Abstract
This paper attempts to present a new analysis for dynamical behavior of twolayer beams with frictional interface which held together in a pressurized environment, including stickslip nonlinear phenomenon. To achieve a proper outlook of twolayer beam structures behavior, it is essential to realize the mechanisms of motion precisely. Coupled equation of transversal and longitudinal vibration of twolayers in the presence of dry friction is derived and nondimensionalized. Furthermore, free and forced vibration of the mentioned system is investigated and the system dynamics is monitored via Poincare maps and Lyapunov exponent analysis. A comparative study with ANSYS is developed to show the accuracy of the proposed approach.
Stickslip phenomenon; Twolayer beam; Interlayer slip; Frictional interfaces; Chaotic motion
Stickslip analysis in vibrating twolayer beams with frictional interface
Hamid M. Sedighi^{I,}^{*} * Author email: hmsedighi@gmail.com ; Kourosh H. Shirazi^{II}; Khosro NaderanTahan^{II}
^{I}Mechanical Engineering Department, Shahid Chamran University, Ahvaz, 6135743337 Iran
^{II}Shahid Chamran University, Ahvaz, Iran
ABSTRACT
This paper attempts to present a new analysis for dynamical behavior of twolayer beams with frictional interface which held together in a pressurized environment, including stickslip nonlinear phenomenon. To achieve a proper outlook of twolayer beam structures behavior, it is essential to realize the mechanisms of motion precisely. Coupled equation of transversal and longitudinal vibration of twolayers in the presence of dry friction is derived and nondimensionalized. Furthermore, free and forced vibration of the mentioned system is investigated and the system dynamics is monitored via Poincare maps and Lyapunov exponent analysis. A comparative study with ANSYS is developed to show the accuracy of the proposed approach.
Keywords: Stickslip phenomenon, Twolayer beam, Interlayer slip, Frictional interfaces, Chaotic motion.
1 INTRODUCTION
Mechanical systems in which components are constrained through friction contact surfaces often lead to complex contact kinematics involving stickslip nonlinear motions. The explanation of stickslip intermittent motion is based on the instantaneous drop from a static friction force to a constant kinetic friction force. Accurate analysis of layered structure dynamical behavior with frictional interface in a pressurized environment requires accurate modeling of dry friction to realize exactly the stickslip regimes. In the case of layered beams, one of the earliest works was developed by Goodman and Klumpp [1], where the emphasis has revolved around the maximum amount of energy dissipation for twolayered beam in the case of dynamic loading. Badrakhan [2] derived the energy dissipated by Coulomb friction and optimum pressure for maximum energy for any number of layers.
Damisa et al. [3] investigated the effect of the nonuniformity in interfacial pressure as well as the frequency of the driving load, in the context of energy dissipation and logarithmic damping decrement. In the mentioned study, they consider the simple governing equation of motion with pure slip assumption. They extended their analysis to incorporate the effect of relative sizes or the material properties of layered beams of two dissimilar materials and laminate thicknesses [4].
Li et al. [5] presented an algorithm for solving the static contact problem in a multi leaf spring, whereas the frictional effect has not been taken into account in their research. Awrejcewicz et al. [6] investigated the chaotic vibrations of multilayered Euler Bernoulli and Timoshenko type beams for a series of boundary conditions. They verified the reliability of the obtained results via Finite Element and Finite Difference Method. In the work by Awrejcewicz and Krysko [7] an iterative algorithm was proposed to solve efficiently onesided interaction between two rectangular plates within the Kirchhoff hypothesis supplemented by physical nonlinearities. Furthermore, a novel iteration procedure for dynamical problems was introduced by Awrejcewicz et al. [8], where in each time step a contacting plate zone are improved. The chaotic dynamics of multilayer mechanical beam structures was studied by Krys'ko et al. [9]. They took into account physical, geometrical, and contact nonlinearities and developed a method for studying phase synchronization using wavelet analysis and the Morlet wavelet method.
There are many mechanical systems with frictionengaged subsystems such as brakes, clutches, gas turbine blade roots and machine tools. Such selfexcited oscillations have been extensively studied as reviewed in [10]. A springblock model with a single degreeoffreedom has explained the behavior of stickslip oscillations [11]. Kang et al. [12] focused on the dynamic pattern during the steadystate oscillation of springblock models, with definition of continuous friction curve for dry friction. Awrejcewicz et al. [13] considered two coupled oscillators with negative Duffing type stiffness which was frictioninduced and externally excited and investigated stickslip chaotic behavior of the system.
