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1 Physica D 238 (2009) Contents lists available at ScienceDirect Physica D journal homepage: Nonlinear dynamics of oscillators with bilinear hysteresis and sinusoidal excitation Tamás Kalmár-Nagy, Ashivni Shekhawat Department of Aerospace Engineering, Texas A&M University, United States a r t i c l e i n f o a b s t r a c t Article history: Received 20 May 2008 Received in revised form 10 January 2009 Accepted 1 June 2009 Available online 23 June 2009 Communicated by G. Stepan Keywords: Bilinear hysteresis Hysteretic oscillator Caughey model Forced response The transient and steady-state response of an oscillator with hysteretic restoring force and sinusoidal excitation are investigated. Hysteresis is modeled by using the bilinear model of Caughey with a hybrid system formulation. A novel method for obtaining the exact transient and steady-state response of the system is discussed. Stability and bifurcations of periodic orbits are studied using Poincaré maps. Results are compared with asymptotic expansions obtained by Caughey. The bilinear hysteretic element is found to act like a soft spring. Several sub-harmonic resonances are found in the system, however, no chaotic behavior is observed. Away from the sub-harmonic resonance the asymptotic expansions and the exact steady-state response of the system are seen to match with good accuracy Elsevier B.V. All rights reserved. 1. Introduction The phenomenon of hysteresis is widely prevalent in nature and has been studied in diverse contexts [1 5]. Of particular engineering interest is the forced response of oscillators with hysteresis. In this paper the dynamic response of bilinear hysteretic oscillators is studied. Many systems of practical interest, like beam-column connections, relay oscillators, riveted and bolted structures, elasto-plastic materials etc. can be modeled by using the bilinear model of hysteresis. This model is inspired by elasto-plasticity and can be considered to be a generalization of the Prandtl models [6,7]. The dynamics of systems with bilinear hysteresis has been a topic of many scholarly studies, including those due to Caughey [8,9], Iwan [10], Masri [11] and Pratap et al. [12,13]. Aside from their occurrence in several natural processes, hysteretic systems have shown to be of significant engineering interest. Several devices that exploit hysteresis (such as shape-memory alloy based active/passive structures, wire-cable isolators, and magnetic dampers) have been proposed and used for vibration isolation and suppression [14 18]. In such systems hysteresis is deliberately incorporated to enhance damping and other properties of the device. NiTi based shape memory alloys present a particularly interesting avenue for research and application in the field of hysteretic systems [19,20,18]. Several approaches and mathematical models are available for the description of hysteresis. Prominent amongst there are the models developed by Preisach [21], Bouc and Wen [22], Masing [23], Duhem [24], Baber and Noori [25], Prandtl [6,7]. Also widely used is the T(x) family of models in magnetism [26] and the models of hysteresis contained in constitutive laws for shape memory alloys [27,28,19]. Models of hysteresis can be grouped into two categories. Models in the first category have their roots in functional analysis and define hysteresis by using operators. Roughly, a hysteretic operator has a rate independent memory [29]. Operator-based models generally make use of differential inequalities or appropriate integral formulations. The models of Bouc-Wen, Preisach and Duhem, amongst others, fall under this category. The second group of models uses piecewise smooth functions to define the hysteresis loop. The T(x) model, the model of Prandtl, the ideal and non-ideal relay, and the bilinear model fall into this category. However, it should be noted that most piecewise smooth models admit equivalent definitions in terms of differential inequalities [30]. The bilinear model is a piecewise linear model of hysteresis which results in several peculiarities of the system response. Even though the response can be found analytically for the distinct regions, the switches between the various modes can make the overall response complicated. In his pioneering studies Caughey [8] used averaging techniques to study the steady-state response of a bilinear hysteretic Corresponding author. Tel.: address: physicad@kalmarnagy.com (T. Kalmár-Nagy) /$ see front matter 2009 Elsevier B.V. All rights reserved. doi: /j.physd

