Instability of the motion of a beam of periodically varying length

Zaja̧czkowski, J.; Lipiński, Janusz

doi:10.1016/0022-460x(79)90373-0

Cited by 29 publications

(7 citation statements)

References 1 publication

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“…A simulation with cantilever beam is designed, which is a time-varying structure whose mass is changing over time. For experimental verification, additional damping caused by moving mass and the limitation of experimental condition make it quite difficult [29], so simulation verification is a frequently used method. In the generation of simulation data, we gave full consideration to boundary conditions, and the Gaussian noise is added into the data to simulate real scene.…”

Section: Simulation Verification Of Time-varying Transient Operationamentioning

confidence: 99%

Moving window self-iteration PCE based OMA for slow linear time-varying structures

Wang¹,

Chen²,

Zhang³

2017

J. vibroeng.

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In order to reduce the time and space complexity of operational modal analysis (OMA) for slow linear time-varying (SLTV) structures based on moving window principal component analysis (MWPCA), this paper proposes a new moving window self-iteration principal component extraction (MWSIPCE) method. Different from getting principal components by singular value decomposition (SVD) or eigenvalue decomposition (EVD) in MWPCA algorithm, MWSIPCE just extracts the first-several-orders principal components by self-iteration. Comparing with MWPCA, MWSIPCE has lower time and space complexity. What's more, this paper explains the reason of modal exchange in some data windows in detail, and gives an illustration to how to set window length. The OMA results on non-stationary vibration response simulation signal of time-varying cantilever beam under white noise excitation show that this method can well identify the time-varying transient modal parameters (natural frequencies and modal shapes) for SLTV structures and has less time and space consume, the algorithm is also more precise than MWPCA.

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Section: Simulation Verification Of Time-varying Transient Operationamentioning

confidence: 99%

Moving window self-iteration PCE based OMA for slow linear time-varying structures

Wang¹,

Chen²,

Zhang³

2017

J. vibroeng.

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“…They derived the equations of motion of a simple cantilever beam model utilizing Newton's second law, and assumed a special velocity profile to obtain a semi-analytic solution for specific axial velocities and approximate solutions for various velocities. A perturbation method was introduced by Zajaczkowski and Lipinski [9] to investigate the parametric instability of the motion of a cantilever beam; however, their model was restricted to the periodically varying length. Wang and Wei [10] studied the vibration problem of a moving slender prismatic beam using a modified Galerkin method with time-dependent basis functions based on Newton's second law.…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, using Hamiltonian dynamic analysis, Wang et al [18] investigated an axially translating elastic Bernoulli-Euler cantilever beam featuring time-variant velocity. Clearly, the stability analysis of dynamical systems is very important; we refer to Zajaczkowski and Lipinski [9], Theodore et al [19], Pakdemirli and Ulsoy [20], and Wang et al [18] for further studies.…”

Section: Introductionmentioning

confidence: 99%

Dynamic analysis of an axially translating viscoelastic beam with an arbitrarily varying length

Wang

Zhong

et al. 2010

Acta Mech

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The nonlinear free vibration of an axially translating viscoelastic beam with an arbitrarily varying length and axial velocity is investigated. Based on the linear viscoelastic differential constitutive law, the extended Hamilton's principle is utilized to derive the generalized third-order equations of motion for the axially translating viscoelastic Bernoulli-Euler beam. The coupling effects between the axial motion and transverse vibration are assessed under various prescribed time-varying velocity fields. The inertia force arising from the longitudinal acceleration emerges, rendering the coupling terms between the axial beam acceleration and the beam flexure. Semi-analytical solutions for the governing PDE are obtained through the separation of variables and the assumed modes method. The modified Galerkin's method and the fourth-order RungeKutta method are employed to numerically analyze the resulting equations. Further, dynamic stabilization is examined from the system energy standpoint for beam extension and retraction. Extensive numerical simulations are presented to illustrate the influences of varying translating velocities and viscoelastic parameters on the underlying dynamic responses. The material viscosity always dissipates energy and helps stabilize the transverse vibration.

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“…Another class of flexible extendible beams fall under the broad topic of axially moving solid continua associated with spacecraft antennas. Elmaraghy & Tabarrok (1975) and Zajaczkowski et al (1979Zajaczkowski et al ( , 1980 studied the stability of Euler-Bernoulli beams subjected to periodic sliding motions. These studies were restricted to linear beams; however, the results revealed the complex nature of the instability of sliding beams.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear dynamics of a sliding beam on two supports under sinusoidal excitation

Somnay

Ibrahim

2006

Sadhana

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This study deals with the nonlinear dynamics associated with large deformation of a beam sliding on two-knife edge supports under external excitation. The beam is referred to as a Gospodnetic-Frisch-Fay beam, after the researchers who reported its static deformation in closed form. The freedom of the beam to slide on its supports imparts a nonlinear characteristic to the force-deflection response. The restoring elastic force of the beam possesses characteristics similar to those of the roll-restoring moment of ships. The Gospodnetic-Frisch-Fay exact solution is given in terms of elliptic functions. A curve fit of the exact solution up to eleventhorder is constructed to establish the governing equation of motion under external excitation. The dynamic stability of the unperturbed beam is examined for the damped and undamped cases. The undamped case reveals periodic orbits and one homoclinic orbit depending on the value of the initial conditions. The response to a sinusoidal excitation at a frequency below the linear natural frequency is numerically estimated for different excitation amplitude and different values of initial conditions covered by the area of the homoclinic orbit. The safe basins of attraction are plotted for different values of excitation amplitude. It is found that the safe region of operation is reduced as the excitation amplitude increases.

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Instability of the motion of a beam of periodically varying length

Cited by 29 publications

References 1 publication

Moving window self-iteration PCE based OMA for slow linear time-varying structures

Moving window self-iteration PCE based OMA for slow linear time-varying structures

Dynamic analysis of an axially translating viscoelastic beam with an arbitrarily varying length

Nonlinear dynamics of a sliding beam on two supports under sinusoidal excitation

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