Galerkin Arnoldi algorithm for stability analysis of time-periodic delay differential equations

Transcription

1 Nonlinear Dyn : DOI 1.17/s ORIGINAL PAPER Galerkin Arnoldi algorithm for stability analysis of time-periodic delay differential equations Zaid Ahsan Anwar Sadath Thomas K. Uchida C. P. Vyasarayani Received: 16 December 214 / Accepted: 16 July 215 / Published online: 1 August 215 Springer Science+Business Media Dordrecht 215 Abstract We present an algorithm for determining the stability of delay differential equations DDEs with time-periodic coefficients and time-periodic delays. The DDEs are first posed as an equivalent system of partial differential equations PDEs along with a nonlinear boundary condition. A Galerkin approximation is then employed to discretize the PDEs into a set of time-periodic ordinary differential equations ODEs. Finally, we apply a modified version of the Arnoldi algoritho extract the dominant eigenvalues of the Floquet transition matrix without computing the entire matrix, thereby reducing the required number of integrations of the ODE system. Five numerical examples demonstrate that our modified Arnoldi algorithm provides reliable approximations of the dominant eigenvalues of the Floquet transition matrix, and does so with substantially less computational effort than the classical Floquet method. The stability charts and bifurcation diagrams generated using our Galerkin Arnoldi method clearly demonstrate the utility of this approach for establishing the stability of a system of DDEs. Z. Ahsan A. Sadath C. P. Vyasarayani B Department of Mechanical and Aerospace Engineering, Indian Institute of Technology Hyderabad, Ordnance Factory Estate, Yeddumailaram, Telangana 5225, India vcprakash@iith.ac.in T. K. Uchida 318 Campus Drive, Stanford, CA 9435, USA Keywords Arnoldi algorithm Delay differential equation Floquet theory Galerkin method Stability Time-periodic 1 Introduction Delay differential equations DDEs are used to model time-delayed systems in a variety of applications, including biology [4], population dynamics [13], control systems [16], manufacturing [2,6,1], and traffic flow modeling [2]. A fundamental and critical task is determining the regions of parameter space for which a system of DDEs is stable. Several methods have been proposed in the literature for studying the stability of DDEs with constant coefficients; however, the stability analysis of DDEs with time-periodic coefficients and time-periodic delays is nontrivial [7,11,14]. We adopt the approach of first posing a system of DDEs as an equivalent system of hyperbolic partial differential equations PDEs with a nonlinear boundary condition. A Galerkin approximation [18,19,22,23] is then used to discretize the PDEs into a system of time-periodic ordinary differential equations ODEs, whereupon standard stability analysis techniques can be applied. A popular method for determining the stability of time-periodic ODEs is to use Floquet theory. We first specify as many independent initial condition vectors as there are degrees of freedom in the system. The system of ODEs is then integrated for one time period

2 1894 Z. Ahsan et al. using each initial condition vector; the final state vectors forhe columns of the Floquet transition matrix. The system is stable if, and only if, all eigenvalues of the Floquet transition matrix have magnitude less than unity. Clearly, forming the Floquet transition matrix becomes intractable for systems with many degrees of freedom. Bauchau and Nikishkov [3] proposed a method for determining the stability of very large timeperiodic systems using the Arnoldi algorithm, an iterative method that approximates the dominant eigenvalues of the system without computing the full Floquet transition matrix. In this work, we extend the method of Bauchau and Nikishkov [3] to study the stability of time-periodic