2 T. Kalmár-Nagy, A. Shekhawat / Physica D 238 (2009) (a) Piecewise affine hysteretic term. (b) Rules for mode transitions. Fig. 1. Bilinear hysteresis. oscillator subject to sinusoidal excitation. The method of averaging is particularly suited for the bilinear oscillator because the averaging results in a smooth slow-flow, despite the overall non-smooth nature of the system. Caughey derived analytical expressions for the frequency response and compared these with simulations using an analog circuit. He found that the system exhibited a soft resonance with no jumps. In Ref. [9] he extended his analysis to random excitations. Masri [11] found the exact solutions for a damped harmonic oscillator with a bilinear hysteretic restoring force. While he reduced the system to one nonlinear algebraic equation which was to be solved using numerical methods he did not report a robust numerical method for solving this equation. Pratap and co-workers [12,13] studied the free response of a bilinear hysteretic oscillator, and later extended their work to include forced response due to impulse loading. They exploited the piecewise linear nature of the system to deduce the analytical solutions for each mode and patched the various solutions to construct the global solution. They found that impulse loading can result in a chaotic response due to a Smale horseshoe. Kalmár-Nagy and Wahi [31] studied the forced response of a relay oscillator by exploiting its piecewise linear nature to obtain numerical as well as analytical results. They found that the harmonically forced relay oscillator has periodic, quasi-periodic and chaotic solutions in the steady-state. They observed chaos due to the transversal intersections of the stable and unstable manifolds of a saddle. The subject of piecewise smooth systems and the techniques developed in this paper are not restricted to hysteresis. Several researchers have investigated the behavior of piecewise smooth systems by using analytical as well as numerical methods. Natsiavas [32] studied the long-term behavior of oscillators with trilinear restoring force by using semi-analytic methods. Like Masri [11] he also found the exact solutions up to the numerical solution of a transcendental equation. Using his analysis he concluded that the system can exhibit periodic as well as chaotic behavior. Luo and Menon [33] used similar techniques to study the dynamics of a periodically forced linear system with a dead-zone restoring force. They investigated global chaos in the system and found that the grazing bifurcation is embedded in the chaotic motion observed in the system. Long and co-workers [34] studied the grazing phenomenon in greater detail. They presented experimental and numerical results for response of a harmonically impacted cantilever structure. In particular, they investigated the corner-colliding bifurcation in non-smooth systems. Other works investigating non-smooth dynamical systems include those of Thota and Dankowicz [35], Lenci and Rega [36 38] and many others. Recently many studies have been directed towards analyzing the complex response of hysteretic systems by using modern tools of nonlinear dynamics. In his 1990 work Capecchi [39] used the Harmonic Balance Method to study the response of a hysteretic oscillator with periodic excitation of period T. He used a hysteresis model based on the Masing rules and analyzed the system by using return maps and found that for full hysteretic loops only 1T periodic steady-state responses existed. Since other responses, like 5T and 7T periodic subharmonic responses have also been reported previously, Capecchi concluded that the higher harmonics play a significant role in the overall dynamics of the oscillator. In Ref. [40] Capecchi and co-workers studied the complex behavior of multiple degree-of-freedom systems with hysteresis. Lacarbonara and Vestroni [41] used Poincaré maps and continuation algorithms to map out the bifurcation sequences of some hysteretic systems using the Masing and Bouc-Wen models. They also found that in general the response is periodic with period one. However, they reported complex behavior for cases with high hysteretic loss. Pilipchuk et al. [42] proposed a nonlinear boundary value problem formulation for computing sub-harmonic orbits of a class of harmonically forced conservative systems based on a non-smooth temporal transformation. This paper presents a study of the response of an oscillator with a hysteretic restoring force with sinusoidal excitation. A robust numerical technique to find the exact (to arbitrary precision) transient and steady-state response of the system is described and discussed. The piecewise linear nature of the system is used to obtain solutions that are piecewise analytical. The discussed technique enables us to robustly and efficiently determine the switching time between the various linear regimes, thus allowing us to patch the local solutions to form the global solution. By using this methodology a map similar to a Poincaré map is constructed for analysis of the steady-state response of the system. The dynamics of this map are studied numerically by using bifurcation and continuation techniques. 2. Problem statement The focus of this study is the following system ẍ + F(x, ɛ) = A cos ωt, t [t 0, t f ], (1) where F(x, ɛ) is a hysteretic term. Here the bilinear hysteretic operator is defined by a hybrid systems approach (a differential formulation is used in [30]). This approach draws on the piecewise smooth functions based description of the model, i.e. F(x, ɛ) is modeled by piecewise linear affine mappings. Fig. 1 shows the input-output graph for a typical bilinear hysteretic operator. The piecewise linear affine mappings have two different slopes, thus the name bilinear hysteresis. As shown in Fig. 1, the bilinear hysteretic operator can be thought of as a hybrid system comprising of four discrete state modes m {I, II, III, IV}. We label these four modes I, II, III and IV. For the individual

3 1770 T. Kalmár-Nagy, A. Shekhawat / Physica D 238 (2009) Table 1 Rules for mode transitions in the bilinear hysteresis model. Transition I II x = x I 2, ẋ < 0 I IV x = x I, ẋ > 0 II III ẋ = 0, ẍ > 0 III IV x = x III + 2, ẋ > 0 III II x = x III, ẋ < 0 IV I ẋ = 0, ẍ < 0 Rule modes F I (x, ɛ) = x + (1 x I )ɛ, F II (x, ɛ) = (1 ɛ)x ɛ, F III (x, ɛ) = x + (x III 1)ɛ, F IV (x, ɛ) = (1 ɛ)x + ɛ, where x I and x III are the values of x at the beginning of modes I and III, respectively. There are six permitted transitions between the modes of the operator, viz., I II, I IV, II III, III II, III IV, and IV I. The transitions between the various modes are governed by rules shown in Fig. 1 and listed in Table 1. Note that the relations given by Eq. (2) and the rules defined in Table 1 ascertain the continuity of F with respect to x across the mode transitions. By using the rules for the mode transitions the mode of the operator at any time t can be uniquely determined given the mode of the operator at a previous time t 0, the value of F at t = t 0, and the history of the input x(t) for t [t 0, t]. Therefore Eq. (1) and the appropriate initial conditions are as follows ẍ + F m (x, ɛ) = A cos ωt, (x(t 0 ), ẋ(t 0 ), F(t 0 ), m(t 0 )) = (x 0, v 0, F 0, m 0 ), t [t 0, t f ]. (3) While Eq. (3) defines the evolution of the continuous state x, the discrete state m evolves in accordance to the mode transition rules listed in Table 1. Remarks 1. The parameter ɛ plays a special role in the bilinear model. For ɛ = 0 the bilinear model reduces to the linear function F(x, 0) = x. Thus, ɛ is a measure of nonlinearity and hysteresis in the system, and for small ɛ the response of the system is expected to be similar to that of the harmonic oscillator. This motivates our use of the averaging technique. 2. The minimum amplitude of a hysteretic loop for the bilinear model is 1, where the amplitude, R = (x max x min )/2, is defined as the half the difference between the maximum and minimum displacement. Thus, for the steady-state response of Eq. (3) to be hysteretic it should have an amplitude greater than For the modes IV, II the stiffness of the system is proportional to 1 ɛ, while for the modes I, III it is equal to 1. It is expected that increasing ɛ will result in a net reduction of the stiffness, thus reducing the resonance frequency. Further, for a fixed ɛ bigger hysteresis loops will correspond to lower net stiffness, thus the system is expected to exhibit a soft resonance. 3. Method of solution The hysteretic nature of F makes solving Eq. (3) challenging. A straightforward, albeit inefficient, approach for solving Eq. (3) is to integrate forward in time by using very small timesteps and checking for mode transitions at every step. This approach has the flaw that no matter how small a timestep is used there is always the chance of missing a mode transition or finding a spurious one (such a case is shown in Fig. 3). In this paper an efficient approach for finding the exact solution of Eq. (3) is presented. By an exact solution it is meant that the solution can be evaluated to a desired precision. In addition, the complexity of the underlying algorithm is T log(1/η), where T = t f t 0 is a measure of the time interval over which the solution of Eq. (3) has to be calculated and η is a measure of the accuracy of the solution. The high fidelity numerical results obtained by using the exact solutions are augmented with asymptotic expansions for the steady-state solution of Eq. (3). The asymptotic results are essentially same as those derived by Caughey [8], and in that sense the present work is an extension of Caughey s efforts Exact solution For any given mode m {I, II, III, IV}, Eq. (3) reduces to that of a simple harmonic oscillator with sinusoidal excitation and can be solved analytically. At the time of transition, t, the quadruplet (x(t), ẋ(t), F(t), m(t)) serves as the initial conditions for the next mode. Proceeding in this manner it is possible to construct the solution of Eq. (3) over any time interval. The problem of solving Eq. (3) then boils down to finding the times for the mode transitions. An efficient algorithm for finding the transition times is presented next. Let the nth transition from mode m n 1 to mode m n take place at t = t n, and let (x n, v n, F n ) = (x(t n ), ẋ(t n ), F(t n )). We define the discrete phase φ n as φ n (ωt n + φ n 1 ) mod 2π, where φ n 1 is the value of the phase for the previous mode. By introducing the phase, time t is essentially reset to 0 at every mode transition, thus reducing the governing equation for any mode to the following canonical form ẍ + F m (x, ɛ) = A cos(ωt + φ n ), (x(0), ẋ(0), F(0), m(0)) = (x n, v n, F n0, m n ), t [0, t n ]. (5) (2) (4)

