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Study on the limit cycle oscillation of the drum brake non-linear vibration model at low frequency

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Abstract

A 5-degree non-linear dynamic model is presented to describe the low frequency vibration of drum brake. The centre manifold theory is applied to reduce the system at the Hopf bifurcation point. Through the calculation of normal form of the reduced system at the Hopf bifurcation point, the limit cycle oscillations (LCOs) amplitude is obtained. By this method, the effect of the drum brake parameter on LCO amplitude is studied, the law of the LCOs amplitude varying with systematic parameters is obtained.

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Authors and Affiliations

  1. School of Mechanical Engineering, Hubei University of Technology, Wuhan, Hubei, 430068, People’s Republic of China

    Minggang Zhou, Yuan Chen & Jingdong Zhou

  2. China Ship Development and Design Center, Wuhan, Hubei, 430064, People’s Republic of China

    Yong Wang

Authors
  1. Minggang Zhou
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  2. Yuan Chen
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  3. Yong Wang
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  4. Jingdong Zhou
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Corresponding author

Correspondence to Minggang Zhou.

Appendices

Appendix A: Parameter values

R = 0.18 m—Radius of shoes

R b  = 0.175 m—radius of base plate

R d  = 0.182 m—Radius of drum

R p  = 0.17 m—Distance between drum center and shoe anle

M b  = 40 kg—Equivalent mass of base plate

M s  = 3 kg—Mass of shoe

M 1 = 4 kg—-Equivalent mass of M 1

M 2 = 4 kg—Equivalent mass of M 2

α = 0.0873 rad—Angle between the axis of symmetry and OO 1

β = 0.8727 rad—Angle between the axis of symmetry and terminate of shoe

α1 = 0.1396 rad—-As shown in Fig. 2.

α2 = 1.4835 rad—As shown in Fig. 2.

K 11 = 2e7 N/m, coefficient of linear term of stiffness K 1

K 12 = 5e6 N/m2—Coefficient of quadratic term of stiffness K 1

K 13 = 5e6 N/m3—Coefficient of cubic term of stiffness K 1

K 21 = 2e7 N/m—Coefficient of linear term of stiffness K 2

K 22 = 5e6 N/m2—Coefficient of quadratic term of stiffness K 2

K 23 = 5e6 N/m3—Coefficient of cubic term of stiffness K 2

K m11 = 2e6 N/m—Coefficient of linear term of stiffness K m1

K m12 = 1e6 N/m2—Coefficient of quadratic term of stiffness K m1

K m13 = 1e6 N/m3—Coefficient of cubic term of stiffness K m1

K m21 = 2e6 N/m—Coefficient of linear term of stiffness K m2

K m22 = 1e6 N/m2—Coefficient of quadratic term of stiffness K m2

K m23 = 1e6 N/m3—Coefficient of cubic term of stiffness K m2

K b1 = 1e7 N/m—Coefficient of linear term of stiffness K b

K b2 = 8e6 N/m2—Coefficient of quadratic term of stiffness K b

K b3 = 8e6 N/m3—Coefficient of cubic term of stiffness K b

C 1 = 5 N/m/s—Coefficient of damping of first mode

C 2 = 5 N/m/s—Coefficient of damping of third mode

C m1 = 5 N/m/s—Coefficient of damping of the lining

C m2 = 5 N/m/s—Coefficient of damping of the lining

C b  = 5 N/m/s—Equivalent coefficient of damping of base plate

f = 0.3—Brake friction coefficient

F b  = 400 N—Brake force

Appendix B: Content of B 0,B 1

$$ \begin{aligned} B_0 &=\left[ {{\begin{array}{llllllllll} -2.2607& 4011.5& & & & & & & & \\ -4011.5&-2.2607& & & & & & & & \\ & & -2.4331& 4053.4& & & & & & \\ & & 4053.4& -2.433& & & & & & \\ & & & & -0.02949& 168.84& & & & \\ & & & & 168.84& -0.02949& & & & \\ & & & & & & & 219.68& & \\ & & & & & & -219.68& & & \\ & & & & & & & & -0.07957& 217.91 \\ & & & & & & & & 217.91& -0.07957 \\ \end{array} }} \right]\\ B_1 &=\left[ {{\begin{array}{llllllllll} 0.57630&-60.699&-0.12115&14.7404&0.32704&-1361.3&2.30433&-9.24361&-0.90154&-82.5647 \\ -0.000401&0.04280&-1.07{\rm e}^{-4}&9.859{\rm e}^{-3}&-0.00025&0.9474& -0.00134&-0.34390&0.00066&-0.30015\\ -0.11468&14.0842&-0.59546&62.2751&-0.15256&274.70&0.38667&-1134.41&0.29922&-1143.22\\ -8.857{\rm e}^{-5}&0.007950&0.000445&-0.04711&9.899{\rm e}^{-6}&0.20662&-0.00094&0.78288&5.616{\rm e}^{-5}&0.81024\\ 0.078012&-8.21545&-0.016831&2.04076&0.04421&-184.28&0.31252&-2.0417&-0.12195&-11.983\\ 1.301{\rm e}^{-6}&-1.578{\rm e}^{-4}& 6.130{\rm e}^{-6}&-6.407{\rm e}^{-4}&1.642{\rm e}^{-6}&-0.0031&-3.53{\rm e}^{-6}&0.01172&-3.27{\rm e}^{-6}&0.01179\\ -0.07190&8.54537&-0.28520&29.7682&-0.08321&171.69&0.12218&-549.60&0.17056&-551.80\\ 3.702{\rm e}^{-3}&-0.43693&0.013729&-1.43227&4.149{\rm e}^{-3}&-8.8351&-4.99{\rm e}^ {-3}&26.5479&-8.60{\rm e}^{-3}&26.6249\\ 0.08984&-10.691&0.36073&-37.6558&0.10458&-214.56&-0.15863&694.747&-0.2140&697.658\\ -3.686{\rm e}^{-3}&0.43504&-0.01366&1.42533&-0.00413&8.7972&4.956{\rm e}^{-3}&-26.420&8.559{\rm e}^{-3}s&-26.496\\ \end{array} }} \right]\\ \end{aligned} $$

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Zhou, M., Chen, Y., Wang, Y. et al. Study on the limit cycle oscillation of the drum brake non-linear vibration model at low frequency. Int J Mech Mater Des 4, 317–324 (2008). https://doi.org/10.1007/s10999-008-9075-1

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