Optimal Design of Omnidirectional Tilt Sensors Based on Magnetic-Gravity Coupled Pendulous Equilibrium

doi:10.3969/j.issn.1004-132X.2026.03.005

Abstract

Abstract:

To address the issues of limited measurement directions and complex structures in existing magnetic-effect tilt sensors， a suspended， gravity-balanced omnidirectional inclinometer was proposed based on a magneto-gravitational coupling design. Finite element simulations were used to determine the optimized engineering parameters for the permanent magnet dimensions and the detection air gap. Under small perturbations， the natural restoring process was analyzed to evaluate the influences of ball material properties and contact surface geometry on the contact behavior， which guided the optimization of the spherical crown concave structures and ball material selection. Calibration experiments were conducted to verify the accuracy of the simulation models. Through systematic characterization， mapping models were established between the X、Y -axis magnetic field signals and displacement， and between displacement and inclination angle. Additionally， an analytical model was constructed between the Z-axis magnetic field and the inclination angle. The combination of these models enabled simultaneous estimation of inclination angle $θ$ and azimuth angle $φ$ . To fully exploit the response characteristics of each magnetic field component across different tilt angle ranges， a segmented tilt angle calculation strategy was proposed. Validation experiments show that within the measurement range of -20° to 20°， the maximum tilt angle error under various azimuth angles does not exceed ±0.3°.

Key words: tilt sensor, omnidirectional tilt measurement, four-spherical-crown concave pair, magnetic-gravitational coupling

摘要：

现有磁效应式倾角传感器测量方向受限、结构复杂，为此提出了一种基于磁-重力耦合设计的悬垂式重力平衡全向倾角传感器。通过有限元仿真确定了永磁体尺寸与检测气隙的工程优化参数；分析了系统小扰动下自然回归过程中滚珠材料和接触面几何参数对接触特性的影响，优化了球冠凹面尺寸设计与滚珠材质选型。在校准实验中，验证了仿真模型的有效性。通过标定实验建立X、Y双轴磁场信号与位移、位移与角度的映射模型，以及Z方向磁场信号与倾角的解析模型，两者结合可实现倾角 $θ$ 与方位角 $φ$ 的同步解析。为充分利用各磁场分量在不同倾角区间的响应特性，提出分段式倾角解算策略。验证实验表明，该传感器在测量范围-20°~20°内，不同方位角下最大倾角测量误差不超过±0.3°。

关键词: 倾角传感器, 全向倾角测量, 四球冠凹面副, 磁-重力耦合

CLC Number:

TH712

WANG Zhongkai, XIE Tao, WANG Hongliang, HU Fangmin. Optimal Design of Omnidirectional Tilt Sensors Based on Magnetic-Gravity Coupled Pendulous Equilibrium[J]. China Mechanical Engineering, 2026, 37(3): 555-563.

王仲凯, 谢涛, 王洪亮, 胡方敏. 磁-重力耦合悬垂平衡的全向倾角传感器优化设计[J]. 中国机械工程, 2026, 37(3): 555-563.

Figures/Tables 15

Fig.1 Structural model of the sensor

Fig.2 Section view of four-spherical-crown concave pair

Fig.3 Simulation results under the condition of 2 mm diameter permanent magnet and 3 mm air gap

Fig.4 By response versus tilt angle for different permanent magnet diameters under different gap conditions

Fig.5 Measurement principle of inclination sensor

Tab.1 Key parameters of ball material

名称	材料	弹性模量/MPa	泊松比	密度/（kg·m^-3）
滚珠1	Al	7100	0.33	2700
滚珠2	PP	1370	0.42	910
滚珠3	PTFE	500	0.40	2200

Tab.2 Simulation results of stress

	球冠高度/mm	最大摩擦应力/kPa	平均摩擦应力/kPa	最大等效应力/kPa	平均等效应力/kPa
PTFE	0.2	0.086 97	0.011 87	8.619	1.295
	0.3	0.116 49	0.008 95	5.422	1.211
	0.4	0.297 07	0.042 90	3.799	1.061
PP	0.2	0.097 82	0.013 28	8.296	1.247
	0.3	0.133 76	0.010 37	5.240	1.170
	0.4	0.290 71	0.042 67	3.594	1.024
Al	0.2	0.097 83	0.013 17	8.753	1.315
	0.3	0.126 15	0.010 08	5.509	1.230
	0.4	0.372 28	0.056 87	3.780	1.076

Fig.6 Frictional stress contour plots of PTFE balls on 4 contact surfaces under different spherical cap heights

Fig.7 Calibration device for inclinometer

Fig.8 Calibration experimental results

Fig.9 Calibration results in X direction and Y direction

Tab.3 Related indicators of the calibration results in X direction and Y direction

方向

线性度/

重复性/

迟滞性/

灵敏度/

（µT·（°）^-1）

Fig.10 Bz average value and inclination angle polynomial fitting

Fig.11 Verification experiment

Fig.12 Maximum errors of validation experiment

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