Classical Vibration Analysis of Axially Moving Continua

Abstract

Axially moving continua, such as high-speed magnetic tapes and band saw blades, experience a Coriolis acceleration component which renders such systems gyroscopic. The equations of motion for the traveling string and the traveling beam, the most common models of axially moving materials, are each cast in a canonical state space form defined by one symmetric and one skew-symmetric differential operator. When an equation of motion is represented in this form, the eigenfunctions are orthogonal with respect to each operator. Following this formulation, a classical vibration theory, comprised of a modal analysis and a Green’s function method, is derived for the class of axially moving continua. The analysis is applied to the representative traveling string and beam models, and exact closed-form expressions for their responses to arbitrary excitation and initial conditions result. In addition, the critical transport speed at which divergence instability occurs is determined explicitly from a sufficient condition for positive definiteness of the symmetric operator.

On the construction of the solution of an equation describing an axially moving string

“…The normalization requirements can be found in [1]. The φ n (x) satisfy the orthonormality relations…”

“…One of these methods is the method of modal analysis based on eigenfunction expansions. This method has been introduced in [6,7] and in [1], and is used nowadays frequently for these types of problems (see for instance [3,4]). To apply this method an operator notation has to be introduced, an inner product has to be defined, an eigenvalue problem has to be solved, and orthonormality relations have to be determined.…”

“…This last method was developed for this problems by Meirovitch in [6,7] and by Wickert and Mote in [1]. Both methods will be compared.…”

Analysis of the thermo-elastic vibration for axially moving Euler beam

“…Simpson [2] analyzed the natural frequencies and mode curves of the axially moving beam with clamped boundary, his calculated results show that modes distorted violently as the moving speed increases. Moreover, Wickert [3] brought forth that axially moving beam belongs to gyroscopic system and researched the forced vibration of axially moving beam by modal analysis and Green's function method. The above articles conducted their investigations on the vibration performance and response of the axially moving beam under a certain uniform temperature field.…”

Analytical Analysis on Nonlinear Parametric Vibration of an Axially Moving String with Fractional Viscoelastic Damping

“…In many cases, due to initial, parametric, and external excitations, the generation of unwanted transverse vibrations may limit their applications. Therefore, the dynamic behaviors of such devices have been widely investigated by numerous scholars for the past decades and are still of interesting today [1][2][3][4][5][6][7][8][9][10]. Moreover, according to the demand of the actual engineering problems, some researchers have applied their theoretical results to design and optimize the axially moving structures [11,12].…”