NON-LINEAR TRANSVERSE VIBRATIONS and 3 : 1 INTERNAL RESONANCES OF A TENSIONED BEAM ON MULTIPLE SUPPORTS

Abstract – In this study, nonlinear transverse vibrations of a tensioned Euler-Bernoulli beam resting on multiple supports are investigated. The immovable end conditions due to simple supports cause stretching of neutral axis and introduce cubic nonlinearity to the equations of motion. Forcing and damping effects are included in the analysis. The general arbitrary number of support case is investigated and 3, 4, and 5 support cases analyzed in detail. A perturbation technique (the method of multiple scales) is applied to the equations of motion to obtain approximate analytical solutions. 3:1 internal resonance case is also considered. Natural frequencies and mode shapes for the linear problem are found for the tensioned beam. Nonlinear frequencies are calculated; amplitude and phase modulation figures are presented for different forcing and damping cases. Frequency-response and force-response curves are drawn. Different internal resonance cases between modes are investigated.


INTRODUCTION
Transverse vibrations of beams are of importance in engineering systems and investigated in detail.A literature survey up to 1979 is done by Nayfeh and Mook [1].Nonlinear free vibrations of multispan beams on elastic supports were studied by Lewandowski [2] using dynamic stiffness method to find frequencies and nonlinear modes of vibrations by considering effects of support flexibility on the frequency amplitude relations.Özkaya [3] discussed the effects of different end conditions for beam-mass systems.More recent works on this type are due to [4][5][6][7][8][9][10][11][12][13][14][15][16].These studies include restrained beams [4,5], in-span support [6,7], different boundary conditions for nonlinear vibrations [8][9][10], stepped beam systems using artificial neural networks [9] and finite element methods [12], single, multiple mass on simply supports [13][14][15][16], and non ideal support cases for three different simply supported beams [16], infinite mode analysis was performed [17].Nonlinear vibrations and 3:1 internal resonances on multiple supports were investigated and excitation frequency-frequency response curves drawn for different support numbers [18] and Tekin et al. studied on three-to one resonance in multi stepped beam systems [19].For slightly curved beams with stretching, one may refer to Rehfield [20].There are also some studies about axially moving beams composed of viscoelastic materials [17][18][19][20][21][22][23].Beams simply-supported in span were discussed and frequency response functions are determined [23].Varadan et al. [24] studied nonlinear behavior of a beam in bending with immovable ends for S.M. Bağdatlı, H.R. Öz and E. Özkaya 204 various loadings and edge conditions.The authors concluded that it was enough to consider only the nonlinearity arising from the axial force in the nonlinear analysis of beams with immovable ends since the hardening effect due to axial force predominates over that resulting from the use of an actual nonlinear expression for curvature.Da Silva [25] derived nonlinear equations for a class of inextensible flexible multibeam structures having arbitrary cross section varying along its span and also having supports and lumped masses.Cheng et al. [26] investigated nonlinear random response of internally hinged beams using finite element method.Main and Jones [27,28] formulated exact analytical solutions for free vibrations of tensioned beams with an intermediate viscous damper and a viscous damper attached transversely near a support using dynamic stiffness method to obtain characteristic equations for both clamped and pinned supports.Mazzilli et al. [29] constructed nonlinear normal modes and nonlinear multi modes using the method of multiple scales for a beam with uniformly distributed axial and a thrust force and compared with finite element method simulations.
In this study, nonlinear transverse vibrations of a tensioned Euler-Bernoulli beam with multiple simple supports are considered.The beam is stretched during vibration due to immovable supports.This introduces cubic nonlinearity to the equations of motion.Transverse forcing and damping are also included in the problem.The equations of motion are derived for general case: arbitrary number of supports, and then solved for 3, and 4 support cases by using the method of multiple scales.Natural frequencies are calculated and mode shapes are presented.The effect of support number on the natural frequencies is investigated for the nonlinear vibrations.Amplitude and phase modulation relations are presented for different forcing and damping cases.3:1 internal resonance cases are investigated between different modes of vibration.w  denotes the transverse displacement of the beam section between supports m and m+1.L is the length of the beam.t * is the time.The total number of supports is n.A is the mass per unit length, EA is longitudinal rigidity, EI is flexural rigidity and P is the axial tension force on the beam.The Lagrangian can be written as follows where dot denotes derivative with respect to t * and prime denotes derivative with respect to x * .The first integral is the kinetic energy of the beam section between any successive supports.The second integral is the elastic energy in bending, the third integral is the elastic energy in extension due to stretching of the neutral axis and the last one is the elastic energy due to axial tension.Applying Hamilton's principle and performing the necessary algebra, the equations of motion and boundary conditions for the general case for the tensioned beam in dimensional form is obtained as follows

