Effect of Viscoelasticity on the Natural Frequencies of Axially Moving Continua

Abstract

Linear models of axially moving viscoelastic beams and viscoelastic pipes conveying fluid are considered. The natural frequencies of the models are calculated. For both models, viscoelasticity terms are assumed to be of order one. Natural frequencies corresponding to various beam and pipe parameters are presented. Effects of viscoelasticity on the natural frequencies are discussed.

1. Introduction

High-speed magnetic tapes, fiber windings, band saw blades, thread lines, power transmission chain and belts, aerial cable tramways, paper sheets, fluid conveying pipes, and so forth are technological examples of axially moving continua. Understanding the dynamics of axially moving continua is very important to design and to prevent such systems from hazards and accidents. Both linear and nonlinear models exist for vibrations of axially moving systems in the literature. A comprehensive review of the literature was presented by Ulsoy et al.[1] and Wickert and Mote Jr. [2]. Transverse vibrations of travelling beam and string models were investigated by Wickert and Mote Jr. [3]. In that study, equations of motion for string and beam were represented in a canonical form. To analyse the models, modal analysis and Green's function method were used. Wickert and Mote Jr. [4] further investigated complex model analysis for axially moving materials. Ulsoy and Mote Jr. [5] studied bandsaw vibrations and their stability. The study emphasized that the natural frequencies of the bands decrease by increasing axial velocity and increase by increasing axial tension or strain. To analyse axially moving continua vibrations, the transport speed can be modeled as constant or variable. The equation of motion of axially moving continua with variable speed was first derived by Miranker [6]. Miranker analysed transverse vibrations of a tape moving between a pair of pulley by using a variational method. The problem of an axially accelerating string with harmonic excitation at one end was investigated by Mote Jr. [7] who determined the stability by using Laplace transformation method. Pakdemirli et al. [8] rederived the equations by using Hamilton's principle. They analysed the stability and response of the system using Floquet theory. Transport velocity variation was assumed to be sinusoidal about a mean velocity of zero. Mockensturm et al. [9] presented an axially moving string vibration model with constant transport velocity and variable tension. Stabilities for different resonance conditions and limit cycle were analysed. Pakdemirli and Batan [10] considered periodic constant acceleration and deceleration type of velocity variations for axially accelerating strips. Öz and Pakdemirli [11] modelled an axially moving Euler-Bernoulli type beam vibration with harmonically varying velocities. The work covered principal parametric resonances, sum and difference type combination resonances. Pakdemirli and Öz [12] gave a detailed resonance analysis for which up to four modes were involved and obtained stability regions for axially moving beams.

Vibrations of pipes conveying fluid possess a similar mathematical model and hence were considered within the context of axially moving continua. Those models were analyzed by many researchers. Païdoussis and Li [13] analyzed in detail pipes conveying fluids. They discussed both linear and nonlinear dynamics of such systems. Öz and Boyacı [14] and Öz [15] examined transverse vibrations of tensioned pipes conveying fluid with fluctuating fluid velocities. In the mentioned papers [14, 15] linear and nonlinear equations of motion were presented.