DDEs. The mathematical model is discussed in Sect. 2. We first convert the system of DDEs into a system of ODEs using a Galerkin approximation, then use the Arnoldi algoritho determine the stability of the resulting ODEs. In the sequel, we refer to the proposed method as the Galerkin Arnoldi algorithm. In Sect. 3, we validate the Galerkin Arnoldi algorithm against the results of classical Floquet theory and those reported in the literature. We present five numerical examples to demonstrate the proposed method. Conclusions are provided in Sect Mathematical modeling In this section, we present the Galerkin Arnoldi algorithm as a method for determining the stability of timeperiodic DDEs. We first briefly describe the use of a Galerkin approximation to convert time-periodic DDEs into time-periodic ODEs; a detailed treatment has been reported by Sadath and Vyasarayani [19]. The Arnoldi algorithm is then applied to determine the stability of the resulting system of ODEs. 2.1 Galerkin approximation for time-periodic DDEs Consider the following DDE: n ẍ + ctẋ + ktx + c p tẋt τp c t + p=1 m q=1 k q txt τq k t = 1 where ct, kt, c p t, and k q t are time-dependent coefficients, and τp ct and τ q k t are time-dependent delays p = 1, 2,...,n and q = 1, 2,...,m. The time-dependent coefficients and delays are assumed to be periodic functions with fundamental period T.We assume the following history functions for the DDE 1: xt = αt, τ t 2a ẋt = βt, τ t 2b where τt = max τp ct, τ q kt. We introduce the following standard transformation [12,15,18]: ys, t = xt + τts 3 whereupon we obtain the following PDE: 2 y t 2 = 1 2 y τt t s τt y τ 2 t s + τt s 2 y τt t s [ τtτt τ 2 ] t s y + τ 2 t s, 1 s 4 and the following boundary condition for 4: 2 y t 2 + ct y s= t + kty, t s= n + c p t y m + k q ty t = p=1 s= τ c p t τt q=1 The initial conditions are as follows: τ k q t τt, t 5 ys, = αs, 1 s 6a ys, = βs, t 1 s 6b We now assume the following approximate solution for the PDE 4: ys, t N φ k sη k t = φ T sηt 7 k=1 where φ = [φ 1 s, φ 2 s,...,φ N s] T are the basis functions we use shifted Legendre polynomials as the basis functions and ηt = [η 1 t, η 2 t,...,η N t] T are the generalized coordinates. Upon substituting the series solution 7 into the PDE 4, pre-multiplying by φs, and integrating over the domain s [ 1, ], we arrive at the following system of second-order ODEs:

3 Galerkin Arnoldi algorithm for stability analysis M ηt = τt C + τt τt D ηt τt + τ 2 t C + τtτt τ 2 t τ 2 D ηt t 8 where matrices M, C, and D are defined as follows: M = C = D = φsφ T s ds φs s φt s ds sφs s φt s ds 9a 9b 9c Note that closed-form expressions can be obtained for matrices M, C, and D when shifted Legendre polynomials are used as basis functions [18]. We define the following matrices for notational convenience: C = 1 τt C + τt τt D 1a K = τt τ 2 t C + τtτt τ 2 t τ 2 D 1b t We must now impose the boundary condition 5. Upon substituting the series solution 7 into the boundary condition 5, we obtain the following ODE: m ηt = c ηt + kηt 11 where m, c, and k are defined as follows: m = φ T n c = ctφ T c p tφ T τ p ct τt p=1 m k = ktφ T k q tφ T τ q kt τt q=1 12a 12b 12c The boundary condition 11 can be imposed during the solution of the ODE system 8 using the Lagrange multiplier method or the tau method; we use the latter. We first replace the last row of 8 with the boundary condition 11 to obtain the following system of ODEs: [ ] [ ] [ ] M C K ηt = ηt + ηt 13 m c k where M, C, and K are the truncated matrices obtained upon removing the last row from M, C, and K, respectively. We now define the following substitutions, again for notational convenience: M Tau = [ ] M, C m Tau = and rewrite 13 asfollows: [ ] C, K c Tau = [ ] K k 14 M Tau ηt = C Tau ηt + K Tau ηt 15 Equation 15 is the final ODE approximation of the original DDE with time-periodic delays 1. The initial conditions