This paper investigates the dynamic responses of multi-bay frames with identical bay lengths subjected to a transverse single moving load and successive moving loads with a constant interval at a constant speed. The effects of the bay length and the speed of the moving load on the response of the multi-bay frame subjected to a single point load are investigated numerically by the finite element method. A computer code is developed by using MATLAB to perform the finite element analysis. The Newmark method is employed to solve for the dynamic responses of the multi-bay frame. With this, the dynamic response of the frame subjected to successive moving loads with a constant interval is investigated. Also, the resonance and cancellation speeds are determined by using the 3D relationship of speed parameter-force span length to beam length ratio-dynamic magnification factor and the associated contour lines. The maximum impact factor of a 1-bay frame and multi-bay frames under single moving load are determined at the specific speed parameters. Those values are independent of elastic modulus, area moment of inertia, beam/column lengths of the frame and also the number of bays forming the frame. It is also found that the first resonance response in the vertical direction of the frame is related to the second mode of vibration.

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In a recent study, the eigenvalue sensitivities of a combined system consisting of a linear structure with several lumped attachments were analyzed. An efficient approach was developed for the eigenvalue sensitivities with respect to the parameters of the attachments and their locations by using the implicit function theorem. In the study mentioned above, the theory was applied to a Bernoulli-Euler beam with several attachments. Motivated by this study, the methodology established there will be applied, in the present paper, to an axially vibrating rod with several lumped attachments. Hence, the utility of the methodology is justified through a quite different type of vibrational system. The method is shown as an efficient tool that can be taught in related graduate-level courses for the eigenvalue estimation of complex linear systems with several lumped attachments.

The high-speed railways require more viaducts than conventional railways. The dynamic interaction effect between train and viaduct are important issue due to the risk of derailment, structural safety and deterioration of the passenger comfort. In this study, viaduct is modelled as a multi-bay frame. The multi-bay frame is modelled by finite element method. The train is idealized as a two-axle system with 4 degrees of freedom. The equations of motions of the coupled vehicle-structure system are determined via generalized Lagrange's equation. The Wilson-theta time integration method is employed to determine the dynamic response of the system. The effective mode shapes are investigated using 3D frequency-velocity-amplitude graphs. The resonant response has been determined at first and second modes of 1 and 2-bay frames.

In this study, the mode shapes dominate the dynamic response of a viaduct under the passage of the high-speed train. The viaduct is modelled as a multi-bay frame and the train is modelled as a two-axle multi-body system with 4 DOF. The finite element method with Lagrange's equation has been used to determine the equations of in-plane motions, based on the Bernoulli-Euler beam theory. The Wilson-theta time integration scheme is employed to determine the dynamic response of the structure. To determine which modes are dominate the dynamic response, 3D frequencyvelocity-amplitude graph is plotted.