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Dynamic response analysis of a motor–gear transmission system considering the rheological characteristics of magnetorheological fluid coupling

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Abstract

This study develops a method to suppress the mechanical vibrations of a motor–gear transmission system (MGTS) due to speed and external load variations. It also achieves soft starting of the drive motor using a magnetorheological fluid coupling (MRFC) with variable stiffness and damping instead of the traditional rigid coupling. Therefore, a mechanical–electromagnetic coupled dynamics model of an MGTS is established, which includes a drive motor, a gear transmission system, an MRFC, and a load. Based on the developed dynamics model, the effects of the MRFC on the dynamic response of the MGTS at different coil currents in the startup and stable operation stages of the drive motor are investigated. The results show that as the coil current increases, the coil current overshoot increases, and the overshoot duration is 0.125 s. When the current is 2.0 A, the coil current overshoot reaches a maximum, and the overshoot rate is 8.84%. Concurrently, when the coil current increases from 0.5 to 2.0 A, the magnetic field intensity in the MRF working area increases from 0.38 to 0.74 T, the torque increases from 70 to 115 N·m, and the response time of the MRFC reduces from 0.125 to 0.002 s. Moreover, the relative vibration center offset rates (RVCORs) in the x, y, and θ directions of nodes 4, 8, 10, and 15 gradually decrease with increasing coil current. However, these RVCORs reach maximum when the coil current is 2.0 A, with those in the x direction being 0.586%, 0.447%, 0.446%, and  − 0.263%, respectively, and in the y direction being 0.586%, 0.451%, 0.497%, and − 0.264%, respectively. The RVCORs of the helical gear meshing node of the MGTS in the θ direction are 0.0722%. Furthermore, the maximum vibration amplitude reduction rates (MVARRs) in the x, y, and θ directions of nodes 4, 8, 10, and 15 gradually increase with increasing coil current. The MVARRs of each node reach the maximum when the coil current is 2.0 A; the MVARRs of nodes 4, 8, 10, and 15 in the x direction are 40.98%, 83.4%, 83.49%, and 2.17%, respectively, and those in the y direction are 64.4%, 83.4%, 83.46%, and 2.16%, respectively. The MVARRs of the helical gear meshing node in the θ direction have an MVARRs of 29.1%. Moreover, the vibration amplitudes of the gear meshing node decay the fastest in the θ direction, and the decay time reduces from 2.8 to 0.3 s when the coil current increases from 0.5 to 2.0 A.

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Abbreviations

V :

Coil voltage

I :

Coil current

L :

Coil inductance

R :

Coil resistance

V 0 :

Constant voltage

t :

Time

\(\tau \) :

Shear stress of MRF

\(\eta \) :

Zero-field viscosity of MRF

\({\tau }_{(H)}\) :

Dynamic yield stress of MRF

\(\mathop \gamma \limits^{.}\) :

Shear strain rate of MRF

B :

Magnetic flux density

H :

Magnetic field intensity

µ 0 :

Vacuum magnetic permeability of MRF

µ r :

Relative magnetic permeability of MRF

R 1 :

Radius of driven cylinder

R 2 :

Radius of driving cylinder

\(\Delta \omega \) :

Speed difference between input and output

l :

Working gap length of MRF

l e :

Effective working gap length of MRF

E M :

Relative shear modulus of MRF

\(E^{\prime }\) :

Storage modulus of MRF

\(E^{\prime \prime }\) :

Loss modulus of MRF

\(\varrho \) :

Volume fraction of MRF

B s :

Magnetization intensity of MRF

λ :

Loss factor

k M :

Torsional stiffness coefficient of MRF

c M :

Torsional damping coefficient of MRF

J M :

Rotational inertia of MRFC

C c :

Critical damping coefficient of MRFC

\({\omega }_{n}\) :

Angular frequency

\({\xi }_{M}\) :

Damping ratio

K M :

Stiffness matrix of MRFC

M M :

Mass matrix of MRFC

C M :

Damping matrix of MRFC

X M :

Displacement matrix of MRFC

K s :

Stiffness matrix of shafting element

M s :

Mass matrix of shafting element

C s :

Damping matrix of shafting element

X s :

Displacement matrix of shafting element

e :

Transmission error

\(x_{j + 1} ,\;y_{j + 1} ,\;z_{j + 1}\) :

Translation displacement of nodes j + 1

\(\theta_{{x_{j} }} ,\;\theta_{{y_{j} }} ,\;\theta_{{z_{j} }}\) :

Rotational angular displacement of nodes j

\(\theta_{{x_{j + 1} }} ,\;\theta_{{x_{j + 1} }} ,\;\theta_{{x_{j + 1} }}\) :

Rotational angular displacement of nodes j + 1

\(m_{j} ,\;I_{{{\text{ij}}}}\) :

Mass and moment of inertia to nodes j

\(m_{j + 1} ,\;I_{{{\text{ij}} + 1}}\) :

Mass and moment of inertia to nodes j + 1

E :

Elastic modulus of shaft material

P :

