
$\inline&space;\dpi{120}&space;\bg_white&space;\large&space;v(t)=-\frac{C}{w}\,&space;cos(wt)$
$\inline&space;\dpi{120}&space;\bg_white&space;\large&space;x(t)=-\frac{C}{w^{2}}\,&space;sin(wt)$

Because vibration acceleration, velocity, displacement, and frequency are all linearly dependent on each other, this relationship can be visualized in the following plots. Energy transferred into a system is most closely related to the velocity, so a vibration or shock engineer must pay special attention to the velocity experienced.