## Yongle Lu , Zhen Qu , Jie Yang , Wenxin Wang , Wenbo Wang and Yu Liu## |

Parameter | Symbol | Value (mm) | |
---|---|---|---|

Supporting beam | Length | [TeX:] $$l_1$$ | 350 |

Width | [TeX:] $$b_1$$ | 550 | |

Height | [TeX:] $$h_1$$ | 100 | |

Tiny sensing beam | Length | [TeX:] $$l_2$$ | 40 |

Width | [TeX:] $$b_2$$ | 55 | |

Height | [TeX:] $$h_2$$ | 10 | |

Mass | Length | [TeX:] $$l_3$$ | 280 |

Height | [TeX:] $$h_3$$ | 200 | |

Supporting frame | Outside length | [TeX:] $$c_1$$ | 800 |

Inside length | [TeX:] $$c_2$$ | 600 | |

Height | [TeX:] $$h_4$$ | 110 |

Under the acceleration from -220,000 g to 220,000 g along +Z axis, the results of maximum deformation and the maximum von-Mises stress are showed in Fig. 4 during the [TeX:] $$\pm 10 \%$$ of measuring range of the accelerometer. The maximum deformation is [TeX:] $$0.57 \mu \mathrm{m}$$ when external acceleration is 220,000 g, which is less than the distance between the proof masses and the bottom floor. The maximum von-Mises stress is 147.85 MPa when external acceleration is [TeX:] $$\pm 220,000 \mathrm{~g}$$, which is less than permissible stress of silicon [22]. So, the measuring range of the accelerometer can reach 200,000 g.

Under the acceleration of 200,000 g along the axis +Z, the theoretical results according to Eq. (7) and the simulation results of maximum deformation from the point "M" ([TeX:] $$0 \mu \mathrm{m}$$ of horizontal axis) to the point "K" ([TeX:] $$730 \mu \mathrm{m}$$ of horizontal axis) are show in Fig. 5.

Resistors are on the surface of tiny sensing beams. The lengths of four resistors are placed in the direction X ([110] crystal orientation of p-type silicon) [23], distributed on the central position of four tiny sensing beams, respectively (as shown in Fig. 6(a)). The size of the resistor is [TeX:] $$40 \mu \mathrm{m} \times 10 \mu \mathrm{m} .$$ Wheatstone bridge circuit is adopted as the induction circuit of the accelerometer (as shown in Fig. 6(b)), with 5 V constant voltage source as the input signal [TeX:] $$U_{i n}$$.

When the accelerometer structure is not affected by acceleration, the tiny beams will not be out of shape. So, the resistance values of resistors will not change and be equal, and the Wheatstone bridge circuit has no output [24]. When the external acceleration input, the stress of [TeX:] $$R_1 \text { and } R_3$$ are equal, the stress of [TeX:] $$R_2 \text { and } R_4$$ are equal, so the output can be obtained by calculating the average stress of the resistors [TeX:] $$R_1 \text { and } R_2$$ [25].

where, [TeX:] $$U_{o u t} \text { and } U_{i n}$$ are respectively, the output and input voltage of Wheatstone bridge circuit; [TeX:] $$\sigma_{Z X 1}$$ and [TeX:] $$\sigma_{Z X 2}$$, respectively, represent the X-axis stress of the resistors [TeX:] $$R_1 \text { and } R_2$$, [TeX:] $$\sigma_{Z Y 1} \text { and } \sigma_{Z Y 2}$$, respectively represent the Y-axis stress of the resistors [TeX:] $$R_1 \text { and } R_2$$ when external acceleration is along Z-axis. [TeX:] $$\pi_{44}$$ is a piezoresistive coefficient of p-type silicon at room temperature and the value is [TeX:] $$138.1 \times 10^{-11} \mathrm{~m}^2 / \mathrm{N}$$ [26].

Different output under different acceleration can be calculated by Eq. (14) and the fitting result is in Fig. 8(c), so the sensitivity of the piezoresistive high-g accelerometer is [TeX:] $$0.88 \mu \mathrm{V} / \mathrm{g} \text {. }$$

When external acceleration along the axis X, the sensitivity along the axis X [TeX:] $$\left(S_X\right)$$ is as follows:

where, [TeX:] $$a_x$$ is acceleration along the axis X; [TeX:] $$\sigma_{X Y 1} \text { and } \sigma_{X Y 2}$$ respectively represent the Y-axis stress of the resistors [TeX:] $$R_1 \text { and } R_2$$, [TeX:] $$\sigma_{X Z 1} \text { and } \sigma_{X Z 2}$$, respectively, represent the Z-axis stress of the resistors [TeX:] $$R_1 \text { and } R_2$$ when external acceleration is along X-axis.

