On the influence of lateral vibrations of supports for an axially moving string

Horssen, Wim T. van

doi:10.1016/s0022-460x(03)00362-6

Cited by 35 publications

(14 citation statements)

References 9 publications

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“…String Model e purpose of this section is to discuss the contour of modal motions of the axially moving string model. e mode functions can be derived by the analytical method [37,38], and the exact solutions of the string model in real mode form has been obtained [39,40]. e solutions to equation (3) are assumed as…”

Section: Travelling Wave Modes Of Axially Movingmentioning

confidence: 99%

On Travelling Wave Modes of Axially Moving String and Beam

Lü

Yang

Zhang

et al. 2019

Shock and Vibration

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The traditional vibrational standing-wave modes of beams and strings show static overall contour with finite number of fixed nodes. The travelling wave modes are investigated in this study of axially moving string and beam although the solutions have been obtained in the literature. The travelling wave modes show time-varying contour instead of static contour. In the model of an axially moving string, only backward travelling wave modes are found and verified by experiments. Although there are n − 1 fixed nodes in the nth order mode, similar to the vibration of traditional static strings, the presence of travelling wave phenomenon is still spotted between any two adjacent nodes. In contrast to the stationary nodes of string modes, the occurrence of galloping nodes of axially moving beams is discovered: the nodes oscillate periodically during modal motions. Both forward and backward travelling wave phenomena are detected for the axially moving beam case. It is found that the ranges of forward travelling wave modes increase with the axially moving speed. It is also concluded that backward travelling wave modes can transform to the forward travelling wave modes as the transport speed surpasses the buckling critical speed.

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Section: Travelling Wave Modes Of Axially Movingmentioning

confidence: 99%

On Travelling Wave Modes of Axially Moving String and Beam

Lü

Yang

Zhang

et al. 2019

Shock and Vibration

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“…One can see that F 1 disappears and G 3 is the reflected wave of F 2 at x = l 0 . Using the boundary condition (7), one has…”

Section: Fixed-fixed Casementioning

confidence: 99%

“…Based on the transfer function formulation and wave propagation, Tan and Ying [6] subsequently derived an exact solution for the response of a translating string with general boundary conditions. Van Horssen [7] used a Laplace transform method instead, constructing exact solutions of the lateral vibrations in travelling strings due to small lateral vibrations of the supports.…”

Section: Introductionmentioning

confidence: 99%

A reflected wave superposition method for vibration and energy of a travelling string

Chen

Luo

Ferguson

et al. 2017

Journal of Sound and Vibration

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This paper considers the analytical free time domain response and energy in an axially translating and laterally vibrating string. The domain of the string is either a constant or variable length, dependent upon the general initial conditions. The translating tensioned strings possess either fixed-fixed or fixed-free boundaries. A reflected wave superposition method is presented as an alternative analytical solution for a finite translating string. Firstly, the cycles of vibration for both constant and variable length strings are provided, which for the latter are dependent upon the variable string length. Each cycle is divided into three time intervals according to the magnitude and the direction of the translating string velocity. Applying d'Alembert's method combined with the reflection properties, expressions for the reflected waves at the two boundaries are obtained. Subsequently, superposition of all of the incident and reflected waves provides results for the free vibration of the string over the three time intervals. The variation in the total mechanical energy of the string system is also shown. The accuracy and efficiency of the proposed method are confirmed numerically by comparison to simulations produced using a Newmark-Beta method solution and an existing state space function representation of the string dynamics.

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“…This idealization has led to assuming ''artificial'' web tension and web speed profiles. In several cases, the idyllic conditions of constant tension and constant speed have been assumed [15][16][17][18][19][20][21][22][23][24][25][26][27]. Initials studies on the axially moving materials were of Sack [15] and Archibald and Emslie [17].…”

Section: Introductionmentioning

confidence: 99%

A parametric study on the dynamics of two-span roll-to-roll microcontact printing system

Ali

Hawwa

2019

Sādhanā

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The recent trend towards micro-patterning of thin film webs brought stricter manufacturing tolerances than traditionally admissible vibration levels. Therefore, one has to consider roll-to-roll (R2R) systems with their system dynamics model encompassing all systems' components, namely, the axially moving web(s), the rotating rolls and other accessories. In this study, the dynamic characteristics of an axially moving web in a two-span R2R system are investigated. Influence of geometrical parameters on the R2R dynamics and the parametric resonances of the moving web is presented. The time variations of web axial speed and webtransmitted tension are rigorously obtained by solving the web tension-roller angular speed equation, which is obtained from the conservation of mass law. The effects of system parameters such as length of the web, inertia of idler roll and radius of the rewinding/unwinding roll on the angular speed and web-transmitted tension have been discussed. The axially moving web was modelled as a string, which was mathematically represented by a second-order hyperbolic partial differential equation. The effect of the frequency of the disturbance at the roll on the web dynamics is also discussed. The transverse vibration response at selected points on the web shows higher frequency fluctuations corresponding to lower transport axial speed. It is noted that instability in the transverse response will occur when the first and second frequencies of the oscillation converge.

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On the influence of lateral vibrations of supports for an axially moving string

Cited by 35 publications

References 9 publications

On Travelling Wave Modes of Axially Moving String and Beam

On Travelling Wave Modes of Axially Moving String and Beam

A reflected wave superposition method for vibration and energy of a travelling string

A parametric study on the dynamics of two-span roll-to-roll microcontact printing system

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