In this paper, a detailed review on the dynamics of axially moving systems is presented. Over the past 60 years, vibration control of axially moving systems has attracted considerable attention owing to the board applications including continuous material processing, roll-to-roll systems, flexible electronics, etc. Depending on the system’s flexibility and geometric parameters, axially moving systems can be categorized into four models: String, beam, belt, and plate models. We first derive a total of 33 partial differential equation (PDE) models for axially moving systems appearing in various fields. The methods to approximate the PDEs to ordinary differential equations (ODEs) are discussed; then, approximated ODE models are summarized. Also, the techniques (analytical, numerical) to solve both the PDE and ODE models are presented. The dynamic analyses including the divergence and flutter instabilities, bifurcation, and chaos are outlined. Lastly, future research directions to enhance the technologies in this field are also proposed. Considering that a continuous manufacturing process of composite and layered materials is more demanding recently, this paper will provide a guideline to select a proper mathematical model and to analyze the dynamics of the process in advance.

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Dynamic models of axially moving systems: A review

This paper investigates the nonlinear free and forced vibration of axially moving simply-supported thin circular cylindrical shells in the sub-harmonic region, considering the effect of viscous structure damping Motion equations of axially moving circular cylindrical shells in cylindrical coordinates are derived with the aid of the Hamilton principle and employing the Donnell–Mushtari nonlinear shells’ theory Three nonhomogeneous nonlinear partial differential equilibrium equations concerning displacements across the three directions of cylindrical coordinate are reduced to two equations as a result of using the definition of the airy function The particular and private solution of airy stress function is obtained utilizing the assumption of the form of the displacement in the radial direction based on the simply-supported boundary conditions and combination of axisymmetric and asymmetric driven and companion modes By gaining the airy function, the third motion equation is discretized using the Galerkin method into the set of coupled nonlinear nonhomogeneous ordinary differential equations The perturbation method and Runge–Kutta 4th order are employed as a semi-analytical and a numerical solution method, which in practice leads to the prediction of the nonlinear frequencies and amplitudes of various modes of vibration at different velocities and external excitation fluctuation domain and frequency The bifurcation analysis of the velocity and external excitation parameters in sub-harmonic regions shows the changes in the instability condition of the system It causes the appearance of the pitchfork bifurcation and the Neimark–Sacker bifurcation points The accuracy of both, the direct normal form and numerical method results are compared against each other and validated in the particular case in the absence of the velocity with available data The system would be more stable at a higher speed near 1:1 external resonance due to the activation of more asymmetric vibration modes and the growth of the nonlinear softening characteristic in the absence of companion vibration modes

Nonlinear vibration of axially moving simply-supported circular cylindrical shell

Fabric membrane, a typical composite material, is widely applied in building structures, agricultural facilities, packaging engineering, and aeronautical engineering, etc. However, it may fail subject to large-amplitude vibration induced by impact due to its lightweight and small stiffness properties. Herein, the nonlinear damped vibration of a pretensioned rectangular orthotropic membrane structure under impact loading is studied by analytical, numerical and experimental methods. The governing equation is derived based on the von Karman large deflection theory, and the analytical solution is obtained by the Bubnov-Galerkin method and the Krylov-Bogolubov-Mitropolsky (KBM) perturbation method. Meanwhile, the numerical and experimental analysis are carried out for validation of analytical model and good agreement is achieved. Furthermore, parametric study is also performed to find the sensitivity of the design parameters to the vibration response. The results obtained in the paper lay solid foundation for the vibration control and dynamic design of orthotropic membrane structures.

Impact-induced nonlinear damped vibration of fabric membrane structure: Theory, analysis, experiment and parametric study

Transverse vibration and instability of axially travelling web subjected to non-homogeneous tension

https://iopscience.iop.org/article/10.1088/1361-6501/aa9046/pdf

Vibration-based estimation of tension for an axially travelling web in roll-to-roll manufacturing

Nonlinear vibration characteristics of a moving membrane with variable velocity have been examined. The velocity is presumed as harmonic change that takes place over uniform average speed, and the nonlinear vibration equation of the axially moving membrane is inferred according to the D’Alembert principle and the von Karman nonlinear thin plate theory. The Galerkin method is employed for discretizing the vibration partial differential equations. However, the solutions concerning to differential equations are determined through the 4th order Runge–Kutta technique. The results of mean velocity, velocity variation amplitude, and aspect ratio on nonlinear vibration of moving membranes are emphasized. The phase-plane diagrams, time histories, bifurcation graphs, and Poincare maps are obtained; besides that, the stability regions and chaotic regions of membranes are also obtained. This paper gives a theoretical foundation for enhancing the dynamic behavior and stability of moving membranes.

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Nonlinear Parametric Vibration and Chaotic Behaviors of an Axially Accelerating Moving Membrane

The transverse vibration characteristics and stability of a moving membrane with several intermediate elastic supports are investigated considering the translating speed. The motion equations of the vibration system are derived according to the membrane vibration theory and the D'Alembert principle. The constraint of the middle printing roller to the membrane is used as the intermediate support which is replaced by the constraint force. And the vibration characteristic equation is established by using Laplace transformation method and the numerical calculation method. Finally, the relationships between the natural frequency of the moving membrane and the translating speed, tension, supporting position and rigidity modulus are analyzed respectively, and the critical speed when the membrane is in a stable state is obtained. The conclusions provide a theoretical basis and effective approaches for improving the work stability of the printing press.

https://journals.sagepub.com/doi/pdf/10.1260/0263-0923.33.1.65

Transverse Vibration Characteristics and Stability of a Moving Membrane with Elastic Supports

Most studies on the membrane vibration are limited to discussing small deflection linear problems, but rarely on the study of nonlinear large deflection problems. In practice, however, membrane def...

https://journals.sagepub.com/doi/pdf/10.1177/0263092317711597

Nonlinear vibration characteristics and stability of the printing moving membrane

Paper board with bending stiffness is usually used as the substrates on cigarette package printing industry. The vibration of these paper boards in high speed affects the printing precision. The dynamic characteristics and stability of moving paper board with finite interior elastic point supports and elastic edges restrained are investigated. First, the energy function of the system is established by using the extended Hamilton’s principle; second, the dimensionless equations of motion for the moving paper board are obtained using the element-free Galerkin method. The equations of motion and the eigenvalue equations of the system are established. The relationship between the first three complex frequencies of the system and the moving speed is then obtained by the numerical calculation. The effects of the elastic point supports, the elastically restrained edges, and the dimensionless speed of the motion on the dynamic stability of the paper board are analyzed. The critical speed when the paper board is i...

https://journals.sagepub.com/doi/pdf/10.1177/1461348418757881

Dynamic stability of an axially moving paper board with added subsystems

An axially moving printing web with variable density in a printing process causes a geometric nonlinear vibration, and a nonlinear vibration system is established using the von Karman nonlinear pla...

https://journals.sagepub.com/doi/pdf/10.1177/1461348419829371

Yan Wang

Papers

Nonlinear Parametric Vibration and Chaotic Behaviors of an Axially Accelerating Moving Membrane

Transverse Vibration Characteristics and Stability of a Moving Membrane with Elastic Supports

Nonlinear vibration characteristics and stability of the printing moving membrane

Dynamic stability of an axially moving paper board with added subsystems

Nonlinear vibration and stability of a moving printing web with variable density based on the method of multiple scales