On the asymptotic approximation of the solution of an equation for a non-constant axially moving string

Malookani, Rajab A.; Horssen, Wim T. van

doi:10.1016/j.jsv.2015.12.043

Cited by 14 publications

(8 citation statements)

References 21 publications

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“…Comparison was made at dimensionless transport velocity of '0.1', ' * ' equal to 0.5 and using the initial conditions given in Eqs. (26), (27), Fig. 7(a): First four terms of the analytical model presented in [6] are plotted (Exact) against the model developed in this study (Numerical).…”

Section: Fixed and Varying-speed Web Vibrationmentioning

confidence: 99%

“…(10)- (12) are augmented with the Eqs. (6), (7), and the initial conditions (26) and (27) are used for solving the coupled system for the time-domain response.…”

Section: The Coupled Web Vibration -R2r Dynamics Problemmentioning

confidence: 99%

“…(ii) Papers focusing on nonlinear oscillations of axially moving strings with constant axial velocity and varying transmitted tension [17][18][19][20][21] where parametric excitations and nonlinear stability were analyzed. (iii) Papers analyzing the periodic, quasi-periodic, chaotic, and transient motions of axially moving materials with axial acceleration and constant transmitted tension [22][23][24][25][26][27][28][29][30][31][32]. (iv) Papers presenting parametrically excited nonlinear responses of axially moving strings with time-harmonic varying axial velocity and constant transmitted tension [33][34][35][36] where the effects of parameters such as mean velocity, web stiffness and damping coefficients, and a middle support on frequency response curves and bifurcation points were investigated.…”

Section: Introductionmentioning

confidence: 99%

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Influence of roll-to-roll system’s dynamics on axially moving web vibration

Hawwa¹,

Ali²,

Hardt³

2019

J. vibroeng.

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In this paper, a model for transverse web vibration in a roll-to-roll system is presented. Web axial tension and web axial speed, decisive parameters in the equation of motion that describes web vibration, are rigorously obtained by considering the two rolls-web coupled system's dynamics, coupled with the equation of motion. According to the present analysis, the idealized simply-supported boundary conditions, commonly used in studies on vibrations of axially moving structures are not needed. Instead, a mathematical model comprised of the governing equation of web transverse vibration and the roll angular velocity-web axial tension relationship is solved as a coupled system. A finite-difference based algorithm is used for solving the coupled system of differential equations. It is worth noting that the web axial speed and web-transmitted tension are not constants when a certain amount of the web material is transferred from the unwinding roll to the winding roll; they vary nonlinearly after a short transient period. The transverse vibration response at selected points on the web span shows higher (lower) frequency fluctuations corresponding to lower (higher) transport axial speed. This behavior is significantly different from that of a vibrating web under constant axial speed and tension.

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Section: Fixed and Varying-speed Web Vibrationmentioning

confidence: 99%

“…(10)- (12) are augmented with the Eqs. (6), (7), and the initial conditions (26) and (27) are used for solving the coupled system for the time-domain response.…”

Section: The Coupled Web Vibration -R2r Dynamics Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Influence of roll-to-roll system’s dynamics on axially moving web vibration

Hawwa¹,

Ali²,

Hardt³

2019

J. vibroeng.

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“…Truncations of the solution to a finite number of oscillation modes is quite common in solving initial-boundary value problem such as in the studies of actuated microand nano-beam problems. However, in some problems such as in [14,15], this truncation cannot be done due to the modes internal resonances. This in general cannot be known in advance.…”

Section: Introductionmentioning

confidence: 99%

On resonances in a weakly nonlinear microbeam due to an electric actuation

2021

Self Cite

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In this paper, the oscillations of an actuated, simply supported microbeam are studied for which it is assumed that the electric load is composed of a small DC polarization voltage and a small, harmonic AC voltage. Bending stiffness and mid-plane stretching are taken into account as well as small viscous or structural damping. No tensile axial force is assumed to be present. By using a multiple time-scales perturbation method, approximations of the solutions of the initial-boundary value problem for the microbeam equation are constructed. This analysis is performed without truncating the infinite series representation in advance as is usually done in the existing literature. It is shown in which cases truncation is allowed for this problem. Moreover, accurate and explicit approximations of the natural frequencies up to order ε 3 of the actuated microbeam are also obtained. Intriguing and new modal vibrations are found when the frequency of the harmonic AC voltage is (near) half or twice a natural frequency of the microbeam, i.e., near a superharmonic or a subharmonic resonance.

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“…Miranker [1] was the first who developed the mathematical equations of motion for laterals oscillations of axially accelerating string. For further studies on vibrations of string and beams, the reader is referred the papers [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. To suppress the vibrations and noise in structures and machines, damping of string material is widely taken into consideration [17].…”

Section: Introductionmentioning

confidence: 99%

On (Non) Applicability of a Mode-Truncation of a Damped Traveling String

Malookani

Sandilo

Hanan³

2019

Mehran Univ. res. j. eng. technol.

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This study investigates a linear homogeneous initial-boundary value problem for a traveling stringunder linear viscous damping. The string is assumed to be traveling with constant speed, while it is fixed at both ends. Physically, this problem represents the vertical (lateral) vibrations of damped axially moving materials. The axial belt speed is taken to be positive, constant and small in comparison with a wave speed, and the damping is also considered relatively small. A two timescale perturbation method together with the characteristic coordinate's method will be employed to establish the approximateanalytic solutions. The damped amplitude-response of the system will be computed under specific initial conditions. The obtained results are compared with the finite difference numerical technique for justification. It turned out that the introduced damping has a significant effect on the amplitude-response.Additionally, it is proven that the mode-truncation is applicable for the damped axially traveling string system on a timescale of order  -1 .

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On the asymptotic approximation of the solution of an equation for a non-constant axially moving string

Cited by 14 publications

References 21 publications

Influence of roll-to-roll system’s dynamics on axially moving web vibration

Influence of roll-to-roll system’s dynamics on axially moving web vibration

On resonances in a weakly nonlinear microbeam due to an electric actuation

On (Non) Applicability of a Mode-Truncation of a Damped Traveling String

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