4 T. Kalmár-Nagy, A. Shekhawat / Physica D 238 (2009) Table 2 Parameters in expression of x(t) for different modes. Mode, m ω 2 m k m B m I 1 (x I 1)ɛ A/(ω 2 I ω 2 ) II 1 ɛ ɛ A/(ω 2 II ω2 ) III 1 (x III + 1)ɛ A/(ω 2 III ω2 ) IV 1 ɛ ɛ A/(ω 2 IV ω2 ) t l->ll X l φ 4 6 Fig. 2. Variation of t I II with x I and φ I for ɛ = 0.3, A = 1.6, ω = 1.5. Eq. (5) can be solved to give the following equation for evolution of x(t) x(t) = ( x n k mn B mn cos φ n ) cos(ωn t) + ( (v n + B mn ω sin(φ n ))/ω mn ) sin(ωmn t) + B mn cos(ωt + φ n ) + k mn, (6) where the variables k, ω, and B for the different modes are listed in Table 2. The mode transitions occur at zeros of equations of the kind x(t) = const. or ẋ(t) = 0 (see Table 1). Consider, for example, the equation governing a I II transition. This equation is given by where x(t I II ) = x I 2, x(t) = (x I k I B I cos φ I ) cos(ω I t) + (B I ω sin(φ I )/ω I ) sin(ω I t) + B I cos(ωt + φ I ) + k I. (8) In Eq. (8), v I = 0 has been used since by definition ẋ is zero at the beginning of mode I. The mode transition I II occurs at the first positive root of Eq. (7), denoted by t I II, that satisfies ẋ(t I II ) < 0. Finding the first positive solution of Eq. (7) (and other similar equations) is a nontrivial task. Even for fixed ɛ, A and ω the solutions are in general discontinuous in x I and φ I. Fig. 2 shows the variation of t I II with x I, φ I for one case. (7) Root finding algorithm As discussed in the previous Section, equations of the following general kind need to be solved for the transition time g(t) = C 1 cos ˆωt + C 2 sin ˆωt + C 3 cos(ωt + 1 ) + C 4 = 0. (9) Eq. (9) can be written as cos( ˆωt + 2 ) + C ˆω ω cos(ωt + 1) + D = 0, where C = ωc 3 /( ˆω C C 2), 2 2 = arctan(c 1 /C 2 ) π/2, and D = C 4 / C C 2 2. The above equation can be further simplified by the following change of variables ( z = ˆω t + ) 1 ω to yield (10) (11) f (z) = cos(z + ) + C cos(ωz) + D = 0, Ω (12) where = 2 1 ˆω/ω, and Ω = ω/ ˆω. Eq. (12) can be solved to any desired precision by using the bisection algorithm if intervals that contain exactly one root are identified. Note that by the Mean Value Theorem all roots of Eq. (12) are either coincident with or lie between the roots of the following equation f (z) = sin(z + ) + C sin(ωz) = 0, (13)

