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This paper studies transverse vibration of nanoscale cantilevered and bridged sensors carrying a nanoparticle. The nanoscale sensors are modelled as Euler–Bernoulli beams with surface effect and nanoparticle as a concentrated mass. Frequency equations of cantilevered and bridged beam-mass system are derived and exact resonance frequencies are calculated. An alternative Fredholm integral equation method is used to obtain an approximate explicit expression for the fundamental frequency for both cases. A comparison between the approximate and analytical results is made and the approximation accuracy is satisfactory. The influences of the residual surface stress, surface elasticity, and attached mass on the resonance frequencies and mode shapes are discussed. These results are useful to illustrate the surface phenomena and are helpful to design micro-/nano-mechanical sensors.

In order to investigate the transverse vibration of eccentric rotor in a 12/8 switched reluctance motor (SRM), a whole nonlinear coupled vibration equation of eccentric rotor is built with finite element method (FEM). Based on single tooth radial force and the key parameters variation with rotor position, an analytical formula of magnetic resultant taking into account of the rotor’s vibration displacement is deduced in detail, which is applied onto the intermediate node of eccentric rotor in form of concentrated force. Once the windings currents obtained either by experiments or numerical simulations is input, the vibration response can be solved numerically by Newmark-$\beta$ method. Six phase windings currents under angle position control (APC) strategy are chosen as an example and the vibration response are discussed to reflect intrinsic vibration characteristics. From radical resultant vector and its amplitude spectrum, it is proved that the magnetic resultant vector presents multi-petals star shape. The frequency components in magnetic resultant are $X$, $5X$, and $24kX\pm X$, $24kX\pm 5X$, related to rotational speed, current waveform and minimum common multiple of stator and rotor teeth. However, from displacement locus and its amplitude spectrum, the frequency component of the rotor vibration displacement is also related to the critical whirl speed of the rotor. Transverse superharmonic resonance of eccentric rotor appears at some particular rotational speed and result in a larger rotor vibration. If the rotor runs at the superharmonic speed of 1/19 of first-order critical whirl speed, the maximum vibration displacement radius of the eccentric rotor reaches almost four times that of the rated speed. The vibration locus at these particular speed show rich diversity.

The transverse vibrations of cracked beams with rectangular cross sections resting on Pasternak and generalized elastic foundations are considered. Both the Euler–Bernoulli (EB) and Timoshenko beam (TB) theories are used. The open edge crack is represented as a rotational spring whose compliance is obtained by the fracture mechanics. By applying the compatibility conditions between the beam segments at the crack location and the boundary conditions, the characteristic equations are derived, from which the nondimensional natural frequencies are solved as the roots. Sample numerical results showing the effects of crack depth, crack location, foundation type and foundation parameters on the natural frequencies of the beam are presented. It is observed that the existence of crack reduces the natural frequencies, whereas the elasticity of the foundation increases the stiffness of the system and thus the natural frequencies. It is also observed that the type of elastic foundation has a significant effect on the natural frequencies of the cracked beam.

Based on the theory of double-beam dynamic stiffness, a dynamic equilibrium equation for an inclined beam segment was established for the analysis of the dynamic characteristics of a double-layer sheathing cable system. The closed-form transverse dynamic stiffness matrix of the system considering the bending stiffness, inclination angle, boundary condition, and the sag of the beam was obtained simultaneously. The distribution rule of its elements and determinant, defined as dynamic stiffness coefficients and the characteristic function separately, were analyzed in the space and frequency domain. By using a numerical case study, the vibration characteristics of the double-beam system were investigated. It shows that in a given frequency range, the modal frequencies of the double-beam system are affected by both beams, and the beam with smaller stiffness contributes more to the system frequency. The contributions of each single beam to the system frequencies may be further determined by the distribution rule obtained. Based on these results, a method was proposed for quick evaluation of the contribution of each single beam to the modal frequency of the system. This will help to identify the frequency and cable force of double-layer sheathing cables.

For a super high-speed elevator running in a hoistway, it will encounter air flows at high speed. The transverse force and pitching moment generated by the air intensify the transverse vibration of the elevator. In this paper, by fully considering the guide rail excitation and air disturbance, the transverse vibration of a super high-speed elevator under different working conditions is examined. Based on the Lagrange principle, a four degree-of-freedom (DOF) model is adopted for the transverse vibration of the elevator. Combined with computational fluid dynamics (CFD), the effects of various parameters corresponding to different working conditions on the aerodynamic forces acting on the transverse surfaces (the surfaces facing the guide rail) of the car is analyzed. Finally, the Newmark-$\beta$ method is employed to analyze the effect of air disturbances on the transverse vibration acceleration of the car under different working conditions. The results show that when the car is symmetrically positioned, the aerodynamic characteristics on both transverse surfaces of the car also appear to be symmetric. The operating speed and the distance between the car’s transverse surface and the hoistway wall (DCH) have a minor effect on the transverse vibration of the car, and the car is basically in a state of forced balance in the transverse direction. However, once the car deviates from the symmetric position, the balance will be violated, and the transverse resultant force and moment of the car will increase with the increase in the deviation amount. Among all these factors, the influence of the rotation angle on the elevator’s vibration acceleration is the most significant.

