Multi-harmonic forced vibration and resonance of simple beams to moving vehicles

Abstract

This study modeled the moving-vehicle-induced forcing excitation on a single-span prismatic bridge as a multiple frequency-multiplication harmonic load on the modal coordinates of a linear elastic simple Euler–Bernoulli beam, and investigated the forced modal oscillation and resonance behavior of this type of dynamic system. The forced modal responses consist of multiple frequency-multiplication steady-state harmonics and one damped mono-frequency complementary harmonic. The analysis revealed that a moving load induces high-harmonic forced resonance amplification when the moving speed is low. To verify the occurrence of high-harmonic forced resonance, numerical tests were conducted on single-span simple beams based on structural modeling using the finite element method (FEM) and a moving sprung-mass oscillator vehicle model. The forced resonance amplification characteristics of the fundamental mode for beam response estimation are presented with consideration to different end restraint conditions. The results reveal that the high-harmonic forced resonance may be significant for the investigated beams subjected to vehicle loads moving at specific low speeds. For the investigated single-span simple beams, the moving vehicle carriage heaving oscillation modulates the beam modal frequency, but does not induce notable variation of the modal oscillation harmonic structure for the cases that vehicle of small mass moves in low speed.

Graphical abstract

Keywords

forced vibration / linear Euler beam / moving load / harmonic structure / frequency modulation / end restraints

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Appendix A: Derivation of Eqs. (7a)–(7c)

Based on Eqs. (4) and (6), the Fourier coefficients can be derived as follows.

\begin{array}{rl}{a}_{0,r}=& \frac{1}{L}{\int }_0^L{\varphi }_r\left(x\right)\mathrm{d}x\\ =& \frac{1}{L}{\int }_0^L\left(C_1\mathrm{s}\mathrm{i}\mathrm{n}\lambda x+C_2\mathrm{c}\mathrm{o}\mathrm{s}\lambda x+C_3\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}\lambda x+C_4\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}\lambda x\right)\mathrm{d}x\\ =& \frac{2C_1{\mathrm{s}\mathrm{i}\mathrm{n}}^2{\displaystyle \frac{\lambda L}{2}}+C_2\mathrm{s}\mathrm{i}\mathrm{n}\lambda L+2C_3{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}}^2{\displaystyle \frac{\lambda L}{2}}+C_4\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}\lambda L}{\lambda L},\end{array}

