High-dimensional Harmonic Balance Analysis for Second-order Delay-differential Equations

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1 Journal of Vibration and Control OnlineFirst, published on May 18, 2010 as doi: / High-dimensional Harmonic Balance Analysis for Second-order Delay-differential Equations LIPING LIU Department of Mathematics, North Carolina Agricultural and Technical State University, Greensboro, NC 27411, USA TAMÁS KALMÁR-NAGY Department of Aerospace Engineering, Texas A & M University, College Station, TX 77845, USA (Received 30 March 2007 accepted 30 March 2008) Abstract: This paper demonstrates the utility of the high-dimensional harmonic balance (HDHB) method for locating limit cycles of second-order delay-differential equations (DDEs). A matrix version of the HDHB method for systems of DDEs is described in detail. The method has been successfully applied to capture the stable and/or unstable limit cycles in three different models: a machine tool vibration model, the sunflower equation and a circadian rhythm model. The results show excellent agreement with collocation and continuation-based solutions from DDE-BIFTOOL. The advantages of HDHB over the classical harmonic balance method are highlighted and discussed. Key words: Circadian rhythm, delay-differential equations, harmonic balance, Hopf bifurcation, machine tool vibrations, sunflower equation. 1. INTRODUCTION Time delays play an important role in many natural and engineering systems. For example, time delay systems have been studied in fields as diverse as biology (MacDonald, 1989), population dynamics (Kuang, 1993), neural networks (Beuter et al., 1993 Shayer and Campbell, 2000), feedback-controlled mechanical systems (Hu and Wang, 2002), lasers (Pieroux et al., 2001) and machine tool vibrations (Stépán, 2000). Delay effects can also be exploited to control nonlinear systems (Pyragas, 1992 Erneux and Kalmár-Nagy, 2007). A good exposition of delay equations can be found in Stépán (1989). While there are rigorous mathematical techniques to study the dynamics of delay systems, e.g. the center manifold method (Hassard et al., 1981), the computations can be quite involved. Approximation methods can also provide good insight into the behavior of delay equations. These include the method of multiple scales (Hu et al., 1998 Wang and Hu, 2003), the Lindstedt Poincaré method (Morris, 1976 Casal and Freedman, 1980), and the harmonic balance (HB) method (MacDonald, 1995). There has been a recent surge of interest in numerical and analytical approximation for delay-differential equations (DDEs). Several researchers have studied linear stability of DDEs (Insperger and Stépán, 2002 Olgac and Sipahi, 2002 Asl and Ulsoy, 2003 Breda et Journal of Vibration and Control, 00(0): 1 20, 2010 DOI: / SAGE Publications Los Angeles, London, New Delhi, Singapore

