Nonlinear dynamics of oscillators with bilinear hysteresis and sinusoidal excitation

8/12/2019 Nonlinear dynamics of oscillators with bilinear hysteresis and sinusoidal excitation

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Physica D 238 (2009) 17681786

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Physica D

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Nonlinear dynamics of oscillators with bilinear hysteresis andsinusoidal excitationTams Kalmr-Nagy , Ashivni ShekhawatDepartment of Aerospace Engineering, Texas A&M University, United States

a r t i c l e i n f o

Article history:Received 20 May 2008Received in revised form10 January 2009Accepted 1 June 2009Available online 23 June 2009Communicated by G. Stepan

Keywords:Bilinear hysteresisHysteretic oscillatorCaughey modelForced response

a b s t r a c t

The transient and steady-state response of an oscillator with hysteretic restoring force and sinusoidalexcitation are investigated. Hysteresis is modeled by using the bilinear model of Caughey with a hybridsystem formulation. A novel method for obtaining the exact transient and steady-state response of thesystem is discussed. Stability and bifurcations of periodic orbits are studied using Poincar maps. Resultsare compared with asymptotic expansions obtained by Caughey. The bilinear hysteretic element is foundto act like a soft spring. Several sub-harmonic resonances are found in the system, however, no chaoticbehavior is observed. Away from the sub-harmonic resonance the asymptotic expansions and the exactsteady-state response of the system are seen to match with good accuracy.

2009 Elsevier B.V. All rights reserved.

1. Introduction

The phenomenon of hysteresis is widely prevalent in nature and has been studied in diverse contexts [15 ]. Of particular engineeringinterest 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 materialsetc. can be modeled by using the bilinear model of hysteresis. This model is inspired by elasto-plasticity and can be considered to be ageneralization 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].

Asidefrom theiroccurrencein several natural processes,hystereticsystemshaveshownto be of significantengineering interest.Severaldevices 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 [ 1418 ]. In such systems hysteresis is deliberately incorporated toenhance damping andotherproperties of thedevice. NiTi based shape memoryalloys present a particularlyinterestingavenuefor researchand 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 modelsdeveloped 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 definehysteresis by using operators. Roughly, a hysteretic operator has a rate independent memory [ 29]. Operator-based models generally makeuse of differential inequalities or appropriate integral formulations. The models of Bouc-Wen, Preisach and Duhem, amongst others, fallunder this category.The second group of modelsuses piecewise smooth functions to define the hysteresis loop. The T ( x) model, the modelof Prandtl, the ideal and non-ideal relay, and the bilinear model fall into this category. However, it should be noted that most piecewisesmooth 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 thoughthe response can be found analytically for the distinct regions, the switches between the various modes can make the overall responsecomplicated. In his pioneering studies Caughey [ 8] used averaging techniques to study the steady-state response of a bilinear hysteretic

Corresponding author. Tel.: +1 860 730 3245.E-mail address: [email protected] (T. Kalmr-Nagy).

0167-2789/$ see front matter 2009 Elsevier B.V. All rights reserved.doi:10.1016/j.physd.2009.06.016
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T. Kalmr-Nagy, A. Shekhawat / Physica D 238 (2009) 17681786 1769

(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 averagingresults in a smooth slow-flow, despite the overall non-smooth nature of the system. Caughey derived analytical expressions for thefrequency response and compared these with simulations using an analog circuit. He found that the system exhibited a soft resonancewith no jumps. In Ref. [ 9] he extended his analysis to random excitations. Masri [11 ] found the exact solutions for a damped harmonicoscillator with a bilinearhysteretic restoringforce. While he reduced thesystem to onenonlinear algebraicequationwhichwas to be solvedusing numerical methods he did not report a robust numerical method for solving this equation. Pratap and co-workers [12 ,13] studiedthe 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 toconstruct the globalsolution. They found that impulse loading can result in a chaotic response due to a Smale horseshoe.Kalmr-Nagy andWahi [31] studied the forced response of a relay oscillator by exploiting itspiecewise linear nature to obtain numerical as well as analyticalresults. 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 smoothsystems 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 oscillatorswith trilinear restoring force by using semi-analytic methods. Like Masri [ 11 ] he also found the exact solutionsup to the numerical solutionof a transcendental equation. Using his analysis he concluded that the system can exhibit periodic as well as chaotic behavior. Luo andMenon [33] used similar techniques to study the dynamics of a periodically forced linear system with a dead-zone restoring force. Theyinvestigated 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 forresponse of a harmonically impacted cantilever structure. In particular, they investigated the corner-colliding bifurcation in non-smoothsystems. Other works investigating non-smooth dynamical systems include those of Thota and Dankowicz [ 35], Lenci and Rega [3638 ]

