High-dimensional Harmonic Balance Analysis for Second...

High-dimensional Harmonic Balance Analysis forSecond-order Delay-differential Equations

LIPING LIUDepartment of Mathematics, North Carolina Agricultural and Technical State University,Greensboro, NC 27411, USA ([email protected])

TAMÁS KALMÁR-NAGYDepartment 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 forlocating limit cycles of second-order delay-differential equations (DDEs). A matrix version of the HDHBmethod for systems of DDEs is described in detail. The method has been successfully applied to cap-ture the stable and/or unstable limit cycles in three different models: a machine tool vibration model, thesunflower equation and a circadian rhythm model. The results show excellent agreement with collocationand continuation-based solutions from DDE-BIFTOOL. The advantages of HDHB over the classical har-monic balance method are highlighted and discussed.

Key words: Circadian rhythm, delay-differential equations, harmonic balance, Hopf bifurcation, machine tool vibra-tions, 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), pop-ulation 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 con-trol nonlinear systems (Pyragas, 1992� Erneux and Kalmár-Nagy, 2007). A good expositionof delay equations can be found in Stépán (1989).

While there are rigorous mathematical techniques to study the dynamics of delay sys-tems, e.g. the center manifold method (Hassard et al., 1981), the computations can be quiteinvolved. Approximation methods can also provide good insight into the behavior of de-lay 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 theharmonic balance (HB) method (MacDonald, 1995).

There has been a recent surge of interest in numerical and analytical approximationfor delay-differential equations (DDEs). Several researchers have studied linear stability ofDDEs (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: 10.1177/1077546309341134

��2010 SAGE Publications Los Angeles, London, New Delhi, Singapore

Journal of Vibration and Control OnlineFirst, published on May 18, 2010 as doi:10.1177/1077546309341134

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 nonlinearresponse (Gilsinn, 2005�Wahi and Chatterjee, 2005). To validate the calculations, analyticalor approximate solutions can be compared with collocation and continuation-based solutionsfrom 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 Duffingoscillation (Liu et al., 2006), to fluid dynamics (Ragulskis et al., 2006), and to complex fluidstructural interactions (Liu and Dowell, 2005). Wu and Wang (2006) developed Mathemat-ica/Maple programs to approximate the analytical solutions of a nonlinear undamped Duffingoscillation.

There are two rather different ways to apply the conventional HB method. One is towork 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 higherharmonics to provide more accurate approximations for practical engineering design (Saupe,1983� Krasnolselskii, 1984� Gilsinn, 2005� Wahi and Chatterjee, 2005). Other variants ofthis frequency domain method include: the HB method (Kim et al., 1991) and the nonlinearfrequency domain (NLFD) method (McMullen et al., 2001).

Higher-order harmonic representation for complex and/or high-dimensional systems isoften difficult. For such systems, the high-dimensional harmonic balance (HDHB) methodhas 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 ofnonlinear ordinary differential equations (ODEs) and has been used successfully in comput-ing the high-speed unsteady aerodynamic flows about elastically deforming aircraft struc-tures (Thomas et al., 2002b, 2003).

In this method, the solutions are sought in terms of time-domain quantities, therebyavoiding the derivation of algebraic expressions for the Fourier coefficients of the nonlinearterms. The calculations in HDHB are performed in the time domain over one period of oscil-lation. 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 higherharmonics in relatively low-dimensional systems.

So far little work has been devoted to the HDHB analysis of DDEs, and the main purposeof this paper to bring this method to the attention of the community of researchers on time-delay systems. DDEs describe systems where the present rate of change of state depends on apast value (or history) of the state. Generally speaking, the theory of DDEs is a generalizationof the theory of ODEs into infinite-dimensional phase spaces. However, this generalizationis not a trivial task (Kuang, 1993).

The structure of the paper is as follows. In Section 2 we describe a matrix version ofthe conventional HB method for general systems of second-order DDEs. Section 3 focuseson the HDHB method with detailed formulas. In Section 4, we illustrate the utility of theHB methods on three systems described by DDEs: a metal cutting model, the sunflowerequation and a circadian rhythm model. The HDHB method has been successfully applied tocapture both stable and unstable limit cycles of these dynamical systems. The results showan excellent agreement with those from continuation using DDE-BIFTOOL. The advantagesof HDHB over the classical HB method are highlighted and discussed. Finally, conclusionsare drawn in Section 5.

