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An angular acceleration sensor can be used for the dynamic analysis of human and joint motions. In this paper, an angular acceleration sensor with novel structure based on the principle of electromagnetic induction is designed. The method involves the construction of a constant magnetic field by the excitation windings of sensor, and the cup-shaped rotor that cut the magnetic field. The output windings of the sensor generate an electromotive force, which is directly proportional to the angular acceleration through the electromagnetic coupling when the rotor has rotational angular acceleration. The mechanical structure and the magnetic working circuit of the sensor are described. The output properties and the mathematical model including the transfer function and state-space model of the sensor are established. The asymptotical stability of the sensor when it is working is verified by the Lyapunov Theorem. An angular acceleration calibration device based on the torsional pendulum principle is designed. The method involves the coaxial connection of the angular acceleration sensor, torsion pendulum and a high-precision angle sensor, and then an initial external force is applied to the torsion pendulum to produce a periodic damping angle oscillation. The angular acceleration sensor and the angle sensor will generate two corresponding electrical signals. The sensitivity coefficient of the angular acceleration sensor can be obtained after processing these two-channel signals. The experiment results show that the sensitivity coefficient of the sensor is about 17.29 mv/Krad·s^{2}. Finally, the errors existing in the practical applications of the sensor are discussed and the corresponding improvement measures are proposed to provide effective technical support for the practical promotion of the novel sensor.

Angular acceleration is a common physical quantity in a rotating system, which can reflect the vibrational state of the rotation angle for the rotation shaft and transmission equipment while working. The responses of the rotation shaft and transmission equipment to various stimulations can be analyzed by measuring the angular acceleration, and especially the dynamic interference of rotary systems can be displayed in the form of angular acceleration. The measurement of rotating mechanical angular acceleration has wide applications, including the fields of automobiles, the military, aerospace, industry, electronics,

Angular acceleration can be measured indirectly by employing rotation angle sensors or angular speed sensors. However, the signal processing of these methods are very troublesome. Especially, the problems of delay characteristics and noise amplification are hard to resolve, which has prompted the development of direct angular acceleration measurement techniques. The applications of linear acceleration sensor are very wide [

In recent years, some experts and scholars have conducted a series of researches on angular acceleration measurement. Moody [^{2} and a bandwidth of 50 Hz was demonstrated [

In this paper, an angular acceleration sensor without the range limitation of the rotation angle is proposed to measure the instantaneous angular acceleration of the rotating system directly, based on the principle of electromagnetic induction. In addition, the installation of the sensor is very convenient, as the sensor just needs to be coaxially connected with the rotating system when it is used. The output signal of the sensor can be led directly without the slip ring to avoid the weakening of the signal. This paper describes the mechanical structure and working principle of this angular acceleration sensor, and constructs the mathematical model of the sensor, including the transfer function and state-space model. The asymptotical stability of the sensor is verified by the Lyapunov theorem when the sensor is working. Finally, the sensor calibration is demonstrated, and the probable existing errors and their impact on the applications of the sensor are analyzed.

The 3D assembly diagram of the angular acceleration sensor based on electromagnetic induction is shown in

The magnetic circuit is shown in _{1} assuming the magnetic circuit is symmetrical, the magnetic flux Φ_{1} of the magnetic circuit can be expressed by the following equation according to the magnetic circuit theorem [

In which:
_{1}
The effective windings number of the excitation windings
_{1}
The effective length through the air-gap Φ_{1}
_{1}
The effective area through the air-gap Φ_{1}
_{0}
The magnetic permeability of the air
_{R}_{R}_{R1}
The effective area through the cup-shaped rotor Φ_{1}
_{fi}_{1}
_{fi}_{1}
_{f}_{ti}_{1}
_{ti}_{1}
_{i}

The cup-shaped rotor can be seen as a number of squirrel-cage rotors. When the magnetic flux is cut counterclockwise by the cup-shaped rotor with a speed of _{e}_{R}_{R}_{R}

Similarly, the magnetic flux Φ_{R}_{R}

In which:
_{R}_{R}_{R}_{R}_{R}_{R2}
The effective area through the cup-shaped rotor of the magnetic flux Φ_{R}_{Fi}_{R}_{Fi}_{R}_{Ti}_{R}_{Ti}_{R}