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 oscillatorwith periodic excitation of period T . He used a hysteresis model based on the Masing rules and analyzed the system by using return mapsand found that for full hysteretic loops only 1 T periodic steady-state responses existed. Since other responses, like 5 T and 7 T periodic sub-harmonic responseshave also been reported previously, Capecchi concludedthat the higher harmonicsplay a significant role in theoveralldynamicsof the oscillator. In Ref. [40] Capecchi and co-workersstudied the complex behavior of multiple degree-of-freedom systems withhysteresis. Lacarbonara and Vestroni [41] used Poincar maps and continuation algorithms to map out the bifurcation sequences of somehysteretic 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 valueproblem formulation for computing sub-harmonic orbits of a class of harmonically forced conservative systems based on a non-smoothtemporal transformation.

This paper presents a study of the response of an oscillator with a hysteretic restoring force with sinusoidal excitation. A robustnumerical technique to find the exact (to arbitrary precision) transient andsteady-state response of the systemis described and discussed.The piecewise linear nature of the system is used to obtain solutions that are piecewise analytical. The discussed technique enables us torobustly and efficiently determine the switching time between the various linear regimes, thus allowing us to patch the local solutionsto form the global solution. By using this methodology a map similar to a Poincar map is constructed for analysis of the steady-stateresponse 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

x + F ( x, ) = Acos t , t [t 0 , t f ], (1)where F ( x, ) is a hysteretic term. Here the bilinear hysteretic operator is defined by a hybrid systems approach (a differential formulationisusedin [30]). This approach draws on thepiecewise smoothfunctions based description of themodel, i.e. F ( x, ) is modeled by piecewiselinear affine mappings. Fig. 1 shows the input-outputgraph 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 ahybrid 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


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1770 T. Kalmr-Nagy, A. Shekhawat / Physica D 238 (2009) 17681786

Table 1Rules for mode transitions in the bilinear hysteresis model.

Transition Rule

I II x = xI 2, x < 0I IV x = xI , x > 0II III x = 0, x > 0III IV x = xIII +2, x > 0III II x = xIII , x < 0IV I x = 0, x < 0modes

F I ( x, ) = x + (1 xI ) ,F II ( x, ) = (1 ) x ,F III ( x, ) = x + ( xIII 1) ,F IV ( x, ) = (1 ) x + , (2)

where xI and xIII are the values of x at the beginning of modes I and III , respectively. There are six permitted transitions between themodes of the operator, viz., I II , I IV , II III , III II , III IV , and IV I . The transitions between the various modes aregoverned 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 ascertainthe continuity of F with respect to x across the mode transitions. By using the rules for the mode transitions the mode of the operator atany 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 theinput x(t ) for t [t 0 , t ].Therefore Eq. (1) and the appropriate initial conditions are as follows

x + F m( x, ) = Acos t , ( x(t 0), x(t 0 ), F (t 0), m(t 0)) = ( x0 , v 0 , F 0 , m0), 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 listedin Table 1 .Remarks

1. The parameter plays a special role in the bilinear model. For = 0 the bilinear model reduces to the linear function F ( x, 0) = x. Thus, is a measure of nonlinearity and hysteresis in the system, and for small the response of the system is expected to be similar to thatof 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 = ( xmax xmin )/ 2, is defined as thehalf the difference between the maximum and minimum displacement. Thus, for the steady-state response of Eq. (3) to be hystereticit should have an amplitude greater than 1.

3. For the modes IV , II the stiffness of the system is proportional to 1

, while for the modes I , III it is equal to 1. It is expected that

increasing will result in a net reduction of the stiffness, thus reducing the resonance frequency. Further, for a fixed bigger hysteresisloops 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 tointegrate forward in time by using very small timesteps and checking for mode transitions at every step. This approach has the flaw thatno 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 isshown 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 solutioncan be evaluated to a desired precision. In addition, the complexity of the underlying algorithm is T log(1/) , where T = t f t 0 is ameasure of the time interval over which the solution of Eq. (3) has to be calculated and 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-statesolution of Eq. (3). The asymptotic results are essentially same as those derived by Caughey [ 8], and in that sense the present work is anextension of Caugheys efforts.

3.1. 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 besolved analytically. At the time of transition, t , the quadruplet ( x(t ), x(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 boilsdown to finding the times for the mode transitions. An efficient algorithm for finding the transition times is presented next.