Lyapunov exponents measure the exponential rates of divergence or convergence of nearby trajectories in state space. If such equations for the system are nonsmooth, the estimation of Lyapunov exponents is not straightforward [14]. A novel dry friction modeling and its impact on Lyapunov exponent estimation was proposed by Awrejcewicz et al. [15]. A simple friction law was implemented by Licsko and Csernak [16] to predict the chaotic regimes of wellknown springblock model. The most efficient methods for estimation the largest lyapunov exponent (LLE) using chaos synchronization have been introduced by Stefanski et al. [14, 17].
The main objective of the present study is to identify the stickslip dynamical behavior of layered beams with frictional interfaces. In this regard, coupled vibrational equation of motion in the transverse and longitude directions is derived. The nonlinear partial differential equations of motion have been reduced by implementation of the BubnovGalerkin method. Quasiperiodic and stickslip chaotic motion of layered beams are investigated. Finally, to indicate the accordance of the proposed approach with finite element analysis, a comparative study with ANSYS is developed.
2 GOVERNING DIFFERENTIA EQUATION OF TWOLAYER BEAM
Fig. 1 shows twolayer cantilever beam with length L, width b, modulus of elasticity E and moment of inertia I for each layer. Top layer is subject to transverse load and the layers are maintained in contact by means of pressure P_{0}. The freebody diagram of a differential element of two layer beam has been illustrated in Fig. 2.
Assume that each layer here is the EulerBernoulli beam and the vertical displacements of two layers is the same. The symbols w, u_{1} and u_{2} denote the displacements of a point in the middle plane of the flexible beam in y and x direction for top and bottom layers, respectively. The lateral dynamic equation for differential segment of the beam can be expressed in the following form:
where m = A_{1}ρ_{1} + A_{2}ρ_{2} denotes the mass per unit length. Neglecting the effect of rotary inertia, the conservation of the angular momentum about the zaxis gives:
the total bending moment M can be formulated as
where d is the center to center distance of two layers. The condition of absent external axial forces leads to
combining Eqs. (1b) and (2b) yields
The longitudinal governing equations of motion for top layer is given by the following
where T(x,t) is the friction force transmitted between the two layers. Furthermore, the constitutive relations can be expressed as:
Substituting Eqs. (7a) and (7b) into Eq. (3) yields
Finally, from Eqs. (5) and (8), transversal governing equation of motion can be derived as follows:
Using Eq. (7b), the dynamic equation (6b) can be simplified as below:
and the vibrational equation of longitudinal motion for the bottom layer can be obtained as
The Coulomb type of friction, where the friction force is proportional to the normal reaction, is considered in this paper. The general model to describe a dry friction force T(x,t) is given by
where
_{rel} is the interlayer slip velocity assumed between the layers at the interface. As indicated in Fig. 3, it can be found as [18]
By introducing the following nondimensional variables
the nondimensional nonlinear equations of motion can be written as
where
and the nondimensional interlayer slip velocity can be written as
Using the BubnovGalerkin [18] principle and the multimode method defined below:
where functions φ_{r}(x), β_{ri}(x) (r = 1,2,...,n) must satisfy the transversal and longitudinal boundary conditions of each layer of the beam, coupled governing equations of twolayer beam in terms of the timedependent variables can be obtained. Lateral and longitudinal mode shapes of the clampedfree beam are expressed as
In the aforementioned equation, λ_{r} is the root of characteristic equation for rth eigenmode. In order to investigate nonlinear dynamical behavior of two layer beam in presence of stickslip phenomenon, coupled governing equations of lateral and longitudinal motion (15) and (17) should be solved, simultaneously.
3 RESULTS AND DISCUSSION
3.1 Free vibration
Let us consider the case of free vibration in order to find how the beam vibrates in lateral and longitudinal directions in the presence of coulomb friction. Primary analysis is limited to the case of constant pressure at the interface.
As shown in Fig. 4 the amplitude of lateral vibration decreases until the slip phase of motion vanishes and two layers of the beam vibrate as a single beam without slipping. It also indicates the details of lateral vibration in the vicinity of two motion phases and the conversion of stickslip phase to pure stick one. Because of interlayer friction existence and energy dissipation, the amplitudes of lateral and longitudinal vibration decrease, as can be observed from Figs. 4 and 5, until a special time that initial slip doesn't occur and layered beam vibrates without slipping.
To extend study this paper brings stickslip phenomenon into consideration. Therefore, the influence of dry friction on the nonlinear vibration of layered beam has been investigated. In the first phase of motion, the frictional force alternatively converts from slip to stick state. However, in the second phase of motion, the interlayer friction is always less than µp(x) and slipping doesn't take place. It should be noted that, in order to obtain actual displacement in longitudinal direction, the following formulation have been taken into account [19]:
where (x,t) is the longitudinal displacement of midline of the beam.
3.2 Forced vibration
Figs. 6 and 7 show the lateral and longitudinal interlayer displacements at the free end of the cantilever beam for the excitation frequency of r_{t} = 0.3, in the presence of friction and without friction.