When external acceleration along the axis Y, the sensitivity along the axis Y [TeX:] $$\left(S_Y\right)$$ is as follows:

where, [TeX:] $$a_Y$$ is acceleration along the axis Y; [TeX:] $$\sigma_{Y X 1} \text { and } \sigma_{Y X 2}$$, respectively, represent the X-axis stress of the resistors [TeX:] $$R_1 \text { and } R_2$$, [TeX:] $$\sigma_{Y Z 1} \text { and } \sigma_{Y Z 2}$$, respectively, represent the Z-axis stress of the resistors [TeX:] $$R_1 \text { and } R_2$$ when external acceleration is along Y-axis.

Cross-axis sensitivity is expressed by the ratio of the sensitivity of the axis X/Y to the sensitivity of the axis Z, so the cross-axis sensitivity of the axis X [TeX:] $$\left(S_{XZ}\right)$$ and the cross-axis sensitivity of the axis Y [TeX:] $$\left(S_{YZ}\right)$$ are as follows:

Fig. 7 shows a common accelerometer structure for comparison. The sensitivity of the axis X, the axis Y and the axis Z of our proposed structure and this comparison structure are shown in Fig. 8.

Mechanical systems have many vibrating modes and corresponding resonant frequencies. By observing the vibrating modes and the resonant frequency of the structure through modal analysis, it is possible to predict whether the resonant frequency of the structure is suitable for high vibration conditions. The first order resonant frequency is natural frequency, other resonant frequencies are higher and higher, so the first to fourth order conditions are often considered for prediction of performance [15]. The design of the high-g accelerometer should have a high natural frequency. Other resonant frequencies should be much higher than the natural frequency, and the first order vibrating mode and second order vibrating mode should be different, so as to ensure that the designed high-g accelerometer can work normally in a vibration environment. In modal analysis, the end of the support beams is fixed, the first-four vibrating modes of the accelerometer sensitive structure are shown in Fig. 9, whose resonant frequencies are shown in Table 2.

Table 2.

The first order | The second order | The third order | The fourth order | |
---|---|---|---|---|

Resonant freq. (kHz) | 453.6 | 906.9 | 907 | 1,460.5 |

In Fig. 9, the first order vibrating mode is characterized with the proof masses perpendicular to the XOY plane and reciprocating along the axis Z, which is the main mode with the natural frequency of 453.6 kHz. If the structure generates vibrations close to the natural frequency, the center of the structure (the red area in Fig. 9(a)) is most vulnerable. The second order and the third order vibrating mode have similar resonant shapes and resonant frequencies. Their resonant frequencies are more than twice of the natural frequency, which is conducive to reducing the mode interference [27]. Last, the fourth order vibrating mode is characterized with two proof masses on one side of the YOZ plane and two proof masses on the other side rotating clockwise and counterclockwise about the axis Y, respectively.

Fig. 10 shows the natural frequency with different widths and lengths of supporting beam, which is consistent with Eq. (12) that shorter and wider supporting beam can improve natural frequency. Since the length of the tiny sensing beam is equal to the width of the cross gap, and in order to avoid the influence of the width of the tiny sensing beam on the position of the resistor, Fig. 11 shows that the thickness of tiny sensing beam has minor effect on natural frequency, which is consistent with Eq. (12), but thicker tiny sensing beam can improve sensitivity. So, combined with the results obtained in Section 4 that the width of supporting beam has minor effect on sensitivity, then the accelerometer structure can weaken the compromise between natural frequency and sensitivity.

Fig. 11 shows the simulation and theoretical results of natural frequency with different widths of supporting beam. The theoretical results of natural frequency can be calculated according to Eq. (12). The simulation results agree with the theoretical ones with 5% bias, which is the average of biases per width sample—it is in total fourteen width samples in Fig. 12. The bias mainly comes from the deviation between the selected mode function and the actual mode function. Since Rayleigh method is an approximate estimation and simplifies the mode function to the expression of sine function, the selected mode function cannot be exactly the same as the actual mode function.

To test the dynamic characteristics of the accelerometer, we carry out an impact simulation. Ansys Workbench LS-DYNA is usually used to solve the time related problem, such as structural collision and impact [28].

In LS-DYNA, 200,000 g impact is given to the accelerometer surface. The impact waveform is a half sine wave, whose duration is [TeX:] $$20 \mu \mathrm{s} .$$ And observation time is [TeX:] $$40 \mu \mathrm{s}$$ after impact. Fig. 13(a) shows that the stress mainly focused on the tiny sensing beams. Fig. 13(b) shows that the tiny sensing beams take 118.42 MPa impact, less than permissible stress of silicon. There is vibration after impact. The results show that the piezoresistive high-g accelerometer has strong resistance to impact.