2 2 L. LIU and T. KALMÁR-NAGY al., 2004 Butcher et al., 2004 Kalmár-Nagy, 2005 Yi et al., 2006 Insperger et al., 2009 Mann et al., 2009). Many papers have been written on numerical approximations of nonlinear response (Gilsinn, 2005 Wahi and Chatterjee, 2005). To validate the calculations, analytical or approximate solutions can be compared with collocation and continuation-based solutions from DDE-BIFTOOL (Engelborghs et al., 2001, 2002) and PDDE-Cont (Szalai et al., 2006). For general dynamical systems, the HB method is widely used from the simplest Duffing oscillation (Liu et al., 2006), to fluid dynamics (Ragulskis et al., 2006), and to complex fluid structural interactions (Liu and Dowell, 2005). Wu and Wang (2006) developed Mathematica/Maple programs to approximate the analytical solutions of a nonlinear undamped Duffing oscillation. There are two rather different ways to apply the conventional HB method. One is to work with only one harmonic to obtain a qualitative understanding of the dynamics (Mac- Donald, 1995 Kalmár-Nagy et al., 2001 Wang and Hu, 2003), while others also use higher harmonics to provide more accurate approximations for practical engineering design (Saupe, 1983 Krasnolselskii, 1984 Gilsinn, 2005 Wahi and Chatterjee, 2005). Other variants of this frequency domain method include: the HB method (Kim et al., 1991) and the nonlinear frequency domain (NLFD) method (McMullen et al., 2001). Higher-order harmonic representation for complex and/or high-dimensional systems is often difficult. For such systems, the high-dimensional harmonic balance (HDHB) method has been developed by Dowell and Hall (2001), Hall et al. (2000) and Thomas et al. (2002a). The HDHB method was developed primarily to deal effectively with very large systems of nonlinear ordinary differential equations (ODEs) and has been used successfully in computing the high-speed unsteady aerodynamic flows about elastically deforming aircraft structures (Thomas et al., 2002b, 2003). In this method, the solutions are sought in terms of time-domain quantities, thereby avoiding the derivation of algebraic expressions for the Fourier coefficients of the nonlinear terms. The calculations in HDHB are performed in the time domain over one period of oscillation. Thus, some have suggested this be called the time-domain harmonic balance method. By either name it is an effective method for considering high-dimensional systems or higher harmonics in relatively low-dimensional systems. So far little work has been devoted to the HDHB analysis of DDEs, and the main purpose of this paper to bring this method to the attention of the community of researchers on timedelay systems. DDEs describe systems where the present rate of change of state depends on a past value (or history) of the state. Generally speaking, the theory of DDEs is a generalization of the theory of ODEs into infinite-dimensional phase spaces. However, this generalization is not a trivial task (Kuang, 1993). The structure of the paper is as follows. In Section 2 we describe a matrix version of the conventional HB method for general systems of second-order DDEs. Section 3 focuses on the HDHB method with detailed formulas. In Section 4, we illustrate the utility of the HB methods on three systems described by DDEs: a metal cutting model, the sunflower equation and a circadian rhythm model. The HDHB method has been successfully applied to capture both stable and unstable limit cycles of these dynamical systems. The results show an excellent agreement with those from continuation using DDE-BIFTOOL. The advantages of HDHB over the classical HB method are highlighted and discussed. Finally, conclusions are drawn in Section 5.

3 HIGH-DIMENSIONAL HARMONIC BALANCE ANALYSIS 3 2. HB FOR SYSTEMS OF DDES In this section, the matrix version of the HB method is described for the general systems of second-order DDEs. The HDHB approach, which is presented in the next section, is derived from the HB system. Periodic solutions of differential equations can be well approximated by Fourier series (Mickens, 1996). The HB method consists of first substituting a truncated Fourier expansion into the governing equations. Then, by using the orthogonal properties of the sine and cosine functions, the coefficients of the harmonic terms cosnt and sinnt are equated to zero and the resulting system of nonlinear algebraic equations are solved for the unknowns. Here n 1 2N,whereN is the number of harmonic terms. The first-order HB method, i.e. including one harmonic in the analysis, is easy to apply to the systems with polynomial nonlinearities with a dominant first harmonic. For motions with evident higher harmonic components, more harmonics have to be included in the analysis which limits the usefulness of this approach. A system of second-order differential equations with delays can be written in the matrix form as M X t B X t KX t f X t X t (1) where M B and K are n n matrices, X t x 1 t x 2 t x n t T is a n 1 variable vector, and the right-hand side is a vector of nonlinear functions: f f 1 X t f 2 X t fn X t T.SinceT -periodic motions are sought, the dimensional time t is normalized as t t, where 2T is the fundamental frequency of the oscillation. Note that the frequency is not known aprioriand is therefore treated as unknown. The equation after the scaling becomes 2 M X t B X t KX t f X t X t (2) where the scaled time delay is. A2-periodic solution of equation 2 can be approximated by the truncated Fourier series expansion x i t a i 0 N j1 a i 2 j1 sin jt a i 2 j cos jt (3) where a i k are the unknown Fourier coefficient variables, and N is the number of overall harmonics used in the Fourier series expansion. The truncated expansion of the nonlinear termmaybeexpressedas where f i X t X t b i 0 N j1 b i 2 j1 sin jt b i 2 j cos jt (4)