Letthe nth transition from mode mn1 to mode mn take place at t = t n , and let ( xn , v n , F n) = ( x(t n), x(t n), F (t n)) . We define the discretephase n asn ( t n + n1) mod 2 , (4)

where n1 is the value of the phase for the previous mode. By introducing the phase, time t is essentially reset to 0 at every modetransition, thus reducing the governing equation for any mode to the following canonical form

x + F m( x, ) = Acos( t + n), ( x(0), x(0), F (0), m(0)) = ( xn , v n , F n0 , mn), t [0, t n]. (5)


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Table 2Parameters in expression of x(t ) for different modes.

Mode, m 2m km Bm

I 1 ( xI 1) A/( 2I 2 )II 1 A/( 2II 2 )III 1 ( xIII + 1) A/( 2III 2 )IV 1 A/( 2IV 2 )

6

4

2

0

2

4

0

24

6

X l

t l - >

l l

Fig. 2. Variation of t I II with xI and I for = 0.3, A = 1.6, = 1.5.

Eq. (5) can be solved to give the following equation for evolution of x(t )

x(t ) = xn kmn Bmn cos n cos( n t ) + (v n + Bmn sin ( n))/ mn sin ( mn t ) + Bmn cos( t + n) + kmn , (6)where the variables k, , 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 x(t ) = 0 (see Table 1 ). Consider, for example, the equation governing a I II transition. This equation is given by

x(t I II ) = xI 2, (7)where

x(t ) = ( xI kI BI cos I ) cos( I t ) + (BI sin ( I )/ I ) sin ( I t ) + BI cos( t + I ) + kI . (8)In Eq. (8), vI = 0 has been used since by definition x is zero at the beginning of mode I . The mode transition I II occurs at the firstpositive root of Eq. (7) , denoted by t I II , that satisfies x(t I II ) < 0. Finding the first positive solution of Eq. (7) (and othersimilar equations)is a nontrivial task. Even for fixed , Aand the solutions are in general discontinuous in xI and I . Fig. 2 shows the variation of t I II with xI , I for one case.

3.1.1. Root finding algorithmAs discussed in the previous Section, equations of the following general kind need to be solved for the transition time

g (t ) = C 1 cos t + C 2 sin t + C 3 cos( t + 1) + C 4 = 0. (9)Eq. (9) can be written as

cos( t + 2) + C

cos( t + 1) + D = 0, (10)

where C = C 3 /( C 21 + C 22 ), 2 = arctan (C 1 / C 2) / 2, and D = C 4 / C

21 + C 22 . The above equation can be further simplified by the

following change of variables

z = t + 1

(11)

to yield

f ( z ) = cos( z + ) + C

cos( z ) + D = 0, (12)where = 2 1 / , and = / . Eq. (12) can be solved to any desired precision by using the bisection algorithm if intervals thatcontain exactly one root are identified. Note that by the Mean Value Theorem all roots of Eq. (12) are either coincident with or lie betweenthe roots of the following equation

f ( z ) = sin ( z + ) + C sin ( z ) = 0, (13)


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(a) f ( z ) versus z with C = 1.8, = 0.5145 , = 0.2. Thefirst root, z = 5.54, is shown by a circle.(b) f ( z ) versus z with C = 1.8, = 0.5143 , = 0.2. Thefirst root, z = 6.99, is shown by a circle.

Fig. 3. First root of f ( z ) for some values of C , , . 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 zerosof sin( z + ) and/or sin( z ), i.e., sin(2( z i + )) sin (2 z i) = 0. Further, if the interval contains two roots then they are separated by theunique zero of d f ( z )/ d z in the interval. The complete proof for Proposition 1 can befoundin Ref. [30], and the proof is outlined in Appendix A . By using Proposition 1 it is possibleto construct intervals that have at most two zeros of f ( z ) and at most one zero of d f / d z . The unique zeros (if they exist) of d f / d z in theseintervals can be calculated to any desired accuracy by using the bisection algorithm [43]. Since the zeros of d f / d z partition the intervalsinto subintervals containing at most one root of f ( z ), this root, if it exists, can be found to any desired accuracy by using the bisectionalgorithm. 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 anydesired accuracy by using the bisection algorithm. Fig. 3 shows the first root of f ( z ) calculated using the proposed method for certainvalues of the parameters. The figure highlights the fact that the location of the root is a discontinuous function of the parameters. It is alsoemphasized that the complexity of the proposed algorithm compares favorably with traditional RungeKutta type integration schemes. Adiscussion of the complexity and performance of the algorithm is presented in Appendix B .

