2.034: Nonlinear Dynamics and Waves. Term Project: Nonlinear dynamics of piece-wise linear oscillators Mostafa Momen

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1 2.034: Nonlinear Dynamics and Waves Term Project: Nonlinear dynamics of piece-wise linear oscillators Mostafa Momen May 2015 Massachusetts Institute of Technology 1

2 Nonlinear dynamics of piece-wise linear oscillators Term Project: Mostafa Momen With main focus on the paper: On the dynamics of oscillators with bi-linear damping and stiffness. Int. J. Non. Linear. Mech by Natsiavas, S., Introduction Many recent studies have investigated the oscillators with piecewise linear and nonlinear damping and restoring forces. Koltter (1953) demonstrated response diagrams for a number of piecewise linear systems, using the Ritz-Galerkin method. Oscillators with bi-linear stiffness were examined by Maezawa et al. (1980), who used a Fourier series expansion approach for determining periodic solutions of general periodic excitations. Later Shaw and Holmes (1983) studied a periodically forced single-degree of freedom nonlinear oscillator with piecewise linear change in the restoring force. Since the nonlinearity was piecewise linear, explicit solutions are known on each side of the point of the discontinuity in slop x 0 and they suggested a procedure to investigate system response at almost everywhere including the discontinuities. The system that they examined is displayed below in Fig.1: Fig. 1. Diagram of the considered dynamical system in Shaw and Holmes (1983) When the stiffness ratio increases, the system is referred to as an impact oscillator and by employing a simple impact rule Shaw and Holmes have analytically found the single impact orbits of period n and examined their stability. They also investigated a limiting case in which one slope approaches infinity. Harmonic, subharmonic, and chaotic 2

3 motions were found to exist and the bifurcations leading to them were analyzed in their paper. When the forcing is periodic the usual expectation is that the long time response of the system will also be periodic. However, many recent studies such as Natsiavas (1988, and 1989) have shown that these systems may demonstrate behavior characterized by many unusual and complex features. Natsiavas (1990) studied the dynamics of the piecewise symmetric linear oscillators. The stiffness and damping in his previous papers were positive but he assumes that the damping coefficient becomes negative for small displacement amplitudes there, resembling the characteristics of the classical van der Pol oscillator. Due to this assumption, he observes some cases with Hopf bifurcation; however in the paper that we will study no Hopf bifurcation is possible since we only consider positive damping. Natsiavas (1990) showed that in the cases that both the damping ratios and the stiffness of the system are variable, the system exhibits a mixture of Duffing and van der Pol behavior characteristics, like jump and beating phenomena. Moreover, note that although the free vibrations of piecewise-linear dynamical systems can be solved exactly by sticking together the solutions in each linear zone, it is not yet possible to acquire a closed-form solution for an excited steady-state vibration. Classical perturbation methods such as Lindstedt Poincare, Krylov Bogoliubov Mitropolsky, or multiple scale method are valid only when the non-linearity of the considered dynamical system is weak and thus some other techniques such as incremental harmonic balance (IHB) or the one that we will describe as the following are suggested by some researchers (Natsiavas 1989, Xu et al 2002 and 2003). Xu et al. (2003) have investigated the bifurcation and chaos of a harmonically excited oscillator with piecewise linearities using IHB method. In their problem, the system exhibits chaos via the route of period-doubling bifurcations that we will discuss here through Natsiavas (1989) paper. Compared with classical approaches, the IHB method is not confined to small exciting parameters and weak non-linearity, and is remarkably effective in computer implementation for obtaining response with a desired accuracy over a wide range of varying parameter, with both stable and unstable solutions, subharmonic, harmonic and superharmonic resonances being traced directly. Here we focus on the Natsiavas (1989) paper and study his proposed analytical approach to these piecewise changes in the restoring and damping forces. The analysis here starts by determining the exact, single-crossing, steady-state solutions by imposing a set of periodicity conditions. The oscillators considered possess restoring and damping forces, which are bi-linear functions of the system velocity and displacement, respectively. Algebraic manipulations reduce this problem to the solution of a single transcendental equation. Then, the stability and bifurcation of these solutions is examined. And finally some numerical simulations are performed to yield more physical understanding of the investigated dynamical system as well as to validate the suggested analytical solutions. 3