It appears from these Figs. that in the absence of friction, trajectories in the lateral and longitudinal are regular and the quasiperiodic motion occurs. In the presence of dry friction, the behavior of the system becomes irregular as the interlayer pressure increases, especially in the longitudinal direction. In other words, the longitudinal interlayer phase plane is more sensitive to frictional interface in comparison with lateral phase plane.
Time histories of interlayer displacements for upper and lower layers of layered beam have been illustrated in Figs. 8 and 9, for different values of exciting force. As can be observed, top layer behavior don't change, however, for bottom layer, as the force amplitude becomes larger, single peak gradually splits into two peaks in perperiod.
From the results of numerical simulation, in the vicinity of initial slip, it is found that the responses of system seem to be extremely sensitive to the exciting force amplitude, as shown in the Fig. 10. When there is a little change in the force amplitude, the motion of layered beam will be transferred from quaiperiodic to the chaotic response suddenly. Fig. 11 shows the Poincare sections for 800 seconds of simulation in this situation for q_{1}  _{1} and q2  _{2} planes, respectively. The points of Poincaré sections are obtained with respect to the period of kinematic excitation 2π/ω. It is obvious from this Fig. that system demonstrates the chaotic motions for distinct aforementioned conditions.
As the state of system becomes far from initial slip, indicated in Fig. 12, when the exciting amplitude increases, nonlinear behavior of lateral and longitudinal motions become more regular than around the initial slip, especially for lateral vibration. However, the phase portrait in the longitudinal direction, as shown in Fig. 12b, exhibits less sensitivity to this change.
To study the effect of friction on the layeredbeam dynamics, phase portraits in lateral direction for different values of friction coefficient are plotted in Fig. 13. As can be expected, the less coefficient of friction causes the less time of sticking phase and more regular dynamic behavior. When the coefficient of friction becomes larger, the phase plane of motion seems to be more irregular and proceed to chaotic behavior.
The effect of exciting frequency have been conducted in this research through varying the nondimensional parameter r_{t}. Figs. 14a to 14d indicate that an increase in the parameter r_{t} from 0.2 to 0.3 leads to a decrease in the uneven vibrational slip velocities at the interface, for studied values of dry friction coefficient. It is clear that the irregular fluctuating of interface slips in the phase space decreases by increasing the parameter r_{t}.
Poincre sections of typical chaotic motions of layered structure for two different values of nondimensional frequencies r_{t} = 0.2 and r_{t} = 0.3 are plotted in Fig. 15. The Poincare sections are projected onto q_{1}  _{1}, q_{2}  _{2} and q_{3}  _{3} planes. While it is difficult to recognize chaotic behavior from the time history, Poincaré maps and Lyapunov exponents clearly show the chaotic nature of the response.
The largest Lyapunov exponent estimation is also used to identify the system dynamic behaviors. For the calculation processes of the largest Lyapunov exponent, the algorithm introduced by Fu and Wang [20] have been employed. In order to find the smallest synchronization value (the largest Lyapunov exponent), bifurcation diagram of the disturbance value against the synchronization parameter, is constructed for investigated situation in Fig 11. In Fig. 16, the bifurcation diagram shows z = x  y versus coupling coefficient k_{s}, so the searched value k_{s}_{,min} as a point on the horizontal axis is obtained where z approaches to zero. Therefore, the largest Lyapunov exponent is λ_{max} = k_{s}_{,min} which implies that the system is evidently chaotic.
Similarly, the largest Lyapunov exponent has been calculated for the situation studied in Figs. 15a and 15b. From bifurcation diagram shown in Fig. 17, the largest Lyapunov exponent in this situation is λ_{max} = k_{s}_{,min} = 0.023.
To ensure the accuracy of the proposed formulations and examine the achieved results a comparative case study with ANSYS is developed. It can be seen from Figs. 18 and 19 that there is a little difference, between stickslip regions and the frequency response obtained from numerical simulations and ANSYS software results. Furthermore, the amplitudes of lateral and interlayer displacement of top layer are in good agreement with finite element results. Nevertheless, the difference between these results has no significant effect on the prediction of distinct motion regimes in the dynamical behavior of layered beams.
4 CONCLUDING REMARKS
In the present study, nonlinear dynamical behavior of twolayer beam with frictional and pressurized interface including stickslip nonlinear phenomenon has been investigated. If the layered beam vibrates freely, the amplitude of vibration decreases until the slip phase of motion vanishes and two layers of the beam vibrate as a single beam without slipping. In the case of forced vibration, quasiperiodic and stickslip chaotic motion of layered beams have been illustrated via Poincare maps and Lyapunov exponent analysis for different values of system parameters. Finally, a comparative study with ANSYS is prepared to demonstrate the accuracy of the presented formulations.