3.2. Heuristic estimates and asymptotic analysis

Before deriving the asymptotic expansions certain useful results can be obtained by simple reasoning and order of magnitudearguments.

It was noted earlier that if the steady-state response amplitude is less than 1, then the steady-state oscillations of the system areconservative. For a given amplitude of excitation, the frequency of excitation required to sustain steady-state oscillations with magnitudemarginally above 1 can be calculated as follows

A1 2 =

1. (14)

The above equation implies that there is a upper and a lower bound on the frequency of excitation required for existence of hystereticoscillations in the steady-state. These can be obtained as follows

upper = 1 + A, low er = 1 A. (15)If A 1 then there is no lower bound on for the existence of hysteretic oscillations. By using the above results the A plane can bedivided 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 1 in Eq. (1) the loading can be treated as quasi-static. In such a casethe steady-state response amplitude, R, can be found as followsR = A 1

, if A > 1, (16)

and conservative oscillations result if A 1. The above relation suggests that for 1 and 1 the steady-state will have very largeamplitude oscillations even if A O (1). This result is expected since for 1 the net stiffness of the system is greatly reduced. Fig. 5shows the variation of R with A for various values of for 1. The asymptotic expansions provided in the next Section can be used toverify the above estimates, however, strictly speaking the asymptotic expansions are valid only for 0, thus the asymptotic estimatescannot be used to justify the above claims for all .3.2.1. 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

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 = x0 + x1 + 2 x2 + the scale is while in the expansion x = x0 + 1/ 2 x1 + x2 + the scale is 1/ 2 . See Refs. [44 ,45] for details.


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0 2 4 6A

F(x)

x

F(x)

x

0

0.5

1

1.5

2

2.5

Fig. 4. Partitions of the A plane corresponding to conservative (shaded) and hysteretic (unshaded) oscillations in the steady-state.

1 2 3 4 5 6

10

20

30

40

A

R

= 0.9

= 0.1

= 0.5

= 0.7

Fig. 5. Variation of response amplitude with amplitude of excitation for

1.

uniformity near resonance. Thus, thederived expansions areexpectedto give good results at frequencies away from theresonantfrequencyof the system.

Following the KBM method Caughey [8] assumed the following form for the steady-state response of Eq. (1)

xss(t ) = Rcos( t + ), R O ( ), O ( ), (17)and obtained the following relations for the frequency response of the system

2 = C (R)

R AR

2

S(R)

R

2 1/ 2

. (18a)

tan = S(R)

C (R) 2R, (18b)

where

S(R) = R

sin 2 I II , (19)

C (R) = R

I II + (1 ) 2

sin2 I II . (20)

In the above equations I II is defined as I II = arccos (1 2/ R), (21)

and it corresponds to the value of t + when the transition I II occurs. Caughey proved that the steady-state response is linearlystable. He also proved that the amplitude at the primary resonance is given by the following expressionR =

44 A

. (22)

Therefore, the primary resonance is bounded if A < 4 / and becomes unbounded for sufficiently large A. Since < 1, the primaryresonance is always unbounded for A > 4/ . The above results can be used to partition the A plane into regions corresponding


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0 0.5 1 1.5 2

A

R

R

0

0.2

0.4

0.6

0.8

1

Fig. 6. Partitions of the A plane corresponding to bounded (shaded) and unbounded (unshaded) primary resonance.

2 4 6 8 10R

0.2

0.4

0.6

0.8

( 1

2 ) /

0

1

Fig. 7. Variation of

with R.

to bounded and unbounded primary resonance. Fig. 6 shows this partition where the shaded region corresponds to bounded primaryresonance and the unshaded region corresponds to unbounded primary resonance.At resonance, Eq. (18a) becomes

2 = C (R)

R, (23)

or, on substituting for C (R) and rearranging

2 1 = 1

I II +

12

sin2 I II . (24)

By defining a new non-dimensional number,

= (1

2)/ , the following can be deduced

= 1

I II

12

sin2 I II . (25)

This represents a simplification because the non-dimensional number is related directly to I II which in turn depends only on R. Fig. 7shows thevariation of with R. By using Eqs. (22) and (25) theamplitude at resonance as well as theresonance frequency canbe estimatedforany given , A,and . Theasymptotic expansions forthe steady-state response of thesystem canalso be used to estimate theequivalentdamping and stiffness properties of the system, see Ref. [ 46] for more details on the issue. We define the equivalent damping, , and theequivalent natural frequency, 0 , of the system such that the steady-state response of the following oscillator is identical to steady-stateresponse of Eq. (1) obtained by using the asymptotic analysis

x + 2 0 x + 20 x = Acos t . (26)The equivalent natural frequency 0 and the damping coefficient can be found to be the following