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 ofsecond-order DDEs. The HDHB approach, which is presented in the next section, is derivedfrom 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 expansioninto the governing equations. Then, by using the orthogonal properties of the sine and cosinefunctions, the coefficients of the harmonic terms cos�n�t� and sin�n�t� are equated to zeroand the resulting system of nonlinear algebraic equations are solved for the unknowns. Heren � �1� 2� � � � � N�, where N is the number of harmonic terms.

The first-order HB method, i.e. including one harmonic in the analysis, is easy to apply tothe systems with polynomial nonlinearities with a dominant first harmonic. For motions withevident higher harmonic components, more harmonics have to be included in the analysiswhich limits the usefulness of this approach.

A system of second-order differential equations with delays can be written in the matrixform 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� �x1

��t� � x2

��t� � � � � � xn

��t��T is a n � 1variable vector, and the right-hand side is a vector of nonlinear functions: f � f1

�X��t��

f2

�X��t�� fn

�X��t��T. Since T -periodic motions are sought, the dimensional time �t

is normalized as t ��t , where � 2�T is the fundamental frequency of the oscillation.Note that the frequency � is not known a priori and is therefore treated as unknown. Theequation after the scaling becomes

�2M �X �t�� B X �t��KX �t� f �X �t� �X �t � �� (2)

where the scaled time delay is ��.A 2� -periodic solution of equation 2 can be approximated by the truncated Fourier series

expansion

xi�t� a�i�0 �N�

j1

�a�i�2 j�1 sin� j t�� a�i�2 j cos� j t�

�� (3)

where a�i�k are the unknown Fourier coefficient variables, and N is the number of overallharmonics used in the Fourier series expansion. The truncated expansion of the nonlinearterm may be expressed as

fi �X �t� �X �t � �� b�i�0 �N�

j1

�b�i�2 j�1 sin� j t�� b�i�2 j cos� j t�

�� (4)

where


b�i�0 1

2�

� 2�

0fi �X �t� �X �t � �� dt�

b�i�2 j�1 1

�

� 2�

0fi �X �t� �X �t � �� sin� j t� dt�

b�i�2 j 1

�

� 2�

0fi �X �t� �X �t � �� cos� j t� dt� (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� j t� and cos� j t� yields a system of n � �2N � 1� algebraic equations forthe Fourier coefficients a�i�j ( j 0� 1� 2� � � � � 2N , i 1� 2� � � � � n). The resulting algebraicsystem of equations can be written in a vector form to determine the n� �2N �1� unknowns�Qx (the hat is used to denote frequency-domain quantities)

��2MJ2 � �BJ�K� �Qx � �Rx 0� (6)

where

�Qx

�

a�1�0 a�2�0 � � � a�n�0

a�1�1 a�2�1 � � � a�n�1

��

��

a�1�2N a�2�2N � � � a�n�2N

�� Rx

�

b�1�0 b�2�0 � � � b�n�0

b�1�1 b�2�1 � � � b�n�1

��

��

b�1�2N b�2�2N � � � b�n�2N

��

and

J

�

0

J1

� � �

JN

�� Jk

�0 �k

k 0

�� k 1� � � � � N � (7)

Solving the system in equation 6 requires analytical expressions for the nonlinear functionsb�i�j in terms of the variables a�i�j (i 1� 2� � � � � n, j 0� 1� 2� � � � � 2N ).

As mentioned previously, the frequency � (or, equivalently, the period T ) is unknown inthe above analysis. The algebraic system in equation 6 for the Fourier coefficients and theresponse frequency � has n � �2N � 1�� 1 variables but n � �2N � 1� equations. For theimposed condition, usually the first harmonic of one of the degrees could be with a fixedphase. This is known as phase-fixing for steady-state solutions. For example, we impose thecondition a�1�1 0 or a�1�2 0 on the phase of the first harmonic of the motion.


3. HDHB FOR SYSTEMS OF DDES

As pointed out by Liu et al. (2006), the major disadvantage of the HB method is the tediousderivations of the algebraic expressions for the Fourier coefficients of the nonlinear termsof the dynamical system. The key aspect of the HDHB method is that instead of working interms of Fourier coefficient variables as in the classical HB approach, the dependent variablesare cast in the time domain and stored at 2N � 1 equally spaced sub-time levels over theperiod of one cycle of motion. The Fourier and time-domain variables are related to oneanother via a constant Fourier transformation matrix. Working in terms of time-domain sub-time level solution variables avoids the harmonic balancing step of the Fourier coefficientsolution variables in the classical HB method. This makes the HDHB method very easy toformulate within the framework of an existing time marching nonlinear solver.

