Dynamic response of clamped axially moving beams: Integral transform solution

The generalized integral transform technique (GITT) is employed to obtain a hybrid analyt-ical–numerical solution for dynamic response of clamped axially moving beams. The use of the GITT approach in the analysis of the transverse vibration equation leads to a coupled system of second order differential equations in the dimensionless temporal variable. The resulting transformed ODE system is then solved numerically with automatic global accuracy control by using the subroutine DIVPAG from IMSL Library. Excellent convergence behavior is shown by comparing the vibration displacement of different points along the beam length. Numerical results are presented for different values of axial translation velocity and flexural stiffness. A set of reference results for the transverse vibration displacement of axially moving beam is provided for future co-validation purposes. Dynamic behavior of axially moving beams has been extensively studied because of its technological relevance in various applications such as data storage tapes [1], textile machines [2], automotive belts [3], band saws [4] and fluid conveying pipes [5,6]. Many experimental, analytical and numerical approaches have been employed to investigate the vibration characteristics and dynamic stability of axially moving beams. By simplifying the tape drive to a fixed–fixed Euler–Bernoulli beam model with axial velocity, Hayes and Bhushan [1] determined the natural frequencies and mode shapes of the gyroscopic system, performed parametric studies on axial velocity, tension, free span length and tape thickness, and compared the numerical results to experimental data measured by both static and dynamic methods. Lee and Mote [5] presented the energy expression of translating tensioned beams and fluid conveying pipes, and discussed the dynamic stability of the translating continua under both symmetric and asymmetric boundary configurations. Ni et al. [6] demonstrated the application of the technique of differential transformation method to the free vibration problem of pipes conveying fluid with several typical boundary conditions, where the natural frequencies and critical flow velocities were obtained. Based on the method of multiple scales, Öz and Pakdemirli [7] and Öz [8] studied the stability boundaries of an axially moving Euler–Bernoulli beam with time-dependent velocity under simply supported and clamped boundary condition, respectively. Chen and Zhao [9] obtained a conserved quantity in the free nonlinear transverse vibration of axially moving nonlinear beams with simple or fixed supports , which was applied to verify the Lyapunov stability of the straight equilibrium configuration of a beam moving with low axial speed. Chen et al. [10] applied the Galerkin method to …