4 4 L. LIU and T. KALMÁR-NAGY b i f i X t X t dt 2 0 b i 2 j f i X t X t sin jtdt b i 2 j f i X t X t cos jtdt (5) Here X t and X t has to be substituted according to equation 3. Substituting the expressions 3 and 4 into equation 1 and collecting terms associated with each harmonic sin jt and cos jt yields a system of n 2N 1 algebraic equations for ( j N, i 1 2n). The resulting algebraic system of equations can be written in a vector form to determine the n 2N 1 unknowns Q x (the hat is used to denote frequency-domain quantities) the Fourier coefficients a i j 2 MJ 2 BJ K Q x R x 0 (6) where Q x a 1 0 a 2 a 1 1 a 2 a 1 2N 0 a n 0 1 a n 1 a 2 2N a n 2N R x b 1 0 b 2 b 1 1 b 2 b 1 2N 0 b n 0 1 b n 1 b 2 2N b n 2N and 0 J J 1 0 k J k k 0 k 1N (7) J N Solving the system in equation 6 requires analytical expressions for the nonlinear functions b i j in terms of the variables a i j (i 1 2n, j N). As mentioned previously, the frequency (or, equivalently, the period T ) is unknown in the above analysis. The algebraic system in equation 6 for the Fourier coefficients and the response frequency has n 2N 1 1 variables but n 2N 1 equations. For the imposed condition, usually the first harmonic of one of the degrees could be with a fixed phase. This is known as phase-fixing for steady-state solutions. For example, we impose the condition a 1 1 0ora on the phase of the first harmonic of the motion.

5 HIGH-DIMENSIONAL HARMONIC BALANCE ANALYSIS 5 3. HDHB FOR SYSTEMS OF DDES As pointed out by Liu et al. (2006), the major disadvantage of the HB method is the tedious derivations of the algebraic expressions for the Fourier coefficients of the nonlinear terms of the dynamical system. The key aspect of the HDHB method is that instead of working in terms of Fourier coefficient variables as in the classical HB approach, the dependent variables are cast in the time domain and stored at 2N 1 equally spaced sub-time levels over the period of one cycle of motion. The Fourier and time-domain variables are related to one another via a constant Fourier transformation matrix. Working in terms of time-domain subtime level solution variables avoids the harmonic balancing step of the Fourier coefficient solution variables in the classical HB method. This makes the HDHB method very easy to formulate within the framework of an existing time marching nonlinear solver Formulation of the HDHB method The Fourier coefficients are related to the time-domain solution by equation 3. The 2N 1 HB Fourier coefficient solution variables are related to the time-domain solution by x i t 1 sint cost sin Nt cosnt Q i x (8) where Q i x denote the ith column of the matrix Q x. The time-domain solution at 2N 1 equally spaced nodes over a period of oscillation can be expressed via the 2N1-dimensional Fourier transformation matrix F.Thatis, Q x F Q x (9) where x 1 t 0 x 2 t 0 x n t 0 x 1 t 1 x 2 t 1 x n t 1 Q x (10) x 1 t 2N x 2 t 2N x n t 2N with t j j22n 1 ( j N), and the 2N 1 2N 1-dimensional transform matrix is 1 sint 0 cos t 0 sin Nt 0 cos Nt 0 1 sint 1 cos t 1 sin Nt 1 cos Nt 1 F (11) 1 sint 2N cos t 2N sin Nt 2N cos Nt 2N Conversely, the HB Fourier coefficients can be expressed in terms of the solution using the inverse of the Fourier transformation matrix, i.e.

6 6 L. LIU and T. KALMÁR-NAGY Q x F 1 Q x (12) F 1 2 2N 1 Similarly R x F R x (or R x F 1 R x ), where sin t 0 sin t 1 sin t 2N cos t 0 cos t 1 cos t 2N (13) sin Nt 0 sin Nt 1 sin Nt 2N cos Nt 0 cos Nt 1 cos Nt 2N f 1 X t 0 X t 0 f 2 X t 0 X t 0 f n X t 0 X t 0 f 1 X t 1 X t 1 f 2 X t 1 X t 1 f n X t 1 X t 1 R x (14) f 1 X t 2N X t 2N f 2 X t 2N X t 2N f n X t 2N X t 2N Equation 6 is then rewritten as 2 MJ 2 BJ KF 1 Q x F 1 R x 0 (15) Multiplying both sides of equation 15 by F gives 2 FMJ 2 F 1 FBJF 1 FKF 1 Q x R x 0 (16) In this study, the above system is referred as the HDHB solution system. Solving equation 16 does not require analytical expressions for the Fourier components. Also, it is easy to combine a HDHB solver with an existing time-marching code such as a computational fluid dynamics software (Thomas et al., 2002a, 2003). Again, the frequency in the system in equation 16 is unknown. Similar to the system in equation 6 in the frequency domain, the system in equation 16 in the time domain also needs one imposed condition. The phase fixing technique could also be applied. Converting the imposing phase condition in the HB analysis (i.e. a 1 1 0ora 1 2 0) into the time domain yields or x 1 t 0 sin t 0 x 1 t 1 sin t 1 x 1 t 2N sin t 2N 0 (17) x 1 t 0 cos t 0 x 1 t 1 cos t 1 x 1 t 2N cos t 2N 0 (18)