3.1. Formulation of the HDHB method

The Fourier coefficients are related to the time-domain solution by equation 3. The 2N � 1HB Fourier coefficient solution variables are related to the time-domain solution by

xi �t� �1 sin �t� cos�t� � � � � � � sin �Nt� cos�Nt�

� � �Q�i�x � (8)

where �Q�i�x denote the i th column of the matrix �Qx . The time-domain solution at 2N � 1

equally spaced nodes over a period of oscillation can be expressed via the 2N�1-dimensionalFourier transformation matrix F. That is,

Qx F �Qx � (9)

where

Qx

�

x1�t0� x2�t0� � � � xn�t0�

x1�t1� x2�t1� � � � xn�t1�

��

��

x1�t2N � x2�t2N � � � � xn�t2N �

�� (10)

with t j j �2��2N �1�� ( j 0� 1� 2� � � � � 2N ), and the �2N �1�� 2N �1�-dimensionaltransform matrix is

F

�

1 sin t0 cos t0 � � � sin Nt0 cos Nt0

1 sin t1 cos t1 � � � sin Nt1 cos Nt1

��

��

��

1 sin t2N cos t2N � � � sin Nt2N cos Nt2N

�� (11)

Conversely, the HB Fourier coefficients can be expressed in terms of the solution using theinverse of the Fourier transformation matrix, i.e.


�Qx F�1Qx � (12)

F�1 2

2N � 1

�

12 12 � � � 12

sin t0 sin t1 � � � sin t2N

cos t0 cos t1 � � � cos t2N

��

��

sin Nt0 sin Nt1 � � � sin Nt2N

cos Nt0 cos Nt1 � � � cos Nt2N

�� (13)

Similarly Rx F �Rx (or �Rx F�1Rx ), where

Rx

�

f1 �X �t0� �X �t0 � �� f2 �X �t0� �X �t0 � �� fn �X �t0� �X �t0 � ��f1 �X �t1� �X �t1 � �� f2 �X �t1� �X �t1 � �� fn �X �t1� �X �t1 � ��

��

��

f1 �X �t2N � �X �t2N � �� f2 �X �t2N � �X �t2N � �� fn �X �t2N � �X �t2N � ��

�� (14)

Equation 6 is then rewritten as

��2MJ2 � �BJ�K�F�1Qx � F�1Rx 0� (15)

Multiplying both sides of equation 15 by F gives

��2FMJ2F�1 � �FBJF�1 � FKF�1�Qx �Rx 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 acomputational fluid dynamics software (Thomas et al., 2002a, 2003).

Again, the frequency � in the system in equation 16 is unknown. Similar to the systemin equation 6 in the frequency domain, the system in equation 16 in the time domain alsoneeds one imposed condition. The phase fixing technique could also be applied. Convertingthe imposing phase condition in the HB analysis (i.e. a�1�1 0 or a�1�2 0) into the timedomain yields

x1�t0� sin t0 � x1�t1� sin t1 � � � � � x1�t2N � sin t2N 0 (17)

or

x1�t0� cos t0 � x1�t1� cos t1 � � � � � x1�t2N � cos t2N 0� (18)


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 work-pieces (turning). Two forms of the nonlinear dependence of the cutting force on the chipthickness are studied, namely a power-law function and its third-order polynomial expan-sion. The second model is the sunflower equation in which the nonlinearity is the transcen-dental function of the delayed term. Even though the nonlinearity is polynomial in our thirdmodel, a model for the circadian rhythm, this example exhibits coexisting stable and unstablelimit cycle oscillations. Direct comparison of the results are provided by using the softwareDDE-BIFTOOL (Engelborghs et al., 2001, 2002). DDE-BIFTOOL is a Matlab package fornumerical bifurcation and stability analysis of delay differential equations using collocationmethods. One of the advantages of the HDHB method is that it is easy and direct to imple-ment without requiring a specific commercial software package. The implementation of theHDHB method is also straightforward to very high-dimensional systems.

For the three models considered in this study, the first harmonic of the solution is dom-inant for parameter values and thus the corresponding limit cycle would look like an ellipsein the x–x � phase plane. Therefore, the phase portraits of the solutions are omitted here. Theexamples given here serve to show the ease and efficacy of the HDHB method. A directcomparison of computational complexity with the original HB method is not provided, sincethe latter requires symbolic calculations. Avoiding such computations is indeed one of themain strengths of the HDHB method.