20 = C (R)R , (27)


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5

10

R

0

15

0.6 0.8 1

Fig. 8. Comparison of exact response curves obtained from the map with the those obtained by the KBM method. Solid lines: -Map, Circles: KBM Method, Dashed line:Locus of resonance frequency and resonance amplitude for 0 .8 < A < 0.1 obtained by the KBM method. = 0.6, A varied from 0 .1 to 0 .8 in increments of 0 .1.

and

= S(R)2R 0 . (28)

Thus, the equivalent natural frequency of the systemand itsresonance frequencymatch at resonance. Also note that the equivalent naturalfrequency and the resonance frequency are not equal in general.

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 uniformlyvalid near resonance and cannotbe readily extended to incorporate sub-harmonic responses. Furthermore, the asymptotic expansions areguaranteed to work only in thelimitof 0. In order to overcome these shortcomings, a constructsimilar to a Poincar map is introducedfor 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 tocharacterize 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 thesystem can be characterized as

(I IV )ni times I II (III II )mi times III IV i such blocks with different ni , mi .

For example, i = 1, n1 = m1 = 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 atleast one IV I transition. By definition xI is zero, i.e., the velocity is zero at the beginning of state I . The beginning of state I is thencharacterized by the time t and the value of x(= xI ) at which the IV I transition occurs. Since the phase variable I was introduced toreplace time, the beginning of state I is characterized by the pair ( xI , I ). A periodic orbit may have many IV I transitions. Any one of these transitions can be used to define a map( xI , I ) i

+1

= ( xI , I ) i. (29)

The map (also referred to as the -map) can be constructed by using Proposition 1 to calculate the exact mode transition times for thesystem and using the analytical expressions for the system response between transitions [ 30]. This map is in general not continuous andits dynamics can be a lot more complicated than its smooth counterparts [47].

The steady-state response of Eq. (3) via the -map was studied by MATCONT [ 48], a MATLAB package for bifurcation and continuationanalysis of continuous and discrete dynamical systems. It was found that for most parameter values the steady-state response of thesystem consists of almost sinusoidal periodic orbits with period equal to the period of excitation. As established earlier, the system canexhibit both bounded and unbounded resonance. All estimates provided in Section 3.2 were seen to hold to a good degree of accuracy fora 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 nounstable orbits were detected.

Fig. 8 shows a comparison of the response curves obtained by the -map and the KBM method for = 0.6 and A varied between 0 .1and 0 .8.The amplitude is obtained from the orbit of the -map by using the relation R = ( xI xIII )/ 2. It was found that the orbits of the -map

are symmetric and R is almost equal to xI . Fig. 8 also shows the so-called backbone curve obtained by using the KBM method. This curveshown by the dotted line in the Figure is the locus of resonance frequencies and the corresponding response amplitudes for varying A. It


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0 0.5 1 1.5

2

4

6

8

10

12

0

14

x I

0.15 0.2 0.25 0.3 0.35

x I

2

2.5

3

3.5

(a) Primary and secondary resonances. (b) A magnified view of the sub-harmonic responses.

Fig. 9. Comparison ofresponsecurves obtainedby using theKBM method(shownby circles)and the -maps(shown by solid line)for a case with sub-harmonicresonance;

= 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 away from resonance. This deviation was expected since the one-term expansions are only uniformly valid awayfrom resonance.

Even though the exact response curves shown in Fig. 8 consist solely 1 T periodic orbits, where T is the period of excitation, it wasalso observed that for many cases the exact response displays several sub-harmonic resonances. Fig. 9(a) and (b) compare the responseamplitude obtained by averaging and our exact method for a case with sub-harmonic resonance. Near sub-harmonic resonances thesteady-state no longer consists of only one frequency component as assumed in the KBM method, and thus this solution does not capturethe steady-state response accurately for such cases. Fig. 10 shows a comparison of the steady-state orbits away from and near thesub-harmonic resonances. It is evident from the Fourier spectrum of x(t ) that multiple frequency components are present in the responsenear the sub-harmonic resonances.

