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Previously, classical boundary conditions (i.e. simple, fixed and cantilevered) have been adopted for the beams under the moving loads in deriving the dynamic response. In reality, most beams cannot be simply classified as the ones with classical supports, but are elastically restrained. For theoretical completeness, a general theory will be developed in this note for the damped beams with elastic restraints modeled by vertical and rotational springs subjected to a moving damped sprung mass. Essential to the present theory is the solution of the transcendental equation for the frequency by the bi-section method. The solution obtained can also be applied to the classical boundary conditions (i.e. simple, fixed and cantilevered) under a moving sprung mass. The reliability of the present theory, along with the bi-section method for solving the frequency and mode shape, is validated by comparison with the solution obtained by the finite element method (FEM) for various damping ratios, mass ratios and running speeds of the sprung mass.

The vertical tyre force is crucial to the study of the dynamics of a tractor semi-trailer. The paper presents an experimental method for determining the vertical tyre force by determining the vertical acceleration of the un-sprung mass and the vertical acceleration of the sprung mass when the tractor semi-trailer moves. The results of this study form the basis for determining the dynamic tyre force without the installing of sensors on road.

The dynamic response of a moving vehicle has been utilized to extract the frequencies of the supporting bridge. In most previous studies, the vehicle was modeled as a single-degree-of-freedom sprung mass moving over a simple beam, which suffers from the drawback that the sprung mass may be affected by the vehicle motion. To overcome this drawback, this paper presents a two-mass vehicle model for extracting the bridge frequencies, which contains a sprung mass (vehicle body) and an unsprung mass (axle mass). By using the response of the unsprung mass, the bridge response can be more realistically extracted. The main findings of the present study are as follows: (1) the use of unsprung mass in the vehicle model can faithfully reveal the dynamic responses of both the vehicle and bridge, (2) the increase in the unsprung mass can effectively help the extraction of bridge frequencies, including the second frequency, (3) under high levels of road roughness, the proposed model can identify the bridge frequencies, while the single-mass model cannot, and (4) in the presence of vehicle damping, the proposed model can identify the bridge frequencies under high levels of road roughness without additional techniques of processing.

In this study, the critical load and natural vibration frequency of Euler–Bernoulli single nanobeams based on Eringen’s nonlocal elasticity theory are investigated. Cantilever nanobeams with attached sprung masses were subjected to compressed concentrated and distributed follower forces. The parameter that determines the direction of nonconservative follower forces was given the positive and negative values, therefore, sub-tangential and super-tangential load were analyzed. The stability analysis is based on dynamical stability criterion and was carried out using a numerical algorithm for solving segmental nanobeams with many boundary conditions. The presented algorithm is based on the exact solutions of motion equations which are derived from equilibrium conditions for each separated segment of the nanobeam. Two comparison studies are conducted to ensure the validity and accuracy of the presented algorithm. The excellent agreement of critical load for Beck’s nano-column on Winkler foundation observed was confirmed as reported by other researchers. The effect of different values of the nonlocality parameter, tangency coefficient, spring stiffness coefficient, location of sprung mass and the greater number of attached sprung masses on a critical load of nanobeams compressed by nonconservative load are discussed. One of the presented results shows that significant differences between local and nonlocal theory appear when the beam subjected to follower forces loses its stability by flutter.