4.1. 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 termsof the present and delayed state:

�x � 2� x � x f �x�t�� x�t � �� (19)

Time is rescaled so that the assumed periodic motion has a period of 2� . The model equationafter the scaling becomes

�2 �x � 2�� x � x f �x�t�� x�t � �� (20)

where � 2�T is the fundamental frequency and the scaled delay time ��. Thefrequency is not known a priori, therefore it is treated as an unknown. A widely acceptedform for the cutting force nonlinearity is a power law. The nondimensional form for thisnonlinearity is (Kalmár-Nagy et al., 1999)

f �x�t�� x�t � �� p2� �

3�

�1�

�1� 3

2� � �x�t � �� x�t��

�� (21)

where p and � are system parameters. The details of the machine tool vibration modelingcan be found in previous studies (Stépán, 1989, 1997� Kalmár-Nagy et al., 1999). In the


following, to contrast the HB and HDHB techniques, we first present the analysis of a seriesexpanded form of this model. The classical HB method cannot be used for the general power-law 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 easilycaptured by the usual time marching methods.

4.1.1. Power Series Force Model

The nonlinear dependence force on the chip thickness may be simply modeled by a truncatedpower series (here x x�t � �):

�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 cut-ting. The existence and nature of a Hopf bifurcation in this tool vibration model was pre-sented and proved analytically with the help of the center manifold and Hopf bifurcationtheory in Kalmár-Nagy et al. (2001).

For this simple model, both the standard HB and HDHB methods are applicable, andboth methods provide predictions in excellent agreement with the numerical solutions. Thedetailed method implementations for the HB and HDHB analysis and comparison with DDE-BIFTOOL results are reported below.

The HB1 ResultsIncluding the zeroth (constant term) and the first harmonic in the motion form (equation 3),i.e. N 1, gives

x �t� a0 � a1 sin �t� � (23)

where the phase condition a2 0 has been imposed. Equation 6 becomes

��2J2 � 2��J� I� �Qx � �Rx 0 (24)

with

J �0 0 0

0 0 �1

0 1 0

��

and

�Qx

�

a0

a1

a2

�� Rx

�

b0

b1

b2

��

�

2p a21 sin2

2�2p sin

2 � 6pa21 sin3

2

�a1 sin

2

� �2p sin 2 � 6pa2

1 sin3 2

�a1 cos 2

��


Writing the above HB system (equation 24) explicitly gives

a0 � 2p a21 sin2

2 0� (25)

�1� �2�a1 ��

2p sin

2� 6pa2

1 sin3

2

�a1 sin

2 0� (26)

2� � a1 ��

2p sin

2� 6pa2

1 sin3

2

�a1 cos

2 0� (27)

Equation 25 immediately gives the result

a0 2p a21 sin2

2� (28)

Equations 26 and 27 can both be solved for a21 to yield

a21

1� p � �2 � p cos

6p sin4 2

(29)

a21 �

2�� p sin

6p sin3 2 cos 2

� (30)

The right-hand sides of equations 29 and 30 should be equal. This is true when

1� �2 � 2�� tan

2 0� (31)

Since ��, this transcendental frequency–delay relationship can only be solved numeri-cally for � for a specific time delay � . The amplitude of the first harmonic (a1) can then besolved from either equation 29 or equation 30, and the amplitude of the zeroth harmonic canbe solved from equation 28 as

a0 1� p � �2 � p cos

3 sin2 2

� (32)

We note that imposing the condition a1 0 would lead to the same results.The algebraic expressions for the HB analysis with more than one harmonics are com-

plicated, thus details of the higher-order calculations are omitted.