5 1772 T. Kalmár-Nagy, A. Shekhawat / Physica D 238 (2009) (a) f (z) versus z with C = 1.8, Ω = , = 0.2. The first root, z = 5.54, is shown by a circle. (b) f (z) versus z with C = 1.8, Ω = , = 0.2. The first root, z = 6.99, is shown by a circle. Fig. 3. First root of f (z) for some values of C, Ω,. Note the discontinuity of the root with respect to the parameters. Proposition 1. f (z) has at the most two roots in the interval (z i, z i+1 ) where z i s are elements of an ordered set consisting of the peaks or zeros of sin(z + ) and/or sin(ωz), i.e., sin(2(z i + )) sin(2ωz i ) = 0. Further, if the interval contains two roots then they are separated by the unique zero of df (z)/dz in the interval. The complete proof for Proposition 1 can be found in Ref. [30], and the proof is outlined in Appendix A. By using Proposition 1 it is possible to construct intervals that have at most two zeros of f (z) and at most one zero of df /dz. The unique zeros (if they exist) of df /dz in these intervals can be calculated to any desired accuracy by using the bisection algorithm [43]. Since the zeros of df /dz partition the intervals into subintervals containing at most one root of f (z), this root, if it exists, can be found to any desired accuracy by using the bisection algorithm. Finally, since the roots of f (z) are either coincident with or separated by the roots of f (z) they can also be evaluated to any desired accuracy by using the bisection algorithm. Fig. 3 shows the first root of f (z) calculated using the proposed method for certain values of the parameters. The figure highlights the fact that the location of the root is a discontinuous function of the parameters. It is also emphasized that the complexity of the proposed algorithm compares favorably with traditional Runge Kutta type integration schemes. A discussion of the complexity and performance of the algorithm is presented in Appendix B Heuristic estimates and asymptotic analysis Before deriving the asymptotic expansions certain useful results can be obtained by simple reasoning and order of magnitude arguments. It was noted earlier that if the steady-state response amplitude is less than 1, then the steady-state oscillations of the system are conservative. For a given amplitude of excitation, the frequency of excitation required to sustain steady-state oscillations with magnitude marginally above 1 can be calculated as follows A 1 ω 2 = ±1. The above equation implies that there is a upper and a lower bound on the frequency of excitation required for existence of hysteretic oscillations in the steady-state. These can be obtained as follows ω upper = 1 + A, ω lower = 1 A. If A 1 then there is no lower bound on ω for the existence of hysteretic oscillations. By using the above results the A ω plane can be divided into regions corresponding to hysteretic and conservative oscillations in the steady-state. Fig. 4 shows these regions. Another interesting insight can be gained by noting that for ω 1 in Eq. (1) the loading can be treated as quasi-static. In such a case the steady-state response amplitude, R, can be found as follows R = A ɛ, if A > 1, 1 ɛ and conservative oscillations result if A 1. The above relation suggests that for ɛ 1 and ω 1 the steady-state will have very large amplitude oscillations even if A O(1). This result is expected since for ɛ 1 the net stiffness of the system is greatly reduced. Fig. 5 shows the variation of R with A for various values of ɛ for ω 1. The asymptotic expansions provided in the next Section can be used to verify the above estimates, however, strictly speaking the asymptotic expansions are valid only for ɛ 0, thus the asymptotic estimates cannot be used to justify the above claims for all ɛ Asymptotic expansion Here the one-term uniform expansion for the steady-state response of the system originally derived by Caughey [8] is presented. It is well-known that the scaling of the system changes near resonance, 1 and expansions that are uniformly valid elsewhere lose their (14) (15) (16) 1 Scaling of the system refers to the exponent of the dominant correction term in the asymptotic expansions for the solution. For example, in the expansion x = x 0 + ɛx 1 + ɛ 2 x 2 + the scale is ɛ while in the expansion x = x 0 + ɛ 1/2 x 1 + ɛx 2 + the scale is ɛ 1/2. See Refs. [44,45] for details.

6 T. Kalmár-Nagy, A. Shekhawat / Physica D 238 (2009) F(x) 2 x ω 1.5 F(x) 1 x A Fig. 4. Partitions of the A ω plane corresponding to conservative (shaded) and hysteretic (unshaded) oscillations in the steady-state. 40 ε = R ε = 0.7 ε = 0.5 ε = Fig. 5. Variation of response amplitude with amplitude of excitation for ω 1. uniformity near resonance. Thus, the derived expansions are expected to give good results at frequencies away from the resonant frequency of the system. Following the KBM method Caughey [8] assumed the following form for the steady-state response of Eq. (1) A x ss (t) = R cos(ωt + θ), Ṙ O(ɛ), θ O(ɛ), (17) and obtained the following relations for the frequency response of the system [ ( ω 2 = C(R) ) 2 ( ) ] A S(R) 2 1/2 R ±. R R where tan θ = S(R) C(R) ω 2 R, S(R) = ɛr π sin2 γ I II, ( ɛγ I II + (1 ɛ)π ɛ 2 sin 2γ I II C(R) = R π In the above equations γ I II is defined as γ I II = arccos(1 2/R), (18a) (18b) ). (20) and it corresponds to the value of ωt + θ when the transition I II occurs. Caughey proved that the steady-state response is linearly stable. He also proved that the amplitude at the primary resonance is given by the following expression R = 4ɛ 4ɛ πa. Therefore, the primary resonance is bounded if A < 4ɛ/π and becomes unbounded for sufficiently large A. Since ɛ < 1, the primary resonance is always unbounded for A > 4/π. The above results can be used to partition the A ɛ plane into regions corresponding (19) (21) (22)

7 1774 T. Kalmár-Nagy, A. Shekhawat / Physica D 238 (2009) R 0.6 ω R ε ω A Fig. 6. Partitions of the A ɛ plane corresponding to bounded (shaded) and unbounded (unshaded) primary resonance (1 ω 2 )/ε R Fig. 7. Variation of ω with R. to bounded and unbounded primary resonance. Fig. 6 shows this partition where the shaded region corresponds to bounded primary resonance and the unshaded region corresponds to unbounded primary resonance. At resonance, Eq. (18a) becomes ω 2 = C(R) R, or, on substituting for C(R) and rearranging ( γ I II + π 1 2 sin 2γ I II ω 2 1 ɛ = 1 π By defining a new non-dimensional number, ω = (1 ω 2 )/ɛ, the following can be deduced ω = 1 ( γ I II π 1 ) π 2 sin 2γ I II. ). (23) (24) (25) This represents a simplification because the non-dimensional number ω is related directly to γ I II which in turn depends only on R. Fig. 7 shows the variation of ω with R. By using Eqs. (22) and (25) the amplitude at resonance as well as the resonance frequency can be estimated for any given ɛ, A, and ω. The asymptotic expansions for the steady-state response of the system can also be used to estimate the equivalent damping and stiffness properties of the system, see Ref. [46] for more details on the issue. We define the equivalent damping, ξ, and the equivalent natural frequency, ω 0, of the system such that the steady-state response of the following oscillator is identical to steady-state response of Eq. (1) obtained by using the asymptotic analysis ẍ + 2ξω 0 ẋ + ω 2 0x = A cos ωt. (26) The equivalent natural frequency ω 0 and the damping coefficient ξ can be found to be the following ω 2 0 = C(R) R, (27)

