The Control of Nonlinear Oscillatory Systems with Delay Upravljanje nelinearnih nihajočih sistemov z zakasnitvami

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1 Anali PAZU - etnik 3, leto 03, številka The Control of Nonlinear Oscillatory Systems with Delay Uravljanje nelinearnih nihajočih sistemov z zakasnitvami Rudolf Pušenjak, *, Maks Oblak Faculty of Industrial Engineering Novo mesto / Šegova ulica, SI-8000 Novo mesto s: rudi.usenjak@teleing.com; maks.oblak@fl.uni-mb.si * Full Prof. Dr. Rudolf Pušenjak, Kidričeva 5, 940 jutomer, Slovenia; Tel.: Abstract: This aer treats the adatation of the Extended indstedt-poincare Method with multile time scales (E-PM) for analysis of stationary and nonstationary resonances of harmonically excited Duffing oscillator with time delay feedback control. The fundamental and ⅓ subharmonic resonance, resectively of Duffing oscillator with feedback control are analyzed in details. The comarison of stationary resonances with nonstationary resonances due to the slowly varying excitation frequency is resented by means of examles, which are comuted by rogramming tool Mathematica. Key words: Duffing oscillator, time delay feedback control, osnovna in subharmonična resonanca, E-P method. Povzetek: Članek obravnava riredbo Razširjene indstedt-poincarejeve Metode z več časovnimi skalami (R-PM) za analizo stacionarnih in nestacionarnih resonanc harmonično vzbujanega Duffingovega nihala z zakasnitvijo ovratno zančnega uravljanja. Analizi osnovne in ⅓ subharmonične resonance Duffingovega nihala z uravljanjem sta rikazani v odrobnostih. V članku je s omočjo zgledov izvedena rimerjava stacionarnih resonanc z nestacionarnimi resonancami, ki jih ovzroča očasno sremenljiva vzbujevalna frekvenca, ri čemer so rezultati dobljeni s omočjo rogramskega orodja Mathematica. Ključne besede: Duffingovo nihalo, uravljanje s časovno zakasnitvijo, osnovna in subharmonična resonanca, R-P metoda.. Introduction The control of nonlinear dynamical systems, which can exhibit eriodic, aeriodic and even chaotic oscillations, reresents a great research challenge due to the theoretical achievements and ractical alications. Realization of feedback control is connected with inevitable delays, which occur in data acquisition and signal rocessing. Mathematical modelling of such dynamical systems is described in the form of nonlinear delayed differential equations (DDE's), which cannot be handled by the standard methods. Insired by the ower of the Extended indstedt-poincare Method with multile time scales (E-PM) [], the goal of this aer is to extend its alicability to the roblems of time delay nonlinear control in the steady-state of dynamical systems, which are harmonically excited. In the aer, the method is alied to the feedback control of the harmonically excited Duffing oscillator.. Nomenclature w u = deflection of the beam = dislacement x = dimensionless satial variable t = time c * = coefficient of viscous daming = axial force K = beam stiffness ω = natural angular frequency ω 0 = linear angular frequency ω = excitation angular frequency β = coefficient of cubic nonlinearity θ = time delay ε = erturbation arameter a = gain of the roortional art of the controller b = gain of the differential art of the controller = excitation amlitude η = subharmonic factor σ = detuning τ = fast time scale τ = slow time scale A(τ ) = amlitude of oscillations as function of the slow time scale Ф 0(τ ) = hase angle of oscillations as function of the slow time scale γ(τ ) = auxiliary variable r = rate of detuning change 5

2 Anali PAZU - etnik 3, leto 03, številka 3. Buckling of the hinged-hinged beam: An examle of derivation of Duffing equation Duffing oscillator is a dynamical system with one degree of freedom and a cubic nonlinearity. Examles of Duffing oscillator can be found elsewhere in science and engineering. In mechanics, for examle, vibrations of active susension system, which takes into account the nonlinearity of tires, vibrations of offshore latforms, vibrations of hinged-hinged beams, clamed-hinged beams [], etc. are modelled by means of Duffing oscillator. In modelling rotordynamics, even couled nonlinear oscillators of Duffing tye are used [3]. In this aer, we consider the buckling henomenon of the hinged-hinged beam as an examle. Deflection