The HDHB1 ResultsHere the nonlinear algebraic system

��2FMJ2F�1 � �FBJF�1 � FKF�1�Qx �Rx 0 (33)

needs to be solved (M 1, B 2� , K 1). For N 1, the time-domain valuesQx

�x0 x1 x2

�Tare sought at three equally spaced sub-time levels over a period of


oscillation, i.e. x0 x�0�, x1 x�2�3� and x2 x�4�3�. The expressions for the Fouriertransformation matrices F and F�1, and the matrix D FJF�1 are

F 1

2

�

2 0 2

2�

3 �1

2 ��3 �1

�� F�1 1

3

�

1 1 1

0�

3 ��3

2 �1 �1

�� D 1�

3

�

0 1 �1

�1 0 1

1 �1 0

��

The linear terms ��2D2 � 2��D� I�Qx in equation 16 become

�

d0

d1

d2

��

�

13�

2 ��2x0 � x1 � x2�� 23��

�3 �x1 � x2�� x0

13�

2 ��2x1 � x2 � x0�� 23��

�3 �x2 � x0�� x1

13�

2 ��2x2 � x0 � x1�� 23��

�3 �x0 � x1�� x2

�� (34)

The basic components in the nonlinear terms are the delay terms x�t � �. From equation 8we obtain x �t � �� x�t� �f �t � �� f�t�� F�1Qx :

�

c0

c1

c2

��

�

13 ��2x0 � x1 � x2� �1� cos ��

�3

3 �x2 � x1� sin

13 ��2x1 � x2 � x0� �1� cos ��

�3

3 �x0 � x2� sin

13 ��2x2 � x0 � x1� �1� cos ��

�3

3 �x1 � x0� sin

�� (35)

where ci x�ti � �� x�ti� �f�ti � �� f�ti�� F�1Qx for �i 0� 1� 2�.The explicit expressions for r0, r1 and r2 in terms of x0, x1 and x2 are as follows

Rx

�

pc0 � p �c2

0 � c30

�pc1 � p

�c2

1 � c31

�pc2 � p

�c2

2 � c32

�� (36)

where c0� c1� c2 are given in equation 35. Note that linear terms (the ci ) also appear in Rx ,owing to the chosen form (equation 2) of the class of DDEs studied. Combining the linearterms (equation 34) with the nonlinear terms (equation 36), the HDHB1 system with theimposed condition in equations 17 or 18 can be written explicitly as

��

d0 � pc0 � p �c20 � c3

0� 0�

d1 � pc1 � p �c21 � c3

1� 0�

d2 � pc2 � p �c22 � c3

2� 0�

2x0 � x1 � x2 0�

or

��

d0 � pc0 � p �c20 � c3

0� 0�

d1 � pc1 � p �c21 � c3

1� 0�

d2 � pc2 � p �c22 � c3

2� 0�

x1 � x2 0�

(37)

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


Figure 1. Machine tool cutting with the power series force: first harmonic amplitude versus the bifurcationparameter p from the HDHB method with one and two harmonics, in comparison with the results fromthe HB method with one harmonic. Open circle: HDHB1� open square: HDHB2� solid line: HB1.

straightforward as one only needs to add solution variables in Qx while the rest is taken careof by the matrix vector multiplications.

Numerical ResultsTo facilitate numerical analysis, we set

0�3� � 0�1� 4�385� pcr 0�22� � 34� (38)

The motions are dominated by the first harmonic with fundamental frequency of 1.095.This frequency is almost constant in the range p � [0�209� 0�22]. The frequency predictionsfrom both HB and HDHB methods are near 1.09 regardless of the number of harmonicsincluded in the analysis.

The amplitude result from the HB method including one harmonic is shown in Figure 1as the solid line. Compared with the bifurcation diagram (the amplitude of the unstable limitcycle oscillations versus p) in (Kalmár-Nagy et al., 2001), the HB method with one harmonicprovides a good approximation (with excellent agreement with the results from the directnumerical solution) for the oscillations near the bifurcation point.

In order to compare the results with those from the HB method, the solutions from theHDHB 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 methodwith 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 isrelatively small and can be neglected. For this model, the amplitude results from using theHDHB method with one harmonic are smaller than the real solutions. Including one moreharmonic in the HDHB analysis, the result for the first harmonic amplitude improves sub-stantially. The HDHB method with two harmonics provides a good approximation (squares)and including more harmonics in the HDHB analysis provides no substantial changes in theresults.


Figure 2. Machine tool cutting with the fraction power force: first harmonic amplitude versus thebifurcation 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 bifurca-tion parameter p from the HDHB method with two harmonics (HDHB2). Solid line: DDE-BIFTOOL� opencircle: HDHB2 with equation 18� star: HDHB2 with equation 17.