8 15 T. Kalmár-Nagy, A. Shekhawat / Physica D 238 (2009) R ω Fig. 8. Comparison of exact response curves obtained from the map Π with the those obtained by the KBM method. Solid lines: Π-Map, Circles: KBM Method, Dashed line: Locus of resonance frequency and resonance amplitude for 0.8 < A < 0.1 obtained by the KBM method. ɛ = 0.6, A varied from 0.1 to 0.8 in increments of 0.1. and ξ = S(R) 2Rωω 0. Thus, the equivalent natural frequency of the system and its resonance frequency match at resonance. Also note that the equivalent natural frequency and the resonance frequency are not equal in general. (28) 4. Steady-state analysis: Harmonic and sub-harmonic response Even though the asymptotic estimate obtained by Caughey is seen to match the exact response of the system well, it is not uniformly valid near resonance and cannot be readily extended to incorporate sub-harmonic responses. Furthermore, the asymptotic expansions are guaranteed to work only in the limit of ɛ 0. In order to overcome these shortcomings, a construct similar to a Poincaré map is introduced for analyzing the long term behavior of the system. We consider only those steady-state responses that consist of orbits periodic in time. First the general structure of the trajectories of the system is discussed and then this structure is used to define a map that can be used to characterize the periodic orbits. Obviously, for any trajectory mode IV is always followed by mode I and mode II is always followed by mode III. Similarly, mode I is always preceded by IV and mode III is always preceded by mode II. Therefore the most general structure of a periodic trajectory of the system can be characterized as ( (I IV) ni times I II (III II) mi times ) III IV. i such blocks with different n i,m i For example, i = 1, n 1 = m 1 = 0 results in the simplest periodic orbit of the system with the following order of mode transitions ( I II III IV ) repeated indefinitely. From the above arguments it is seen that every periodic orbit of the system has at least one IV I transition. By definition ẋ I is zero, i.e., the velocity is zero at the beginning of state I. The beginning of state I is then characterized by the time t and the value of x(= x I ) at which the IV I transition occurs. Since the phase variable φ I was introduced to replace time, the beginning of state I is characterized by the pair (x I, φ I ). A periodic orbit may have many IV I transitions. Any one of these transitions can be used to define a map (x I, φ I ) i+1 = Π(x I, φ I ) i. The map Π (also referred to as the Π-map) can be constructed by using Proposition 1 to calculate the exact mode transition times for the system and using the analytical expressions for the system response between transitions [30]. This map is in general not continuous and its dynamics can be a lot more complicated than its smooth counterparts [47]. The steady-state response of Eq. (3) via the Π-map was studied by MATCONT [48], a MATLAB package for bifurcation and continuation analysis of continuous and discrete dynamical systems. It was found that for most parameter values the steady-state response of the system consists of almost sinusoidal periodic orbits with period equal to the period of excitation. As established earlier, the system can exhibit both bounded and unbounded resonance. All estimates provided in Section 3.2 were seen to hold to a good degree of accuracy for a wide range of parameters. The amplitude of the single-harmonic response was found to be single valued for all the cases investigated. The stability of the orbits was determined by computing the Floquet multipliers via MATCONT [49]. For the parameter ranges studied no unstable orbits were detected. Fig. 8 shows a comparison of the response curves obtained by the Π-map and the KBM method for ɛ = 0.6 and A varied between 0.1 and 0.8. The amplitude is obtained from the orbit of the Π-map by using the relation R = (x I x III )/2. It was found that the orbits of the Π-map are symmetric and R is almost equal to x I. Fig. 8 also shows the so-called backbone curve obtained by using the KBM method. This curve shown by the dotted line in the Figure is the locus of resonance frequencies and the corresponding response amplitudes for varying A. It (29)

9 1776 T. Kalmár-Nagy, A. Shekhawat / Physica D 238 (2009) x I ω (a) Primary and secondary resonances. x I ω (b) A magnified view of the sub-harmonic responses. Fig. 9. Comparison of response curves obtained by using the KBM method (shown by circles) and the Π-maps (shown by solid line) for a case with sub-harmonic resonance; ɛ = 0.3, A = 1.6. can be seen that the curve obtained by the KBM method approximates the exact resonance frequency and amplitudes to a good degree of accuracy for values of ω away from resonance. This deviation was expected since the one-term expansions are only uniformly valid away from resonance. Even though the exact response curves shown in Fig. 8 consist solely 1T periodic orbits, where T is the period of excitation, it was also observed that for many cases the exact response displays several sub-harmonic resonances. Fig. 9(a) and (b) compare the response amplitude obtained by averaging and our exact method for a case with sub-harmonic resonance. Near sub-harmonic resonances the steady-state no longer consists of only one frequency component as assumed in the KBM method, and thus this solution does not capture the steady-state response accurately for such cases. Fig. 10 shows a comparison of the steady-state orbits away from and near the sub-harmonic resonances. It is evident from the Fourier spectrum of ẋ(t) that multiple frequency components are present in the response near the sub-harmonic resonances. Since the Π-map is discontinuous, the higher period orbits are not born via any classical bifurcations of the map. For this reason MATCONT was not able to continue the fixed points over certain parameter values in several cases. The response curve shown in Fig. 9(a) corresponds to higher period orbits for some parameter values and was completed by using orbit diagrams. Fig. 11 shows the 2-period orbit of the Π-map that exists near the first sub-harmonic resonance. Fig. 12 shows the response amplitude obtained from the orbit diagram and the orbit diagram itself near the sub-harmonic resonances for ɛ = 0.3, A = 1.6. The sub-harmonic resonances in the system are not limited to 1:5 or 1:7 resonances. In fact, for larger excitation magnitudes an entire series of sub-harmonic resonances can be observed. In order to observe a range of sub-harmonic resonances it is required that the amplitude of excitation be greater than 1, the reason being that if A < 1 then the steady-state consists of hysteretic oscillations only for ω > 1 A, and some sub-harmonic resonances corresponding to ω < 1 A will not be observed. Fig. 13 shows the response curves for ɛ = 0.4 and A varied between 2 and 10 in steps of 2. A comparison of the exact response curves and those obtained from the KBM analysis is shown in Fig. 14. As expected, the exact response and the KBM method are in good agreement away from the resonances. Figs show the steady-state orbits and the Fourier spectrum for a sequence of sub-harmonic resonances for ɛ = 0.4, A = 6. The figures show 1:3, 1:5, 1:7 and 1:9 type of sub-harmonic resonances. It is clear from the Fourier spectrum that the steady-state contains multiple frequency components. 5. Transient response and higher period orbits The transient response of the system can have a complex sequence of mode transitions. Due to the presence of grazing bifurcations the mode transition sequence in the transient response can have a sensitive dependence on initial conditions for some parameter values and initial conditions. Fig. 19 shows the transient response for two trajectories starting from nearby initial conditions. In the Figure Case I corresponds to the initial conditions (x(0), ẋ(0), φ(0), m(0)) = (2, 0, , I) while Case II corresponds to (x(0), ẋ(0), φ(0), m(0)) = (2, 0, , I). Even though the two cases differ only in the starting value of the discrete variable φ at the fourth place of decimal, the two initial conditions lead to different mode transition sequences. The figure clearly shows that this difference is due to the presence of a grazing bifurcation. However, while the sequence of mode transitions is significantly different, the long-term time history of x(t) is nearly the same for both cases. In the many simulations that were run it is found that this behavior is a general rule rather than an exception. Further, it was noted that the steady-state response of the system is unique (irrespective of initial conditions). Thus, in spite of the transient differences all initial conditions lead to the same steady-state response. Fig. 20 shows the identical steady-state response for Cases I and II. Figs. 21 and 22 show transient responses of the system. The figures show that the grazing phenomenon is not necessarily encountered in all cases or for all initial conditions. The figures also highlight the fact that the (x, ẋ) space is not the true phase space of the system as trajectories here intersect due to the explicit time dependence of the governing equation. In these figures the plots of F(x) versus x contain illustrations where the actual plot is suitably rotated and scaled for better demonstration of the dynamics. Typically higher period orbits of the Π-map occur when there is a significant difference in the frequency of excitation and the natural frequency of the system. In such cases the response of the system comprises two frequency components, one significantly higher than the other. The high frequency component often results in frequent changes in the modes, however, the overall character of the solution is still sinusoidal with one dominant frequency, and thus these cases are adequately captured by the KBM analysis. Figs show the graphs for some of these higher period orbits.