of the beam is resented on the Figure and for the sake of simlicity, the beam is analyzed through the dimensionless satial variable x=l/ on the interval 0 x, where is the length of the beam and l is the variable distance, measured from its left end. The beam is at both ends subjected to the comressive axial force and externally forced by a harmonic excitation P(x,t), which deends on the satial variable x and on the time t. Excitation P(x,t) can be realized in ractice, when the beam suorts a machine with a rotating imbalance. Viscous daming of the beam is characterized by the coefficient c * and the beam stiffness is denoted by K. By using Hamilton rincile [4], we can derive the governing PDE of small deflections w(x,t), where longitudinal motions of the beam are neglected: where 4 w w K t w c w w P x t 4 x 0 x x t t *, d, 0,, w t w t w 0, t w, t 0, 0 x x, () () are the corresonding boundary conditions of the hinged-hinged beam at its suorts. By simlifying the analysis assuming the symmetry of P(x,t) around the satial coordinate x=0.5, the beam is in the first mode oscillation and both excitation P(x,t) and deflection w(x,t), resectively have a sinusoidal shae in x. However, excitation P(x,t) additionally varies cosinusoidal in deendence on the time. Taking both satial and temoral variations into account, we seek the solution of Eq. () by means of ansatz:, cos sin,, sin, (3) P x t t x w x t u t x By substituting Eq. (3) into Eq. (), by searation of variables and integration involved and by using boundary conditions (), we can derive the Duffing equation: * 4 3 u c u u K u cost, (4) which governs the temoral vibration u(t) of the beam dislacement. By using notations: k * 4 k c c,, K, k k0, (5) which can be used to introduce small quantities, deending on the small erturbation arameter ε (with an excetion of introducing the natural angular frequency ω ), Eq. (4) can be rewritten on the standard form: k 3 k u cu u u k cost k0, (6) which is analyzed in the rest of aer in the general case. 6

3 Anali PAZU - etnik 3, leto 03, številka Fig.. Deflection w of the hinged-hinged beam 4. Harmonically excited Duffing oscillator with time delay feedback control Desite of simle harmonically excitation, Duffing oscillator is able to roduce various eriodic, aeriodic or even chaotic oscillations in deendence on arameter values. Chaotic oscillations are in general unwanted and therefore give rise to the use of active feedback control in order to revent them. Active control is advantageous over the assive control in many resects, however it oens numerous roblems of technical nature. Due to the active control, the dynamics of the system becomes more comlicated as the consequence of inevitable time delays in controllers, transducers and actuators such as analogue filters, hydraulic and neumatic actuators. Time delays can cause instability of the system, which is controlled by the use of feedback loo. In the resence of time delays, the classical control theory has a limited use in linear systems, however it fails in the case of nonlinear systems. In order to facilitate the analysis of the nonlinear dynamical systems with time delay feedback, the Extended indstedt-poincare method with multile time scales (E-PM) [] is roosed in this aer in the case of Duffing oscillator. The analysis of Duffing oscillator with time delay control is erformed by assumtion of a small modal daming, weak cubic nonlinearity and small gain coefficients of controller. Characteristic for the use of E-PM is, that all of these arameters are exressed in deendence on a small erturbation arameter ε. In the case of fundamental resonance, the external harmonic excitation is also small and deending on the erturbation arameter ε, however, this restriction is removed in the case of subharmonic resonance. In accordance with this assumtions, the governing equation of Duffing oscillator with harmonic excitation and cubic nonlinearity (6), which alies the time delay feedback control, is rearranged in the form: d u t du t 3 d k k c cos u t u t au t b u t k t dt dt dt k0, (7) where u(t) denotes dislacements of the dynamical system, εc reresents modal daming of the system, ω means natural (angular) frequency, β means coefficient of cubic nonlinearity, a and b mean the gain coefficients of the roortional and