Since the -map is discontinuous, the higher period orbits are not born via any classical bifurcations of the map. For this reasonMATCONT 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 higherperiod orbitsfor some parametervalues andwas completedby using orbit diagrams. Fig. 11 showsthe 2-periodorbitof the -map that exists near the first sub-harmonic resonance. Fig. 12 shows the response amplitude obtained from the orbit diagramand the orbit diagram itself near the sub-harmonic resonances for = 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 anentire 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 onlyfor > 1 A, and some sub-harmonic resonances corresponding to < 1 A will not be observed. Fig. 13 shows the responsecurves for = 0.4 and A varied between 2 and 10 in steps of 2. A comparison of the exact response curves and those obtained from theKBM analysis is shown in Fig. 14. As expected, the exact response and the KBM method are in good agreement away from the resonances.Figs. 1518 show the steady-state orbits and the Fourier spectrum for a sequence of sub-harmonic resonances for = 0.4, A = 6. Thefigures 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 containsmultiple 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 themode transition sequence in the transient response can have a sensitive dependence on initial conditions for some parameter values andinitial conditions. Fig. 19 shows the transient response for two trajectories starting from nearby initial conditions. In the Figure Case Icorresponds to the initial conditions ( x(0), x(0),( 0), m(0))

= (2, 0, 3.8228 , I ) while Case II corresponds to ( x(0), x(0),( 0), m(0))

=(2, 0, 3.8229 , I ).Even though the two cases differ only in the starting value of the discrete variable at the fourth place of decimal, the two initialconditions lead to different mode transition sequences. The figure clearly shows that this difference is due to the presence of a grazingbifurcation.However,while thesequence of mode transitions is significantly different, thelong-termtime history of x(t ) is nearlythe samefor both cases. In the many simulations that were run it is found thatthisbehavioris a general rule rather than an exception. Further, it wasnoted that the steady-state response of the system is unique (irrespective of initial conditions). Thus, in spite of the transient differencesall 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 encounteredin all cases or for all initial conditions. The figures also highlight the fact that the ( x, x) space is not the true phase space of the system astrajectories here intersect due to the explicit time dependence of the governing equation. In these figures the plots of F ( x) versus x containillustrations where the actual plot is suitably rotated and scaled for better demonstration of the dynamics.

Typically higher period orbits of the -map occur when there is a significant difference in the frequency of excitation and the naturalfrequency of the system. In such cases the response of the system comprises two frequency components, one significantly higher than theother. 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. 2325 show the graphsfor some of these higher period orbits.


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(a) Steady-state phase portrait. Input frequency, = 0.5. x = 2.9598 , = 0.2042 at the IV I transition.(b) Steady-state frequency spectrum of x(t ) .

(c) Steady-state phase portrait. Input frequency, = 0.3. x = 3.1265 , = 6.0182 at the IV I transition.(d) Steady-state frequency spectrum of x(t ) .

(e) Steady-state phase portrait. Input frequency, = 0.187. x = 2.2772 , = 5.8248 at the IV I transition.(f) Steady-state frequency spectrum of x(t ).

Fig. 10. Steady-state response and corresponding frequency spectrum away and near the sub-harmonic resonance. = 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 are marked by solid circles.

6. Conclusions

Theresponseof a bilinear hysteretic oscillator with sinusoidal excitation wasstudied in a hybridsystemsframework. The exact transientand steady-state response of a 1-DOF oscillator with bilinear hysteresis and sinusoidal excitation was obtained in this paper for the firsttime. The exact solution was constructed by calculating the transition times exactly, and by solving the equations of motion analyticallybetween mode switches. It was shown that the complexity of the proposed algorithm is T log(1/) , where T is the length of interval of integration and is thedesired accuracy.As a comparison thecomplexity of RungeKutta type algorithms is typically T / p with0 < 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 sensitivedependence on initial conditions for some parameter values and initial conditions. In spite of a complicated transient response, thenon-trivial steady-state of the system consists of periodic orbits of period 2 / , where is the frequency of excitation. The systemexhibits 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. Thelong term behavior of the system was studied by using various tools like orbit diagrams, time domain simulations, fixed point calculationsand 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


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(a) x(t ) versus t . (b) Steady-state frequency spectrum of x(t ) . First few oddmultiples of are marked by solid circles.

(c) Magnified view of some mode transitions. (d) x(t ) versus x(t ).Fig. 11. A 2-period orbit for = 0.3, = 0.3041 , A = 1.6. Transitions are denoted as I II : solid circle, I IV : solid square, II III : empty triangle, III II : emptysquare, III IV : empty circle, IV I : solid triangle. x = 3.1676 , = 0.1101 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. = 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 = 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.


