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Structural Engineering and Mechanics Volume 59, Number 6, September25 2016 , pages 1139-1153 DOI: https://doi.org/10.12989/sem.2016.59.6.1139 |
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Effects of deformation of elastic constraints on free vibration characteristics of cantilever Bernoulli-Euler beams |
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Tong Wang, Tao He and Hongjing Li
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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, kv and kr, 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 kv and kr. Then the mode shape functions are given. Finally, effects of kv and kr on free vibration characteristics of the beam with different slenderness ratios are calculated and analyzed. The results indicate that the effects of kv 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 kv can be neglected for the fundamental frequency characteristics, while cannot for higher-order ones. However, the effects of kr are large on both higher- and lower-order free vibration characteristics, and cannot be neglected no matter the slenderness ratio is large or small. | ||
| Key Words | ||
| deformation; elastic constraint; free vibration characteristic; cantilever beam; Bernoulli-Euler beam; frequency equation; mode shape function | ||
| Address | ||
| Tong Wang : College of Civil Engineering, Shanghai Normal University, Shanghai, 201418, China Tao He : College of Civil Engineering, Shanghai Normal University, Shanghai, 201418, China; School of Civil Engineering, University of Birmingham, Birmingham, B15 2TT, UK Hongjing Li : College of Civil Engineering, Nanjing Tech University, Nanjing, 211816, China | ||