4.1.2. Power-law Force Model

The model given by equations 20 and 21 provides a description of the nonlinear force de-pendence on the chip thickness also valid farther from the bifurcation point (provided thatcontact 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 harmonicsin Figure 3 and three harmonics in Figure 4. Power spectrum analysis of x �t� reveals thatthe oscillations are dominated by the fundamental frequency. The results from the HDHBanalysis further verify that the amplitudes of the constant term, the second- and higher-orderharmonics are negligible compared with the amplitude of the first harmonic. Therefore, onlythe results of the first harmonic amplitudes are shown here.


Figure 4. Machine tool cutting with the fraction power force: first harmonic amplitude versus the bifur-cation 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 theHDHB system, and the results may be different as shown in Figures 2–4. Nonetheless, theresults from the HDHB analysis with either imposing condition converge to the real solu-tions as the number of harmonics increases from one to three. For the motions away fromthe bifurcation point, e.g. for p less than 0.215, the results from imposing two conditions de-viate slightly from the real solutions. Further numerical simulations show that adding moreharmonics into the HDHB analysis eliminates the discrepancies.

In conclusion, imposing different phase angles in the HDHB analysis may lead to dif-ferent predictions if a small number of harmonics is included. As the number of harmonicsincluded in the HDHB analysis increases, the discrepancies between the predictions fromimposing either condition become small and both converge to the real solutions. Further-more, the farther away the bifurcation parameter is from the bifurcation point, the larger thenumber of harmonics included in the HDHB analysis is needed for the same accuracy in themotion prediction.

Comparing the results from the HDHB method in Figures 2–4 for the fraction powerforce with those in Figure 1 for the power series force, for sufficiently accurate results thenumber of high harmonics needed in the HDHB analysis for the fraction power force issimilar to that for the power series force.

4.2. 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 aperturbation expansion for the frequency and amplitude.

In the sunflower equation 39, the delay also appears in the coefficients. By usingsome special treatment, the results from the HB analysis with one harmonic may be obtained


Figure 5. Sunflower model: fundamental frequency versus the bifurcation parameter from the HDHBmethod with one harmonics, in comparison with the real solutions. Solid line: DDE-BIFTOOL� opencircle: HDHB1 with equation 18� star: HDHB1 with equation 17.

Figure 6. Sunflower model: first harmonic amplitude versus the bifurcation parameter from the HDHBmethod 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 con-ventional HB method is virtually impossible to implement. The HDHB method does notencounter any special difficulty for this model. The implementation of the HDHB methodfor this equation model is similar to that for the machine tool vibrations, and the details ofthe formulations are omitted here.

As shown in Figure 5 for the frequency predictions, the results from the HDHB analysiswith one harmonic with either imposing conditions (equation 17 or equation 18) deviateslightly from the real solutions for large bifurcation values. Including one more harmonicin the HDHB analysis, both results from imposing conditions equation 17 and equation 18converge 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 thatthe oscillations are dominated by the fundamental frequency. Therefore, only the results forthe first harmonic amplitude are shown in the figures. The results from the HDHB analysisincluding one harmonic are shown in Figure 6, and for two harmonics in Figure 7. In thesefigures the results from DDE-BIFTOOL are shown by solid lines.


Figure 7. Sunflower model: first harmonic amplitude versus the bifurcation parameter from the HDHBmethod 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 phaseconditions is very small. The predictions from the HDHB analysis with one harmonic matchthe real solutions well for the motions near the bifurcation point as shown in Figure 6. Forthe motions away from the bifurcation point, including one more harmonic in the HDHBanalysis improves the accuracy. As shown in Figure 7, the HDHB results from imposingdifferent conditions are identical and match the real solutions well for the motions beyondthe bifurcation point. Further numerical simulations reveal that adding more harmonics tothe HDHB analysis provides no substantial changes in the results.

4.3. A Circadian Rhythm Model

A model for circadian rhythm is given as (Ohlsson, 2006)

y�� 2y� � y �� y�� 2 y2�� 3 y3�� (40)

with

�2 �23 �� 1� �3� ��13 �

�3 �1

3

�3�2 � 6�� 1

��13 �3� ��23 �

where the bifurcation parameter is �, and the delay time is . A similar model is studied byVerdugo and Rand (2009). This example was chosen to demonstrate the ability of HDHBto correctly capture coexisting limit cycles. Since the right-hand side of equation 40 is athird-order polynomial of the delayed term, the conventional HB method can in principlebe applied with no problem. However, owing to the presence of the constant term in theharmonic approximation, these results are relatively complicated. The implementation ofthe HDHB method encounters no difficulty and the high-order results are in an excellentagreement with the results from DDE-BIFTOOL.