A thermoelastic coupling model of axially moving beam based on Timoshenko beam model is established, which comprehensively takes into account the length–diameter ratio, axial load and moving speed, and unify the axial tension and compression load into one governing equation, and study the influence of the length–diameter ratio, axial load, motion speed and thermal load of the beam model. The differential governing equations of transverse vibration of axially moving beam with considering the axial tensile and compressive loads are established based on the Timoshenko beam theory and Hamilton’s principle. The dynamic characteristics of different slender beams with axial load and pinned–pinned, clamped–clamped and clamped-free boundary conditions are investigated, respectively. The dimensionless frequencies of beam calculated numerically with the differential quadrature method (DQM) are compared with analytical solutions for some special cases. For axially moving beams working in the thermal environment, the temperature field inside the beam is simulated by the heat conduction equations, and the thermal effect on dynamic characteristics of beam is studied under different thermal heating cases.

Both functionally graded materials (FGMs) and fluid-conveying pipes have wide applications in engineering communities. In this paper, the transverse vibration and stability of multi-span viscoelastic FGM pipes conveying fluid are investigated. Volume fraction laws including power law, sigmoid law and exponential law are introduced to describe the variations of material properties in FGM pipes. A hybrid method which combines reverberation-ray matrix method and wave propagation method is developed to calculate the natural frequencies, and the results determined by present method are compared with the existing results in literature. Then, a comparative study is performed to investigate the effects of fluid velocity, volume fraction laws and internal damping on transverse vibration and stability of the FGM pipes conveying fluid. The results demonstrate that the present method has high precision in dynamic analysis of multi-span pipes conveying fluid. It is also found that natural frequencies of FGM pipes can be adjusted by devising the volume fractions laws. This particular feature can be tailored to fulfill the special applications in engineering.

The transverse vibration of axially travelling web may lead to poor quality of product in roll-to-roll manufacture, especially for the web with long span, which is usually installed with some elastic supports intermediately. This paper presents a mechanical model for predicting the dynamic behaviors of axially travelling web with intermediate elastic supports. Both of the axial movement and intermediate support are considered in this proposed model. The web is modeled as a thin plate and the elastic supports are modeled as monaxial springs. The derived equations of in-plane motion are linear and uncoupled with the transverse motion, whereas the equation of transverse motion is nonlinear and coupled with the in-plane motions. The Galerkin method is used to discretize the governing equations and compute natural frequencies. The numerical results agree well with the finite element results for the natural frequencies. The dynamic responses of the web are computed by employing the generalized-$\alpha$ time integration method. A single intermediate elastic support case as well as multiple supports case are considered in this paper. The effects of the support position and stiffness on the natural frequency, stability and dynamic response of the travelling web are discussed.

This paper addresses the transverse vibration of a nematic elastomer (NE) beam embedded in soft viscoelastic surroundings with the aim to clarify a new dissipation mechanism caused by dynamic soft elasticity of this soft material. Based on the viscoelasticity theory of NEs in low-frequency limit and the Timoshenko beam theory, the governing equation of motion is derived by using the Hamilton principle and energy method, and is solved by the complex modal analysis method. The dependence of vibration property on the intrinsic parameters of NEs (director rotation time, rubber relaxation time, anisotropic parameter) and foundation (spring, shear and damping constants) are discussed in detail. The results show that dynamic soft elasticity leads to anomalous anisotropy of energy transfer and attenuation. The relative stiffer foundation would restraint the rubber dissipation of viscoelastic beams, but has less influence on the director rotation dissipation, which is particular for NE beams. This study would provide a useful guidance in the dynamic design of NE apparatus embedded in soft viscous media.

In this paper, a fractional differential equation is introduced to describe the transverse vibrations of an axially moving viscoelastic string. An iterative algorithm is constructed to analyze the dynamical behavior. By conveying the memory effect of the fractional differential terms step by step, the computation cost can be greatly reduced. As a numerical example, the effects of the viscoelastic parameters on a moving string are investigated.

The temperature field distribution and the effect of thermal heating on vibration characteristics of axially moving Timoshenko beam are investigated. The governing equation of transverse vibration of axially moving beam considering the axial load is established based on Timoshenko beam theory and Hamilton’s principle. The effect of thermal heating and ratio of length to height on natural frequencies is studied. This research shows that when the load formed by thermal heating reaches the critical load of the beam, the first-order buckling mode will be excited; the effect of thermal heating can decrease the natural frequencies of the beam, the beam undergoes coupled-mode flutter with the increase of the axial velocity, the beam is easier to achieve the state of instability. The temperature plays an important role of the modal characteristics of the beam.