Axially moving viscoelastic continua vibrations were studied by many researchers. Chen et al. [16] investigated bifurcation and chaos in transverse vibrations of axially moving Kelvin type viscoelastic strings. They analysed the bifurcation and chaos in 1-term and 2-term truncated systems using Galerkin's method. Regular and chaotic vibrations of axially moving strings were investigated for 1st, 2nd, 3rd, and 4th order truncated systems by the same authors [17]. Asymptotic analysis of axially accelerating strings was done by Chen et al. [18]. The authors analysed Mote's and Kirschoff's nonlinear models with variable velocity using asymptotic perturbation method. Natural frequencies of the systems were found and principal parametric resonances were analyzed. Effects of stiffness, viscosity, initial stress, and axial speed fluctuation on the string vibrations were treated and compared for two models. Lee and Oh [19] considered dynamics of axially moving beams subjected to axial tension. They developed spectral element method and investigated the effects of viscoelasticity and moving speed. First, they found the lowest five eigenvalues of simply supported, stationary, one and two span beams using spectral element method, finite element method, and exact theory. Then they obtained the effects of moving speed and viscoelasticity on the variation of the lowest two eigenvalues. The Method of Multiple Scales (a perturbation method) was used by Chen and Yang [20]. They conducted a stability analysis for parametric resonances of axially moving viscoelastic beams. Stability boundaries were investigated for different boundary conditions. The method of averaging was applied to the 2-term Galerkin truncation of the equation of motion of axially accelerating viscoelastic beam by Chen et al. [21]. They analyzed effects of dynamic viscosity, the mean axial speed, and the tension in parametric vibrations of beams. The Method of Multiple Scales was also applied to the nonlinear vibration model of axially moving viscoelastic string supported by a partial viscoelastic guide by Ghayesh [22]. Ghayesh obtained stability and bifurcation point variations on frequency response curves for different string and guide parameters (i.e., viscosity, guide length, speed, nonlinearity coefficient, damping coefficient, and guide stiffness). Panda and Kar [23] studied nonlinear dynamics of viscoelastic pipes conveying fluids. They analyzed combination, principal, parametric, and internal resonances of pipes.

Özhan and Pakdemirli [24, 25] proposed a general solution algorithm to solve an arbitrary cubic nonlinear vibration model which covers gyroscopic systems. In the mentioned studies, axially moving beam and axially moving viscoelastic beam models were taken as application problems. Forced vibrations of axially moving continua were analyzed. Primary resonances and three-to-one internal resonances of forced vibrations were obtained. Özhan and Pakdemirli [26] obtained primary parametric resonances of a generalized cubic nonlinear vibration model. Pipe conveying fluid model was taken as an application problem.

The objective of this study is to show the effect of viscoelasticity on the natural frequencies of axially moving beams and pipes conveying fluids. For both models, the linear equations of motion are considered. Numerical results are discussed and the natural frequencies are obtained. The effects of various beam and pipe parameters on the natural frequencies of the system are investigated in detail in the figures. A detailed analysis of the natural frequencies in the case of viscoelasticity is presented in this study for the first time.

2. Equations of Motion

In this section, equations of motion of axially moving viscoelastic beams and viscoelastic pipes conveying fluid will be presented.

2.1. Axially Moving Viscoelastic Beam Model

Equation of motion of a uniform axially moving viscoelastic beam and boundary conditions in linear nondimensional form are [11, 27]

\overset{‥}{w} + (v_{0}^{2} - 1) w^{′′} + 2 v_{0} {\dot{w}}^{'} + v_{f}^{2} w^{I V} + ∊ α {\dot{w}}^{I V} = 0,

(1)

w (0, t) = w (1, t) = w^{′′} (0, t) = w^{′′} (1, t) = 0 .

(2)

In (1)w(x, t) is transverse displacement of the beam. v₀ is the constant axial transport velocity, dot denotes differentiation with respect to time, and prime denotes differentiation with respect to the spatial variable. ∊ is a small perturbation parameter. The dimensionless parameters are

x = \frac{\bar{x}}{L}, w = \frac{\bar{w}}{L}, t = \bar{t} \sqrt{\frac{P}{ρ A L^{2}}}, v_{0} = \frac{{\bar{v}}_{0}}{v_{c}}, v_{c} = \sqrt{\frac{P}{ρ A}}, v_{f} = \sqrt{\frac{E I}{P L^{2}}}, α = \frac{I η}{L^{3} \sqrt{ρ A P}},

(3)

where x and t are nondimensional spatial and time variables, $\bar{x}$ and $\bar{t}$ are dimensional forms of spatial and time variables. $\bar{w} (\bar{x}, \bar{t})$ is dimensional form ofdisplacement. η represents viscosity. Viscoelastic material obeys the Kelvin model. α is the dimensionless parameter related to the viscosity. L is the beam length, A is the cross sectional area, ρ is the density of the beam, ρA is the mass per unit length of the beam, and ${\bar{v}}_{0}$ and v_c are the dimensional and critical axial velocities of the beam, respectively. P is the axial tension force and I is the second moment of area of the beam cross section.