Table 3 summarizes the comparison of the proposed piezoresistive high-g accelerometer with existing devices [3,15,16,18,29]. The piezoresistive high-g accelerometer with the proposed structure has higher natural frequency and lower transverse effect. Only the structure is optimized, so this accelerometer can be produced by using silicon and basic processing. It is suitable for mass production. However, compared with triaxial accelerometers, there are some limitations when using proposed Z-aixs accelerometer. Using three single-axis devices to measure three vectors may increase the volume and complexity of the measurement system.

Table 3.

Structure | Sensitivity | Natural frequency | Transverse effect, X/Y (%) | Work axis | Materials |
---|---|---|---|---|---|

A single-cantilever structure and two dual-cantilever [3] | [TeX:] $$0.80-0.88 \mu \mathrm{V} / \mathrm{g} / 5 \mathrm{~V}(\mathrm{X} / \mathrm{Y}), 1.36 \mu \mathrm{V} / \mathrm{g} / 5 \mathrm{~V}(\mathrm{Z})$$ | 134.7 kHz | - | Three | Silicon |

Eight supporting beams and four sensing beams [18] | [TeX:] $$4.15 \mu \mathrm{V} / \mathrm{g} / \mathrm{V}$$ | 5,607 Hz | 1.67/0.82 | Single | Silicon |

Our-beams and central-island mass [15] | [TeX:] $$0.5611 \mu \mathrm{V} / \mathrm{g}$$ | 408.19 kHz | - | Single | Silicon |

Four self-supporting micro beams [16] | [TeX:] $$1.586 \mu \mathrm{V} / \mathrm{g} / 3 \mathrm{~V}$$ | 445 kHz | - | Single | SOI |

Eight beams (EBs) [29] | [TeX:] $$131.4 \mu \mathrm{V} / \mathrm{g} / \mathrm{V}$$ | 14.8 kHz | 1.04/0.76 | Single | Silicon |

Proof masses with a cross gap | [TeX:] $$0.88 \mu \mathrm{V} / \mathrm{g} / 5 \mathrm{~V}$$ | 453.6 kHz | 0.25/0.11 | Single | Silicon |

To improve natural frequency and to reduce transverse effect of piezoresistive high-g accelerometer, we introduce the design and simulation of a piezoresistive high-g accelerometer. Special-shaped proof masses system with a cross gap can shorten the distance between the gravity center of the proof system and the neutral layer of supporting beams to restrain the transverse effect. Four tiny sensing beams are bonded above the cross gap, which can weaken the compromise between natural frequency and sensitivity. The simulation results show that the natural frequency of the piezoresistive high-g accelerometer is 453.6 kHz and the sensitivity is [TeX:] $$0.88 \mu \mathrm{V} / \mathrm{g} \text {. }$$ The cross-axis sensitivity of the axis X axis and the axis Y is respectively, 0.25% and 0.11%. Contrastively, the cross-axis sensitivity of X-axis and Y-axis is respectively, reduced by 93.2% and 96.9%. It has a higher natural frequency and a lower transverse effect and the results of impact simulation meet the requirements. Our research provides a support for the application of design and fabrication of piezoresistive high-g accelerometers.

He received M.S. degree from Chongqing University of posts and telecommunications in 2012. He received the Ph.D. degree in instrument science and technology at Chongqing University in 2015. He is associate professor at Chongqing University of Posts and Telecommunications and a director of Chinese Institute of Inertial Technology. His research interests include solid state vibration gyro and photoelectric sensing technology.

He received B.S. degree in integrated circuit design and integrated system from Chongqing University of Posts and Telecommunications in 2021. Since September 2021, he is purchasing M.S. degree in School of Optoelectronic Engineering of Chongqing University of Posts and Telecommunications. His current research interests include inertial components and system, navigation and positioning.

He received B.S. degree in integrated circuit design and integrated system from Chongqing University of Posts and Telecommunications in 2019. Since September 2019, he is purchasing M.S. degree in School of Optoelectronic Engineering of Chongqing University of Posts and Telecommunications. His current research interests include inertial components and system, navigation and positioning.

He received B.S. degree in electronic science and technology from Chaohu college in 2018. Since September 2019, he is purchasing M.S. degree in School of Optoelectronic Engineering of Chongqing University of Posts and Telecommunications. His current research interests include quantum optics and optical fiber sensing.

He received M.S. and Ph.D. degrees in Chongqing University in 2000 and 2006, respectively. He is a professor and doctoral supervisor at Chongqing University of Posts and Telecommunications. He has been researching sensor parts and systems, motion detection, attitude measurement, navigation and positioning for a long time.

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