Dynamic Modeling and Analysis of the Central Rigid Body-Timoshenko Beam Model Based on Unconstrained Modes
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摘要:
梁的横向变形会导致梁纵向缩短，建模过程中考虑梁横纵变形二次耦合项则存在动力刚化现象，这说明梁的纵向变形会对模型的广义刚度造成影响。对于做旋转运动的梁结构，旋转运动时还会受到离心力的作用而产生轴向拉力，轴向拉力同样也会引起梁的轴向变形，这种影响对粗短梁更加明显。以大范围运动中心刚体-Timoshenko梁模型为研究对象：首先，运用Timoshenko梁理论以及Hamilton原理建立含离心力的动力学模型；其次，引入非约束模态概念，采用Frobenius方法求解非约束模态振型函数以及固有频率；最后，通过数值仿真探究不同恒定转速时非约束模态与约束模态广义刚度的差异和非约束模态条件下离心力对模型的影响。
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关键词:
- 刚柔耦合 /
- 动力刚化 /
- Timoshenko梁 /
- 非约束模态
Abstract:The transverse deformation of the beam will lead to the longitudinal shortening deformation, and this transverse-longitudinal deformation coupling will bring the dynamic stiffening effect term on the generalized rigidity of the beam model. For the rotating beam structure, the centrifugal force will cause axial tension, with coupling axial and transverse deformation of the beam and bring additional geometric stiffness, which is more obvious for the thick short beam. The central rigid body-Timoshenko beam model with a large-range-motion center was investigated. Firstly, the dynamic model with centrifugal forces was established by means of the Timoshenko beam theory and the Hamilton principle. Secondly, the unconstrained mode concept was introduced, and the unconstrained mode shape functions and natural frequencies were solved with the Frobenius method. Finally, numerical simulations were carried out to explore the difference of generalized stiffness between the unconstrained mode and the constrained mode at different constant speeds, and the effects of centrifugal forces on the model under unconstrained mode condition were discussed.
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Key words:
- rigid-flexible coupling /
- dynamic stiffening /
- Timoshenko beam /
- unconstrained mode
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图 1 中心刚体-Timoshenko梁系统结构图
Figure 1. The structural diagram for the central rigid body-Timoshenko beam system
图 3 恒定转速为0.1 rad/s的广义刚度
Figure 3. The generalized stiffness at a constant speed of 0.1 rad /s
图 4 恒定转速为0.5 rad/s的广义刚度
Figure 4. The generalized stiffness at a constant speed of 0.5 rad /s
图 5 恒定转速为1 rad/s的广义刚度
Figure 5. The generalized stiffness at a constant speed of 1 rad /s
图 6 恒定转速为5 rad/s的广义刚度
Figure 6. The generalized stiffness at a constant speed of 5 rad /s
图 7 恒定转速为0.1 rad/s的广义刚度
Figure 7. The generalized stiffness at a constant speed of 0.1 rad /s
图 8 恒定转速为0.5 rad/s的广义刚度
Figure 8. The generalized stiffness at a constant speed of 0.5 rad / s
图 9 恒定转速为10 rad/s含离心力模型的广义刚度
Figure 9. The generalized stiffness of the model with centrifugal forces at a constant speed of 10 rad / s
图 10 恒定转速为10 rad/s不含离心力模型的广义刚度
Figure 10. The generalized stiffness of the model without centrifugal forces at a constant speed of 10 rad / s
图 11 恒定转速为0.1 rad/s时，梁末端振动响应
Figure 11. Vibration responses of the beam end at a constant speed of 0.1 rad / s
图 12 恒定转速为0.5 rad/s时，梁末端振动响应
Figure 12. Vibration responses of the beam end at a constant speed of 0.5 rad / s
图 13 恒定转速为10 rad/s时，梁末端振动响应
Figure 13. Vibration responses of the beam end at a constant speed of 10 rad / s
表 1 Timoshenko 梁参数
Table 1. Parameters of the Timoshenko beam
beam parameter length L / m cross-sectional
area Ab / m2cross-sectional moment
of inertia I / m4Young’s modulus
E/GPadensity
$\rho /({\rm{kg} }/{ {\rm{m} }^{ 3} })$Poisson’s
ratio $\upsilon $shear
coefficient $\kappa $value 0.5 1.6E−3 2.56E−6 163.8 7 850 0.3 5/6 
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