10 T. Kalmár-Nagy, A. Shekhawat / Physica D 238 (2009) (a) Steady-state phase portrait. Input frequency, ω = 0.5. x = , φ = at the IV I transition. (b) Steady-state frequency spectrum of ẋ(t). (c) Steady-state phase portrait. Input frequency, ω = 0.3. x = , φ = at the IV I transition. (d) Steady-state frequency spectrum of ẋ(t). (e) Steady-state phase portrait. Input frequency, ω = x = , φ = at the IV I transition. (f) Steady-state frequency spectrum of ẋ(t). Fig. 10. Steady-state response and corresponding frequency spectrum away and near the sub-harmonic resonance. ɛ = 0.3, A = 1.6. Transitions are denoted as: I II: solid circle, II III: empty triangle, III IV : empty circle, IV I: solid triangle. On the frequency spectra the first few odd multiples of ω are marked by solid circles. 6. Conclusions The response of a bilinear hysteretic oscillator with sinusoidal excitation was studied in a hybrid systems framework. The exact transient and steady-state response of a 1-DOF oscillator with bilinear hysteresis and sinusoidal excitation was obtained in this paper for the first time. The exact solution was constructed by calculating the transition times exactly, and by solving the equations of motion analytically between mode switches. It was shown that the complexity of the proposed algorithm is T log(1/η), where T is the length of interval of integration and η is the desired accuracy. As a comparison the complexity of Runge Kutta type algorithms is typically T/η p with 0 < p < 1. Thus, the proposed algorithm was found to be superior to the traditional small time increment type algorithms. It was found that the transient response of the system is complicated, and the sequence of mode transitions showed sensitive dependence on initial conditions for some parameter values and initial conditions. In spite of a complicated transient response, the non-trivial steady-state of the system consists of periodic orbits of period 2π/ω, where ω is the frequency of excitation. The system exhibits a soft-resonance, which could be bounded or unbounded depending on the amplitude of excitation. The system did not display a jump phenomenon. Several sub-harmonic resonances were observed in the system, however, no chaotic behavior was encountered. The long term behavior of the system was studied by using various tools like orbit diagrams, time domain simulations, fixed point calculations and continuation analysis of a discrete map. The asymptotic expansions first obtained by Caughey [8] were used to obtain estimates for the resonance frequency and the amplitude at resonance. These estimates were seen to match well with the exact response away from the primary and secondary resonances. Heuristic estimates were provided for the response amplitude in case of low frequency excitation. Heuristic arguments were also used to

11 1778 T. Kalmár-Nagy, A. Shekhawat / Physica D 238 (2009) (a) x(t) versus t. (b) Steady-state frequency spectrum of ẋ(t). First few odd multiples of ω are marked by solid circles. (c) Magnified view of some mode transitions. (d) ẋ(t) versus x(t). Fig. 11. A 2-period orbit for ɛ = 0.3, ω = , A = 1.6. Transitions are denoted as I II: solid circle, I IV : solid square, II III: empty triangle, III II: empty square, III IV : empty circle, IV I: solid triangle. x = , φ = at one IV I transition. (a) Response amplitude. (b) Orbit diagram. Fig. 12. Response curve obtained from the orbit diagram in case of sub-harmonic resonance. ɛ = 0.3, A = 1.6. (a) Response curves for A between 2 and 10. (b) Magnified view of sub-harmonic resonances for A = 6. Fig. 13. Response curves for ɛ = 0.4 and A varied between 2 and 10 in steps of 2 obtained by using orbit diagrams. Note that there are several sub-harmonic resonances.

12 30 T. Kalmár-Nagy, A. Shekhawat / Physica D 238 (2009) R ω Fig. 14. Comparison of exact response curves obtained from the Π-map with the those obtained by the KBM method. Solid lines: Π-map, Dashed lines: KBM Method. ɛ = 0.4, A varied from 2 to 10 in increments of 2. (a) Steady-state orbit phase portrait. I II: solid circle, II III: empty triangle, III IV : empty circle, IV I: solid triangle. x = , φ = at the IV I transition. (b) Frequency spectrum of steady-state ẋ(t). Fig :3 Sub-harmonic response for ɛ = 0.4, A = 6, ω = (a) Steady-state orbit phase portrait. I II: solid circle, II III: empty triangle, III IV : empty circle, IV I: solid triangle. x = , φ = at the IV I transition. (b) Frequency spectrum of steady-state ẋ(t). Fig :5 Sub-harmonic response for ɛ = 0.4, A = 6, ω = partition the A ω space into parts corresponding to regions that exhibit hysteretic oscillations in the steady-state and those that exhibit conservative oscillations. Both estimates were seen to match well with the exact solution. Acknowledgements The authors thank the Department of Aerospace Engineering, Texas A&M University and the US Air Force Office of Scientific Research (Grant No. AFOSR ) for providing financial support for this study.