7 HIGH-DIMENSIONAL HARMONIC BALANCE ANALYSIS 7 4. APPLICATION OF THE HDHB METHOD The use of HB methods is demonstrated on three different second-order DDEs in this section. The first model describes machine tool vibrations in cutting of revolving cylindrical workpieces (turning). Two forms of the nonlinear dependence of the cutting force on the chip thickness are studied, namely a power-law function and its third-order polynomial expansion. The second model is the sunflower equation in which the nonlinearity is the transcendental function of the delayed term. Even though the nonlinearity is polynomial in our third model, a model for the circadian rhythm, this example exhibits coexisting stable and unstable limit cycle oscillations. Direct comparison of the results are provided by using the software DDE-BIFTOOL (Engelborghs et al., 2001, 2002). DDE-BIFTOOL is a Matlab package for numerical bifurcation and stability analysis of delay differential equations using collocation methods. One of the advantages of the HDHB method is that it is easy and direct to implement without requiring a specific commercial software package. The implementation of the HDHB method is also straightforward to very high-dimensional systems. For the three models considered in this study, the first harmonic of the solution is dominant for parameter values and thus the corresponding limit cycle would look like an ellipse in the x x phase plane. Therefore, the phase portraits of the solutions are omitted here. The examples given here serve to show the ease and efficacy of the HDHB method. A direct comparison of computational complexity with the original HB method is not provided, since the latter requires symbolic calculations. Avoiding such computations is indeed one of the main strengths of the HDHB method Machine Tool Vibrations The model describes the nonlinear vibrations arising in machine tool cutting (Stépán, 1989, 1997). The general nondimensional form is of the harmonic oscillator with nonlinear terms of the present and delayed state: x 2 x x f xt xt (19) Time is rescaled so that the assumed periodic motion has a period of 2. The model equation after the scaling becomes 2 x 2 x x f xt xt (20) where 2T is the fundamental frequency and the scaled delay time.the frequency is not known apriori, therefore it is treated as an unknown. A widely accepted form for the cutting force nonlinearity is a power law. The nondimensional form for this nonlinearity is (Kalmár-Nagy et al., 1999) 2 f xt xt p xt xt (21) 3 2 where p and are system parameters. The details of the machine tool vibration modeling can be found in previous studies (Stépán, 1989, 1997 Kalmár-Nagy et al., 1999). In the

8 8 L. LIU and T. KALMÁR-NAGY following, to contrast the HB and HDHB techniques, we first present the analysis of a series expanded form of this model. The classical HB method cannot be used for the general powerlaw model, as the Fourier coefficients of this nonlinearity cannot be obtained in closed form. The unstable limit cycle oscillations occur below the critical value of the parameter p. The type of the bifurcation is subcritical, and unstable limit cycle oscillations are not easily captured by the usual time marching methods Power Series Force Model The nonlinear dependence force on the chip thickness may be simply modeled by a truncated power series (here x xt ): 2 x 2 x x p x x x x 2 x x 3 (22) This model is simple and explains some observed nonlinear machine tool vibrations in cutting. The existence and nature of a Hopf bifurcation in this tool vibration model was presented and proved analytically with the help of the center manifold and Hopf bifurcation theory in Kalmár-Nagy et al. (2001). For this simple model, both the standard HB and HDHB methods are applicable, and both methods provide predictions in excellent agreement with the numerical solutions. The detailed method implementations for the HB and HDHB analysis and comparison with DDE- BIFTOOL results are reported below. The HB1 Results Including the zeroth (constant term) and the first harmonic in the motion form (equation 3), i.e. N 1, gives x t a 0 a 1 sin t (23) where the phase condition a 2 0 has been imposed. Equation 6 becomes 2 J 2 2J I Q x R x 0 (24) with and Q x a 0 a 1 R x J b 0 b 1 2pa1 2 sin2 2 2p sin 2 6pa2 1 sin3 a1 sin 2 2 a 2 b 2 2p sin 2 6pa2 1 sin3 2 a1 cos 2

