#### 2.1. Principle of Quasi-Static Calibration of High-g Accelerometer

It is hard to realize the static calibration of high-g accelerometer, but it can usually be conducted in a quasi-static state. Quasi-static calibration excites the calibrated system using an extremely low amount of high-frequency of excitation signals, in the hope that the proportion occupied by the inherent frequency component of the calibrated system in its response output can be small and within a pre-specified range, namely, exciting the accelerometer with wide-pulse acceleration signals. Meanwhile, the excitation signals (obtained by standard instruments) that can be traced to the source function as the input signals, and the impact sensitivity and relative uncertainty of the accelerometer are calculated after multiple fittings. We implement this calibration method in this study.

Figure 1 shows half-sine excitation signals with amplitude of 1 V and pulse widths being 500 μs and 200 μs, respectively. The resonant frequency of the calibrated accelerometer is 40 kHz, its resonance peak is 40 dB and its normalized amplitude-frequency characteristics are shown in

Figure 2.

Figure 3 shows the calibrated accelerometer’s frequency domain response to the two pulsed excitations, and

Figure 4 demonstrates the calibrated accelerometer’s time domain response to the two pulse excitations gained by Fourier inversion.

As shown in

Figure 4, the response and excitation signals change with the same trend, and the oscillation curve superimposed on the response signal is the response error. The narrower the pulse width of the excitation signal, the wider the frequency range covered by the main lobe width of the corresponding frequency spectrum, and thus the higher the modal frequency the accelerometer can excite. This then results in a larger amplitude error between the response and excitation signals. As shown in

Figure 4, the amplitude error of the half-sine signal with a pulse width of 200 μs reaches 7.4%, while that of the half-sine signal with a pulse width of 500 μs reaches 1.9%.

Since there are a large variety of high-g accelerometers with different resonant frequencies, abundant tests were required to determine the pulse width of the excitation signal so as not to arouse the resonant frequency of the accelerometer, which increases the workload of the calibration test. Therefore, to achieve quasi-static calibration of high-g accelerometers, the minimum pulse width of excitation signals with different resonant frequencies must be theoretically determined within a specific response error [

17].

#### 2.2. Minimum Pulse Width of Excitation Signal Required for Quasi-Static Calibration

High-g accelerometers can be divided into two types, namely the piezoelectric type and the piezoresistive type. Although they have different principles and structures, both can be approximately equivalent to the spring-mass system, namely, the response characteristics of the accelerometer can be described by a second-order mathematical model.

The mathematical model of the accelerometer can be expressed as in Equation (1):

where

$m$ denotes mass weight;

$c$ denotes the damping coefficient;

$k$ denotes the rigidity coefficient;

$y$ is the displacement of the mass relative to the base;

$\ddot{x}$ denotes the excitation acceleration. Let

$\alpha =\frac{\ddot{x}}{{\ddot{x}}_{m}}$,

$\theta =\frac{t}{\tau}$,

$\zeta =\frac{{c}_{c}}{c}$,

${c}_{c}=2\sqrt{km}$,

$\delta =-\frac{ky}{m{\ddot{x}}_{m}}$,

$R=\frac{{T}_{n}}{\tau}$,

${T}_{n}=2\pi \sqrt{m/k}=\frac{2\pi}{{\omega}_{n}}$, then Equation (1) can be rewritten as:

where α denotes the dimensionless excitation acceleration;

${\ddot{x}}_{m}$ is the peak value of the excitation acceleration;

$\ddot{x}$ denotes the instantaneous value of the excitation acceleration;

$\theta $ refers to the dimensionless time;

$\tau $ denotes the pulse duration of the excitation acceleration;

${c}_{c}$ is the critical damping coefficient;

$\varphi $ is the dimensionless response acceleration, which is relative to the dimensionless input

$\alpha $;

${T}_{n}$ represents the undamped natural period of the accelerometer; and

$R$ stands for the ratio of the undamped natural period of the accelerometer to the pulse duration of the excitation acceleration.

As indicated by Equation (2), when the pulse duration of the excitation acceleration is much larger than the undamped natural period of the accelerometer, namely $R\to 0$, the first two terms of the equation can be ignored and, therefore, $\varphi \approx \alpha $; when the pulse duration of the excitation acceleration decreases and $R$ increases, the first two terms will be influenced. The first term will cause the oscillation of $\alpha $ and the second term will lead to a time lag.

Assume the input signal of the accelerometer as:

Then, the accelerometer’s response to the half-sine excitation signal can be expressed as:

When

$0\le t\le \tau $, the response can be expressed as:

By solving Equation (5), we could obtain:

where

${a}_{1}=\frac{A{\omega}_{n}^{2}{d}_{1}}{{d}_{1}^{2}+{d}_{2}^{2}}$,

${b}_{1}=-\frac{A{\omega}_{n}^{2}{d}_{2}}{{d}_{1}^{2}+{d}_{2}^{2}}$,

${c}_{1}=-\frac{A{\omega}_{n}^{2}{d}_{1}}{{d}_{1}^{2}+{d}_{2}^{2}}$,

${c}_{2}=-\frac{A{\omega}_{n}^{2}(\zeta {\omega}_{n}{d}_{1}-\frac{\pi}{\tau}{d}_{2})}{{\omega}_{n}\sqrt{1-{\zeta}^{2}}({d}_{1}^{2}+{d}_{2}^{2})}$,

${d}_{1}=\frac{2\pi \zeta {\omega}_{n}}{\tau}$,

${d}_{2}={\omega}_{n}^{2}-{\left(\frac{\pi}{\tau}\right)}^{2}$.

Under circumstances when damping is not considered, Equation (6) can be simplified as:

where

${a}_{1}=0$,

${b}_{1}=\frac{{\omega}_{n}^{2}}{{\omega}_{n}^{2}-{(\frac{\pi}{\tau})}^{2}}$,

${c}_{1}=0$,

${c}_{2}=\frac{{\omega}_{n}\frac{\pi}{\tau}}{{\omega}_{n}^{2}-{\left(\frac{\pi}{\tau}\right)}^{2}}$.

By simplifying Equation (7), the following equation can be obtained:

Assume the response error as

$\epsilon $, and the system’s resonant oscillation superimposed on the response waveform amplitude is less than

$\epsilon $. Then:

As indicated by Equation (10):

If

${\omega}_{n}=2\pi {f}_{x}$ and

${f}_{x}$ denotes the resonant frequency of the accelerometer, then:

Equation (12) is the formula to solve the minimum width of the excitation pulse required by the quasi-static calibration of high-g accelerometers with different resonant frequencies. When

$\tau $, the pulse width of the excitation signal, is greater than the minimum width determined by Equation (12), the calibration error of the accelerometer is within the given range (

Table 1).