This paper presents formulations for a Timoshenko beam subjected to an accelerating mass using spectral element method in time domain (TSEM). Vertical displacement and bending rotation of the beam were interpolated by Lagrange polynomials supported on the GaussLobattoLegendre (GLL) points. By using GLL integration rule, the mass matrix was diagonal and the dynamic responses can be obtained efficiently and accurately. The results were compared with those obtained in the literature to verify the correctness. The variation of the vibration frequencies of the Timoshenko and moving mass system was researched. The effects of inertial force, centrifugal force, Coriolis force, and tangential force on a Timoshenko beam subjected to an accelerating mass were investigated.
Dynamic response of structures subjected to a moving force or moving mass is an important issue in engineering problems. For example, the trains have experienced great advances characterized by increasingly higher speeds and weights of vehicles. As a result, the dynamic response, as well as stresses, can be significantly higher than that before or static loads. The problem arose from the observations is a structure subjected to moving masses. Many researchers studied these problems and many studies are presented in the literatures. For examples, references [
The studies mentioned are based on BernoulliEuler beam, while the moment of inertia and shear deformation should be taken into account when ratio of the height to span is large. References [
Two different kinds of spectral element method (SEM) have been developed to analyse various problems of engineering, namely, spectral element method in frequency domain (FSEM) and TSEM. The FSEM proposed by Doyle [
In this paper, a Timoshenko beam spectral element has been developed on the basis of Legendre polynomialsbased spectral finite element. The beam is discretized into a very small number of elements with ten degrees of freedom each. The equations of motion in matrix form for a Timoshenko beam due to a moving mass are derived by using the Hamilton principle. The shape functions of the vertical displacement and the bending rotation for a Timoshenko beam element are formulated by employing Lagrange interpolation supported on the GLL points. The element mass matrix, stiffness matrix, and damping matrix are obtained by GLL integration rule. The effects of inertial forces are considered by the added mass, stiffness, and damping matrix. By assembling element matrices and element nodal vectors, respectively, the global equations of motion for a Timoshenko beam subjected to a moving mass are obtained. The vibration frequencies of the Timoshenko and a moving mass system were analyzed; in addition, the effects of the various forces considering the inertia of the mass will be investigated in detail. The presented formulations can be applied to solve the eigenvalue problem and dynamic responses for a Timoshenko beam with various boundary conditions by direct integration using generalized
A uniform Timoshenko beam acted on by a moving mass
Schematic figure of a Timoshenko beam and a moving mass system.
According to the Timoshenko’s beam theory, the deformed beam can be described by the rotation of the acrossarea and the shear deformation [
As the shear strain
The Timoshenko beam is discretized into a very small number of spectral elements with equal length,
Figure
The
The vertical displacement
The equation of motion for the Timoshenko beam and moving mass system can be obtained by using the Hamilton principle, expressed as follows:
When the moving mass is located at
By substituting (
Where
The matrices
By assembling element matrices and element nodal vectors, respectively, the global equations of motion for a Timoshenko beam subjected to a moving mass can be obtained as follows:
Consider the following:
In order to obtain the dynamic response of the Timoshenko beam and the moving mass system, the generalized
The updated equations for the displacement velocity and acceleration vectors are expressed as
To illustrate this method and verify its effectiveness and correctness, examples 1 and 2 are solved; the solutions obtained by the proposed method are compared with those obtained by other methods (including exact analytical solutions). Example 3 is solved to present the variation of the vibration frequencies of the system during the mass moves on the beam. Example 4 is solved to present the effects of inertial force, centrifugal force, Coriolis force, and tangential force on a Timoshenko beam subjected to an accelerating mass.
Three types of prismatic Timoshenko beams considering the natural vibration problems are solved by the present method. The parameters of the beams are the same with different boundaries, simplesimple, clampedfree, and clampedsimple. The properties of the beam are modulus of elasticity
To test the convergence and validity of the method, the first nineorder natural frequencies obtained by presented method are compared with the analytical solution presented by Huang [
Free vibration frequencies for simplesimple Timoshenko beam.








 


6838.8338  23190.888  43444.615  64925.614  86738.326  108481.00  111981.29  120647.23  129920.78 

6838.8336  23190.827  43443.498  64939.222  86711.057  108431.78  111981.29  120647.23  130004.36 

6838.8336  23190.827  43443.493  64939.186  86710.905  108431.36  111981.29  120647.23  130003.65 

6838.8336  23190.827  43443.493  64939.185  86710.899  108431.34  111981.29  120647.23  130003.61 
Exact [ 
6838.8336  23190.827  43443.493  64939.185  86710.889  108431.34  111981.29  120647.23  130003.61 
Tables
Free vibration frequencies for clampedfree Timoshenko beam.