9 HIGH-DIMENSIONAL HARMONIC BALANCE ANALYSIS 9 Writing the above HB system (equation 24) explicitly gives a 0 2pa 2 1 sin2 0 2 (25) 1 2 a 1 2p sin 2 6pa2 1 sin3 a 1 sin (26) 2 a 1 2p sin 2 6pa2 1 sin3 a 1 cos (27) Equation 25 immediately gives the result Equations 26 and 27 can both be solved for a1 2 to yield a 0 2pa 2 1 sin2 (28) 2 a p 2 p cos 6p sin 4 2 (29) a p sin 6p sin 3 cos (30) 2 2 The right-hand sides of equations 29 and 30 should be equal. This is true when 1 2 2tan 2 0 (31) Since, this transcendental frequency delay relationship can only be solved numerically for for a specific time delay. The amplitude of the first harmonic (a 1 ) can then be solved from either equation 29 or equation 30, and the amplitude of the zeroth harmonic can be solved from equation 28 as a 0 1 p 2 p cos (32) 3sin 2 2 We note that imposing the condition a 1 0wouldleadtothesameresults. The algebraic expressions for the HB analysis with more than one harmonics are complicated, thus details of the higher-order calculations are omitted. The HDHB1 Results Here the nonlinear algebraic system 2 FMJ 2 F 1 FBJF 1 FKF 1 Q x R x 0 (33) needstobesolved(m 1, B 2, K 1). For N 1, the time-domain values Q x x 0 x 1 x 2 T are sought at three equally spaced sub-time levels over a period of

10 10 L. LIU and T. KALMÁR-NAGY oscillation, i.e. x 0 x0, x 1 x23 and x 2 x43. The expressions for the Fourier transformation matrices F and F 1,andthematrixD FJF 1 are F F D The linear terms 2 D 2 2D IQ x in equation 16 become d 0 d 1 d x 0 x 1 x x 3 1 x 2 x x 1 x 2 x x 3 2 x 0 x 1 (34) x 2 x 0 x x 0 x 1 x 2 The basic components in the nonlinear terms are the delay terms xt. From equation 8 we obtain x t xt f t ft F 1 Q x : c 0 c 1 c 2 1 2x 3 0 x 1 x 2 1 cos 3 x 3 2 x 1 sin 1 2x 3 1 x 2 x 0 1 cos 3 x 3 0 x 2 sin (35) 1 2x 3 2 x 0 x 1 1 cos 3 x 3 1 x 0 sin where c i xt i xt i ft i ft i F 1 Q x for i The explicit expressions for r 0, r 1 and r 2 in terms of x 0, x 1 and x 2 are as follows pc 0 p c0 2 c3 0 R x pc 1 p c1 2 c3 1 pc 2 p (36) c2 2 c3 2 where c 0 c 1 c 2 are given in equation 35. Note that linear terms (the c i ) also appear in R x, owing to the chosen form (equation 2) of the class of DDEs studied. Combining the linear terms (equation 34) with the nonlinear terms (equation 36), the HDHB1 system with the imposed condition in equations 17 or 18 can be written explicitly as d 0 pc 0 pc0 2 c3 0 0 d 1 pc 1 pc1 2 c3 1 0 d 2 pc 2 pc2 2 c x 0 x 1 x 2 0 or d 0 pc 0 pc0 2 c3 0 0 d 1 pc 1 pc1 2 c3 1 0 d 2 pc 2 pc2 2 c3 2 0 x 1 x 2 0 A similar procedure can be applied to include high harmonics for high-order approximations. For the high-order HDHB analysis, the expressions are long, however, the implementation is (37)

