Exact free vibration analysis of axially moving viscoelastic plates
Introduction
Axially moving plates, such as power transmission belts, polymer sheets in publishing industries and magnetic tapes are commonly used in industry. In some instances, the mechanical properties of these polymeric or composite plates cannot be described by elasticity or viscosity theories alone. To model the mechanical behavior of these plates, recourse to more general theories such as viscoelasticity is required.
In viscoelastic materials, the state of strain in an element at a particular time not only depends on the state of stress at the time but also on the history of stresses. Similarly, the state of stress of an element depends on the history of strains. This property of viscoelastic materials, is called the Memory Effect [1].
Viscoelastic modeling of moving objects was originally performed by Fung et al. [2]. They studied the transient motion of an axially moving viscoelastic string constituted by the Boltzmann superposition principle. Later, Fung et al. [3] extended their previous work to investigate the effect of damping properties of material on the nonlinear free vibration of moving belts. Zhang and Zu [4], [5] considered the nonlinear free and forced vibrations and parametrically excited oscillations of a moving belt by adopting a Kelvin model of differential type for the viscoelastic material. Based on the beam belt, the dynamic stability of a moving web with uniform initial tension was investigated by Marynowski and Kapitaniak [6] using two different rheological models, namely, the Kelvin–Voigt and Burgers models. Marynowski [7] investigated the nonlinear vibration of a beam-like model of an axially moving web with time-dependent tension. Chen et al. [8] studied the transverse vibration of an initially stressed viscoelastic string obeying a fractional differential constitutive law. Later on, Chen and Yang [9] investigated the stability and vibration of axially accelerating viscoelastic beams. Finally, Lee and Oh [10] derived a spectral element model for the dynamics and stability of axially moving viscoelastic beams subjected to axial tension.
In all of the above studies, the researchers have used one-dimensional string or beam models to model the behavior of axially moving objects. The use of one-dimensional models instead of two-dimensional models for axially moving objects simplifies the analysis and often renders acceptable results when the objects are long and narrow. When the width of the object is appreciable however, it is often better to employ 2D models. Although there have been several 2D models of elastic moving objects using membrane and plate theories [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], the 2D modeling of viscoelastic moving objects is limited to the work of Marynowski [22].
Marynowski [22] used a two-dimensional rheological element to model an axially moving viscoelastic web material. Using the rheological model and the plate theory, he carried out some numerical investigations of dynamic stability of the paper web.
In the present study, an exact solution for the free vibration of viscoelastic moving plates subjected to in-plane forces is presented. The solution is based on the classical theory of plates and the exact finite strip method (FSM), which has been presented previously by the first author and his colleagues [16], [17], [18] for elastic isotropic, orthotropic and composite moving plates. In this paper, using the governing differential equation that defines the dynamics of a moving plate (the viscoelastic properties of which follow rheological models), the exact stiffness matrix of the plate is extracted. This stiffness matrix is a complex matrix and contains axial speed, viscoelastic properties, in-plane forces and the unknown free vibration frequencies. Assembling the stiffness matrices of several strips of this kind, taking into account the conventional compatibility conditions at the common nodes, yields the stiffness matrix of the whole plate which can be analyzed for free vibration and stability analysis. The exact vibration characteristics obtained here can be used as a benchmark to determine the accuracy of non-exact numerical methods.
Section snippets
Theory
The basic hypothesis of linear viscoelasticity, that the stress tensor is linearly dependent upon the past history of the strain tensor, is formally expressed aswhere σij(t) is the stress history tensor; εij(t) is the strain history tensor and Rijkl is the fourth order tensor of relaxation function. Using Laplace transform, it is seen that this hypothesis is equivalent to the existence of a complex tensor such thatFrom a practical
Numerical results
In order to extract the numerical results, the three-parameter standard viscoelastic solid model of Fig. 3 is utilized to model the shear behavior of the moving plate, whereas the hydroelastic behavior of the material is considered to be elastic, so that we havein which μ0 and κ0 are the shear modulus and the bulk modulus of material at the frequency of zero, and c1 and βc1 are two relaxation times. Therefore the engineering moduli that are dependent on the
Conclusions
The behavior of many two-dimensional moving plates such as those made from polymer composites is more accurately modeled assuming viscoelasticity. In this paper, an exact finite strip model is presented for the free vibration analysis of axially moving viscoelastic plates. In this model, the exact stiffness matrix of a viscoelastic strip moving with a constant speed and subjected to in-plane forces which are constant across the strip width is extracted using the classical theory of thin plates
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