differential art of the controller, resectively, θ means the time delay, ω denotes the excitation (angular) frequency and where k, (k=0,) are excitation amlitudes, which are adated in such a way, that 0=0, = holds in the case of fundamental resonance and 0=, =0 holds in the case of subharmonic resonance. In the sequel, E-PM with multile time scales [] will be alied in the analysis of nonstationary resonances of Duffing oscillator with time delay control. Nonstationary resonances are the henomenon, which occur at slowly varying excitation frequency or excitation amlitude, resectively. In this aer, only slowly varying excitation frequency will be taken into account. Stationary resonances will be also studied as the secial cases, when the excitation frequency has a role of an adjustable, but otherwise constant arameter. In order to master both fundamental as well as subharmonic resonances, one introduces the so called subharmonic factor, which has the value = at the fundamental (or even at the suerharmonic) resonance, but it becomes an adequate natural number, such as =3 at the ⅓ subharmonic resonance, for examle. Deviation of the excitation frequency ω from the linear frequency ω 0 can be exressed by equation: 0, (8) where the arameter σ stands for the detuning. On the basis of Eq. (8), the fast time scale τ and the slow time scale τ, resectively, are introduced: 7

4 Anali PAZU - etnik 3, leto 03, številka 0 t t (9.a), (9.b) which give rise to the relacement of time derivatives d/dt in d / dt with differential oerators: d dt 0 d 0 0 dt (0.a). (0.b) Using artial derivatives (0.a,b), Eq. (7) is rewritten into nonlinear artial differential equation of the form: u u u u u c 0 u u au, u, b u, cos in the case of fundamental resonance and into equation u u u u u c 0 u u au 0, b0 u 0, b u 0, cos (.a) (.b) in the case of subharmonic resonance. The aroximate solution of Eqs. (.a,b) in deendence of time scales τ and τ can be found by alying the erturbation rocedure, when one exresses dislacements u(t) in the form of the ower series: u k u k,. () k0 5. Analysis of nonstationary fundamental resonance of Duffing oscillator with delay feedback control When ower series () is introduced into Eq. (.a) and terms of like owers of ε on both sides of equation are equated, we get the following set of artial differential equations, that are linear, if are successively solved: 0 : : u0 0 u0 0 The general solution of Eq. (3) is sought in the form: u u0 u u cos 0 c u0 au0 0, b0 u0 0,, A cos, (5) u0 0 where both the amlitude A(τ ) and hase Φ 0(τ ), resectively, are modulated in deendence on the slow time scale and are currently undetermined. They will be determined in the next ste of the erturbation rocedure by elimination of so called secular terms in the solution u (τ,τ ) of Eq. (4). Due to the modulated (3) (4) amlitude A(τ ) and hase Φ 0(τ ) in Eq. (5), oscillations of Duffing oscillator are aeriodic. By substitution of solution (5) into Eq. (3) and by considering the fact, that frequencies ω 0, ω can take only ositive values, the following imortant relation is obtained: 0 (6) By substitution of the general solution (5) into Eq. (4) and by considering relation (6) one has: 8

5 Anali PAZU - etnik 3, leto 03, številka u u cos 0 cos 0 sin 0 sin 0 da d0 ca sin 0 A cos 0 d d 3 A 3cos 0 cos 3 0 aa 4 cos 0 cos aa sin 0 sin b A sin 0 cos b A cos 0 sin (7) The right hand side of Eq. (7) contains secular terms, which cause the unlimited growth of the solution. In order to obtain an uniform solution, secular terms must be eliminated by assembling terms, which aear at cos[τ -Φ 0(τ )] and sin[τ -Φ 0(τ )], resectively and by equating the obtained exressions with zero: By introduction of the new variable: d0 3 3 cos 0 A A aa cos b A sin 0 d 4 Eqs. (8) and (9) are rewritten on the form: da sin 0 ca aa sin b A cos 0 d, (0) 0 d d 3 3 cos A A aa cos b A sin 0 d d 4 da sin ca aa sin b A cos 0 d, (8). (9), (). () Equations () and () reresent the system of ordinary nonlinear differential equations. Solution of both equations give us the course of amlitude and hase, resectively of nonstationary fundamental resonance of Duffing oscillator with time delay feedback control. Solution of the system of Eqs. () and () can be obtained by various methods of numerical integration, where often the Runge-Kutta method is used. 5.. Stationary fundamental resonance of Duffing oscillator with delay feedback control Stationary oscillations of Duffing oscillator aear, when A(τ )=A=const, Φ 0(τ )= Φ 0=const, γ(τ )= γ = da d const. Consequently, it holds 0, 0, d d d 0 and Eqs. (), () are simlified to the d nonlinear algebraic equations: By squaring Eqs. (3) and (4) and adding, we get the following equation of the amlitude-detuning resonse of Duffing oscillator with time delay feedback control: 3 3 cos A A aacos b Asin 0 4, (3) sin c A aasin b Acos 0. (4) 3 A a cos b sin c a sin b cos A 8 4. (5) 9