EQUATIONS OF MOTION
(2) The equations are made dimensionless using the following definitions where R is the radius of gyration of the beam cross-section with respect to the neutral axis.Substituting the dimensionless parameters into the equations of motion yields

METHOD OF MULTIPLE SCALES
The method of multiple scales will be applied to the partial differential equation system and boundary conditions directly.There is no quadratic non-linearities, that's why one can write an expansion of the form where  is a small book-keeping parameter representing that the deflections are small.This procedure models a weak non-linear system.T 0 =t and T 2 = 2 t are the fast and slow time scales.Here only the primary resonance case is considered.The forcing and damping terms are ordered as shown below so that they are included in the cubic order of expansion, , the time derivatives are written as , where . After expansion, one obtains equations of motion and boundary conditions at different orders as follows Order (): S.M. Bağdatlı, H.R. Öz and E. Özkaya 206 Order ( 3 ): Solution of the first order of expansion gives natural frequency values and a solvability condition is obtained from the second order of expansion.

Exact Solution to the Linear Problem
For Eq. ( 8) one can assume solutions of the form for any beam segment where cc stands for complex conjugate of the preceding terms.Eqs. ( 7) and ( 9) give The solution of the equations can be sought by assuming the following shape function for any beam segment Frequency equations can be obtained when the boundary conditions are applied.

Approximate Solution to the Non-linear Problem
Solution of nonlinear Eq. ( 8) gives corrections to the problem.They will have a solution only if a solvability condition is satisfied as explained in reference [30].The secular and nonsecular terms are separated to find the solvability condition by assuming a solution of the form and inserting it into Eq.( 8), the terms related with secularities are discarded.Here stands for the solution related with non-secular terms.One obtains Non-Linear Transverse Vibrations and 3:1 Internal Resonances of a Tensioned Beam on Multiple Supports 207 Assuming that excitation frequency is close to one of the natural frequencies of the system as shown below where  is a detuning parameter of order 1, the solvability condition for Eqs. ( 13) and ( 14) is obtained as follows where The complex amplitude A in Eq. ( 16) can be written in terms of a real amplitude a and a phase  Then amplitude and phase modulation equations can be obtained as follows where, . Eq. ( 18) will be solved for steady-state case in the next section and variation of nonlinear amplitude with forcing will be discussed.

NUMERICAL RESULTS
In this section numerical examples for frequencies will be presented for different cases.Firstly, the linear natural frequencies for different tension forces and support locations () will be calculated.Then, the non-linear frequencies for free, undamped vibrations will be calculated.For this case, by taking =f= =0, one obtains D 2 a=0 and a = a 0 (constant) (19) from Eq. (19).Here a 0 is the steady-state real amplitude.The non-linear frequency is where is the correction coefficient due to nonlinear terms.Up to the second order of approximation, the non-linear frequencies have a parabolic relation with the maximum amplitude of vibration.The linear frequencies and nonlinear correction terms for the first five frequencies are presented in Table 1 for 3 support case, in Table 2 for 4 support case for different axial tension values.The effects of locations of supports and axial tension are given.For the same support locations, the tension increases the frequencies, as expected, as shown in Table 1.For a given value of tension, the effect on the frequencies at different modes varies depending on the location of support.As the support is moved to the center of the beam, the frequencies increase in the first modes for all tension values.In other modes, it depends on the location of the node and closeness of the support to the node.Nonlinear frequency versus amplitude curves are plotted in Figs.2-3 for v p = 1, 10 respectively.In these figures, the variation of nonlinear frequency is plotted for threesupport case for the first mode when =0.1, 0.2, 0.3, 0.4 and 0.5.Nonlinearities are of      At the steady state, a and  become zero.The frequency detuning parameter is as follows For three-support case, frequency response curves, are plotted in Fig. 6 v p =1, respectively.This term is a for nearness of the forcing frequency to the natural frequency.In all figures f=1, µ=1 are assumed.In Fig. 6 for v p =1, vibration decrease as the support becomes closer to the midpoint and the behavior is of hardening type.Maximum vibration amplitude is obtained when σ>0 for all mid support locations but at different σ values.Their maxima are obtained at lower frequencies when the support location changed towards ¼ of the beam, then it is obtained at higher frequencies when the support is moved towards midpoint.The jump region becomes smaller as the mid support is located towards middle of the beam.Comparison of Fig. 6 shows that, maximum vibration amplitude is obtained at higher frequencies when the support location is close to the ends or midpoint.In Figs.7-8 and in Fig. 9 the curves are shown for four-support and five-support cases respectively.The behavior is of hardening type in all figures and tension decreases the jump region.
Inserting this into the O( 3 ), one obtains Solution of the equations at this order is as follows The solvability conditions can be calculated as follows, where, The complex amplitudes can be written in polar form One finally obtains frequency modulation equations for steady state solutions, 0 , then Eq. ( 26) becomes where

