Significance Statement
Most engineering fields (bridges, railroads, pavement etc.) are subjected to moving forces. This moving force creates stress in one dimensional flexural structure. There are many analytical solution methods on moving force problems which include the integral transform methods, mode superposition methods, Green function methods, transfer matrix methods etc.
As analytical solutions are not always available, various numerical methods have been developed. The most widely used finite element method (FEM) requires a huge number of degrees of freedom whereas the spectral element method (SEM) requires only a minimum number of degrees of freedom to provide accurate solutions at any frequency ranges. Some researchers have applied the SEM to the moving force problems in an inefficient way and their solutions were not accurate. Hence they should employ static Green’s function.
Researchers led by Professor Usik Lee from Department of Mechanical Engineering at Inha University in Republic of Korea proposed a new spectral element analysis method in which they stated that no static Green’s function or structural discretization method is required to improve the accuracy of the solution. They compared the results from their newly proposed spectral element analysis method with the results from standard FEM and found their proposed method provides extremely accurate solutions. The study is published in the peer-reviewed journal, International Journal of Mechanical Sciences.
When a cantilever beam is subjected to moving force, it produces vibration. In order to predict the vibration, an SEM-based solution technique was developed. Researchers assumed the beam is made of an elastic material and represented it by the Timoshenko beam model. They used discrete Fourier transform theory to represent the time history of a moving force as a series of stationary point forces distributed on the beam in the frequency domain. First, in the frequency domain, they obtained individual vibration responses due to each stationary point force. Each individual vibration responses were obtained by representing the whole beam by the two-element model (TEM). Next, they summed these individual vibrations to obtain total vibration responses. Finally they obtained vibration responses in the time domain from total vibration responses obtained in the frequency domain by using the inverse FFT algorithm.
They considered various speeds of a moving force: a constant moving speed and a time-varying moving speed. When the critical speed is more than the speed of the moving force, the deformed shape tend to return to its original shape. The deformation of the beam tends to increase with time when the moving point force moves at the critical speed. This deformation keeps increasing until the moving point force arrives at right end. The transverse displacement was simulated at various locations as the point force moves forward to the free end.
The authors found out that when the beam is subjected to a moving force for a longer duration, the transverse displacements at lower moving speed are larger than at higher moving speeds. They found out that even at lower speeds, the dynamic response analysis cannot be replaced with static analysis. The dynamic responses increase with time when the point force moves at a speed equal or greater than the lowest critical speed.
By comparing numerical results obtained by the newly proposed TEM-based SEM with those obtained by other solution techniques including FEM, they verified that the newly proposed solution technique provides much more accurate solutions than the other methods.
Journal Reference
Younghoon Song, Taehyun Kim, Usik Lee, Vibration of a beam subjected to a moving force: Frequency-domain spectral element modeling and analysis. International Journal of Mechanical Sciences, Volume 113, 2016, Pages 162–174.
Department of Mechanical Engineering, Inha University, 100 Inharo, Nam-gu, Incheon 402-751, Republic of Korea
Go To International Journal of Mechanical Sciences