11 HIGH-DIMENSIONAL HARMONIC BALANCE ANALYSIS 11 Figure 1. Machine tool cutting with the power series force: first harmonic amplitude versus the bifurcation parameter p from the HDHB method with one and two harmonics, in comparison with the results from the HB method with one harmonic. Open circle: HDHB1 open square: HDHB2 solid line: HB1. straightforward as one only needs to add solution variables in Q x while the rest is taken care of by the matrix vector multiplications. Numerical Results To facilitate numerical analysis, we set p cr (38) The motions are dominated by the first harmonic with fundamental frequency of This frequency is almost constant in the range p [ ]. The frequency predictions from both HB and HDHB methods are near 1.09 regardless of the number of harmonics included in the analysis. The amplitude result from the HB method including one harmonic is shown in Figure 1 as the solid line. Compared with the bifurcation diagram (the amplitude of the unstable limit cycle oscillations versus p) in (Kalmár-Nagy et al., 2001), the HB method with one harmonic provides a good approximation (with excellent agreement with the results from the direct numerical solution) for the oscillations near the bifurcation point. In order to compare the results with those from the HB method, the solutions from the HDHB method are converted into a corresponding Fourier representation by equation 12. The results from using the HDHB method with one (stars) and two harmonics (open squares) are displayed in Figure 1, in comparison with the result obtained by using the HB method with one harmonic (solid line). In the figure, the amplitude of the first harmonic is shown, while the amplitude of the second harmonic from the HDHB method with two harmonics is relatively small and can be neglected. For this model, the amplitude results from using the HDHB method with one harmonic are smaller than the real solutions. Including one more harmonic in the HDHB analysis, the result for the first harmonic amplitude improves substantially. The HDHB method with two harmonics provides a good approximation (squares) and including more harmonics in the HDHB analysis provides no substantial changes in the results.

12 12 L. LIU and T. KALMÁR-NAGY Figure 2. Machine tool cutting with the fraction power force: first harmonic amplitude versus the bifurcation parameter p from the HDHB method with one harmonic (HDHB1). Solid line: DDE-BIFTOOL open circle: HDHB1 with equation 18 star: HDHB1 with equation 17. Figure 3. Machine tool cutting with the fraction power force: first harmonic amplitude versus the bifurcation parameter p from the HDHB method with two harmonics (HDHB2). Solid line: DDE-BIFTOOL open circle: HDHB2 with equation 18 star: HDHB2 with equation Power-law Force Model The model given by equations 20 and 21 provides a description of the nonlinear force dependence on the chip thickness also valid farther from the bifurcation point (provided that contact loss does not occur between the tool and the material). In equation 21, the power is a fraction ( 1). The implementation of the HDHB method is also straightforward. The set of system parameters given in the previous section is used here. The results from using the HDHB method with one harmonic are displayed in Figure 2, two harmonics in Figure 3 and three harmonics in Figure 4. Power spectrum analysis of x t reveals that the oscillations are dominated by the fundamental frequency. The results from the HDHB analysis further verify that the amplitudes of the constant term, the second- and higher-order harmonics are negligible compared with the amplitude of the first harmonic. Therefore, only the results of the first harmonic amplitudes are shown here.

13 HIGH-DIMENSIONAL HARMONIC BALANCE ANALYSIS 13 Figure 4. Machine tool cutting with the fraction power force: first harmonic amplitude versus the bifurcation parameter p from the HDHB method with three harmonics (HDHB3). Solid line: DDE-BIFTOOL open circle: HDHB3 with equation 18 star: HDHB3 with equation 17. Note that either phase condition (equation 17 or equation 18) could be imposed in the HDHB system, and the results may be different as shown in Figures 2 4. Nonetheless, the results from the HDHB analysis with either imposing condition converge to the real solutions as the number of harmonics increases from one to three. For the motions away from the bifurcation point, e.g. for p less than 0.215, the results from imposing two conditions deviate slightly from the real solutions. Further numerical simulations show that adding more harmonics into the HDHB analysis eliminates the discrepancies. In conclusion, imposing different phase angles in the HDHB analysis may lead to different predictions if a small number of harmonics is included. As the number of harmonics included in the HDHB analysis increases, the discrepancies between the predictions from imposing either condition become small and both converge to the real solutions. Furthermore, the farther away the bifurcation parameter is from the bifurcation point, the larger the number of harmonics included in the HDHB analysis is needed for the same accuracy in the motion prediction. Comparing the results from the HDHB method in Figures 2 4 for the fraction power force with those in Figure 1 for the power series force, for sufficiently accurate results the number of high harmonics needed in the HDHB analysis for the fraction power force is similar to that for the power series force The Sunflower Equation Israelsson and Johnsson (1967) proposed the following equation (a b 0) x a x b sin x t 0 (39) to explain the helical movements of the growing tip (circumnutation) of sunflower plants. The bifurcation parameter is the time delay. Casal and Somolinos (1982) computed a perturbation expansion for the frequency and amplitude. In the sunflower equation 39, the delay also appears in the coefficients. By using some special treatment, the results from the HB analysis with one harmonic may be obtained