6 Anali PAZU - etnik 3, leto 03, številka The corresonding course of the hase of Duffing oscillator is obtained, when Eq. (4) is divided by Eq. (3): sin sin b cos c a tan. (6) cos 3 A a cos b sin 8 When the amlitude A and hase angle γ, resectively are determined, the steady-state solution of the zeroth order for dislacements u(t) can be comuted in the form: cos u t A t O. (7) From this solution we can see, that oscillations of Duffing oscillator are nearly harmonic (a more accurate analysis would be show, that oscillations are in fact eriodic due to the contribution of higher harmonics in the solution u (τ,τ )). When the control of Duffing oscillator is not alied, that is, when, then Eqs. (5), (6) reduce on relationshis: a b0 3 A c A 8 4, (8) c tan. (9) 3 A 8 The obtained result describes the stationary fundamental resonance of Duffing oscillator that is in accordance with result, obtained by Nayfeh in Mook [5], which use multile scales as well as with result of eung in Fung [6], which aly the incremental harmonic balance method. On the basis of this comarison we can conclude that zeroth order aroximation of nonlinear oscillations of Duffing oscillator by E-PM with two time scales offers results, which are accurate enough. 6. Analysis of nonstationary subharmonic resonance of Duffing oscillator with time delay feedback control Analysis of subharmonic resonances is erformed by the similar rocedure as in the case of fundamental resonance starting from Eq. (.b). In this aer, we restrict oneself to the ⅓ subharmonic resonance, where the variable excitation frequency ω can be exressed by means of detuning σ as follows: 3. (30) 0 In the case of ⅓ subharmonic resonance, Eq. (.b) is rewritten on the form: u u u u u c 0 u u cos 3 au 0, b0 u 0, b u 0, (3) By substituting the ower series () into Eq. (3) and by equating terms of like owers of ε on both sides of equation, one obtains the following system of linear artial differential equations (PDE's): 0 : : u0 0 0 cos 3 u, (3) u u0 u , 0 0 0, u c u au b u, (33) The general solution of Eq. (3) is equal to the sum of the general solution of the corresonding homogenous PDE and the articular solution u0, A cos 0 E0 cos 0, (34) 0

7 Anali PAZU - etnik 3, leto 03, številka where A cos 0 describes the solution of the homogeneous equation and E 0 cos 0 is the corresonding articular solution. When the solution A cos 0 is substituted into homogeneous equation, relation (6) is obtained again. Because the articular solution satisfies Eq. (3), too, we can comute the unknown amlitude E 0(τ ) by substituting the articular solution into Eq. (3): E0 8 6 By means of Eq. (34) we ascertain that the amlitude of articular solution E 0(τ ) is modulated in the case of nonstationary resonance (due to the variability of. (35) 3 excitation frequency ω), while it is constant in the case of stationary resonance. Solution (34) can be rewritten by considering Eq. (35) on the form: u0, A cos 0 cos3 8 6 By substituting Eq. (36) into Eq. (33) and by considering relation (6) we get:. (36) u da d0 sin 0 cos 0 u A d d 3 30 d cos3 sin 3 sin 0 c A d sin 3 A cos 0 cos a A cos 0 cos (37) 3 b sin sin 3 3 A Equation (37) contains secular terms on its right hand side. By using a similar rocedure as in the case of fundamental resonance, secular terms are eliminated, when terms, which aear at cos[τ -Φ 0(τ )] and sin[τ - Φ 0(τ )], resectively, are collected and equated by zero: da 3 A ca sin 3 0 aa sin cos 0 d b A d A 3 A cos 3 0 d b acos sin 0, (38). (39) By introducing a new variable 3 0, (40) Eqs. (38) and (39) can be rewritten on the form: da 3 A a ca sin A sin ba cos d 8 8 6, (4)