13 1780 T. Kalmár-Nagy, A. Shekhawat / Physica D 238 (2009) (a) Steady-state orbit phase portrait. I II: solid circle, II III: empty triangle, III IV : empty circle, IV I: solid triangle. x = , φ = at one IV I transition. (b) Frequency spectrum of steady-state ẋ(t). Fig :7 Sub-harmonic response for ɛ = 0.4, A = 6, ω = (a) Steady-state orbit phase portrait. I II: solid circle, II III: empty triangle, III IV : empty circle, IV I: solid triangle. x = , φ = at the IV I transition. (b) Frequency spectrum of steady-state ẋ(t). First few odd multiples of ω are marked by solid circles. Fig :9 Sub-harmonic response for ɛ = 0.4, A = 6, ω = (a) Case I: (x(0), ẋ(0), φ(0), m(0)) = (2, 0, , I). (b) Case II: (x(0), ẋ(0), φ(0), m(0)) = (2, 0, , I). Fig. 19. Comparison of transient response for two close-by initial conditions. Note that in spite of different mode transition sequences the time response of x is almost the same for both cases. I II: solid circles, II III: empty triangles, III II: empty squares, III IV : empty circles, IV I: solid triangles. Appendix A. Root isolation Here the problem of isolating the roots of the following equation is considered g(z) = sin(z + ) + A sin(ωz) = 0, Root isolation means finding open intervals (z 1, z 2 ) such that g(z) has at most one zero in interval. Thus, within these intervals the roots do not have any neighboring roots and are isolated. Once the roots are isolated it is a routine matter to find them to any desired accuracy by using the bisection algorithm. Consider two curves S 1 and S 2 defined as follows S 1 : sin(z + ), S 2 : A sin(ωz). (30) (31) (32)

14 T. Kalmár-Nagy, A. Shekhawat / Physica D 238 (2009) (a) x(t) versus t. (b) ẋ versus x(t). Fig. 20. Steady-state solution for ɛ = 0.3, A = 1.6, ω = 1.5. I II: solid circle, II III: empty triangle, III IV : empty circle, IV I: solid triangle. At the IV I transition x = , φ = (a) x(t) versus t. (b) ẋ(t) versus t. (c) ẋ versus x. (d) F(x) versus x. Fig. 21. Transient response with ɛ = 0.3, A = 3.65, ω = 1.2. Initial conditions: x(t 0 ) = 1, ẋ(t 0 ) = 0, t 0 = π/2ω, starting mode = I. The first 20 mode transitions are shown. I II: solid circles, II III: empty triangles, III IV : empty circles, IV I: solid triangle. Eq. (30) can then be written as S 1 = S 2. Note that the structure of Eq. (30) is invariant under the operation of shifting S 1 and/or S 2 along the z-axis and/or multiplying them by scalars. Eq. (30) can have a root z 1 of multiplicity 2 if sin(z 1 + ) + A sin(ωz 1 ) = 0, cos(z 1 + ) + Aω cos(ωz 1 ) = 0. However, the case with multiplicity of the root equal to greater than 3 is not interesting since it would demand that the following should hold sin(z 1 + ) + A sin(ωz 1 ) = 0, cos(z 1 + ) + Aω cos(ωz 1 ) = 0, sin(z 1 + ) + Aω 2 sin(ωz 1 ) = 0. The necessary conditions for existence of solutions of Eq. (35) can be found to be ω = 1, A = 1 (assuming ω > 0). In this case, Eq. (30) can be solved in closed form, thus the problem of root isolation is resolved. Next, the more interesting cases where the set (35) has no solutions are considered. (33) (34) (35)

15 1782 T. Kalmár-Nagy, A. Shekhawat / Physica D 238 (2009) (a) x(t) versus t. (b) ẋ(t) versus t. (c) ẋ versus x. (d) F(x) versus x. Fig. 22. Transient response with ɛ = 0.7, A = 150, ω = 10. Initial conditions: x(t 0 ) = 1, ẋ(t 0 ) = 0, t 0 = π/2ω, starting mode = I. The first 20 mode transitions are shown. I II: solid circles, I IV : solid squares, II III: empty triangles, III II: empty squares, III IV : empty circles, IV I: solid triangle. (a) x(t) versus t. (b) Steady-state frequency spectrum of ẋ(t). First few odd multiples of ω are marked by solid circles. (c) Magnified view of some mode transitions. (d) ẋ(t) versus x(t). Fig. 23. A 2-period orbit for ɛ = 0.3, ω = 0.01, A = 20. I II: solid circle, I IV : solid square, II III: empty triangle, III II: empty square, III IV : empty circle, IV I: solid triangle. x = , φ = at one IV I transition.

16 T. Kalmár-Nagy, A. Shekhawat / Physica D 238 (2009) (a) x(t) versus t. (b) Steady-state frequency spectrum of ẋ(t). First few odd multiples of ω are marked by solid circles. (c) Magnified view of some mode transitions. (d) ẋ(t) versus x(t). Fig. 24. An 11-period orbit for ɛ = 0.3, ω = 0.002, A = 20. I II: solid circle, I IV : solid square, II III: empty triangle, III II: empty square, III IV : empty circle, IV I: solid triangle. x = , φ = at one IV I transition. (a) x(t) versus t. (b) Steady-state frequency spectrum of ẋ(t). First few odd multiples of ω are marked by solid circles. (c) Magnified view of some mode transitions. (d) ẋ(t) versus x(t). Fig. 25. A 3-period orbit for ɛ = 0.3, ω = 0.03, A = 10. I II: solid circle, I IV : solid square, II III: empty triangle, III II: empty square, III IV : empty circle, IV I: solid triangle. x = , φ = at one IV I transition.