for 15 are obtained by substituting the series solution 7 into the initial conditions given by 6a and 6b, whereupon we obtain the following: η = M 1 η = M φsατs ds φsβτs ds 16a 16b The approximate solution xt of the original DDE 1 can be obtained frohe series solution 7: xt = y, t = φ T ηt 17 We define a state vector x = [ η T, η T] T and express the system of ODEs 15 in state-space form: ẋ = Atx 18 Matrix At is periodic with period T : [ ] I At = MTau 1 K Tau MTau 1 C Tau 19 where I is an identity matrix. Floquet theory can be used to determine the stability of the time-periodic ODE system 18. Let Φ denote the Floquet transition matrix, which can be obtained by integrating the following matrix differential equation: Φ = AtΦ 2

4 1896 Z. Ahsan et al. We integrate 2 to time t = T with initial condition Φ = I to obtain ΦT. We can then write the following: xt = ΦT x 21 where xt is the state of the ODE system 18 after time T when given initial conditions x. From21, we see that the stability of the system can be determined by computing the eigenvalues of the Floquet transition matrix ΦT : the system is stable if, and only if, all eigenvalues of ΦT have magnitude less than unity. In the sequel, we refer to this method of determining the stability of time-periodic DDEs as the Galerkin Floquet method. The major disadvantage of this method is that, for very large systems [i.e., when many terms are used in the Galerkin approximation 7], the calculation of ΦT becomes prohibitively expensive. 2.2 Arnoldi algorithm We now discuss the Arnoldi algorithm, which upon modification can be used to determine the stability of time-periodic DDEs with fewer integrations than the Galerkin Floquet method. We first provide a brief theoretical overview of the Arnoldi algorithm [1,5] following the review of Bauchau and Nikishkov [3]; a complete mathematical development of the algorithm has been presented by Saad [17]. We then present a simple numerical example to demonstrate the practical use of the algorithm. Finally, we discuss the application of the algoritho the stability analysis of time-periodic DDEs Theoretical overview Consider an arbitrary square matrix P R N P N P whose eigenvalues must be found. Using an iterative construction of a Krylov subspace [17], P is reduced to an upper Hessenberg matrix H, the eigenvalues of which approximate those of the original matrix P. With each iteration j of the Arnoldi algorithm, we compute a new column of H and obtain increasingly accurate approximations of the j dominant eigenvalues of P. A new vector in the Krylov subspace, x j+1 R N P, is constructed using the following recurrence relation [17]: h j+1, j x j+1 = Px j j h i, j x i =: r j 22 i=1 where h i, j is the i, j entry of H. Vectors x are constructed so as to form an orthonormal basis of the Krylov subspace. It is critical for x j+1 to be orthogonal to all x i, i = 1, 2,..., j to maintain numerical stability; we orthogonalize using the modified Gram Schmidt process. Rearranging 22 and expressing it in matrix form yields the following: h 1, j Px j = [ ] h x 1, x 2,...,x 2, j j + h... j+1, j x j+1 23 h j, j We define matrix Q j as the sequence of vectors x : Q j = [ ] x 1, x 2,...,x j 24 whereupon the recurrence relation 23 can be written at all iterations in the following compact form: PQ j = Q j H j + h j+1 x j+1 e T j 25 where e T j = [,...,, 1] R j. The upper Hessenberg matrix H j is defined as follows: h 1,1 h 1,2... h 1, j h 2,1 h 2,2... h 2, j h 3,2... h 3, j H j =... h 4, j h j, j 26 Note that the size of H increases with each iteration j of the Arnoldi algorithm. We now consider the eigenvalue problem: Pu = λu 27 where λ denote the exact eigenvalues of matrix P.Inthe jth iteration of the Arnoldi algorithm, the matrix Q j is used to map the vector u onto the Krylov subspace: u = Q j s 28

