Structural Engineering and Mechanics

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Effects of deformation of elastic constraints on free vibration characteristics of cantilever Bernoulli-Euler beams

  • Wang, Tong (College of Civil Engineering, Shanghai Normal University) ;
  • He, Tao (College of Civil Engineering, Shanghai Normal University) ;
  • Li, Hongjing (College of Civil Engineering, Nanjing Tech University)
  • Received : 2016.01.30
  • Accepted : 2016.06.10
  • Published : 2016.09.25
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Abstract

Elastic constraints are usually simplified as "spring forces" exerted on beam ends without considering the "spring deformation". The partial differential equation governing the free vibrations of a cantilever Bernoulli-Euler beam considering the deformation of elastic constraints is firstly established, and is nondimensionalized to obtain two dimensionless factors, $k_v$ and $k_r$, describing the effects of elastically vertical and rotational end constraints, respectively. Then the frequency equation for the above Bernoulli-Euler beam model is derived using the method of separation of variables. A numerical analysis method is proposed to solve the transcendental frequency equation for the continuous change of the frequency with $k_v$ and $k_r$. Then the mode shape functions are given. Finally, effects of $k_v$ and $k_r$ on free vibration characteristics of the beam with different slenderness ratios are calculated and analyzed. The results indicate that the effects of $k_v$ are larger on higher-order free vibration characteristics than on lower-order ones, and the impact strength decreases with slenderness ratio. Under a relatively larger slenderness ratio, the effects of $k_v$ can be neglected for the fundamental frequency characteristics, while cannot for higher-order ones. However, the effects of $k_r$ are large on both higher- and lower-order free vibration characteristics, and cannot be neglected no matter the slenderness ratio is large or small.

Keywords

Acknowledgement

Supported by : National Natural Science Foundation of China

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