Received 23 Jul 2012
In revised form 08 Jan 2013
 [1] Goodman L.E., Klumpp J.H. Analysis of slip damping with reference to turbine blade vibration, Journal of Applied Mechanics, 23, 1956, pp. 421429.
 [2] Badrakhan F. Slip Damping in Vibrating Layered Beams and Leaf Springs: Energy Dissipated and Optimum Considerations, Journal of Sound and Vibration, 174(1), 1994, pp. 91103.
 [3] Damisa O., Olunloyo V.O.S., Osheku C.A., Oyediran A.A. Dynamic analysis of slip damping in clamped layered beams with nonuniform pressure distribution at the interface, Journal of Sound and Vibration, 309, 2008, pp. 349374.
 [4] Olunloyo V.O.S., Damisa O., Osheku C.A., Oyediran A.A. Analysis of the effects of laminate depth and material properties on the damping associated with layered structures in a pressurized environment, Transactions of the Canadian Society for Mechanical Engineering, 34(2), 2010, pp. 165196.
 [5] Li Q., Li W. A contact finite element algorithm for the multileaf spring of vehicle suspension systems, Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering, 218, 2004, pp. 305314. DOI: 10.1243/095440704322955821.
 [6] Awrejcewicz, J., Krysko, A.V., Zhigalov, M.V., Saltykova, O.A., Krysko, V.A. Chaotic vibrations in flexible multilayered BernoulliEuler and Timoshenko type beams, Latin American Journal of Solids and Structures, 5, 2008, 319363.
 [7] Awrejcewicz, J., Krysko, A.V., An iterative algorithm for solution of contact problems of beams, plates and shells, Mathematical Problems in Engineering, 2006, 113, DOI 10.1155/MPE/2006/71548.
 [8] Awrejcewicz, J., Krysko, A.V., Ovsiannikova, O. Novel procedure to compute a contact zone magnitude of vibrations of twolayered uncoupled plates, Mathematical Problems in Engineering 2005:4 (2005) 425435, DOI: 10.1155/MPE.2005.425.
 [9] Krys'ko V.A., Koch M.I., Zhigalov M.V., Krys'ko A.V., Chaotic phase synchronization of vibrations of multilayer beam structures, Journal of Applied Mechanics and Technical Physics, 53(3), 2012, pp. 451459, DOI: 10.1134/S0021894412030182.
 [10] Ibrahim R.A. Frictioninduced vibration, chatter, squeal, and chaos, part II: dynamics and modeling, Appl. Mech. Rev. ASME, 47, 1994, pp. 227253.
 [11] Brockley C.A., Ko P.L. Quasiharmonic frictioninduced vibration, J. Lubr. Tech. Trans. ASME, 89, 1970, pp. 550556.
 [12] Kang J., Krousgrill C.M., Sadeghi F. Oscillation pattern of stickslip vibrations, International Journal of NonLinear Mechanics, 44, 2009, pp. 820828.
 [13] Awrejcewicz J., Sendkowski D., Stickslip chaos detection in coupled oscillators with friction, International Journal of Solids and Structures, 42, 2005, pp. 56695682.
 [14] Stefanski A., Kapitaniak T., Using Chaos Synchronization to Estimate the Largest Lyapunov Exponent of Nonsmooth Systems, Discrete Dynamics in Nature and Society, 2000, 4(3), 207215.
 [15] Awrejcewicz J., Grzelczyk D., Pyryev Yu. A novel dry friction modeling and its impact on differential equations computation and Lyapunov exponents estimation, Journal of Vibroengineering, 10(4), 2008.
 [16] Licskó G., Csernák G. Chaos in a simply formulated dryfriction oscillator, Proceedings of 4th Chaotic Modeling and Simulation International Conference, 31 May 3 June 2011, Agios Nikolaos, Crete Greece.
 [17] Wojewoda J., Stefanski A., Wiercigroch M., Kapitaniak T. Estimation of Lyapunov exponents for a system with sensitive friction model, Arch Appl Mech, 2009, 79, 667677.
 [18] Heuer R., Adam C. Piezoelectric vibrations of composite beams with interlayer slip, Acta Mechanica, 140, 2000, pp. 247263.
 [19] Han Q., Zheng X, Chaotic response of a large deflection beam and effect of the second order mode, European Journal of Mechanics A/Solids, 24, 2005, pp. 944956.
 [20] Fu S., Wang Q., Estimating the largest Lyapunov exponent in a multibody system with dry friction by using chaos synchronization, Acta Mechanica Sinica, 22, 2006, pp. 277283. DOI 10.1007/s104090060004y.
Publication Dates

Publication in this collection
16 Apr 2013 
Date of issue
Sept 2013
History

Received
23 July 2012 
Accepted
08 Jan 2013