8 Anali PAZU - etnik 3, leto 03, številka d 9 A 9 3a. (4) A cos cos 3 sin d b d d 9 Equations (4) and (4) reresents the amlitude resonse and hase resonse equation, resectively, of ⅓ nonstationary subharmonic resonance of Duffing oscillator with time delay control. The corresonding amlitude resonse and hase resonse equation, resectively, of ⅓ stationary subharmonic resonance is obtained, when amlitude and hase does not change in deendence on slow time scale τ, that is, when A(τ )=A=const, Φ 0(τ )= Φ 0=const, γ(τ )=γ=const. Consequently, d A /d 0, d /d 0, d /d 0 and Eqs. (4) and (4) are reduced on the form: 3 A a ca sin Asin cos 0 ba 8 8 6, (43) A 3a A cos cos 3 sin 0 8 b (44) By squaring Eqs. (43) and (44) and adding, we get the equation of the amlitude resonse of stationary subharmonic resonance with time delay control a 9 9 3a 9 c sin bcos A cos 3 sin 8 b A A Equation (45) ossesses a trivial solution A=0 and a nontrivial solution, which is obtained by solving the equation: 9 9 3a A cos 3 b sin A 3a 3c sin 3 cos 0 b When Duffing oscillator is not controlled, that is, when a=b=0, then amlitude resonse of stationary subharmonic resonance is reduced on the form: A A 9c (45). (46). (47) 7. Comutational asects The resent method for both fundamental and ⅓ subharmonic resonance of Duffing oscillator with time delay control is imlemented in rogramming environment Mathematica to aly symbolic as well as numeric comutational caabilities. For examle, stationary resonances (Eqs. (5), (8) for fundamental and (46), (47) for ⅓ subharmonic resonance, resectively) are comuted algebraically. On contrary, the corresonding nonstationary resonances (Eq. (), () for fundamental and Eq. (4), (4) for ⅓ subharmonic resonance, resectively) are comuted by

9 Anali PAZU - etnik 3, leto 03, številka numerical integration, using embedded Runge-Kutta method. The question is, why not use the numerical integration directly on Duffing equation (7), however there are many reasons, why we not use this aroach. The first reason is that we are looking for the steadystate oscillations and thus we must integrate Eq. (7) each time until the transient henomenon dies. Because we must reeat this long lasting comutation for each articular value of detuning (and time delay as well), the aroach is evidently imractical. This inefficiency becomes even more aarent in comutation of nonstationary resonances, where detuning is slowly varied and a secial strategy must be alied to achieve the elimination of the transient henomenon. The second reason is, that numerical integration of Eq. (7) is able to comute stable solutions only. Therefore, unstable branches of resonance curves cannot be obtained by numerical integration of Eq. (7), however they are easily comuted by the resent method. If we limit only on stationary resonances with delay control, we can alternatively aly the incremental harmonic balance (IHB) method [3], which will be resented by authors for a wide kinds of dynamic structures in a forthcoming aer. The reliminary results shows a good agreement between E-P and IHB method in the case of Duffing oscillator with delay control. 8. Results Now look on results of E-P analysis of fundamental and ⅓ subharmonic resonance of Duffing oscillator with time delay control. Both analysis are erformed by using rogramming envinronment Mathematica, where the following values of arameters c=0.05, ω =, β=0.05 are choosen in both cases. The excitation amlitude in the case of fundamental resonance is choosen to be equal =0.5, while in the case of subharmonic resonance it is =30. Analysis of stationary resonances in both cases are comuted at first, while the corresonding analysis of nonstationary resonances follows to bring out imortant differences. Analysis of stationary resonance without control, that is, with values a=0, b=0 of gain arameters is erformed in accordance with Eqs. (8),(9) in the case of fundamental resonance and by using Eq. (47) in the case of ⅓ subharmonic resonance. The stationary resonance of Duffing oscillator with time delay control is comuted with values a=0.05, b= of gain arameters and various time delays taking values θ=0, θ=π/4 and θ=π, resectively, where Eq. (5) is used in the case of fundamental and Eq. (46), resectively in the case of ⅓ subharmonic resonance. Courses of stationary fundamental resonance of Duffing oscillator without control and with