5 Galerkin Arnoldi algorithm for stability analysis 1897 Algorithm 1 Arnoldi algorithm Given: Matrix P R N P N P whose eigenvalues must be found; arbitrary starting vector r R N P ; maximum number of iterations j max Find: Upper Hessenberg matrix H whose eigenvalues approximate those of P h 1, r x 1 1/h 1, r for j from 1 to j max do z Px j r z for i from 1 to j do h i, j x T i z r r h i, j x i end for h j+1, j r x j+1 1/h j+1, j r #Orthogonalize using the modified Gram Schmidt process for i from 1 to j do x j+1 x j+1 x T i x j+1 xi end for end for Upon substitution of 28into27 and premultiplying by Q T j, we obtain the following: Q T j PQ js = ˆλQ T j Q js 29 where ˆλ denote approximations of the j dominant eigenvalues of P. We now substitute 25 into29, note that Q T j Q j = I, and recall that x j+1 is orthogonal to all x i, i = 1, 2,..., j, whereupon we obtain the following simplified expression: H j s = ˆλs 3 Thus, the eigenvalues of H j approximate those of the original matrix P. The Arnoldi algorithm is summarized in Algorithm 1. Note that the original matrix P appears only in the operation z Px j. As such, the Arnoldi algorithm is ideally suited for determining the dominant eigenvalues of large, sparse matrices Numerical example Consider the following matrix: 981 P = The eigenvalues of P are approximately 14.77, 4.14, and The Arnoldi algorithm begins by selecting a random vector r: r T = [2, 3, 5] 32 We set the maximum number of Arnoldi iterations j max = N P = 3. After the first iteration, we obtain the following Hessenberg matrix H 1 and corresponding eigenvalue ˆλ 1 : H 1 = [ ], ˆλ 1 = { } 33 Note that ˆλ 1 approximates the dominant eigenvalue of P. The second iteration results in the following: H 2 = [ ] , ˆλ = { } where the approximation of the dominant eigenvalue has been refined. The third and final iteration provides the exact eigenvalues of P: H 3 = , ˆλ 3 = With each iteration of the Arnoldi algorithm, we obtain an approximation of one additional eigenvalue, and the approximations frohe previous iteration are refined. The dominant eigenvalues are the first to converge, and upon completing N P iterations, all eigenvalues are obtained exactly Application to time-periodic DDEs We introduce two modifications to the Arnoldi algorithm to facilitate its application to the stability analysis of time-periodic DDEs. First, as noted above, the original matrix P appears only in the operation z Px j. Since we are interested in computing the dominant eigenvalues of the Floquet transition matrix ΦT, we replace P with ΦT in Algorithm 1. From21, it is clear that the product ΦT x j is equivalent to the response obtained upon integrating the system ODEs 18 for time T using the initial conditions x j : xt = ΦT x j 36

6 1898 Z. Ahsan et al. This modification to the Arnoldi algorithm allows us to identify the dominant eigenvalues of the Floquet transition matrix ΦT without computing the entire matrix. Clearly, we must perform one time integration for each iteration of our proposed Galerkin Arnoldi scheme. Note that we are interested in only the single dominant eigenvalue of ΦT, since it is this eigenvalue that will determine the stability of the underlying DDE. The number of iterations of the Arnoldi algorithm required to achieve convergence of this eigenvalue is problem dependent. Thus, using a predetermined number of iterations j max can result in poor convergence in some cases and unnecessary computation in others. Rather than set the maximum number of iterations j max explicitly, we instead monitor the convergence of the dominant eigenvalue in our Galerkin Arnoldi algorithm. The termination criterion is defined as follows: ˆλ dom ˆλ dom <λ tol 37 j j 1 where ˆλ dom k is the dominant eigenvalue obtained after the kth iteration and λ tol is the iteration tolerance. As will be shown in the next section, even a strict termination criterion results in substantial computational savings compared to the classical Floquet method. 