2.2. Viscoelastic Pipe Conveying Fluid Model

The nondimensional linear equation of motion and boundary conditions of viscoelastic pipes conveying fluid are [15, 23]

\overset{‥}{w} + w^{I V} + (v_{0}^{2} - P_{0}) w^{′′} + 2 \sqrt{β} v_{0} {\dot{w}}^{'} + ∊ α {\dot{w}}^{I V} = 0,

(4)

w (0, t) = w (1, t) = w^{′′} (0, t) = w^{′′} (1, t) = 0,

(5)

where

\begin{matrix} x = \frac{\bar{x}}{L}, w = \frac{\bar{w}}{L}, t = ω_{0} \bar{t}, \\ ω_{0} = \frac{1}{L^{2}} \sqrt{\frac{E I}{m_{fluid} + m_{pipe}}}, v_{0} = {\bar{v}}_{0} L \sqrt{\frac{m_{fluid}}{E I}}, \\ α = \frac{E^{*}}{L^{2}} \sqrt{\frac{I}{(m_{fluid} + m_{pipe}) E}}, β = \frac{m_{fluid}}{m_{fluid} + m_{pipe}} . \end{matrix}

(6)

w(x, t) is the transverse displacement of pipe, x and t are the nondimensional spatial and time variables, respectively. $\bar{x}$ and $\bar{t}$ are the dimensional forms of spatial and time variables. $\bar{w} (\bar{x}, \bar{t})$ is the dimensional form ofthe displacement. P₀ is the tensile force, L is the length, EI is the flexural rigidity, A is the area, and ρ is the density of the pipe. E^* is the coefficient of internal dissipation, m_fluid and m_pipe are the masses of fluid and pipe, and α is the dimensionless parameter related to the viscosity.

3. Natural Frequencies of Beam and Pipe Models

In (1) and (4) $∊ α {\dot{w}}^{I V}$ represent internal damping for beams and pipes. Although, in the usual modeling, this term is assumed to be small in the literature (i.e., ∊ ≪ 1); in this study, viscoelasticity term is assumed to be of order one. Therefore, the influence of the viscoelasticity on the natural frequencies can be directly analyzed for a wide range of parameters.

To find the natural frequencies of beams and pipes, the linear models with O(1) viscoelasticity are considered

\begin{matrix} \overset{‥}{w} + (v_{0}^{2} - 1) w^{′′} + 2 v_{0} {\dot{w}}^{'} + v_{f}^{2} w^{I V} + α {\dot{w}}^{I V} = 0, \\ w (0, t) = w (1, t) = w^{′′} (0, t) = w^{′′} (1, t) = 0, \\ \overset{‥}{w} + (v_{0}^{2} - P_{0}) w^{''} + 2 v_{0} \sqrt{β} {\dot{w}}^{'} + w^{I V} + α {\dot{w}}^{I V} = 0, \\ w (0, t) = w (1, t) = w^{′′} (0, t) = w^{′′} (1, t) = 0, \end{matrix}

(7)

for which a solution of the below form can be given [11],

w (x, t) = A_{n} (t) e^{i ω_{n} t} Y_{n} (x) + {\bar{A}}_{n} (t) e^{- i ω_{n} t} {\bar{Y}}_{n} (x),

(8)

where ω_n and Y_n(x) represent natural frequencies and mode shapes, respectively. The mode shapes are [11]