alied control are lotted on the Fig..a and corresonding nonstationary resonances on the Fig..b. The variable excitation frequency ω and corresonding detuning σ are assumed to be linear functions of the slow time scale: 0 r, (48) where σ 0 denotes the detuning in the time τ =0 and r denotes the rate of detuning change. In the analysis of both fundamental and ⅓ subharmonic resonance, resectively, the rate of detuning change r=±0.05 is used (the sign + corresonds to the assage with increasing of detuning (or increasing of excitation frequency) and the sign corresonds to the assage with decreasing of detuning (or decreasing of the excitation frequency). From both Figures.a,b is aarent, that a small time delay (such as θ=π/4, for examle) causes the reduction of the eak amlitude of the fundamental resonance curve, while a great time delay (θ=π in the study) causes the magnification of the eak amlitude. Thus for the case of fundamental resonance we can conclude, that the desired eak amlitude can be controlled by the roer choice of delay. However, the courses of nonstationary fundamental resonances show a drastic difference, which is characterized in change of amlitude and aearance of many amlitude eaks. After the first amlitude eak, heights of subsequent amlitude eaks are smaller and smaller. Stationary fundamental resonances of Duffing oscillator are loted in Fig..b for comarison. Resonse curves obtained for stationary and nonstationary ⅓ subharmonic resonances are loted on Figures 3.a,b. Fig. 3.a shows, that delays in the case of stationary ⅓ subharmonic resonance of Duffing oscillator cause a different behavior of the controlled system in resect to the fundamental resonance. When the feedback control of Duffing oscillator is alied and the delay is equal zero, the resonse curve has a slightly larger amlitude as in the uncontrolled case. On contrary, when the feedback control of Duffing oscillator is alied and the delay has a great value, such as θ=π, the resonse curve has a slightly smaller amlitude as in the uncontrolled case. Therefore, changing of delay cannot be so effective in reduction of amlitude as in the case of fundamental resonance. Nonstationary ⅓ subharmonic resonances of Duffing oscillator show a drastic change in resect to corresonding stationary resonances, which are loted in Fig. 3.b for comarison. 3

10 Anali PAZU - etnik 3, leto 03, številka a) b) Fig. Fundamental resonance of Duffing oscillator with time delay control. a) stationary resonance without control (a=b=0) and with alied control at different values of delay θ. b) the corresonding nonstationary resonances with rate of detuning change r = ±0.05. a) b) Fig 3. ⅓ subharmonic resonance of Duffing oscillator with time delay control. a) stationary resonance without control (a=b=0) and with alied control at different values of delay θ. b) the corresonding nonstationary resonances with rate of detuning change r = ±0.05. In both Figures.b and 3.b, nonstationary resonances are drawn with continuous lines when detuning increases and by broken lines, when detuning decreases. 9. Conclusion In the aer it was shown, that E-PM can be successfuly alied to the control of Duffing oscillator with time delay and therefore can be extended to roblems of nonlinear theory of control. Moreover, it comes out, that E-PM roves as useful in solving delayed nonlinear differential equations. The ractical outcome of resent analysis is that delays, which are troublesome in general, in fundamental resonance can be used with benefit, when the eak amlitude of resonance should be maintained in the rescribed range. References dynamical systems with cubic nonlinearities, J. Sound and Vib. 34, 94-6, R. Pušenjak, Razvejitve ri Van der Pol- Duffingovem nihalu, Stroj. vestnik 49, št. 7/8, , R. R. Pušenjak, M. M. Oblak. Incremental harmonic balance method with multile time variables for dynamical systems with cubic nonlinearities, Int. j. numer. Methods eng., vol. 59, iss., 55-9, A. Preymount, Twelve ectures on Structural Dynamics, Sringer, Dordrecht, A. H. Nayfeh, D. T. Mook, Nonlinear Oscillations, Wiley, New York, A. Y. T. eung, T. C. Fung, Phase increment analysis of damed Duffing Oscillators, Int. J. Numer. Meth. Eng. 8, 93 09, R. R. Pušenjak, Extended indstedt-poincare method for non-stationary resonances of 4