The total effective windings of the output windings are _{2}. According to the principle of electromagnetic induction, the induction electromotive force in the output windings induced by the magnetic flux Φ_{R}

By the combination of Equations (

From the above equation, we can find that the induction electromotive force of output windings _{o}

The equivalent circuit of the working sensor is shown in

In _{1}
The resistance of excitation windings
_{R}_{R}_{R}_{R}_{o}_{2}
The current of output windings
_{x}_{2}
The resistance of output windings

For the cup-shaped rotor, the voltage balance equation is:

For the output windings, the voltage balance equation is:

The output voltage of sensor, namely the voltage across load is:

The Laplace transform is demonstrated for the Equations (

It is seen that the intrinsic oscillation frequency (cutoff frequency) is
_{x}_{1}, that is

Appropriate state variables are selected, the diagonal form of the system state-space model can be obtained through the transfer function of the system (10):

The corresponding state variables scheme is shown in

The stability of the system can be divided into external stability and internal stability. The external stability is defined as the influence of the external input on the response of the system at the initial state, which can be judged according to the equation root situation of the closed-loop characteristic equation for the system. The internal stability is only related to the impact of the system itself at the initial state, also called Lyapunov asymptotical stability.

Assuming

Owing to _{1} > 0 and

The calibration equipment of the sensor is shown in

The bracket is used to fix angular acceleration sensor ➁ and high-precision angle sensor ➇; The coupling ➂ is used for the coaxial connection of the angular acceleration sensor ➁ and high-precision angle sensor ➇; The mass ➃ is a self-made iron hoop with holes and has variable quality for selection and combination. The moment of inertia for the calibration system can be changed to realize calibration at different frequencies through selecting and combining masses; the combination of a large mass and a small mass is shown in the Figure; the coil spring ➄, the torsion bar ➅ and the base ➆ form a torsional pendulum, which is the power source of torsional vibration for the entire system. The torsion bar ➅ passes the mass and connects with the angular acceleration sensor ➁ through the upper end of the coupling ➃, and its bottom end is connected with the end of the angle sensor ➇; The dual channel input of the digital oscilloscope ➈ connects with the angular acceleration sensor and the angle sensor, respectively, for the acquisition and preservation of the output waveform of the two sensors during the experiments.

An initial force is applied to the torsion bar ➅, resulting in the coil spring ➄ rotating an angle _{1}_{s}

Assuming
_{n}

The solution for above equation is
_{0} is the initial amplitude of the pivot angle,
_{0} and the amplitude _{N}

The rotation cycle of the system is

The DC resistance of the excitation windings for the angular acceleration sensor is 838 Ω. After a long-time test at the working temperature, excitation windings voltage _{1} = 40 V is selected for the angular acceleration sensor. In this case, the temperature of the sensor is about 25 to 28 Celsius degrees; the initial angle is about 180° when the external force is applied on the coil spring, the output signals of angular acceleration sensor and the angle sensor are shown in

Matlab is used to fit the data in ^{2}).

There will exist a phase difference when the angular acceleration sensor and the angle sensor are working in practice. From the calibration principle, we can see that the peak value of the angular acceleration is the corresponding amplitude position of the pendulum. Therefore, the peak voltages of the two sensors are needed in the calibration process and the phase difference shows little impact on the calibration results. Additionally, the aerostatic bearing can be considered to reduce the influence of bearing friction on calibration results; the calibration range of the system is limited and only suitable for the calibration of the angular acceleration in the lower frequency owing to the constraint of the mechanical structure and the limitation of variation range, even though the calibration frequency can be changed by the mass to a certain extent; the obtained calibration frequency is not continuous because the calibration system depends on the mass.

In order to solve the issues of non-continuous calibration point and lower frequency band, an auxiliary power source can be added to the electromagnetic force to realize a forced vibration state when the calibration equipment is working. The linear regulator of the vibration frequency can be realized by changing the rotational speed of the power source, resulting in the extension of the calibration frequency. These improvement ideas will be demonstrated in our future experiments.