3 Numerical examples In this section, we present five examples to demonstrate the efficacy of the Galerkin Arnoldi algorithm. We study the stability of DDEs with time-periodic coefficients and time-periodic delays using the proposed approach, and compare the results to those obtained using the classical Floquet method and those reported in the literature. In all cases, we choose starting vector r T = [1,,...,]. The simulations were performed using Matlab R212b on a 2.6-GHz Intel Xeon E5-267 processor using 12 parallel threads. The time response of each ODE 18 was obtained using the built-in ode15s integrator with absolute and relative error tolerances of Impedance-modulated turning model with time-periodic coefficients We first consider the following impedance-modulated turning model [21]: b Ω=1/ω Fig. 1 Stability diagram for the impedance-modulated turning model 38 obtained using the Galerkin Arnoldi algorithm red dots and reported by Segalman and Butcher [21] blue lines. Dots indicate points at which the system is stable. Color figure online ẍt + 2ζωẋt + ω b + ɛ cost/2 xt = bω 2 xt τ 1 38 which is a second-order DDE with time-periodic coefficients. Using the Galerkin approximation technique, we convert 38 intoasystemofodesoftheform shown in 15. A convergence analysis reveals that N = 15 terms are required in the Galerkin approximation 7 when parameter values ζ =.1, ɛ =.2, and τ 1 = 2π are used. As illustrated in Fig. 1, the Galerkin Arnoldi method generates a stability diagram that is in good agreement with the results reported in the literature [21] when using an iteration tolerance of λ tol = 1 1. We also find that, on average, the Galerkin Arnoldi algorithm requires 7 iterations and, therefore, 7 time integrations to evaluate the stability at each point, which represents a 77 % decrease in computational cost when compared to the 3 integrations required by the Galerkin Floquet method [18]. As shown in Table 1, the Galerkin Arnoldi approach maintains its accuracy and performance over a range of parameters Ω and b. 3.2 Damped Mathieu s equation with time-periodic coefficients In our second example, we consider the stability of the following time-delayed Mathieu s equation [8]: ẍt + cẋt + δ + ɛ cosωt xt + k 1 xt τ 1 + k 2 xt τ 2 = 39

7 Galerkin Arnoldi algorithm for stability analysis 1899 Table 1 Comparison between Galerkin Floquet and Galerkin Arnoldi algorithms for the impedance-modulated turning model 38 Parameters Dominant eigenvalue Integrations required Galerkin Floquet Galerkin Arnoldi Galerkin Floquet Galerkin Arnoldi Ω =.36, b = i i 3 7 Ω =.38, b = i i 3 7 Ω =.6, b = i i 3 6 Ω =.55, b = i i 24 7 The Galerkin Arnoldi algorithm provides an excellent approximation of the dominant eigenvalue, and is at least three times more efficient than the Galerkin Floquet method. A convergence analysis is performed for each set of parameter values shown here to determine the number of terms required in the Galerkin approximation Fig. 2 Stability diagram for the damped Mathieu s equation 39 obtained using the Galerkin Arnoldi algorithm red dots and reported by Sadath and Vyasarayani [18] blue circles. Dots indicate points at which the system is stable. Color figure online We use parameters c =.5, ω = π, k 1 =.2, k 2 =.1, τ 1 = 2π, and τ 2 = π, and determine the stability of 39forδ [, 12] and ɛ [ 6, 6].Using these parameters, a convergence analysis indicates that N = 15 terms are required in the Galerkin approximation 7 in order to generate the complete stability plot, again using an iteration tolerance of λ tol = 1 1.As shown in Fig. 2, the Galerkin Arnoldi method generates a stability diagrahat is identical to that produced by the Galerkin Floquet method, but is 5 % more efficient, requiring an average of 15 integrations rather than 3 to ascertain the stability at each point. Shown in