The Linear Stability of the Responses of Axially Moving Beams Supported by an Intermediate Spring

In the present paper, the stability investigation of the linear responses of axially travelling beams supported by an intermediate linear spring is performed numerically by means of searching linear complex eigenvalues. The boundary conditions and the dynamic equation of motion are obtained by means of the Extended Principle of Hamilton for the two-span Euler-Bernoulli beam. Assuming the harmonic solution of the governing differential equations of motion, the dispersion relation is derived. To have a nontrivial solution in conjunction with the dispersion relation, the frequency equation is obtained by letting the determinant of the matrix representing the coefficients of the equations of the boundary conditions be equal to zero. The aim of this contribution is to compute the complex values of the natural frequencies from the nonlinear frequency equation by means of numerical methods for the beam supported by an intermediate spring. The real and imaginary parts of the complex eigenvalues versus the constant axial transport speed curves are depicted in order to show the linear instability. Depending on the signs of the real and imaginary parts of the complex eigenvalues, the boundaries of the instability regions were observed.


Introduction
The industrial applications of moving materials classified as axially travelling stringlike, beamlike structures are numerous [1][2][3][4] such as magnetic tapes drives, band saws, power transmission belts, robot arms, serpentine belts and aerial cable tramways.The purpose of the current research paper is to investigate the linear instability of the beam.The dynamic equation of motion and obtained utilizing the Extended Principle of Hamilton for the two-span Euler-Bernoulli beam.The complex mode method [5,6] is used to obtain the nonlinear frequency equation.The boundaries of the instability regions were determined according to the signs of the real and imaginary parts of the eigenvalues.

Formulation of the beam problem
The axially moving continua with an intermediate spring support is illustrated schematically in Fig. 1.

Fig. 1. Geometry of the axially travelling beam.
The equations of motion can be obtained by following the procedure presented in the reference [7] using the Extended Hamilton's Principle.The Euler-Bernoulli beam is moving axially at a slightly fluctuating harmonic velocity * c .After writing the kinetic and potential energies of the system, the Lagrangian for the axially moving beam can be formulized through the Hamilton's Principle as shown in Eq. ( 1).In x should be written as 0 and L respectively.It is assumed that the beam has immovable boundaries at the outer ends.The assumption introduces nonlinearity because of the stretching of neutral fibres.
where P is the total tensional force, which is composed of an applied artificial pre-tension and the tension caused by the extensional and flexural deformations of the centreline, in the axial direction.

Dimensionless formulation
For convenience, the equations and all accompanying boundary conditions are presented in dimensionless form for the two sides.The following non-dimensional variables and parameters introduced in Ref. [8] are employed.
In addition to those given in Eq. 2, the dimensionless pan-wise variable, which represents a dimensionless location constant of the intermediate support and may take a value bigger than 0 and smaller than 1, is introduced: presents the longitudinal rigidity parameter, and f v denotes the flexural rigidity parameter.The axial velocity is converted to dimensionless quantity by dividing it with the critical velocity.Substituting Eq. 2 in the governing equation of the motion, performing required operations and adding damping term ( μ ) yield the dimensionless equations of motion and relevant non-dimensional boundary conditions.
where the following abbreviations are introduced.

A systematic way
In the initial steps we tacitly used the method of multiple scales [9] to separate and obtain the linear parts of the equations in a systematic way [10].The axial transport velocity varying about an arbitrary constant value, 0 c , with a small amplitude fluctuation frequency is given as where ε denotes a small variation of the velocity.The terms 1 w and 2 w representing displacements in Eq. 3 and 4 may be supposed in the form . Thus, these terms satisfy that the longitudinal stiffness terms coming from nonlinear effects will stay in higher orders of expansions.Also, viscous damping term εμ will be in ε order as εμ μ = .The displacement functions

Complex Mode Method
In this research, linear differential problem are taken into consideration, the nonlinear terms are not included in the subsequent sections.Assuming harmonic motion, Wickert and Mote [11] expressed the solution to Eq. ( 9) in the following span-wise characteristic functions: where ω represents the natural frequency, A and A are complex amplitudes and their conjugates, respectively.By means of substituting Eqs. 10 into Eqs.9, the equations of motion can be written for each span with appropriate non-dimensional end boundary conditions and compatibility conditions at the intermediate support.
where 1 W and 2 W is lateral displacement, (ˊ) denotes differentiation with respect to the spatial variable ( x ), 0 c is the axial transport velocity, and f v denotes the flexural rigidity of the beam.The deflections can be expressed in the following form [11]: where 8 1 d d − denote constants.These integration constants must be determined via the boundary conditions given with Equations 12.By substituting the functions given with Eqs. 13 into Eqs.11, the dispersion relation, , can be easily obtained.It is clear that there must be the relationships, , among the eigenvalues.Using the boundary conditions given with Eqs. 13, the matrix of the coefficients can be obtained.We are interested in a sufficient condition for the nontriviality of the solution.The determinant of the matrix of the coefficients must vanish.Thus, the natural frequencies can be evaluated by means of Secant Method from this equality.

Stability
The beam loses its stability if the transport speed exceeds a certain value.16) by inserting 0 = ω [12].Therefore, after assuming the solution function as an exponetial function, a dispersion relation can be obtained.From the the roots of these equations, an artificial parameter can be evaluated as follow.
Using the boundary conditions given with Eq. 17, the matrix of the coefficients of the solution function can be obtained.The determinant of the matrix must vanish.After inserting f v into the determinant, r can be evaluated numerically from this determinant by means of the Newton-Raphson Method.By substituting the calculated values of r into Eq.( 18), the divergence speeds can be evaluated.The lowest divergence speed is defined as the critical speed [12][13][14].

Numerical results
Our numerical investigations were conducted on the beam with the mechanical properties which are listed in Table 1.The flexural rigidity parameter was chosen as 173 .0 = f v [15].The real and imaginary parts of the complex eigenvalues versus the constant axial transport speed curves are plotted in Fig. 2 in order to show the linear instability.Depending on the signs of the real and imaginary parts of the complex eigenvalues, the boundaries of the instability regions are observed in Fig. 2. The divergence speeds are analytically derived.The real and imaginary parts of the complex eigenvalues versus constant axial transport speed curves were calculated in order to show the linear instability regions.The signs of the real and imaginary parts of the eigenvalues are used to determine the boundaries of the instability regions.

Fig. 1 u
denote respectively the transverse and longitudinal displacements, * c is the velocity of the axial transport of the beam, * t denotes time, where the variables with asterisk denote dimensional ones.Properties of the beam include the beam length L, the cross-section area A, I is the cross-sectional inertia of the beam, the mass density ρ and E is the Young's modulus of the beam.k is the translational stiffness of the linear spring.The values of the spatial variable at the usual fast-time scale and ε t T = 1 is the slow-time scales.The time derivatives are as follow:

From.3
When the real part of the eigenvalues λ is negative it indicates damping and consequently the variety of the amplitude.The imaginary parts of the eigenvalues λ characterize the natural frequencies.Depending on the signs of ) Re(λ and ) Im(λ the stability of the behaviour of the beam can be classified as[12]: Divergence and critical speeds Critical speeds are readily derived from Eqs. (

Fig. 2 .
Fig. 2. The real parts (dashed line) and imaginary parts (solid line) of eigenvalues λ versus the transport speed for the first three modes

Table 1 .
Mechanical properties of the system The location of the intermediate support ηThe first twelve speeds obtained from Eq. (18) for various η values are presented inTable 2.