Motion & force formulas

$R_p$ is the total track resistance and $R_L$ the load across the wiper. For $R_L\to \infty$ the output is $V_{s}k$, which is linear. A finite load makes it nonlinear. For $R_L\gg R_p$ the error (unloaded minus loaded output, as a fraction of $V_s$) is approximately $k^2(1-k)R_p/R_L$, largest near $k=2/3$, where it is about $0.148R_p/R_L$.

The first result is for a change of gap: sensitivity rises as the gap shrinks. The second is for a centre plate moved by $x$ from the middle of two fixed plates, each at gap $d_0$ when centred, so $C_1={\epsilon }_0{\epsilon }_{r}A/(d_0-x)$ and $C_2={\epsilon }_0{\epsilon }_{r}A/(d_0+x)$.

$N$ is the number of slots (pulses per revolution per channel). Resolution is the smallest angle that can be distinguished.

$\omega$ in rad/s, $k_t$ in V per rad/s, $n$ in rpm.

For an encoder, $f$ is the pulse frequency of one channel and $N$ the pulses per revolution of that channel; with $\times 4$ decoding, use the count rate and $4N$. For a variable-reluctance (magnetic) pickup facing a toothed ferromagnetic wheel, each tooth gives one pulse, so $N$ is the number of teeth: $f=Nn/60$.

$Z$ and $Y$ are amplitudes. For $r\gg 1$, $Z/Y\to 1$ and the relative motion is about 180° out of phase with the base motion (vibrometer).

$Z$ is proportional to the acceleration amplitude $Y{\omega }^2$ when $r\ll 1$, with static sensitivity $1/{\omega }_n^2$ (relative displacement per unit acceleration). For $\zeta =1/\sqrt{2}$ the second factor is $1/\sqrt{1+r^4}$.

The angle by which the relative displacement lags the input acceleration, taken between 0° and 180° (90° at $r=1$). For $\zeta \approx 0.7$ and $r<1$, $\varphi$ is nearly proportional to frequency, so a waveform with several frequency components is delayed but not distorted.

$d$ is the charge sensitivity in C/N. $C_p$ is the crystal capacitance, $C_c$ the cable capacitance and $C_a$ the amplifier input capacitance. Cable length changes the voltage sensitivity, so a charge amplifier ($V_o=-q/C_f$) is used to remove that dependence.

$g$ is the voltage coefficient in V m/N, $t$ the crystal thickness and $F/A$ the applied stress. This is the open-circuit voltage of the crystal alone, $q/C_p$ with $C_p={\epsilon }_0{\epsilon }_{r}A/t$.

For a voltage amplifier, $R$ is the combined leakage and input resistance and $C$ the total capacitance of sensor, cable and amplifier input. With a charge amplifier, the feedback resistor $R_f$ and capacitor $C_f$ set the −3 dB point.

$X$ is the displacement amplitude and $f$ the vibration frequency in Hz.

$\nu$ is Poisson's ratio of the gauge material and $\Delta \rho /\rho$ the fractional change in its resistivity (the piezoresistive effect). For metal-foil gauges $G\approx 2$; semiconductor gauges have a much larger, temperature-sensitive $G$.

Axial load $F$ on cross-section $A$ of Young's modulus $E$, within the elastic limit. A gauge bonded across the column sees the transverse strain $-\nu \epsilon$.

Equal arms, bridge balanced at zero strain, signs of the active arms arranged so their effects add. The half- and full-bridge results with equal and opposite strains are exactly linear. Temperature effects cancel when identical gauges at the same temperature sit in adjacent arms (a dummy gauge does this for one active gauge).

$d$ is the shaft diameter in metres, $T$ in N m, $G_s$ in Pa. Mechanical power transmitted is $P=T\omega$.