Table 2 are the four most dominant eigenvalues of 39 when δ = 3 and ɛ =, as computed by the two algorithms. Although the Galerkin Arnoldi approximation of λ 4 deviates substantially from its true value, the approximation of λ 1 is reliable and is deter- δ Table 2 Comparison between Galerkin Floquet and Galerkin Arnoldi algorithms for the damped Mathieu s equation 39 at the point δ = 3andɛ = infig.2 Eigenvalue Galerkin Floquet Galerkin Arnoldi λ i i λ i i λ i i λ i i At this point, the Galerkin Arnoldi algorithm requires 15 iterations to converge and provides a relatively coarse approximation of λ 4 ; however, it is the dominant eigenvalue that governs system stability, and the approximation of λ 1 is reliable mined to sufficient precision to comment on the stability of the system. 3.3 Single-degree-of-freedourning model with time-periodic delay In our third example, we consider the following singledegree-of-freedom turning model with a sinusoidal time-varying delay [1]: ẍ t + 2ζ ẋ t + x t = wx t τ t x t 4 where ζ is the relative damping factor, w is the depth of cut, and tildes denote nondimensional quantities. The time-periodic delay is defined as τ t = τ τ 1 cos ω m t. We generate stability diagrams in the plane of the nondimensional spindle speed Ω and depth of cut w. Thus, for each point in our stability diagrams, the values of Ω and w are given. We also specify numerical values for modulation parameter R P = 2π/ τ ω m and modulation amplitude ratio R A = τ 1 / τ. Parameters τ, τ 1, and ω m are computed as follows:

8 19 Z. Ahsan et al. Table 3 Comparison of computational cost of the Galerkin Floquet and Galerkin Arnoldi algorithms for the single-degree-of-freedom turning model 4 Figure Average number of integrations per parameter Computation time per parameter s Galerkin Floquet Galerkin Arnoldi Galerkin Floquet Galerkin Arnoldi 3a b The Galerkin Arnoldi algorithm requires substantially fewer integrations and, therefore, substantially less computation time than the Galerkin Floquet method to generate the stability diagrams shown in Fig. 3 Fig. 3 Stability diagrams for the single-degree-offreedom turning model 4 obtained using the Galerkin Arnoldi algorithm red dots and reported by Insperger and Stépán [1] blue lines using two parameter sets: a ζ =.2, R P = 2, and R A =.1; and b ζ =.5, R P =.4, and R A =.2. Dots indicate points at which the system is stable. Color figure online a w Ω b w Ω τ = 2π Ω 41a τ 1 = R A τ 41b ω m = 2π 41c R P τ In this example, we generate two stability diagrams using parameter sets reported by Insperger and Stépán [1] with an iteration tolerance of λ tol = 1 1. In both cases, the Galerkin Arnoldi algorithm establishes the same stability bounds as classical techniques, but does so substantially more efficiently. As shown in Table 3, the Galerkin Arnoldi algorithm is, on average, 81 % more efficient than the Galerkin Floquet method when generating Fig. 3a, and 77 % more efficient when generating Fig. 3b. 3.4 Two-degree-of-freedom milling equation Next, we consider the following two-degree-of-freedom milling model [9]: ωn 2 + wh xxt xt ẍt + 2ζω n ẋt + whxy t + yt whxx t = xt τ+ whxy t yt τ 42a wh yx t ÿt + 2ζω n ẏt + xt + ωn 2 + wh yyt yt wh yx t wh yy t = xt τ+ yt τ 42b where ω n is the natural angular frequency, ζ is the relative damping, and is the modal mass of the tool. The tool is assumed to be symmetric and, therefore, the parameters in the x- and y-directions are assumed to be equal. The coefficients h xx t, h xy t, h yx t, and h yy t denote the four projections of the specific cutting force coefficient: N t h xx t = gφ j t sinφ j tk n sinφ j t j=1 +K t cosφ j t N t h xy t = gφ j t cosφ j tk n sinφ j t j=1