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0 0.5 1 1.5 2

5

10

15

20

25

R

0

30

Fig. 14. Comparison of exact response curves obtained from the -map with the those obtained by the KBM method. Solid lines: -map, Dashed lines: KBM Method.

= 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 = 12 .3724 , = 6.0230 at the IV I transition.

(b) Frequency spectrum of steady-state x(t ) .

Fig. 15. 1:3 Sub-harmonic response for = 0.4, A = 6, = 0.275.


(b) Frequency spectrum of steady-state x(t ).


partition the A space into parts corresponding to regions that exhibit hysteretic oscillations in the steady-state and those that exhibitconservative 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-06-0787) for providing financial support for this study.


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(a) Steady-state orbit phase portrait. I II : solid circle,II III : empty triangle, III IV : empty circle, IV I :solid triangle. x = 10 .4801 , = 0.0980 at one IV I transition.

(b) Frequency spectrum of steady-state x(t ) .



(b) Frequency spectrum of steady-state x(t ). First few oddmultiples of are marked by solid circles.

Fig. 18. 1:9 Sub-harmonic response for

= 0.4, A

= 6,

= 0.0940.

(a) Case I: ( x(0),

x(0),( 0), m(0))

= (2, 0, 3.8228 , I ). (b) Case II: ( x(0),

x(0),( 0), m(0))

= (2, 0, 3.8229 , 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 thesame 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 + ) + Asin ( z ) = 0, (30)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 rootsdo not have any neighboring roots and are isolated. Once the roots are isolated it is a routine matter to find them to any desired accuracyby using the bisection algorithm.

Consider two curves S1 and S2 defined as follows

S1

: sin ( z

+ ), (31)

S2 : Asin ( z ). (32)


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(a) x(t ) versus t . (b) x versus x(t ).Fig. 20. Steady-state solution for = 0 .3, A = 1 .6, = 1 .5. I II : solid circle, II III : empty triangle, III IV : empty circle, IV I : solid triangle. At the IV I transition x = 1.2375 , = 3.0972.

(a) x(t ) versus t . (b) x(t ) versus t .

(c) x versus x. (d) F ( x) versus x.Fig. 21. Transient response with = 0 .3, A = 3.65 , = 1.2. Initial conditions: x(t 0 ) = 1 , x(t 0 ) = 0 , t 0 = / 2, starting mode = I . The first 20 mode transitions areshown. I II : solid circles, II III : empty triangles, III IV : empty circles, IV I : solid triangle.

Eq. (30) can then be written as S1 = S2 . Note that the structure of Eq. (30) is invariant under the operation of shifting S1 and/or S2 alongthe z -axis and/or multiplying them by scalars.Eq. (30) can have a root z 1 of multiplicity 2 if

sin ( z 1 + ) + Asin ( z 1) = 0, (33)cos( z 1 + ) + A cos( z 1) = 0. (34)

However, the case with multiplicity of the root equal to greater than 3 is not interesting since it would demand that the following shouldhold

sin ( z 1 + ) + Asin ( z 1) = 0,cos( z 1 + ) + A cos( z 1 ) = 0,sin ( z 1 + ) + A2 sin ( z 1) = 0.

(35)

The necessary conditions for existence of solutions of Eq. (35) can be found to be = 1, | A| = 1 (assuming > 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 nosolutions are considered.


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(a) x(t ) versus t . (b) x(t ) versus t .

(c) x versus x. (d) F ( x) versus x.Fig. 22. Transient response with = 0.7, A = 150 , = 10. Initial conditions: x(t 0 ) = 1, x(t 0 ) = 0, t 0 = / 2, starting mode = I . The first 20 mode transitions areshown. 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 x(t ). First fewodd multiples of are marked by solid circles.

(c) Magnified view of some mode transitions. (d) x(t ) versus x(t ) .Fig. 23. A 2-period orbit for

= 0.3,

= 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 = 28 .0901 , = 6.2163 at one IV I transition.


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(a) x(t ) versus t . (b) Steady-state frequency spectrum of x(t ) . First few oddmultiples of are marked by solid circles.

(c) Magnified view of some mode transitions. (d) x(t ) versus x(t ).

Fig. 24. An 11-period orbit for = 0 .3, = 0 .002 , A = 20. I II : solid circle, I IV : solid square, II III : empty triangle, III II : empty square, III IV : emptycircle, IV I : solid triangle. x = 27 .8754 , = 6.1453 at one IV I transition.

(a) x(t ) versus t . (b) Steady-state frequency spectrum of x(t ). First fewodd multiples of are marked by solid circles.

