# Focal Vibration Therapy: Vibration Parameters of Effective Wearable Devices
## 2 Materials and Methods
### Relating Vibration Amplitude and Acceleration  
Typically, past works have quoted vibration intensity in terms of displacement (or vibration amplitude). It is not clear why this has been a trend, since acceleration is relatively easier to measure, especially in the constrained vibration case. Perhaps this has been a trend because vibration systems and accelerometers have been miniaturized to feasible costs and sizes in only the past two decades. Vibration technology itself has been around for much longer, i.e., well over a century. In the past, bulkier and more expensive technologies, e.g., high-speed cameras and linear variable differential transformers (LVDT), were the most feasible options. These technologies can directly measure displacement (or vibration amplitude), possibly explaining why vibration intensity has been widely reported in the literature. We believe that using acceleration as a metric for vibration delivered is more logical and economical, given the widespread availability of low-cost micro-electro-mechanical systems (MEMS) accelerometers. In the spirit of making the comparison with vibration displacement (or amplitude) easier, we present a conversion between displacement (or vibration amplitude) and peak-to-peak acceleration. Consider the motion of a vibration device as is vibrates in the normal direction to the body. We can safely assume a sinusoidal displacement pattern with amplitude \(A/2\) (m) (peak-to-peak amplitude \(A\)). The sinusoidal pattern will have angular frequency \(\omega\) (rad/s). It follows that the displacement s (m) from the mean position, as a function of time t (s), is defined by the sinusoidal wave equation [cite [[https://en.wikipedia.org/wiki/Sine_wave](https://en.wikipedia.org/wiki/Sine_wave)]([https://en.wikipedia.org/wiki/Sine_wave](https://en.wikipedia.org/wiki/Sine_wave)) (accessed date 9 March 2021)]:  
\[\mathrm{s}=\ \frac{A}{2}\sin(\omega\mathrm{t}) \tag{2}\]  
Alternatively, \(\omega\) can also be defined in terms of frequency \(f\) in Hz:  
\[\mathrm{s}=\ \frac{A}{2}\sin(2\pi f\mathrm{t}) \tag{3}\]From basic kinematics, velocity is the first derivative of displacement s (3):  
\[\frac{\mathrm{ds}}{\mathrm{dt}}=\ A\pi f\cos(2\pi f\mathrm{t}) \tag{4}\]  
From basic kinematics, acceleration is the second derivative of displacement s (3):  
\[\frac{\mathrm{d}^{2}\mathrm{s}}{\mathrm{dt}^{2}}=\ -2A\pi^{2}f^{2}\sin(2\pi f \mathrm{t}) \tag{5}\]  
The displacement s is maximum when \(\mathrm{ds}/\mathrm{dt}=0\). Thus, from (4),  
\[\cos(2\pi f\mathrm{t})=0 \tag{6}\]  
\[2\pi f\mathrm{t}=\frac{\pi}{2},or\ \frac{3\pi}{2},\ or\ \frac{5\pi}{2}\ldots \tag{7}\]  
We also know that when displacement is maximum, the acceleration is maximum (in the opposite direction). Taking the first value \(\pi/2\), we get the maximum acceleration from (5):  
\[\mathrm{acceleration}_{\mathrm{Max}}=\ -2A\pi^{2}f^{2}\sin\left(\frac{\pi}{2}\right) \tag{8}\]  
Rearranging, we have a relation where we can compute peak-to-peak amplitude \(A\) (m) given maximum acceleration (ms\({}^{-2}\)) and vibration frequency \(f\) (Hz):  
\[A=-\frac{\mathrm{acceleration}_{\mathrm{Max}}}{2\pi^{2}f^{2}} \tag{9}\]  
If acceleration is instead in g's, and 1 g is 9.81 ms\({}^{-2}\), then the peak-to-peak amplitude \(A\) (m) is given by:  
\[A=-\frac{4.905\mathrm{acceleration}_{\mathrm{Max}}}{\pi^{2}f^{2}} \tag{10}\]  
Maximum acceleration is half of peak-to-peak acceleration. Vibration amplitudes are typically in the order of mm. In terms of peak-to-peak acceleration in g's, the peak-to-peak amplitude \(A\) (mm) is given by:  
\[A=-\frac{2452.5\mathrm{acceleration}_{\mathrm{PKPK}}}{\pi^{2}f^{2}} \tag{11}\]  
Therefore, to convert between peak-to-peak vibration amplitude \(A\) (mm) and peak-to-peak vibration acceleration, \(\mathrm{acceleration}_{\mathrm{PKPK}}\) (g's), (11) may be used, if the frequency of vibration\(f\) (Hz) is known.