4 2. Problem formulation: The bilinear damping and spring system is defined according to the Fig.2 below and the equations of motion for this system may be written as: m!!x + g (!x,x ) = f sin( ωt), g (!x,x ) = c 1!x + k 1 x x x c c 2!x + k 2 x + ( k 1 k 2 )x c x > x c (1) where c 2 = c 1 + c 2 and k 2 = k 1 + k 2 and both k 1 and k 2 are positive but arbitrary constants. Note that g(dx/dt,x) could include piecewise nonlinear functions of damping and spring as well but that makes the problem more challenging and so first we consider the effect of piecewise linear functions and then one can extend similar approach for piecewise nonlinear functions. In this paper only the piecewise linear damper and spring coefficients have been investigated. Furthermore, the exciting force is set in the form fsin(ωt+φ) where φ is an unknown phase angle. Fig. 2. Schematic view of the considered physical system Let x 2 (t) be the system displacement history, starting at time t = 0 from a value x c and staying in the range x>x c up until time t c (Fig. 3). Then let the displacement continue into the x < x c range up until the next crossing with x c with history x 1 (t). Now one could normalize equation (1) and rewrite its non-dimensional form as: (2) where Now one could solve equation (2) and for under-damped cases obtain: (3) 4

5 where Fig. 3. General form of periodic solutions sought 3. Periodic solutions of the piecewise linear system: According to some experimental observations (Masri 1978), the predominant type of response for the system examined is one in which the vibrating mass contacts the elastic stop once per cycle and the conditions of the system are repeated once per forcing cycle. Taking this into account, one could look for and determine n-periodic, single-crossing solutions of this pricewise linear system by imposing the following initial conditions: (4) where ψ c = 2πn θ c. Using equation (4), one should solve this problem for six unknowns A 1, A 2, A 3, A 4, ϕ, and θ c. These coefficients may be found as the following: (5) where all A ni, D s, and E s are known functions of the system parameters and θ c only. Now we could solve equation (5) and using the trigonometry identities to obtain: Periodic solutions with multiple crossings within a response period can also be obtained (6) 5

6 by extending the above procedure, however the algebraic complexity will increase in that way. It is clear that from all the possible solutions, only those which satisfy the form of the solution depicted in Fig. 3, i.e. single-crossing solutions between 0 < θ < 2πn, are acceptable solutions due to the imposed conditions on the form of the desired solutions. 4. Stability analysis: A method proposed by Masri and Caughey (1966) is chosen in this paper to analyze the stability of the periodic solutions, in place of the classical Floquet s theory. This is because as one could observe from equation (2), the damping and spring coefficients of the system considered are not differentiable functions at the displacement values y = 1. The idea here is to perturb the set of the initial conditions leading to a periodic steadystate solution and examine the behavior of the system in time. Consider a periodic solution with history y 2 (θ) and impose the initial conditions y 2 (0)=1 and!y 2 (t = 0) =!y 0 to get: Lets perturb θ as θ = θ Δθ 0 so the initial conditions become: (7) Now assume that the perturbed solution will be equal to 1 again at time equal to θ c + Δθ c according to Fig.3 and define Thus the solution from the perturbed initial conditions would be: (8) (9) Imposing the initial conditions for the perturbed solution and using equation (9) one could acquire: Now one may linearize equation (10) and up to first order get the following equation: Similarly we could differentiate equation (9) in time and apply the initial conditions to obtain: (10) (11) Again linearizing this equation results in the following equation: Now we can summarize equations (11-12) and by using (8) write the following matrix: (12) (13) 6

7 (14) Now one could follow similar procedure for the other piecewise part so that the perturbations at the end of the first response period may be written in the following matrix form: (15) Then, considering only first order terms, the perturbations from the periodic solution after k x n forcing cycles could be related to the initial perturbations by: ξ k = Π k ξ 0 (16) Π QR Hence, the requirement for a periodic solution to be asymptotically stable is that the solutions do not grow in time i.e.: limπ k = 0 (17) k If λ is the eigenvalue of Π with the largest modulus, the above condition implies: λ < 1 λ > 1 Stable Unstable 5. Bifurcation analysis of the piecewise linear systems: The bifurcation may occur as the parameters of the system change, since the modulus of λ can pass through 1 for some combinations of these parameters. These variations can be explained by modern theories on dynamical systems (e.g. Guckenheimer and Holmes 1983). For example, when the change of a system parameter results in complex eigenvalues of Π with modulus of λ equal to 1, a Hopf bifurcation happens. The determinant of the stability matrix in equation (16) could be written after some manipulations as: The presence of positive damping is a sufficient condition for Π <1 and excludes Hopf bifurcations. Furthermore, there are two possibilities for systems with positive damping: One is the case with λ =-1, which results in flip or period doubling bifurcations, while λ =+1 gives rise to three different types of bifurcations (saddle-node, pitchfork and transcritical) according to Guckenheimer and Holmes (1983). λ = +1 flip or period doubling bifurcations λ = 1 saddle node, pitchfork and transcritical (18) 7