\begin{matrix} Y_{n} (x) \end{matrix} = c_{1} {e^{i β_{1 n} x} - \frac{(β_{4 n}^{2} - β_{1 n}^{2}) (e^{i β_{3 n}} - e^{i β_{1 n}})}{(β_{4 n}^{2} - β_{2 n}^{2}) (e^{i β_{3 n}} - e^{i β_{2 n}})} e^{i β_{2 n} x} - \frac{(β_{4 n}^{2} - β_{1 n}^{2}) (e^{i β_{2 n}} - e^{i β_{1 n}})}{(β_{4 n}^{2} - β_{3 n}^{2}) (e^{i β_{2 n}} - e^{i β_{3 n}})} e^{i β_{3 n} x} + [- 1 + \frac{(β_{4 n}^{2} - β_{1 n}^{2}) (e^{i β_{3 n}} - e^{i β_{1 n}})}{(β_{4 n}^{2} - β_{2 n}^{2}) (e^{i β_{3 n}} - e^{i β_{2 n}})} + \frac{(β_{4 n}^{2} - β_{1 n}^{2}) (e^{i β_{2 n}} - e^{i β_{1 n}})}{(β_{4 n}^{2} - β_{3 n}^{2}) (e^{i β_{2 n}} - e^{i β_{3 n}})}] e^{i β_{4 n} (x)}} .

(9)

In (9) β_in are the eigenvalues of the vibrated systems which were obtained from frequency equations and support conditions as later.

3.1. Axially Moving Viscoelastic Beam Model

Consider the following:

\begin{matrix} (v_{f}^{2} + i ω_{n} α) β_{i n}^{4} + (1 - v_{0}^{2}) β_{i n}^{2} - 2 v_{0} ω_{n} β_{i n} - ω_{n}^{2} = 0 \\ i = 1,2, 3,4, n = 1,2, \\ (e^{i (β_{1 n} + β_{2 n})} + e^{i (β_{3 n} + β_{4 n})}) (β_{1 n}^{2} - β_{2 n}^{2}) (β_{3 n}^{2} - β_{4 n}^{2}) + (e^{i (β_{1 n} + β_{3 n})} + e^{i (β_{2 n} + β_{4 n})}) \times (β_{2 n}^{2} - β_{4 n}^{2}) (β_{3 n}^{2} - β_{1 n}^{2}) + (e^{i (β_{2 n} + β_{3 n})} + e^{i (β_{1 n} + β_{4 n})}) \times (β_{1 n}^{2} - β_{4 n}^{2}) (β_{2 n}^{2} - β_{3 n}^{2}) = 0 . \end{matrix}

(10)

3.2. Viscoelastic Pipe Conveying Fluid Model

Consider the following:

\begin{matrix} (1 + i ω_{n} α) β_{i n}^{4} + (P_{0} - v_{0}^{2}) β_{i n}^{2} + 2 v_{0} \sqrt{β} ω_{n} β_{i n} - ω_{n}^{2} = 0 i = 1,2, 3,4, n = 1,2, \\ (e^{i (β_{1 n} + β_{2 n})} + e^{i (β_{3 n} + β_{4 n})}) \times (β_{1 n}^{2} - β_{2 n}^{2}) (β_{3 n}^{2} - β_{4 n}^{2}) + (e^{i (β_{1 n} + β_{3 n})} + e^{i (β_{2 n} + β_{4 n})}) \times (β_{2 n}^{2} - β_{4 n}^{2}) (β_{3 n}^{2} - β_{1 n}^{2}) + (e^{i (β_{2 n} + β_{3 n})} + e^{i (β_{1 n} + β_{4 n})}) \times (β_{1 n}^{2} - β_{4 n}^{2}) (β_{2 n}^{2} - β_{3 n}^{2}) = 0, \end{matrix}

(11)

where ω_n represents the natural frequencies. Numerical values of ω_n as well as β_in can be calculated by using (10) for viscoelastic beamsand (11) for viscoelastic pipes.

4. Numerical Solutions

Numerical solutions of the natural frequencies appear in the complex form:

ω_{n} = {ω_{n}}_{R} + i {ω_{n}}_{I} .

(12)

If the internal damping is assumed to be of order ∊, imaginary parts of (12) will be negligibly small. Thus, the natural frequencies of the system can be obtained from the real part (ω_n = ω_{n
R}). However, if the internal damping is assumed to be of order one, imaginary part of (12) will not be negligible. In that case, real parts can be considered as natural frequencies and imaginary parts can be considered as damping effects.

