Parametric resonance of axially moving Timoshenko beams with time-dependent speed

Tang, You-Qi; Chen, Liqun; Yang, Xiao‐Dong

doi:10.1007/s11071-009-9512-1

Cited by 47 publications

(10 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…11). Also earlier mentioned approach given in [21] gives considerable lower resonant amplitudes in comparison with amplitudes calculated with authors own program solution.…”

Section: Amplitudes Of Response Of the Travelling Beam At Principal Pmentioning

confidence: 81%

See 1 more Smart Citation

Influence of the longitudinal displacement on the nonlinear principal parametric resonance of the woodworking bandsaw

2017

View full text Add to dashboard Cite

Original scientific paper The transverse vibrations of axially moving Timoshenko beam, as suitable mathematical models for woodworking bandsaws, are investigated. Special attention is paid to the influence of longitudinal displacement effect, as opposed to most models which can be usually encountered in the literature. This influence is introduced through the integro-partial differential equations. The expressions for the mode shapes in the case of hybrid supports with different torsion spring stiffness on the support points are also derived. The influence of mean beam velocity and axial tension on its natural frequencies and mode shapes is also investigated. Based on the nonlinear model, the amplitudes of the steady-state response are calculated for the case of principal parametric resonance. Developed program solution was tested on a number of earlier known examples. Present theoretical considerations, with the help of the program solution, is also used to analyse an example from industrial practice. Keywords: hybrid boundary conditions; method of multiple scales; numerical examples; principal parametric resonance; travelling Timoshenko beam Utjecaj uzdužnog pomaka na nelinearne glavne parametarske rezonancije tračne pileIzvorni znanstveni članak U radu su istražene poprečne vibracije aksijalno pomične Timoshenkove grede, kao podesnog modela za matematičko modeliranje tračnih pila. Posebno je obrađen utjecaj aksijalnog pomaka koji je, za razliku od većine modela koji se susreću u literaturi, uveden preko integro-parcijalne diferencijalne jednadžbe. Također su izvedeni izrazi za forme vibriranja za slučaj hibridnih oslonaca s različitim krutostima torzijskih opruga u točkama oslonaca. Ispitivan je utjecaj srednje brzine gibanja grede kao i aksijalnog naprezanja na vlastite frekvencije i oblike vibriranja. Razvijen je i nelinearni model na temelju kojega su izračunati amplitudni odzivi nelinearnih ravnomjernih vibracija za slučaj parametarske rezonancije. Izrađeno programsko rješenje testirano je na određenom broju ranije poznatih primjera. Iznijeta teorijska razmatranja, uz pomoć programskog rješenja, poslužila su i za analizu jednog od dostupnih primjera iz industrijske prakse.

show abstract

“…11). Also earlier mentioned approach given in [21] gives considerable lower resonant amplitudes in comparison with amplitudes calculated with authors own program solution.…”

Section: Amplitudes Of Response Of the Travelling Beam At Principal Pmentioning

confidence: 81%

“…(20) is differentiated with respect to x, earlier mentioned derivations should be finally inserted in. The result of this procedure is the following uncoupled equation (21) In the similar manner the uncoupled Eq. (22) …”

Section: Exact Solution Of the Linear Problemmentioning

confidence: 99%

Influence of the longitudinal displacement on the nonlinear principal parametric resonance of the woodworking bandsaw

2017

View full text Add to dashboard Cite

show abstract

“…(19) into Eq. (18), multiplying the resulting equation by ϕ i (ξ), and integrating over the spatial domain gives…”

Section: Methods Of the Solutionmentioning

confidence: 99%

“…Equations (3a)-(3c) form a set of linear partial differential equations which can be solved via the method of separation of variables [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27]. As such, the general solution for the displacement field of each subsystem can be expressed as a series expansion of the following form…”

Section: Analytical Solutionmentioning

confidence: 99%

Free vibrations of beam-mass-spring systems: analytical analysis with numerical confirmation

2012

View full text Add to dashboard Cite

Free vibrations of a beam-mass-spring system with different boundary conditions are analyzed both analytically and numerically. In the analytical analysis, the system is divided into three subsystems and the effects of the spring and the point mass are considered as internal boundary conditions between any two neighboring subsystems. The partial differential equations governing the motion of the subsystems and internal boundary conditions are then solved using the method of separation of variables. In the numerical analysis, the whole system is considered as a single system and the effects of the spring and point mass are introduced using the Dirac delta function. The Galerkin method is then employed to discretize the equation of motion and the resulting set of ordinary differential equations are solved via eigenvalue analysis. Analytical and numerical results are shown to be in very good agreement.

show abstract

“…Tang et al [14] investigated nonlinear vibrations under weak and strong external excitations of axially moving beams based on the Timoshenko model. Tang et al [15] studied parametric resonance of axially moving Timoshenko beams with time-dependent speed.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear free transverse vibrations of axially moving Timoshenko beams with two free ends

Tang

Chen

2011

Sci. China Technol. Sci.

View full text Add to dashboard Cite

In this paper, nonlinear transverse vibrations of axially moving Timoshenko beams with two free ends are investigated. The governing equations and the associated boundary conditions are derived by the extended Hamilton principle. The method of multiple scales is applied to analyze the nonlinear partial differential equation. The natural frequencies and modes are investigated by performing the complex mode approach. The effect of natural frequencies with the stiffness and the axial speeds are numerically demonstrated. The solvability conditions are established for the cases of without and with 3:1 internal resonances. The relationships between the nonlinear frequencies and the initial amplitudes at different axial speeds and the nonlinear coefficients are showed for the case of without internal resonances. The effects of the related coefficients are demonstrated for the case of 3:1 internal resonances. nonlinear transverse vibration, Timoshenko beam, complex mode approach, method of multiple scales, internal resonances Citation:Li B, Tang Y Q, Chen L Q. Nonlinear free transverse vibrations of axially moving Timoshenko beams with two free ends.

show abstract

Parametric resonance of axially moving Timoshenko beams with time-dependent speed

Cited by 47 publications

References 22 publications

Influence of the longitudinal displacement on the nonlinear principal parametric resonance of the woodworking bandsaw

Influence of the longitudinal displacement on the nonlinear principal parametric resonance of the woodworking bandsaw

Free vibrations of beam-mass-spring systems: analytical analysis with numerical confirmation

Nonlinear free transverse vibrations of axially moving Timoshenko beams with two free ends

Contact Info

Product

Resources

About