8 6. Oscillations with no gap: For the case of oscillators with no gap, which x c = 0, an equivalent critical length is chosen for the problem from x c = f mω 2. The same analysis could be presented in a similar fashion and the only differences appear in equation (5), where all the An 0, D 0, and E 0 terms become zero and equation (6) will change to F E s D c E c D s = 0. Moreover, the undamped system with no gap possesses a natural frequency, which is independent of the amplitude of the oscillation and is expressed by: ω n ω 1 = ω n = 2 k 2 k 1 1+ k 2 k Numerical Results: The author provides some numerical results that are obtained by applying the preceding shown analysis. Fig. 4 demonstrates the response diagram for a softening system where k 1 /k 2 = 4, P = 0.5 and with two levels of damping. As the damping level increases the response amplitude decreases. Furthermore, as it clear is for a softening case the response results to frequency intervals with multiple (two stable and one unstable) solutions at some frequency ratios. Fig. 5 indicates the response diagram for a hardening system that was studied before by Masri (1978). This case shows good agreement between the analytical and experimental values and validates the previous preformed analysis, even though the differences are probably due to errors in measurements and contribution of modes other than the mode considered for the structure tested. The next studied problem validates the results for an oscillator with no gap presented by Shaw and Holmes (1983). They had numerically shown that a flip bifurcation would happen for a value of 2.40 < ω ω 1 < Here one could observe from Fig. 6 that this bifurcation occurs at about 2.408, which confirms their result. 8

9 Fig. 4. Response diagram of a softening case Fig. 5. Response diagram of a hardening case compared with experimental data 9

10 Fig. 6. Maximum eigenvalue of equation (18) vs. frequency ratio for an oscillator with no gap. From now, highly non-linear systems are examined in this paper. Figure 7 shows the response diagram for the Masri oscillator examined above, over a frequency interval where harmonic and subharmonic solutions are possible. In this case, the forcing amplitude is increased to P = 1. For this forcing level (P = 1), the resonances occur at forcing frequencies with values close to integer multiples ω n. There are many unique characteristics emerging from this figure that will be discussed in the following paragraphs. Fig. 7. Positive displacement peak versus frequency ratio in highly nonlinear system 10

11 The n = 4 subharmonic case demonstrates a jump phenomena that is due to the system hardening. However, jump phenomena do not seem to occur in systems with negligible gap, according to numerical results of Thompson et al. (1983) and Shaw and Holmes (1983). There exist frequency intervals such as between the n = 2 and n = 3 where no stable, single-crossing periodic solution is possible. Period doubling sequences happen between the n = 2 and n = 3, single-crossing, stable, steady state orbits. Furthermore, It is clear from Fig. 7 that the upper left branch of the n = 3 and n = 4 subharmonic solutions, where only one single-crossing periodic solution is possible, loses its stability and regains it again after a relatively short frequency interval. Similar behavior has been captured previously for a symmetric system examined in Natsiavas (1989). Only one period doubling was obtained when increasing frequency ratio from left, in contrast to what happens from the right, where period doubling up to 48 was captured. Hence, chaotic motions can take place since cascades of period doubling may reach an accumulation point involving subharmonics of infinite order. Figure 8 displays the Poincare map for the chaotic motion occurring for frequency ratio of Fig. 8. Poincare map at frequency ratio of The non-periodicity of this solution is shown further in Figs 9 and 10. In Fig. 9 the forcing period can be easily identified in the response, nonetheless the presence of subharmonic components is obvious and can be verified further by investigating the response for longer periods of time (T). 11