14 14 L. LIU and T. KALMÁR-NAGY Figure 5. Sunflower model: fundamental frequency versus the bifurcation parameter from the HDHB method with one harmonics, in comparison with the real solutions. Solid line: DDE-BIFTOOL open circle: HDHB1 with equation 18 star: HDHB1 with equation 17. Figure 6. Sunflower model: first harmonic amplitude versus the bifurcation parameter from the HDHB method with one harmonic. Solid line: DDE-BIFTOOL open circle: HDHB1 with equation 18 star: HDHB1 with equation 17. (MacDonald, 1995). In general, due to the transcendental nature of the nonlinearity, the conventional HB method is virtually impossible to implement. The HDHB method does not encounter any special difficulty for this model. The implementation of the HDHB method for this equation model is similar to that for the machine tool vibrations, and the details of the formulations are omitted here. As shown in Figure 5 for the frequency predictions, the results from the HDHB analysis with one harmonic with either imposing conditions (equation 17 or equation 18) deviate slightly from the real solutions for large bifurcation values. Including one more harmonic in the HDHB analysis, both results from imposing conditions equation 17 and equation 18 converge to the real solutions for the considered range of the bifurcation parameter. From fast Fourier transform (FFT) analysis of the motions it can be established that the oscillations are dominated by the fundamental frequency. Therefore, only the results for the first harmonic amplitude are shown in the figures. The results from the HDHB analysis including one harmonic are shown in Figure 6, and for two harmonics in Figure 7. In these figures the results from DDE-BIFTOOL are shown by solid lines.

15 HIGH-DIMENSIONAL HARMONIC BALANCE ANALYSIS 15 Figure 7. Sunflower model: first harmonic amplitude versus the bifurcation parameter from the HDHB method with two harmonics. Solid line: DDE-BIFTOOL open circle: HDHB2 with equation 18 star: HDHB2 with equation 17. For this case, the discrepancy between the results obtained by imposing different phase conditions is very small. The predictions from the HDHB analysis with one harmonic match the real solutions well for the motions near the bifurcation point as shown in Figure 6. For the motions away from the bifurcation point, including one more harmonic in the HDHB analysis improves the accuracy. As shown in Figure 7, the HDHB results from imposing different conditions are identical and match the real solutions well for the motions beyond the bifurcation point. Further numerical simulations reveal that adding more harmonics to the HDHB analysis provides no substantial changes in the results A Circadian Rhythm Model A model for circadian rhythm is given as (Ohlsson, 2006) with y 2y y y 2 y 2 3 y 3 (40) where the bifurcation parameter is, and the delay time is. A similar model is studied by Verdugo and Rand (2009). This example was chosen to demonstrate the ability of HDHB to correctly capture coexisting limit cycles. Since the right-hand side of equation 40 is a third-order polynomial of the delayed term, the conventional HB method can in principle be applied with no problem. However, owing to the presence of the constant term in the harmonic approximation, these results are relatively complicated. The implementation of the HDHB method encounters no difficulty and the high-order results are in an excellent agreement with the results from DDE-BIFTOOL.

16 16 L. LIU and T. KALMÁR-NAGY Figure 8. Circadian model: first harmonic amplitude versus the bifurcation parameter from the HDHB method with one harmonic. Solid line: DDE-BIFTOOL open circle: HDHB1 with equation 18 star: HDHB1 with equation 17. Figure 9. Circadian rhythm model: first harmonic amplitude (a) and the frequency (b) versus the bifurcation parameter from the HDHB method with two harmonics. Solid line: DDE-BIFTOOL open circle: HDHB2 with equation 18 star: HDHB2 with equation 17. For 105 the Hopf bifurcation occurs at cr The collocation results are shown in Figures 8 10 as solid lines for the first harmonic amplitude and the fundamental frequency. As the motions are dominated by the first harmonic, the results for the first harmonic amplitude and the fundamental frequency are presented and discussed here. In the bifurcation diagrams, both the amplitude and frequency curves are folded back, and the turning point is at The coexistence of motions occurs for the bifurcation parameter between the bifurcation point and the turning point, i.e. [ ].