CONCLUDING REMARKS
The transverse vibrations of a tensioned Euler-Bernoulli beam having multiple supports are investigated.The non-linear equations of motion including stretching of the neutral axis due to immovable end conditions are derived.The method of multiple scales is applied to obtain approximate solutions.Exact solutions and numerical values for natural frequencies are given for linear problem.For the non-linear problem, correction terms to linear problem are obtained.Non-linear free and forced vibrations are investigated in detail.The effects of the positions of supports and axial tension are determined.The corrections increase as the number of supports increase and natural frequencies increase always.Stretching of the neutral axis causes a non-linearity of hardening type.For forced and damped vibrations, since the non-linearity is of hardening type, the frequency-response curves are bent to right, causing an increase in the multi-valued regions.When support number is increased, the multi-valued regions and maximum amplitude decrease.Then 3:1 internal resonances are investigated.Support locations and tension values producing internal resonances for three-support case are determined.No 3:1 internal resonance is possible for ω 3 /ω 2 and ω 5 /ω 3 for v p values between 0.1 and 10 For other cases, it is possible for some values.Frequencyresponse and force-response curves are plotted.External excitation frequency is applied to the first mode and responses are calculated for the first and second modes.Finally stability analysis is performed and the borders are drawn.
Transverse Vibrations and 3:1 Internal Resonances of a Tensioned Beam on Multiple Supports 205

Figure 2 .Figure 3 .
Figure 2. Nonlinear frequency-amplitude variation for v p = 1 and for different  values.

Figure 6 .
Figure 6.Forcing frequency-amplitude variation for v p =1 and for different  values.

Figure 10 .
Figure 10.Ratios of different frequencies for v p =1.Figure11.Ratios of different frequencies for v p =5.

3
Transverse Vibrations and 3:1 Internal Resonances of a Tensioned Beam on Multiple Supports 213By evaluating the eigenvalues of the jacobian matrix, stability can be determined.Eigenvalues should not have positive real parts to maintain stability.External excitation frequency response graphs are given in Figs.15, 16for  1 =29.613795, 2 =8.16346, =0.075 and f=1 for three-support case (3:1 internal resonance:  2 / 1 = 3.0027).External excitation frequency is applied to the first mode and responses are calculated for the first and second modes.The second mode is activated for   0.The frequency amplitude response curves are shown in Figs.17, 18for  1 =29.613795, 2 =88.921622, =0.075 and =0.2 for three-support case.As shown in the figures, different forcing amplitudes are applied to the first mode, the second mode amplitude is determined.In these figures v p =1 and η=0.344.The parameters defined in Eq. (22) and Eq.(27) and used here are =0.0802,b 1 =41.2196,b 2 =59.4933, b 3

Table 1 .
The first five frequencies and corrections due to nonlinear terms for v p =1, 10 and different values for three support case.

Table 2 .
The first five frequencies and corrections due to nonlinear terms for v p =1, 10 and different values for four support case.
As the third support is located close to the midpoint the frequencies increase which means the beam becomes stiff.In Figs.2-3, v p values changed between 1 and 10, an increase in v p caused an increase in nonlinear frequencies.The behavior in all figures is of hardening type.In Figs 4 and Figs. 5, four-support case and five-support cases are shown respectively for v p = 1, 10. Again the behaviors are of hardening type.