\begin{array}{rl}{a}_{n,r}=& {\displaystyle \frac{2}{L}}{\int }_0^L{\varphi }_r\left(x\right)\mathrm{c}\mathrm{o}\mathrm{s}{\displaystyle \frac{2n\text{π}x}{L}}\mathrm{d}x={\displaystyle \frac{2}{L}}{\int }_0^L(C_1\mathrm{s}\mathrm{i}\mathrm{n}(\lambda x)+C_2\mathrm{c}\mathrm{o}\mathrm{s}(\lambda x)+C_3\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}(\lambda x)+C_4\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}(\lambda x))\mathrm{c}\mathrm{o}\mathrm{s}{\displaystyle \frac{2n\text{π}x}{L}}\mathrm{d}x\\ =& {\displaystyle \frac{2}{L}}{\int }_0^L{C}_1\mathrm{s}\mathrm{i}\mathrm{n}(\lambda x)\mathrm{c}\mathrm{o}\mathrm{s}{\displaystyle \frac{2n\text{π}x}{L}}\mathrm{d}x+{\displaystyle \frac{2}{L}}{\int }_0^L{C}_2\mathrm{c}\mathrm{o}\mathrm{s}(\lambda x)\mathrm{s}\mathrm{i}\mathrm{n}{\displaystyle \frac{2n\text{π}x}{L}}\mathrm{d}x+{\displaystyle \frac{2}{L}}{\int }_0^L{C}_3\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}(\lambda x)\mathrm{c}\mathrm{o}\mathrm{s}{\displaystyle \frac{2n\text{π}x}{L}}\mathrm{d}x\\ & +{\displaystyle \frac{2}{L}}{\int }_0^L{C}_4\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}(\lambda x)\mathrm{c}\mathrm{o}\mathrm{s}{\displaystyle \frac{2n\text{π}x}{L}}\mathrm{d}x={\displaystyle \frac{2C_1}{L}}{\left[-{\displaystyle \frac{\mathrm{c}\mathrm{o}\mathrm{s}\left(\lambda +{\displaystyle \frac{2n\text{π}}{L}}\right)x}{2\left(\lambda +{\displaystyle \frac{2n\text{π}}{L}}\right)}}-{\displaystyle \frac{\mathrm{c}\mathrm{o}\mathrm{s}\left(\lambda -{\displaystyle \frac{2n\text{π}}{L}}\right)x}{2\left(\lambda -{\displaystyle \frac{2n\text{π}}{L}}\right)}}\right]}_0^L\\ & +{\displaystyle \frac{2C_2}{L}}{\left[{\displaystyle \frac{\mathrm{s}\mathrm{i}\mathrm{n}\left(\lambda +{\displaystyle \frac{2n\text{π}}{L}}\right)x}{2\left(\lambda +{\displaystyle \frac{2n\text{π}}{L}}\right)}}+{\displaystyle \frac{\mathrm{s}\mathrm{i}\mathrm{n}\left(\lambda -{\displaystyle \frac{2n\text{π}}{L}}\right)x}{2\left(\lambda -{\displaystyle \frac{2n\text{π}}{L}}\right)}}\right]}_0^L+{\displaystyle \frac{2C_3}{L}}{\left[{\displaystyle \frac{\lambda \mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}(\lambda x)\mathrm{c}\mathrm{o}\mathrm{s}{\displaystyle \frac{2n\text{π}x}{L}}}{{\lambda }^2+{\left({\displaystyle \frac{2n\text{π}}{L}}\right)}^2}}+{\displaystyle \frac{{\displaystyle \frac{2n\text{π}}{L}}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}(\lambda x)\mathrm{s}\mathrm{i}\mathrm{n}{\displaystyle \frac{2n\text{π}x}{L}}}{{\lambda }^2+{\left({\displaystyle \frac{2n\text{π}}{L}}\right)}^2}}\right]}_0^L\\ & +{\displaystyle \frac{2C_4}{L}}{\left[{\displaystyle \frac{\lambda \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}(\lambda x)\mathrm{c}\mathrm{o}\mathrm{s}{\displaystyle \frac{2n\text{π}x}{L}}}{{\lambda }^2+{\left({\displaystyle \frac{2n\text{π}}{L}}\right)}^2}}+{\displaystyle \frac{{\displaystyle \frac{2n\text{π}}{L}}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}(\lambda x)\mathrm{s}\mathrm{i}\mathrm{n}{\displaystyle \frac{2n\text{π}}{L}}}{{\lambda }^2+{\left({\displaystyle \frac{2n\text{π}}{L}}\right)}^2}}\right]}_0^L\\ =& 2\lambda L\left[{\displaystyle \frac{C_1\mathrm{c}\mathrm{o}\mathrm{s}(\lambda L)-C_1-C_2\mathrm{s}\mathrm{i}\mathrm{n}(\lambda L)}{4n^2{\text{π}}^2-{\lambda }^2{L}^2}}+{\displaystyle \frac{2C_3{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}}^2{\displaystyle \frac{\lambda L}{2}}+C_4\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}(\lambda L)}{4n^2{\text{π}}^2+{\lambda }^2{L}^2}}\right]\end{array}