(c) Magnified view of some mode transitions. (d) x(t ) versus x(t ) .

Fig. 25. A 3-period orbit for

= 0.3,

= 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 = 13 .9895 , = 0.0756 at one IV I transition.


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(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 > 1 then Eq. (30) has at most two roots between adjacent peaks and zeros of sin( z ), if < 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 sinusoidalcurve 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 2lies 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 shiftconstruction. This construction will be used to prove the claim using a contradiction.

Suppose > 1 (the case with < 1 can he handled similarly). Let if possible S1 and S2 intersect three times between an adjacent peakand zero of S2 (see Fig. 27). Then one can use the scale and shift construction to construct another curve S3 such that S1 and S3 intersectthree times between an adjacent peak and zero of S3 and the intersections are closer to each other as compared to the intersections of S1 and S2 . 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 S1 and S2 . Since S2 is monotonic in the considered interval, without the loss of generality itcan be assumed to be decreasing. It follows that if z l < z m < z r then S2 ( z l) < S2( z m) < S2( z r ). By construction S3 satisfies the following

S3( z l) > S2( z l),S3( z m) = S2( z m),S3( z r ) < S2 ( z r ).

(36)

Since the deformation of S2 into S3 can be continuous, it follows that S3 can be chosen to be such that S3 and S2 are on the same side of

S1 in a sufficiently small interval around z m . Then, by continuity S3 and S1 should have at least one intersection between z m and z l, and atleast one intersection between z m and z r . Finally, since it is possible to construct S3 such that there are no intersections other than z m (by


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Proposed AlgorithmTypical Runge-Kutta Scheme

10 10 10 0

10 3

10 2

10 1 C o m p u

t a t i o n a

l E f f o r t

10 4

10 0

Fig. 28. Comparison of computational effort of the proposed algorithm with a typical RungeKutta algorithm. p = 0.25 is used for the RungeKutta algorithm and a timeinterval 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 S1 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 likebisection 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 S1 and S2 in a quarter period of the curve with the higher frequency thenthese intersections can be made to come arbitrarily close to each other by shifting one of the curves. This construction, called the shiftconstruction, is depicted in Fig. 26(b). Thus, if there are two intersections of the curves between an adjacent zero and peak of the curvewith the higher frequency then there exists some shift for which Eq. (30) has a root of multiplicity 2. Therefore, there can be two rootsof 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 toshow that the following are necessary conditions for existence of a root of multiplicity 2 of Eq. (30)

A2 > 1, 1 > A2 2

; for < 1,

A2 < 1, 1 < A2 2; for > 1. (37)If Conditions (37) are not satisfied then the roots of Eq. (30) can be isolated using Proposition 2 . In that case the following intervals containunique roots of Eq. (30)

I n = ( z n , z in ), z n = + n2

; for < 1,I n = ( z n , z in ), z n =

n2 ; for > 1.

(38)

If Conditions (37) are indeed satisfied then Proposition 2 needs to be strengthened. Note that the derivative of Eq. (30) vanishes when

cos( z + ) + A cos( z ) = 0. (39)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 theintervals 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) A22 > 1, 1 > A24; for < 1, A22 < 1, 1 < A24; for > 1.

(40)

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 , theEq. (39) has a unique root in those intervals. Finally, since the roots of Eq. (30) are necessarily separated by roots of Eq. (39) , therefore theindividual 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 3.1.1 is discussed here. The number of intervals containing at mosttwo roots of d f / d z (see Section 3.1.1 ) is a linear function of the total time over which the solution is sought. Within each of these intervalsthe 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/))

function evaluations, where is the ratio of the length of the final bisection interval to the initial interval. By definition < 1, and it is ameasure of the accuracy of the solution. Thus, the overall complexity of the algorithm is T log(1/) , where T is the length of the interval


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over which the solution is sought. As a comparison, the complexity of typical RungeKutta-type small time increment methods is T / p

with 0 < p < 1. Fig. 28 shows the comparison of computational effort involved in achieving similar solution accuracy with the proposedalgorithm and a typical RungeKutta one with p = 0.25. It is seen that for achieving any reasonable accuracy the RungeKutta algorithmneeds several orders more function calls. Even though the figure shows a region of lower accuracy where the traditional method is moreefficient than the proposed one, this region is of little practical interest.

In this section the typical transient and steady-state response of Eq. (1) is presented. The response is obtained by using A comparisonof the exact results with those obtained by the asymptotic analysis is also presented.

References

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Nonlinear dynamics of oscillators with bilinear hysteresis and sinusoidal excitation

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