17 HIGH-DIMENSIONAL HARMONIC BALANCE ANALYSIS 17 Figure 10. Circadian rhythm model: first harmonic amplitude (a) and the frequency (b) versus the bifurcation parameter from the HDHB method with three harmonics. Solid line: DDE-BIFTOOL open circle: HDHB3 with equation 18 star: HDHB3 with equation 17. The bifurcation is subcritical, and the global amplitude/frequency curve in the bifurcation diagram consists of the two branches for the stable and the unstable motions. The HDHB method is capable of capturing both branches for the stable and unstable oscillations. The caveat is however to initialize the nonlinear solver with different initial conditions to find all possible solutions. The results are displayed in Figures 8 10 for various numbers of harmonics and different phase conditions. The first harmonic amplitudes obtained from the HDHB method including only one harmonic, as shown in Figure 8, are quite different when the condition in either equation 17 or equation 18 is imposed. Although both approaches capture the coexistence of the oscillations, the agreement is only qualitative for this order of the approximation. The results from the HDHB analysis with one harmonic deviate substantially from the real solutions even near the bifurcation point. Including one more harmonic in the analysis improves the results dramatically, as shown in Figure 9. The accuracy of the frequency results, however, is not completely consistent with that of the amplitude, as shown in Figure 9b. Including one more harmonic in the HDHB analysis provides accurate solutions for both the amplitude and the frequency as shown in Figure 10a and b. Further numerical analysis shows that including more harmonics in the HDHB analysis does not change the results substantially. It also appears that the agreement between results from the two different phase conditions could be used to establish bounds on the accuracy of the solutions.

18 18 L. LIU and T. KALMÁR-NAGY 5. CONCLUSIONS This study has focused on the HDHB method for determining limit cycle oscillations of second-order DDEs. The variables in the HDHB method are the function values of the solution at discrete times. The nonlinear terms of the system equations are evaluated at these discrete times only, thus avoiding the process of deriving the algebraic expressions for the Fourier coefficients needed for the classical HB method. A detailed formulation of the HDHB method for systems of second-order DDEs has been provided. The utility of the HDHB analysis is illustrated on three examples: a machine tool vibration model, the sunflower equation and a circadian rhythm model. It is demonstrated that it is easy and straightforward to implement the HDHB method on dynamical systems with various types of nonlinearities. The present study includes fractional power and transcendental nonlinearities for which the conventional HB method is virtually impossible to implement. To better explain the method, the results from the HB method are also compared with the results from the HDHB method for the first example. The results from the DDE-BIFTOOL package are used to verify the accuracy of the HB/HDHB methods. For the machine tool cutting model, the HB method is applied to the power series expanded nonlinear force model, and the results from the HB method with one harmonic match those from DDE-BIFTOOL. However, the HB method cannot be directly applied to the fractional power force law. The HDHB method can be directly applied to both versions of the nonlinear force and the implementation is straightforward with no difficulty. The HDHB method with two harmonics provides accurate results. For the sunflower equation, although the results from the HB analysis with one harmonic can be obtained with some special treatment (MacDonald, 1995), in general, the HB method cannot be applied directly because of the transcendental nonlinearity. Again, there is no difficulty in the implementation of the HDHB method for this model and the results with one harmonic match the real solutions well. For the circadian rhythm model, the HDHB method is able to capture coexisting stable and unstable limit cycles, thereby demonstrating the utility of the technique for such cases. With different specific imposing conditions in the HDHB analysis, the results, which may vary for small number of harmonics, converge to the same solutions when a sufficient number of harmonics are included. Acknowledgements. The authors wish to thank Professor Earl Dowell for valuable comments and Dr Pankaj Wahi for valuable help with the DDE-BIFTOOL package. REFERENCES Asl, F. M. and Ulsoy, A. G., 2003, Analysis of a system of linear delay differential equations, Journal of Dynamic Systems, Measurement, and Control 125, Beuter, A., Bélair, J., and Labrie, C., 1993, Feedback and delays in neurological disease: a modeling study using dynamical systems, Bulletin of Mathematical Biology 55, Breda, D., Maset, S., and Vermiglio, R., 2004, Computing the characteristic roots for delay differential equations, IMA Journal of Numerical Analysis 24, Butcher, E. A., Ma, H., Bueler, E., Averina, V., and Szabó, Zs., 2004, Stability of time-periodic delay-differential equations via Chebyshev polynomials, International Journal for Numerical Methods in Engineering 59,

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