12 Fig. 9. Displacement history after first forcing cycles (frequency ratio=4.35). According to Hall (1983) a Fourier transform of the displacement history is carried out to illustrate the effects of the forcing and subharmonic components better. Figure 10 demonstrates a peak right above the forcing frequency, but most of the energy is distributed over broad frequency intervals over the low frequency range, corresponding to subharmonic components of the response history examined. Fig. 10. Fourier transform of displacement amplitude (frequency ratio=4.35). 12

13 Now the author focuses to determine the effect of the damping on the system response. The value of the first damping coefficient is increased substantially to higher levels to study its effects. According to Fig. 11 for ζ 1 =0.15, the response amplitude is reduced for both the harmonic and subharmonic solutions and the unstable branches of n = 1 and n = 2 near frequency ratio=0.71, as well as the n = 2 branch at about 2.5 disappear. Also, no jump phenomena occurs for the n = 3 response. For ζ 1 =0.4, the highest order subharmonic (n = 4) solution disappears completely, while the amplitude of the other subharmonics becomes considerably smaller. Furthermore, their maxima are shifted towards smaller frequencies. Increasing ζ 1 to 0.75 again, completely disappears all the subharmonic solutions, which are substituted by a linear harmonic response. Fig. 11. Effect of damping on displacement response of the system. 8. Conclusion: The long time periodic response of a class of harmonically excited, single degree-offreedom strongly non-linear oscillators have been analyzed. The nonlinearity here appears in both the damping and the restoring forces and are bilinear functions of the system velocity and displacement. Such oscillators provide models for mechanical systems in which components make intermittent contact. The presented procedure allows for an exact and appropriate method in determining the crossing times as well as the periodic response history for such oscillators. The analysis begins with determining the 13

14 exact, single-crossing, steady-state solutions by imposing a set of periodicity conditions and then a suitable analytical procedure is presented for determining the stability of the located periodic orbits. This investigation was carried out by examining the response diagram of that system and studying several frequency ranges where harmonic and subharmonic response, coexistence of multiple solutions of the same or different order, jump phenomena, period doubling sequences and apparently chaotic response were observed. Characteristics like jump phenomena are identified for both harmonic and subharmonic responses of hardening as well as softening systems. The oscillator examined exhibits a regular behavior for some combinations of its parameters. Nevertheless, chaotic response is possible for certain sets of parameters in which the system undergoes bifurcations, which may lead to loss of stability of periodic solutions. Finally, the effects of the system parameters, including damping ratio, forcing level and stiffness change, on some of the characteristics of the system response were analyzed. The presented analysis is also validated by many performed numerical solutions from other papers. Furthermore, notice that the free vibrations of piecewise-linear dynamical systems can be solved exactly by gluing together the solutions in each linear part, nonetheless it is not yet feasible to obtain a closed-form solution for an excited steady-state vibration. Classical perturbation methods like Lindstedt Poincare, multiple scale methods and etc. are valid only when the non-linearity of the considered dynamical system is weak and hence some other techniques such as the one described here are required to tackle these kinds of problems. This work could be extended further to piecewise nonlinear functions as well, even though the formulation and required analysis might be more challenging. 9. Further Developments and Applications These types of solutions with piecewise linear and nonlinearities do exist and could be applied for mechanical vibrating systems such as lightly loaded spur gears, rotor systems, relaxation oscillator systems, cam/follower systems, linkage joints and robotic components, bearings and impact print hammers due to clearances, gaps and impacting components, etc. Fig. 12. Applications in rotor systems and gearboxes. In other fields of electronics, biology, economy, the theoretical models of many nonlinear dynamical problems are also found to be systems having piecewise features. These types of functions may be also used in fluid mechanics and in highly turbulent flows, 14