\begin{array}{rl}{b}_{n,r}=& {\displaystyle \frac{2}{L}}{\int }_0^L{\varphi }_r\left(x\right)\mathrm{s}\mathrm{i}\mathrm{n}{\displaystyle \frac{2n\text{π}x}{L}}\mathrm{d}x={\displaystyle \frac{2}{L}}{\int }_0^L\left(C_1\mathrm{s}\mathrm{i}\mathrm{n}(\lambda x)+C_2\mathrm{c}\mathrm{o}\mathrm{s}(\lambda x)+C_3\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}(\lambda x)+C_4\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}(\lambda x)\right)\mathrm{s}\mathrm{i}\mathrm{n}{\displaystyle \frac{2n\text{π}x}{L}}\mathrm{d}x\\ =& {\displaystyle \frac{2}{L}}{\int }_0^L{C}_1\mathrm{s}\mathrm{i}\mathrm{n}(\lambda x)\mathrm{s}\mathrm{i}\mathrm{n}{\displaystyle \frac{2n\text{π}x}{L}}\mathrm{d}x+{\displaystyle \frac{2}{L}}{\int }_0^L{C}_2\mathrm{c}\mathrm{o}\mathrm{s}(\lambda x)\mathrm{s}\mathrm{i}\mathrm{n}{\displaystyle \frac{2n\text{π}\mathrm{x}}{\mathrm{L}}}\mathrm{d}x+{\displaystyle \frac{2}{L}}{\int }_0^L{C}_3\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}(\lambda x)\mathrm{s}\mathrm{i}\mathrm{n}{\displaystyle \frac{2n\text{π}x}{L}}\mathrm{d}x\\ & +{\displaystyle \frac{2}{L}}{\int }_0^L{C}_4\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}(\lambda x)\mathrm{s}\mathrm{i}\mathrm{n}{\displaystyle \frac{2n\text{π}x}{L}}\mathrm{d}x={\displaystyle \frac{2C_1}{L}}{\left[-{\displaystyle \frac{\mathrm{s}\mathrm{i}\mathrm{n}\left(\lambda +{\displaystyle \frac{2n\text{π}}{L}}\right)x}{2\left(\lambda +{\displaystyle \frac{2n\text{π}}{L}}\right)}}+{\displaystyle \frac{\mathrm{s}\mathrm{i}\mathrm{n}\left(\lambda -{\displaystyle \frac{2n\text{π}}{L}}\right)x}{2\left(\lambda -{\displaystyle \frac{2n\text{π}}{L}}\right)}}\right]}_0^L\\ & +{\displaystyle \frac{2C_2}{L}}{\left[-{\displaystyle \frac{\mathrm{c}\mathrm{o}\mathrm{s}\left(\lambda +{\displaystyle \frac{2n\text{π}}{L}}\right)x}{2\left(\lambda +{\displaystyle \frac{2n\text{π}}{L}}\right)}}+{\displaystyle \frac{\mathrm{c}\mathrm{o}\mathrm{s}\left(\lambda -{\displaystyle \frac{2n\text{π}}{L}}\right)x}{2\left(\lambda -{\displaystyle \frac{2n\text{π}}{L}}\right)}}\right]}_0^L+{\displaystyle \frac{2C_3}{L}}{\left[{\displaystyle \frac{\lambda \mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}(\lambda x)\mathrm{s}\mathrm{i}\mathrm{n}{\displaystyle \frac{2n\text{π}x}{L}}}{{\lambda }^2+{\left({\displaystyle \frac{2n\text{π}}{L}}\right)}^2}}-{\displaystyle \frac{{\displaystyle \frac{2n\text{π}}{L}}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}(\lambda x)\mathrm{c}\mathrm{o}\mathrm{s}{\displaystyle \frac{2n\text{π}x}{L}}}{{\lambda }^2+{\left({\displaystyle \frac{2n\text{π}}{L}}\right)}^2}}\right]}_0^L\\ & +{\displaystyle \frac{2C_4}{L}}{\left[{\displaystyle \frac{\lambda \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}(\lambda x)\mathrm{s}\mathrm{i}\mathrm{n}{\displaystyle \frac{2n\text{π}x}{L}}}{{\lambda }^2+{\left({\displaystyle \frac{2n\text{π}}{L}}\right)}^2}}-{\displaystyle \frac{{\displaystyle \frac{2n\text{π}}{L}}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}(\lambda x)\mathrm{c}\mathrm{o}\mathrm{s}{\displaystyle \frac{2n\text{π}x}{L}}}{{\lambda }^2+{\left({\displaystyle \frac{2n\text{π}}{L}}\right)}^2}}\right]}_0^L\\ =& -4n\text{π}\left[{\displaystyle \frac{C_1\mathrm{s}\mathrm{i}\mathrm{n}(\lambda L)+C_2\mathrm{c}\mathrm{o}\mathrm{s}(\lambda L)-C_2}{4n^2{\text{π}}^2-{\lambda }^2{L}^2}}+{\displaystyle \frac{C_3\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}(\lambda L)+2C_4{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}}^2{\displaystyle \frac{\lambda L}{2}}}{4n^2{\text{π}}^2+{\lambda }^2{L}^2}}\right].\end{array}

Acknowledgements

This study was supported by the SLDRCE Independent Research Fund of the Ministry of Science and Technology of China (Nos. SLDRCE14-B-24 and SLDRCE19-B-33).

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Conflict of Interest

The authors declare that they have no conflict of interest.

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