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Vibration model for multi-span beam with arbitrary complex boundary conditions
LIU Xiangyao, NIE Hong , WEI Xiaohui
State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
Abstract:The transverse free vibration equations for Timoshenko beam were derived. Based on these equations, the vibration model for a multi-span beam with arbitrary complex boundary conditions was given by the transfer matrix method. Without considering the shear deformation and moment of inertia of the neutral axis, the model was simplified as the analogous model for Bernoulli-Euler beam. Three simplified models of some engineering significance were given. They are the free vibration model for a two-span beam, a cantilever with a lumped mass, and a beam with arbitrary lumped masses and translational springs. A comparison between the frequency equations derived by the three simplified models and those by the previous studies shows good consistency of the two, and it is thus concluded that the model developed in this paper is reasonable and feasible.
Key words: vibration analysis     boundary conditions     transfer matrix method     Timoshenko beam     multi-span beam

1 Timoshenko梁的振动微分方程

Bernoulli-Euler梁适合于描述细长梁以低阶固有振动为主的振动.随着固有振动阶数的提高及边界条件的增加,梁被节点分成了若干短粗的小段.此时,梁的剪切变形及绕中性轴的转动惯量的影响变得突出.计入这2种因素的梁模型叫做Timoshenko梁模型.它对变形的基本假设是:梁截面在弯曲变形后仍保持平面,但未必垂直于中性轴.

Timoshenko梁微段的变形与受力分析如图 1所示.图中,M为弯矩；Q为剪力.取坐标x处的梁微段dx为分离体.由于剪切变形,梁横截面的法线不再与梁轴线重合.法线转角θ由轴线转角∂w/∂x和剪切角γ两部分组成,即

 图 1 Timoshenko梁微段的变形与受力分析Fig. 1 Deformation and force analysis of micro-segment of Timoshenko beam

w(x,t)=W(x)·q(t)

Timoshenko梁的振形函数为

2 传递矩阵和递推公式

2.1 场传递矩阵的推导

2.2 点传递矩阵的推导

2.3 递推公式和频率方程

2.4 模型验证

3 Bernoulli-Euler梁的自由振动 3.1 Bernoulli-Euler梁的振动微分方程

Bernoulli-Euler梁的振形函数为

W(x)=a1ch(λx)+a2sh(λx)+a3cos(λx)+a4sin(λx)

3.2 场传递矩阵

4 特 例

4.1 双跨梁情形

k=1、n=f=h=0,可将模型转换成双跨梁的情形,即梁中间只有一个铰支座,位置是αi,将梁分成了两段.

αi→1,式(56)变成

αi→0,式(56)变成

αi→0时,式(58)是个0/0型的不等式,由三角函数的性质可知,式(58)右边的分子是比分母更高阶的无穷小量,则

4.2 悬臂梁带有集中质量的情形

n=1、k=f=h=0,可将模型转换成带有集中质量的情形,即梁中间只有一个集中质量,位置为bi,质量为mi,转动惯量为Ji.

q=0,可得

4.3 带有任意拉压弹簧和集中质量的情形

k=h=0,可将模型转换成带有任意个拉压弹簧和集中质量的情形,即梁中间有n个集中质量,位置为bi(i=1，2，…，n),质量为mi(i=1,2,…,n),转动惯量为Ji(i=1,2,…,n);并且梁中间有f个拉压弹簧,每个坐标为ci(i=1,2,…,f),刚度为ki(i=1,2,…,f).

5 结 论

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#### 文章信息

LIU Xiangyao, NIE Hong, WEI Xiaohui

Vibration model for multi-span beam with arbitrary complex boundary conditions

Journal of Beijing University of Aeronautics and Astronsutics, 2015, 41(5): 841-846.
http://dx.doi.org/10.13700/j.bh.1001-5965.2014.0315