From the sensor principle, it is recognized that magnetic flux Φ_{1} generated by the excitation current includes the variation component ΔΦ_{1} if the excitation voltage has fluctuations. The cup-shaped rotor cuts the magnetic flux Φ_{1} with a constant rotation speed, so its motional electromotive force _{R}_{R}_{R}_{R}_{R}_{1}.

The properties of the excitation windings, the output windings, the resistance of the cup-shaped rotor and the magnetic material may change because of the change in ambient temperature and the heating from extended use of the sensor. These property changes show influences on the output induced electromotive force of the sensor, resulting in the instability of the output performance. The character of the temperature-sensitivity coefficient of the sensor is shown in

In order to decrease the impact of temperature, temperature compensation should be applied. A negative temperature coefficient thermistor can be concatenated in the excitation and the output circuit to compensate the temperature variation influence.

The cup-shaped rotor of actual sensor is a first-order circuit composed of a resistance and inductance, so the change of the current _{R}_{R}_{R}_{R}_{R}_{R}_{1}, causing the change of the magnetic flux Φ_{1}. As a result, linear relationships of the output potential and the angular acceleration are destroyed, resulting in the presence of a non-linear error.

The angular acceleration sensor presented in this paper shows strong practicality without limitation on the rotation angle range. It only needs to connect with the measured rotating system coaxially when working. The structure and principle of the calibration equipment are described, and the calibration curves and performance indicators of the sensor are obtained by experiments. In our following work efforts will be made to examine the practical applications for this sensor in view of the extensive applications of angular acceleration sensors, such as transmission noise measurement and fault diagnosis for complex gearboxes or the internal combustion engine, fuel injection timing and fuel injection amount control for automotive engines, as well as the kinetic analysis for mechanical arm joints and so on.

The work was sponsored in part by the Jiaxing Science and Technology Research Project of China under Grant (2012AY1021) and in part by the Zhejiang Provincial Department of Education Scientific Research Project of China under Grant (Y201226082) and in part by the Zhejiang Province Major Science and Technology Project of China under Grant (2008C01011-4) and in part by the Jiaxing University Self-made Experimental Teaching Instrument and Equipment Research Project (ZS201202) and in part by the Jiaxing University Nanhu College Key Scientific Research Project.

The authors declare no conflict of interest.

3D assembly diagram of the angular acceleration sensor.

3D sectional view of the sensor.

Sectional sketch view of the sensor.

The magnetic circuit of Φ_{1}.

The magnetic circuit of Φ_{R}

The equivalent circuit of the working sensor.

The state variables scheme of sensor.

The calibration equipment of angular acceleration sensor.

Output signals of angular acceleration sensor.

Output signals of angle sensor.

The de-noised signals by Rigrsure wavelet.

Calibration curve of angular acceleration sensor.

Temperature-sensitivity coefficient characteristics of the sensor.

Experimental calibration data.

The peak voltage of the angular acceleration sensor (mV) | 7.32 | 7.15 | 6.44 | 5.44 | 5.06 | 4.48 | 4.04 | 3.61 | 3.09 |

The peak voltage of angle sensor (V) | 2.30 | 2.24 | 1.90 | 1.66 | 1.50 | 1.32 | 1.18 | 1.08 | 0.94 |

The time corresponding peak (s) | 0.071 | 0.586 | 1.098 | 1.606 | 2.12 | 2.626 | 3.138 | 3.656 | 4.158 |

Oscillation period T/(s) | (0.515 + 0.512 + 0.508 + 0.494 + 0.506 + 0.512 + 0.518 + 0.502)/8 = 0.5084 | ||||||||

Angle values (rad) | 2.89 | 2.815 | 2.388 | 2.086 | 1.885 | 1.659 | 1.483 | 1.357 | 1.181 |

System damping ratio coefficient β | [ln(3.091÷1.181)] ÷8÷ | ||||||||

Angular acceleration value (rad/s^{2}) |
−441.5 | −430 | −364.8 | −318.6 | −287.9 | −253.4 | −226.5 | −207.3 | −180.4 |

Sensitivity coefficient (mv/Krad/s^{2}) |
16.58 | 16.63 | 17.65 | 17.08 | 17.58 | 17.67 | 17.83 | 17.37 | 17.23 |