 


2529.4927  13279.912  31045.083  50829.910  71550.753  91995.694  110969.19  119289.42  131460.96 

2529.4927  13279.905  31044.792  50825.847  71565.121  91995.100  110976.56  119244.54  131607.53 

2529.4927  13279.905  31044.791  50825.835  71565.050  91994.834  110976.00  119244.58  131606.55 

2529.4927  13279.905  31044.791  50825.843  71565.047  91994.824  110975.98  119244.57  131606.52 
Exact [ 
2529.4927  13279.905  31044.791  50825.843  71565.047  91994.824  110975.98  119244.57  131606.52 
Free vibration frequencies for clampedsimple Timoshenko beam.








 


9741.9478  26150.351  45547.079  66197.391  87399.691  108645.85  114294.62  128761.33  131483.10 

9741.9469  26150.251  45545.516  66212.036  87376.814  108601.58  114295.46  128739.40  131611.56 

9741.9469  26150.251  45545.510  66211.995  87376.651  108601.16  114295.44  128739.41  131610.66 

9741.9469  26150.251  45545.510  66211.993  87376.644  108601.14  114295.44  128739.40  131610.63 
Reference [ 
9741.9469  26150.251  45545.510  66211.993  87376.644  108601.14  114295.44  128739.40  131610.63 
As the proposed method has the advantage of the finite elements and the spectral method, it can analyse free vibration problem with various boundary conditions and good accuracy.
For the purpose of verification, a Timoshenko beam and a moving mass system neglecting the damping effect with boundary condition simplesimple were considered. The beam is discretized to 16 elements by fiveorder Timoshenko beam spectral element. The dynamic response of the system is obtained by using generalized
The vertical displacement of the beam is normalized by
Figure
Normalized displacement under the equivalent moving loads with the initial speed given by
With only the equivalent force vector that is zero, the natural frequencies of the beam and moving masses system are obtained. For this case, we rewrite (
The properties and parameters of the beam are the same as Example 1. Figures
The first two normalized natural frequencies of simplesimple Timoshenko beam subjected to a moving mass with different velocities.
The first two normalized natural frequencies of clampedfree Timoshenko beam subjected to a moving mass with different velocities.
The first two normalized natural frequencies of clampedsimple Timoshenko beam subjected to a moving mass with different velocities.
The first two normalized natural frequencies of simplesimple Timoshenko beam subjected to a moving mass with different masses.
The first two normalized natural frequencies of clampedsimple Timoshenko beam subjected to a moving mass with different masses.
The first two normalized natural frequencies of clampedsimple Timoshenko beam subjected to a moving mass with different masses.
Figure
To analyse the effect of the different forces considering the inertia of the moving mass, the Timoshenko beam and a moving mass system are introduced with the same parameters as Example 2. The inertial force, Coriolis force, centrifugal force, and tangential force at the moving mass point are, respectively, shown in Figure
Normalized force under the equivalent moving loads with the initial speed given by
Motion equations show that all the forces are the important members of the loads considering the effect of the moving mass and cannot be neglected. Figure
Normalized Displacement under the equivalent moving loads with the initial speed given by
It shows that the displacements with the tangential force neglected agreed with the displacement without neglected any forces, because the property of the Timoshenko beam is that the ratio of the height to span is large; therefore, the deformation of the beam is smaller, and
The equation of motion in matrix form has been formulated for the dynamic response of a Timoshenko beam subjected to a moving mass by using TSEM. The inertia effect of the moving mass can easily be taken into account by assembling added mass, damping, and stiffness matrices to the global mass, damping, and stiffness matrices. The TSEM has the advantages of the spectral method and finite element method; the degrees of freedom can be less with the characteristic of high accuracy; the eigenvalue and dynamic problem can be obtained efficiently with the accurate diagonal mass matrix.
The variation of the vibration frequencies of the Timoshenko and a moving mass system was obtained. Numerical results for the Timoshenko beam and a moving mass system indicate that the inertia effect of the moving load cannot be neglected, the Coriolis, inertia, and centrifugal forces take a more important role than tangential force in the moving mass system, Coriolis, inertia, centrifugal, and tangential forces take the role of the damping, mass, and stiffness matrices, respectively.
The authors declare that there is no conflict of interests regarding the publication of this paper.