Figure 8. Circadian model: first harmonic amplitude versus the bifurcation parameter � from the HDHBmethod 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 thebifurcation parameter � from the HDHB method with two harmonics. Solid line: DDE-BIFTOOL� opencircle: HDHB2 with equation 18� star: HDHB2 with equation 17.

For 1�05 the Hopf bifurcation occurs at �cr 2�614. The collocation results areshown in Figures 8–10 as solid lines for the first harmonic amplitude and the fundamentalfrequency. As the motions are dominated by the first harmonic, the results for the first har-monic amplitude and the fundamental frequency are presented and discussed here. In thebifurcation diagrams, both the amplitude and frequency curves are folded back, and the turn-ing point is at � 2�643. The coexistence of motions occurs for the bifurcation parameter� between the bifurcation point and the turning point, i.e. � � [2�614� 2�643].


Figure 10. Circadian rhythm model: first harmonic amplitude (a) and the frequency (b) versus thebifurcation parameter � from the HDHB method with three harmonics. Solid line: DDE-BIFTOOL� opencircle: HDHB3 with equation 18� star: HDHB3 with equation 17.

The bifurcation is subcritical, and the global amplitude/frequency curve in the bifur-cation diagram consists of the two branches for the stable and the unstable motions. TheHDHB 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 tofind all possible solutions. The results are displayed in Figures 8–10 for various numbers ofharmonics and different phase conditions.

The first harmonic amplitudes obtained from the HDHB method including only oneharmonic, as shown in Figure 8, are quite different when the condition in either equation 17 orequation 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 theHDHB analysis with one harmonic deviate substantially from the real solutions even nearthe bifurcation point.

Including one more harmonic in the analysis improves the results dramatically, as shownin Figure 9. The accuracy of the frequency results, however, is not completely consistent withthat of the amplitude, as shown in Figure 9b.

Including one more harmonic in the HDHB analysis provides accurate solutions for boththe amplitude and the frequency as shown in Figure 10a and b. Further numerical analysisshows that including more harmonics in the HDHB analysis does not change the resultssubstantially. It also appears that the agreement between results from the two different phaseconditions could be used to establish bounds on the accuracy of the solutions.


5. CONCLUSIONS

This study has focused on the HDHB method for determining limit cycle oscillations ofsecond-order DDEs. The variables in the HDHB method are the function values of thesolution at discrete times. The nonlinear terms of the system equations are evaluated atthese discrete times only, thus avoiding the process of deriving the algebraic expressionsfor the Fourier coefficients needed for the classical HB method. A detailed formulation ofthe HDHB method for systems of second-order DDEs has been provided. The utility ofthe HDHB analysis is illustrated on three examples: a machine tool vibration model, thesunflower equation and a circadian rhythm model. It is demonstrated that it is easy andstraightforward to implement the HDHB method on dynamical systems with various typesof nonlinearities. The present study includes fractional power and transcendental nonlinear-ities for which the conventional HB method is virtually impossible to implement. To betterexplain the method, the results from the HB method are also compared with the results fromthe HDHB method for the first example. The results from the DDE-BIFTOOL package areused to verify the accuracy of the HB/HDHB methods.

For the machine tool cutting model, the HB method is applied to the power series ex-panded nonlinear force model, and the results from the HB method with one harmonic matchthose from DDE-BIFTOOL. However, the HB method cannot be directly applied to the frac-tional power force law. The HDHB method can be directly applied to both versions of thenonlinear force and the implementation is straightforward with no difficulty. The HDHBmethod with two harmonics provides accurate results.

For the sunflower equation, although the results from the HB analysis with one harmoniccan be obtained with some special treatment (MacDonald, 1995), in general, the HB methodcannot be applied directly because of the transcendental nonlinearity. Again, there is nodifficulty in the implementation of the HDHB method for this model and the results with oneharmonic match the real solutions well.

For the circadian rhythm model, the HDHB method is able to capture coexisting stableand unstable limit cycles, thereby demonstrating the utility of the technique for such cases.

With different specific imposing conditions in the HDHB analysis, the results, whichmay vary for small number of harmonics, converge to the same solutions when a sufficientnumber of harmonics are included.

Acknowledgements. The authors wish to thank Professor Earl Dowell for valuable comments and Dr Pankaj Wahi forvaluable help with the DDE-BIFTOOL package.

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