The dynamic behavior of a combined bridge–vehicle system in which the bridge is modeled as a Timoshenko beam and the vehicle as a half-car planar model is investigated using the finite element method. The governing equations of motion of the Timoshenko beam with the attached tuned mass damper (TMD) traversed by a moving vehicle are obtained. Adesign optimization algorithm is developed in which the analysis module based on the derived finite element formulation has been combined with the optimization module using the sequential programming technique. The objective is to determine the optimum values of the parameters (frequency and damping ratios) of a TMD, in order to minimize the maximum frequency response of the beam midspan when traversed by a moving vehicle. Results obtained illustrate that by attaching an optimally TMD to the Timoshenko beam a significantly faster vibration control can be achieved.

In this study the finite element formulation for the dynamics of a bridge traversed by moving vehicles is presented. The vehicle including the driver and the passenger is modelled as a half-car planner model with six degree of freedom, travelling on the bridge with constant velocity. The bridge is modelled as a uniform beam with simply supported end conditions that obeys the Timoshenko beam theory. The governing equations of motion are derived using the extended Hamilton principle and then transformed into the finite element format by using the weak-form formulation. The Newmark-β method is utilized to solve the governing equations and the results are compared with those reported in the literature. Furthermore, the maximum values of deflection for the Timoshenko and Euler-Bernoulli beams have been compared. The results illustrated that as the velocity of the vehicle increases, the difference between the maximum beam deflections in the two beam models becomes more significant. Downloaded from where ξ k , c, ρ, and ω k represent the kth mode damping ratio, the damping coefficient, the linear density, and the kth natural frequency of the beam, respectively. Accordingly, the damping ratio for the first vibration mode of the beam is equal to 3.3 per cent.

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In this paper, the chatter phenomenon is investigated through a single degree of freedom model of the milling process. In this regard, the non-linear equation of motion obtained from modeling of the milling process, which is a time-periodic delay differential equation, is simulated, and by changing the parameters: spindle speed and depth of cut, and assuming constant quantities for other parameters of the system the stable and instable points for the system are gained according to these two parameters by numerical method. In the end, the stability chart for this system is plotted and the approximate boundaries between the stability and instability regions are obtained numerically.