17 1784 T. Kalmár-Nagy, A. Shekhawat / Physica D 238 (2009) (a) Scale and shift construction. (b) Moving 2 roots closer by shifts. Fig. 26. Constructions 1 and 2. (a) Case 1. (b) Case 2. Fig. 27. Moving three roots closer by using the scale and shift construction. Proposition 2. If ω > 1 then Eq. (30) has at most two roots between adjacent peaks and zeros of sin(ωz), if ω < 1 then Eq. (30) has at most two roots between the adjacent peaks and zeros of sin(z + ). Proof. As shown in Fig. 26, given a sinusoidal curve C 1 and a point o on the curve it is always possible to construct another sinusoidal curve C 2 with the same frequency such that the two curves intersect at o and within the quarter period of C 2 that contains o the curve C 2 lies above C 1 on one side of o and below it on the other side. The construction shown in Fig. 26 will be referred to as the scale and shift construction. This construction will be used to prove the claim using a contradiction. Suppose ω > 1 (the case with ω < 1 can he handled similarly). Let if possible S 1 and S 2 intersect three times between an adjacent peak and zero of S 2 (see Fig. 27). Then one can use the scale and shift construction to construct another curve S 3 such that S 1 and S 3 intersect three times between an adjacent peak and zero of S 3 and the intersections are closer to each other as compared to the intersections of S 1 and S 2. By continuing this construction the intersections can be made to come arbitrarily close to each other, thus creating a root of Eq. (30) with multiplicity 3, which is a contradiction since the construction of scaling and shifting does not alter the structure of Eq. (30). These arguments can be put in rigorous terms as follows. Let z l, z m, z r be the three intersections of S 1 and S 2. Since S 2 is monotonic in the considered interval, without the loss of generality it can be assumed to be decreasing. It follows that if z l < z m < z r then S 2 (z l ) < S 2 (z m ) < S 2 (z r ). By construction S 3 satisfies the following S 3 (z l ) > S 2 (z l ), S 3 (z m ) = S 2 (z m ), S 3 (z r ) < S 2 (z r ). (36) Since the deformation of S 2 into S 3 can be continuous, it follows that S 3 can be chosen to be such that S 3 and S 2 are on the same side of S 1 in a sufficiently small interval around z m. Then, by continuity S 3 and S 1 should have at least one intersection between z m and z l, and at least one intersection between z m and z r. Finally, since it is possible to construct S 3 such that there are no intersections other than z m (by

18 T. Kalmár-Nagy, A. Shekhawat / Physica D 238 (2009) Proposed Algorithm Typical Runge-Kutta Scheme Computational Effort η Fig. 28. Comparison of computational effort of the proposed algorithm with a typical Runge Kutta algorithm. p = 0.25 is used for the Runge Kutta algorithm and a time interval of unit length (T = 1) is considered for both cases. choosing a large enough scaling factor), therefore, by continuity, there should exist a scaling at which the intersections are arbitrarily close (after which two of them collide and annihilate each other). This, however, leads us to a contradiction, and hence the claim must be true. Note: Similar arguments can be made for other types of intersections of curves. See Fig. 27(b) for example. In this case S 1 is scaled and shifted. Using Proposition 2 it is possible to isolate roots of Eq. (30) in pairs of two. Even though it is an enormous simplification, methods like bisection can be used only if the individual roots can be isolated. A methodology for isolating the individual roots is presented next. Note that if there are two intersections of the curves S 1 and S 2 in a quarter period of the curve with the higher frequency then these intersections can be made to come arbitrarily close to each other by shifting one of the curves. This construction, called the shift construction, is depicted in Fig. 26(b). Thus, if there are two intersections of the curves between an adjacent zero and peak of the curve with the higher frequency then there exists some shift for which Eq. (30) has a root of multiplicity 2. Therefore, there can be two roots of Eq. (30) in the quarter period of the curve with the higher frequency only if Eq. (30) has a root of multiplicity 2 for some. It is easy to show that the following are necessary conditions for existence of a root of multiplicity 2 of Eq. (30) A 2 > 1, 1 > A 2 ω 2 ; for ω < 1, A 2 < 1, 1 < A 2 ω 2 ; for ω > 1. If Conditions (37) are not satisfied then the roots of Eq. (30) can be isolated using Proposition 2. In that case the following intervals contain unique roots of Eq. (30) I n = (z n, z i n ), z n = + n π; for ω < 1, 2 I n = (z n, z i n ), z n = nπ ; for ω > 1. 2ω If Conditions (37) are indeed satisfied then Proposition 2 needs to be strengthened. Note that the derivative of Eq. (30) vanishes when cos(z + ) + Aω cos(ωz) = 0. Eq. (39) has the same structure as Eq. (30) and thus its zeros can be isolated (at least in pairs) using the same arguments. Note that the intervals I n are the same for Eqs. (30) and (39). Further, it is easy to show that the following are necessary conditions for the existence of a root of multiplicity 2 of Eq. (39) A 2 ω 2 > 1, 1 > A 2 ω 4 ; for ω < 1, A 2 ω 2 < 1, 1 < A 2 ω 4 ; for ω > 1. It easily follows that Conditions (37) and (40) cannot hold simultaneously. Thus, if there exist two roots of Eq. (30) in the intervals I n, the Eq. (39) has a unique root in those intervals. Finally, since the roots of Eq. (30) are necessarily separated by roots of Eq. (39), therefore the individual roots of Eq. (30) can be isolated by using Proposition 2 and the above stated arguments. Appendix B. Complexity of exact solution algorithm The complexity of the exact solution algorithm presented in Section is discussed here. The number of intervals containing at most two roots of df /dz (see Section 3.1.1) is a linear function of the total time over which the solution is sought. Within each of these intervals the bisection algorithm is to be used to find all the zeros of f (z) in the interval. Each use of the bisection algorithm requires O(log(1/η)) function evaluations, where η is the ratio of the length of the final bisection interval to the initial interval. By definition η < 1, and it is a measure of the accuracy of the solution. Thus, the overall complexity of the algorithm is T log(1/η), where T is the length of the interval (37) (38) (39) (40)