Transverse Vibration of Two Axially Moving Beams Connected by an Elastic Foundation

Abstract

Transverse vibration of two axially moving beams connected by a Winkler elastic foundation is analyzed analytically. The system is a model of paper and paper-cloth (wire-screen) used in paper making. The two beams are tensioned, translating axially with a common constant velocity, simply supported at their ends, and of different materials and geometry. Due to the effect of translation, the dynamics of the system displays gyroscopic motion. The Euler-Bernoulli beam theory is used to model the deflections, and the governing equations are expressed in the canonical state form. The natural frequencies and associated mode shapes are obtained. It is found that the natural frequencies of the system are composed of two infinite sets describing in-phase and out-of-phase vibrations. In case the beams are identical, these modes become synchronous and asynchronous, respectively. Divergence instability occurs at the critical velocity; and, the frequency-velocity relationship is similar to that of a single traveling beam. The effects of the mass, flexural rigidity, and axial tension ratios of the two beams, as well as the effects of the elastic foundation stiffness are investigated.

Dynamic responses of a damaged double Euler–Bernoulli beam traversed by a ‘phantom’ vehicle

“…Existing literature indicates that there are works that have considered certain aspects of this problem, but they are in a disjointed manner and without the impact of moving loads. Takahashi 37 and Takahashi and Yoshioka 38 presented a double‐beam model with follower force and estimated its modal properties with cracks, as has been by Li and Sun 39 for arbitrary boundary conditions, similar to others 18 with various solution methods 40,41 . Stojanović et al 30 have provided closed‐form solutions for Timoshenko and Rayleigh models (Stojanović and Kozić 26 and Zhao et al 24 ), as has been by Škec et al 42 for brittle and quasi‐brittle surfaces.…”

“…For lower range of velocities around (1-30) km/h, the control of RMS of displacement for the primary beam, secondary beam and the vehicle is not significant using either single or multiple TMDs. For velocity range of around (30)(31)(32)(33)(34)(35)(36)(37)(38)(39)(40)(41)(42)(43)(44)(45) km/h, the TMDs detune in this region and adversely effect the displacement response by around 2%. The TMDs start performing better for higher range of velocity around (60-180) km/h where the displacement response control is around 4% using multiple TMDs and around 1% for a single TMD.…”

“…Figure 23 shows the velocity range where the TMDs are failing to control the acceleration response of the vehicle. In a narrow range of (38)(39)(40)(41)(42)(43)(44)(45) km/h, the TMDs adversely effect the primary system and increase the responses by approximately 0.5%.…”

“…Takahashi 37 and Takahashi and Yoshioka 38 presented a double‐beam model with follower force and estimated its modal properties with cracks, as has been by Li and Sun 39 for arbitrary boundary conditions, similar to others 18 with various solution methods. 40 , 41 Stojanović et al 30 have provided closed‐form solutions for Timoshenko and Rayleigh models (Stojanović and Kozić 26 and Zhao et al 24 ), as has been by Škec et al 42 for brittle and quasi‐brittle surfaces. For composites, similar work exist (Liu and Yang 22 and Rezaiee‐Pajand and Hozhabrossadati 28 ), including buckling (Bochicchio et al 43 and Deng et al 25 ).…”

Transverse Response of an Axially Moving Beam with Intermediate Viscoelastic Support

“…Several studies have attempted to analyze axially moving beams with elastic foundations. Dynamic analysis of coupled moving beams through a Winkler elastic support was investigated by Gaith and Müftü [17]. Yang et al [18] studied the free vibration of axially moving elastic beams resting on an elastic foundation.…”