Moment of inertia of section

ς :

Shear influence factor

a :

Length of beam element

G m :

Shear elastic modulus of shaft material

A :

Cross-sectional area of beam element

j :

Polar moment of inertia

n :

Section influence coefficient

p :

Mass scaling coefficients

q :

Stiffness scaling coefficients

\(\zeta_{1} ,\;\zeta_{2}\) :

Damping coefficient

ω 2 :

Intrinsic frequency

α :

Pressure angle

γ :

Angle (between the gear meshing line and the y-axis)

β b :

Helical angle

r 1, r 2 :

Base radiuses of driving gear and driven gear

K G :

Stiffness matrix of gear meshing element

C G :

Damping matrix of gear meshing element

X G :

Displacement matrix of gear meshing element

M G :

Mass matrix of gear meshing element

\(x_{{{\text{xi}}}} ,\;x_{{{\text{yi}}}} ,\;x_{{{\text{zi}}}}\) :

Translational displacement of the driving gear

\(\theta_{Z1} ,\;\theta_{Z2}\) :

Rotational angular displacement

\(x_{2} ,\;y_{2} ,\;z_{2}\) :

Translational displacement of the driven gear

δ :

Relative total deformation

\(x_{j} ,\;y_{j} ,\;z_{j}\) :

Translation displacement of nodes j

V G :

Meshing matrix of helical gear pair

m 1, m 2 :

Mass of the driving and driven gear

\(I_{x1} ,\;I_{y1} ,\;I_{z1} ,\;I_{x2} ,\;I_{y2} ,\;I_{z2}\) :

Rotational inertia of the driving and driven gear

f s :

Normal force of the helical gear pair

k m :

Helical gear pair meshing stiffness

c m :

Helical gear pair meshing damping

F G :

External excitation force column vector

K B :

Stiffness matrix of bearing element

C B :

Damping matrix of bearing element

M B :

Mass matrix of bearing element

X B :

Displacement matrix of bearing element

k xx, k yy :

Radial stiffness

k zz :

Axial stiffness

\(k_{\theta x\theta x} ,\;k_{\theta y\theta y}\) :

Torsional stiffness

G S :

Gyroscopic matrix of shaft element

K C :

Stiffness matrix of connecting element

C C :

Damping matrix of connecting element

M C :

Mass matrix of connecting element

X C :

Displacement matrix of connecting element

T in :

Input torque of drive motor

T load :

Load torque

M :

Mass matrix of MGTS

K :

Stiffness matrix of MGTS

C :

Damping matrix of MGTS

X(t):

Whole displacement vector of MGTS

F(t):

System external load column vector

T :

MRF transmission torque of MRFC

T L :

Load torque

Xx :

Amplitude of x-direction

Xy :

Amplitude of y-direction

X θ :

Amplitude of θ-direction

x o :

Vibration center of x-direction

y o :

Vibration center of y-direction

θ o :

Vibration center of θ-direction

\(\varepsilon \) :

Relative vibration center offset rate

x M :

Maximum vibration amplitude of x-direction

y M :

Maximum vibration amplitude of y-direction

θ M :

Maximum vibration amplitude of θ-direction

\(\phi \) :

Amplitude reduction rate

ρ :

Density of shaft material

G G :

Gyroscopic matrix of helical gear pair

\(\Omega \) :

Spin speed

\({r}_{s}\) :

Radius of gyration

G :

Whole gyroscopic matrix of MGTS

\(\Omega_{1} ,\;\Omega_{2}\) :

Rotation angular speed

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Acknowledgements

The authors would like to gratefully acknowledge the National Natural Science Foundation of China (51905060); Open Fund of the State Key Laboratory of Mechanical Transmission (SKLMT-MSKFKT-202119); China’s National Natural Science Foundation (52275052/51905064/52105245/51875068); China Postdoctoral Science Foundation (2021M700619/2022MD713697); Natural Science Foundation Project of Chongqing Science and Technology Commission (cstc2020jcyj-msxmX0346/cstc2021jcyj-msxmX0361);Science and Technology Research Program of Chongqing Municipal Education Commission (KJQN201901107).

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Authors and Affiliations

  1. Department of Mechanical Engineering, Chongqing University of Technology, Chongqing, 400054, People’s Republic of China

    Hang Gong, Ruizhi Shu, Zhengqiu Xie, Jin Huang, Rulong Tan, Qin Yin & Zheng Zou

  2. The State Key Laboratory Mechanical Transmissions, Chongqing University, Chongqing, 400054, People’s Republic of China

    Ruizhi Shu

  3. Department of Mechanics and Engineering Science, Shanghai University, Shanghai, 200444, People’s Republic of China

    Yang Xiong

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  1. Hang Gong
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Gong, H., Shu, R., Xiong, Y. et al. Dynamic response analysis of a motor–gear transmission system considering the rheological characteristics of magnetorheological fluid coupling. Nonlinear Dyn 111, 13781–13806 (2023). https://doi.org/10.1007/s11071-023-08585-6

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