In Figure 1, the first three natural frequencies versus transport velocities are shown. Natural frequencies of the beam decrease by increasing transport velocity. As seen in the figure, second and third natural frequency (high frequency modes) curves do not cross the horizontal axis. For specific values of velocities (i.e., critical velocity) divergence instability occurs. Figure 2 shows the variation of the real parts of the first three fundamental frequencies with respect to transport velocities. In Figures 3, 4, and 5, real and imaginary parts of the first three natural frequencies are shown. In Figures 4 and 5, the real parts of the fundamental frequencies reach to zero by iterations.

Figure 1:

First three fundamental frequencies versus axial transport velocities of axially moving viscoelastic beam (v_f = 0.2, α = 0.001).

Figure 2:

Real parts of the first three fundamental frequencies versus axial transport velocities of axially moving viscoelastic beam (v_f = 0.2, α = 0.001).

Figure 3:

Real (solid line) and imaginary parts (dashed line) of the first fundamental frequencies versus axial transport velocities of the viscoelastic beams (v_f = 0.2, α = 0.001).

Figure 4:

Real (solid line) and imaginary parts (dashed line) of the second fundamental frequencies versus axial transport velocities of theviscoelastic beams (v_f = 0.2, α = 0.001).

Figure 5:

Real (solid line) and imaginary parts (dashed line) of third fundamental frequencies versus axial transport velocities of the viscoelastic beams (v_f = 0.2, α = 0.001).

In Figure 6, the effect of the viscoelasticity coefficient on the natural frequencies is shown. The coefficient related to viscoelasticity α is chosen as 0, 0.001, 0.005, 0.008, 0.01, and 0.05. To distinguish the curves, a small fraction of the figure is zoomed. By increasing the viscoelasticity coefficients, natural frequencies decrease for the same transport velocity. Note that α = 0 refers to an elastic Euler-Bernoulli beam.

Figure 6:

First fundamental frequencies versus axial transport velocities of the viscoelastic beams for different α values (small areas are zoomed) (v_f = 0.2).

Results of pipes conveying fluids are given in Figures 7 –11. Figures 7 and 8 show the variation of the natural frequencies with fluid velocities for the first two modes. Natural frequencies of the pipes decrease by increasing fluid velocity. In Figures 9 and 10, first and second natural frequencies versus fluid velocity including real and imaginary parts that are shown. Figure 11 depicts the effect of the viscoelasticity in detail. In order to distinguish the curves, a small area is zoomed and shown. Similar to the axially moving beams, natural frequencies decrease for increasing α values.

Figure 7:

First two fundamental frequencies versus fluid velocities of the pipes (α = 0.001, β = 0.3, and P₀ = 25).

Figure 8:

Real parts of first two fundamental frequencies versus fluid velocities of the pipes (α = 0.001, β = 0.3, and P₀ = 25).

Figure 9:

Real (solid line) and imaginary parts (dashed line) of the first fundamental frequencies versus fluid velocities of the pipes (α = 0.001, β = 0.3, and P₀ = 25).

Figure 10:

Real (solid line) and imaginary parts (dashed line) of the second fundamental frequencies versus fluid velocities of the pipes (α = 0.001, β = 0.3, and P₀ = 25).

Figure 11:

First fundamental frequencies versus fluid velocities of the pipes for different α values (small areas are zoomed) (β = 0.3, P₀ = 25).

The detailed variations of the natural frequencies in the case of viscoelasticity are given therefore in this study.

5. Concluding Remarks

Vibrations of axially moving beams and pipes conveying fluid are analyzed. Linear problem is considered to calculate the natural frequencies of the vibrated systems. Contrary to some previous work which considered viscoelasticity to be small, viscoelasticity terms are assumed to be of order one in this study. Natural frequencies for different beam and pipe parameters are shown. Effects of viscoelasticity on the natural frequencies are discussed. It is found that the viscoelasticity reduces the natural frequencies. Effects of dissipation mechanism on the frequency and critical velocities are discussed also by others [28 –31]. The qualitative behaviours of our results and theirs agree with each other. For a wide range of parameters, the natural frequencies are calculated and can be used in further studies.

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