# Research of giant magnetostrictive actuator’s nonlinear dynamic behaviours

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## Abstract

The multi-coupled nonlinear factors existing in the giant magnetostrictive actuator (GMA) have a serious impact on its output characteristics. If the structural parameters are not properly designed, it is easy to fall into the nonlinear instability, which has seriously hindered its application in many important fields. The electric–magnetic-machine coupled dynamic mathematical model for GMA is established according to J-A dynamic hysteresis model, ampere circuit law, nonlinear quadratic domain model and structure dynamics equation. Nonlinear dynamic analysis method is applied to study the nonlinear dynamic behaviour of the key structure parameters to reveal their influence on the system stability. The design principle of structural parameters is obtained by studying stability of GMA, which provides theoretical basis and technical support for the structural stability design.

## Keywords

GMA J-A model Nonlinear dynamic behaviours Stability## 1 Introduction

## 2 GMA model

*F*is the output force of GMM rod, \(M_{e}\) is the equivalent mass, \(C_{e}\) is the equivalent damping coefficient, \(F_{z}\) is the restoring force of disc spring, whose computing method is in appendix A,

*S*is the GMM rod’s cross-sectional area,

*x*is the output displacement of GMA, \( M_{M}\) is the mass of GMM rod, \( M_{L }\) is mass of the load.

*L*is the length of GMM rod.

*E*is GMM’s elasticity modulus,

*M*is the magnetization intensity, which can be achieved by magnetic field intensity

*H*. The relationship between

*H*and

*M*is described in appendix A.

*H*is

*I*as input and displacement

*x*as output.

*M*(

*I*) is the magnetization when input current is

*I*.

## 3 Research of nonlinear dynamic behaviour in GMA

The effects of sensitive parameters on the system stability are revealed by studying the dynamics characteristics for \(\varphi _1 \), \(\varphi _2 \) and \(f_\varphi \), which provide the theoretical foundation for GMA structure design based on stability.

### 3.1 Research on the damping coefficient

Taking the dimensionless damping coefficient \(\varphi _1 \) as the reference variable, the bifurcation diagram is shown in Fig. 2 when \(\Omega =3\). The response characteristics are shown in Fig. 3 when \(\varphi _1 =10^{-4}\). Spectrum contains multi-frequency, Poincaré mapping is the distribution of random points in the global scope, phase diagram is composed of many circular curves without overlapping, and time-domain waveform is messy without periodicity, which indicates that system is in chaotic state. Figure 4 shows the response characteristics when \(\varphi _1 =0.15\). Spectrum shows that the main frequency is the integer times of 1/2 \(\Omega \) and Poincaré mapping only has two points, which indicates that the system goes into period-doubling bifurcation. Figure 5 shows that the system is in the stable periodic motion when \(\varphi _1 =0.25\).

### 3.2 Research on the square stiffness coefficient

Taking the dimensionless square stiffness coefficient \(\varphi _2 \) as the variable, the nonlinear dynamic behaviour characteristics of the GMA are studied when \(\Omega =2\) and \(\Omega =3\). It can be seen from Fig. 6 that the system maintains the same bifurcation characteristic when \(\varphi _2 \) varies from 0 to 1 under different \(\Omega \). Therefore, \(\varphi _2 \) has nothing to do with the dynamic behaviour in the GMA.

### 3.3 Research on the coupling stiffness coefficient

## 4 Parameters design based on the stability

It is concluded from Sect. 3 that the larger \(\varphi _1 \) and smaller \(f_\varphi \) can improve the stability according to nonlinear dynamic behaviour characteristics of GMA. According to \(\varphi _1 =C_e /\sqrt{M_e K_{e1} }\), larger \(C_e \) and smaller \(K_{e1} \) can increase \(\varphi _1 \). But, smaller \(K_{e1} \) leads to smaller natural frequency \(\omega _0 =\sqrt{K_{e1} /M_e }\). According to \(f_\varphi =\sqrt{K_{e3} /K_{_{e1} }^3 }\), increasing \(K_{e1} \) and decreasing \(K_{e3} \) can decrease \(f_\varphi \).Therefore, increasing \(C_e \), \(K_{e1} \) and decreasing \(K_{e3} \) can improve the system stability.

- 1.
Increasing the GMM rod’s

*S*/*L*will increase \(K_{e1} \) and enhance the stability. So, slender GMM rod is not favourable for stability. - 2.
The coupling stiffness coefficient \(f_\varphi \) of A, B, C disc springs increases successively in the same diameter, and A disc spring has higher stability.

- 3.
Increasing the diameter

*D*of disc spring can increase \(K_{e1} \) and decrease \(K_{e3} \), which can enhance the system stability. - 4.
The larger number of disc springs in overlap can reduce \(f_\varphi \) and improve stability, while the result of involution is just the opposite.

## 5 Test verification

- 1.
The quiet working environment must be ensured, and any outside noise may affect the results.

- 2.
The laser beam should be perpendicular to the cross section of output shaft in GMA. Otherwise, it will affect the measurement result.

- 3.
Temperature control system should keep working in the experiment to maintain constant working temperature for GMM rod.

**1. Mathematical model verification**

Figure 10 is the simulation and experiment curve of GMA in different frequency, loads, combination methods, series of disc springs and minor loop. The good fitting proves the correctness and validity of the GMA model.

**2. Stability verification**

## 6 Conclusion

- 1.
Adding structural factors to the GMA mathematical model can effectively improve the accuracy of the model.

- 2.
It is possible to cause the system to fall into instability with lower structure rigidity and damping.

- 3.
Disc spring plays an important role in stability for GMA. The larger diameter

*D*, higher number of \(n_{\mathrm{c}}\) in overlap and A disc spring can effectively improve the stability

## Notes

### Acknowledgements

This work was supported by the National Natural Science Foundation of China (11272026).

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