Purpose This paper presented a generalized equation of motions, where the effects of damping forces of the entire vibrating system and inertial force of each moving load were considered, and the forced vibration problem regarding the uniform beams subjected to multiple moving loads can be easily solved. Methods To the above end, each mode of vibration was considered as one degree of freedom (DOF) of the entire vibrating system. Then, the mode-superposition method (MSM) and Rayleigh damping theory were incorporated, and the equation of motion: <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="42417_2024_1650_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="420" /> </InlineMediaObject> <EquationSource Format="TEX">${\left[m\right]}_{{n}^{\prime}\times {n}^{\prime}}{\left\{\ddot{\eta }\right\}}_{{n}^{\prime}\times 1}+{\left[c\right]}_{{n}^{\prime}\times {n}^{\prime}}{\left\{\dot{\eta }\right\}}_{{n}^{\prime}\times 1}+{\left[k\right]}_{{n}^{\prime}\times {n}^{\prime}}{\left\{\eta \right\}}_{{n}^{\prime}\times 1}={\left\{f\right\}}_{{n}^{\prime}\times 1}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mfenced close="]" open="["> <mi>m</mi> </mfenced> <mrow> <msup> <mrow> <mi>n</mi> </mrow> <mo>′</mo> </msup> <mo>×</mo> <msup> <mrow> <mi>n</mi> </mrow> <mo>′</mo> </msup> </mrow> </msub> <msub> <mfenced close="}" open="{"> <mover accent="true"> <mi>η</mi> <mo>¨</mo> </mover> </mfenced> <mrow> <msup> <mrow> <mi>n</mi> </mrow> <mo>′</mo> </msup> <mo>×</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mfenced close="]" open="["> <mi>c</mi> </mfenced> <mrow> <msup> <mrow> <mi>n</mi> </mrow> <mo>′</mo> </msup> <mo>×</mo> <msup> <mrow> <mi>n</mi> </mrow> <mo>′</mo> </msup> </mrow> </msub> <msub> <mfenced close="}" open="{"> <mover accent="true"> <mi>η</mi> <mo>˙</mo> </mover> </mfenced> <mrow> <msup> <mrow> <mi>n</mi> </mrow> <mo>′</mo> </msup> <mo>×</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mfenced close="]" open="["> <mi>k</mi> </mfenced> <mrow> <msup> <mrow> <mi>n</mi> </mrow> <mo>′</mo> </msup> <mo>×</mo> <msup> <mrow> <mi>n</mi> </mrow> <mo>′</mo> </msup> </mrow> </msub> <msub> <mfenced close="}" open="{"> <mi>η</mi> </mfenced> <mrow> <msup> <mrow> <mi>n</mi> </mrow> <mo>′</mo> </msup> <mo>×</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mfenced close="}" open="{"> <mi>f</mi> </mfenced> <mrow> <msup> <mrow> <mi>n</mi> </mrow> <mo>′</mo> </msup> <mo>×</mo> <mn>1</mn> </mrow> </msub> </mrow> </math></EquationSource> </InlineEquation>, was obtained. Finally, solving the matrix equation for <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="42417_2024_1650_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="32" /> </InlineMediaObject> <EquationSource Format="TEX">$\{ \eta \}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <mi>η</mi> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> by using the Newmark’s direct integration method, one may obtain the vertical deflections of the beam at any position x and time t. Results The numerical examples revealed that the presented method is easy to tackle the dynamic problem regarding the uniform beams under any number of moving loads with effects of damping forces and inertial forces considered (or neglected), and the obtained results are in good agreement with those obtained from the FEM. Conclusion Although the forgoing “generalized” equation of motions for the “analytical” method is similar to the “conventional” one for the “numerical” FEM, the CPU time required by the former was much less than that required by the latter, because the total mode numbers <InlineEquation ID="IEq01"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="42417_2024_1650_Article_IEq01.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">${n}^{\prime}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi>n</mi> </mrow> <mo>′</mo> </msup> </math></EquationSource> </InlineEquation> (in general <InlineEquation ID="IEq02"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="42417_2024_1650_Article_IEq02.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">${n}^{\prime}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi>n</mi> </mrow> <mo>′</mo> </msup> </math></EquationSource> </InlineEquation> ≤ 15) considered by the presented method is much smaller than the total DOFs (in general <InlineEquation ID="IEq004"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="42417_2024_1650_Article_IEq004.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">${n}$</EquationSource> <EquationSource Format="MATHML"><math> <mi>n</mi> </math></EquationSource> </InlineEquation> ≥ 80) considered by the FEM.

A Generalized Equation of Motion for Dynamic Analysis of a Uniform Beam Subjected to Multiple Moving Loads

期刊论文
发表日期：

2025年2月25日

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Chia-Chin Wu,
Tsung-Han Hsieh,
Ting-Yu Chang

摘要

Purpose

This paper presented a generalized equation of motions, where the effects of damping forces of the entire vibrating system and inertial force of each moving load were considered, and the forced vibration problem regarding the uniform beams subjected to multiple moving loads can be easily solved.

Methods

To the above end, each mode of vibration was considered as one degree of freedom (DOF) of the entire vibrating system. Then, the mode-superposition method (MSM) and Rayleigh damping theory were incorporated, and the equation of motion: ${\left[m\right]}_{{n}^{\prime}\times {n}^{\prime}}{\left\{\ddot{\eta }\right\}}_{{n}^{\prime}\times 1}+{\left[c\right]}_{{n}^{\prime}\times {n}^{\prime}}{\left\{\dot{\eta }\right\}}_{{n}^{\prime}\times 1}+{\left[k\right]}_{{n}^{\prime}\times {n}^{\prime}}{\left\{\eta \right\}}_{{n}^{\prime}\times 1}={\left\{f\right\}}_{{n}^{\prime}\times 1}$ ${[m]}_{n^{'} \times n^{'}} {\{\ddot{η}\}}_{n^{'} \times 1} + {[c]}_{n^{'} \times n^{'}} {\{\dot{η}\}}_{n^{'} \times 1} + {[k]}_{n^{'} \times n^{'}} {\{η\}}_{n^{'} \times 1} = {\{f\}}_{n^{'} \times 1}$ , was obtained. Finally, solving the matrix equation for $\{ \eta \}$ ${η}$ by using the Newmark’s direct integration method, one may obtain the vertical deflections of the beam at any position x and time t.

Results

The numerical examples revealed that the presented method is easy to tackle the dynamic problem regarding the uniform beams under any number of moving loads with effects of damping forces and inertial forces considered (or neglected), and the obtained results are in good agreement with those obtained from the FEM.

Conclusion

Although the forgoing “generalized” equation of motions for the “analytical” method is similar to the “conventional” one for the “numerical” FEM, the CPU time required by the former was much less than that required by the latter, because the total mode numbers ${n}^{\prime}$ $n^{'}$ (in general ${n}^{\prime}$ $n^{'}$ ≤ 15) considered by the presented method is much smaller than the total DOFs (in general ${n}$ $n$ ≥ 80) considered by the FEM.