15 where the magnitude of the turbulent viscosity or damping is dependent on the imposed geostrophic forcing. In such systems, for example according to the following equation α, which is a function of eddy-viscosity, depends on the magnitude of the forcing and also some other variables that might change diurnally: u!! + α!u + ku = F Gestrophic u here is the wind velocity vector on the atmosphere and k is a constant that depends on the system s parameters like Coriolis frequency. One might use a similar argument here by dividing the whole problem to some subdomains in which in each zone a specific turbulent viscosity is defined and then solve the piecewise linear or nonlinear problem using similar procedures shown in this paper. Hence, the idea of these piecewise changes in the damping and restoring forcing functions have a wide applicability and may be employed in many fields. Moreover, results of the present work as well as some other recent studies reveal that piecewise linear oscillators may exhibit behavior characterized by many unusual and complicated features, which are also encountered for systems with continuous non-linearities. There have been some recent studies about piecewise linear dynamical systems which have considered multiple degrees of freedom and well potentials. For instance, Fig. 13 demonstrates two cases that have been examined recently. Fig. 13. Yu (2012) [left] and Han et al. (2014) [right] considered physical systems. Yu (2012) studied some efficient computational methods for determining vibrational responses of piecewise-linear dynamical systems with multiple degrees of freedom and an arbitrary number of gap-activated springs. Responses of a single DOF system with gap-activated spring and a 3-DOF system with three gap-activated springs under harmonic excitations were obtained in their paper. Then their proposed method was applied to a piecewise linear dynamical system with 1000 DOF s and 1000 gap-activated springs under harmonic excitations as is shown in the left panel of Fig. 13. Han et al. (2014) investigated a nonlinear mechanical model with a lump mass and a pair of springs pinned to rigid supports that is displayed in the right panel of the Fig. 13. Their results suggested that the oblique springs provide irrational non-linearity behavior with smooth and discontinuous characteristics and showed the complicated dynamics for piecewise linear discontinuous system of the subharmonic solutions, chaotic solutions and the coexistence of multiple solutions for the single well system, double well system and the triple well dynamics. These studies show that the piecewise oscillators have attracted a lot of attentions recently due to their distinctive nonlinear behavior and wide range of applicability in the real-world problems. More studies are required for piecewise nonlinear functions as well as multiple crossings in a linear/nonlinear system and also non-periodic problems to gain 15

16 better and more comprehensive understanding of this nonlinear oscillatory behavior. 10. References: Guckenheimer, J. and Holmes, P., 1983: Non-linear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer, New York (1983). Hall, J. F., 1982: An FFT algorithm for structural dynamics. J. Earth. engng Struct. Dyn. 10, Klotter, K., 1953: Steady state vibrations in systems having arbitrary restoring and damping forces. Symp. On Van-Iinenr Circuit Analysis. Polytechnic Institute of Brooklyn, New York Han, Y., Q. Cao, Y. Chen, and M. Wiercigroch, 2015: Chaotic thresholds for the piecewise linear discontinuous system with multiple well potentials. Int. J. Non. Linear. Mech., 70, , doi: /j.ijnonlinmec Maezawa, S., Kumano, H. and Minakuchi, Y., 1980: Forced vibrations in an unsymmetric piecewiselinear system excited by general periodic force functions. Buff. JSME 23, Masri, S. F. and Caughey, T. K., 1966: On the stability of the impact damper, J. appi. Mech. 33, Masri, S. F., 1978 : Analytical and experimental studies of a dynamic system with a gap, J. mech. Des Natsiavas, S. and Babcock, C. D., 1988: Behavior of unanchored fluid-filled tanks subjected to ground excitation. J. appl. Mech. 55, Natsiavas, S. 1989, Periodic response and stability of oscillators with symmetric trilinear restoring force. J. Sound Vib Natsiavas, S., 1989: On the dynamics of oscillators with bi-linear damping and stiffness. Int. J. Non. Linear. Mech., 25, , doi: / (90) Natsiavas, S., 1990: Dynamics of Piecewise Linear Oscillators with Van der Pol type damping. Int. J. Non. Linear. Mech., 26, Xu, L., M. W. Lu, and Q. Cao, 2002: Nonlinear vibrations of dynamical systems with a general form of piecewise-linear viscous damping by incremental harmonic balance method. Phys. Lett. A, 301, 65 73, doi: /s (02)00960-x. Xu, L., M. W. Lu, and Q. Cao, 2003: Bifurcation and chaos of a harmonically excited oscillator with both stiffness and viscous damping piecewise linearities by incremental harmonic balance method. J. Sound Vib., 264, , doi: /s x(02)01194-x. Shaw, S. W., and P. J. Holmes, 1983: A periodically forced piecewise linear oscillator. J. Sound Vib., 90, , doi: / x(83) J. M. T. Thompson, A. R. Bokaian and R. Ghaffari, 1983: Subharmonic resonances and chaotic motions of a bi-linear oscillator, J. appl. Math. 31, Yu, S. D., 2012: An efficient computational method for vibration analysis of unsymmetric piecewise-linear dynamical systems with multiple degrees of freedom. Nonlinear Dyn., 71, , doi: /s