9 Galerkin Arnoldi algorithm for stability analysis 191 +K t cosφ j t N t h yx t = gφ j t sinφ j tk n cosφ j t j=1 K t sinφ j t N t h yy t = gφ j t cosφ j tk n cosφ j t j=1 K t sinφ j t where N t is the number of teeth in the tool, φ j t is the angular position of tooth j, and K n and K t are the normal and tangential linearized cutting force coefficients. The function gφ j t is a screen function that determines whether tooth j is in the cut or out of the cut: gφ j t = { 1 ifφ st <φ j t <φ ex otherwise 43 where φ st and φ ex are the start and exit angles of tooth j, respectively. When up-milling, φ st = and φ ex = arccos1 2a/D; when down-milling, φ st = arccos2a/d 1 and φ ex = π, where a/d is the radial depth of cut ratio. The angular position φ j t of tooth j at time t is given by the following: φ j t = 2πΩ/6 t + j2π/n t 44 where Ω is the spindle speed in revolutions per minute. In the milling process, due to the tooth pass excitation effect, the projections of the cutting force coefficient are time-periodic, with period equal to the time delay τ. This time delay is equal to the tooth-passing period, and is given by τ = 6/ N t Ω. In this example, we consider the down-milling process and generate the stability diagram of 42. w mm Ω 1 3 rpm Fig. 4 Stability diagram for the two-degree-of-freedom milling model 42 with radial depth of cut ratio a/d = 1. The red dots indicate the stable points obtained using the Galerkin Arnoldi method; the blue line indicates the stability boundary reported by Insperger and Stépán [9]. Color figure online We use the following parameter values: N t = 2, ω n = 5793 rad/s, ζ =.11, =.3993 kg, K t = N/m 2, and K n = N/m 2. A convergence analysis reveals that, for these parameter values, N = 25 terms are required in the Galerkin approximation 7. Figure 4 illustrates the stability diagram as the depth of cut w and spindle speed Ω are varied, using an iteration tolerance of λ tol = 1 1. We can clearly see that the stability region predicted by the Galerkin Arnoldi method agrees with that reported by Insperger and Stépán [9]. The Galerkin Arnoldi method requires, on average, only 12 integrations to generate the stability diagram compared to 1 integrations required by the Galerkin Floquet method; as shown in Table 4, the Galerkin Arnoldi method is substantially faster than the Galerkin Floquet method. Table 4 Comparison between Galerkin Floquet and Galerkin Arnoldi algorithms for the two-degree-of-freedom down-milling process 42 Parameters Dominant eigenvalue Computation time s Galerkin Floquet Galerkin Arnoldi Galerkin Floquet Galerkin Arnoldi Ω = 14,w = i i Ω = 185,w = Ω = 2,w = i i Ω = 23,w = i i For the parameters shown, the Galerkin Arnoldi method requires an average of 13 integrations compared to 1 integrations required by the Galerkin Floquet method

10 192 Z. Ahsan et al. a b c Fig. 5 Bifurcation types for time-periodic systems: a secondary Hopf or Neimark Sacker bifurcation; b period-one bifurcation; and c period-doubling or flip bifurcation. The unit circle is shown in blue. Color figure online 3.5 Bifurcation and chatter frequency diagrams We now apply the Galerkin Arnoldi method to generate bifurcation and chatter frequency diagrams for the single-degree-of-freedourning model with a sinusoidal time-varying delay 4. Recall that, in stable regions, the magnitude of the dominant eigenvalue of the Floquet transition matrix Φ is no greater than 1; in unstable regions, it is greater than 1. At the stability boundary, the magnitude of the dominant eigenvalue is exactly equal to 1, and therefore lies on the unit circle in the complex plane. We denote a dominant eigenvalue at the stability boundary as λ b. The location of λ b on the unit circle gives rise to different types of bifurcation and determines the chatter frequencies arising in cases where the cutting process is unstable. In Fig. 5, we illustrate three bifurcation types: Imλ b = : In this case, as shown in Fig. 5a, the eigenvalues escape the unit circle in the form of a complex pair. This situation is known as a secondary Hopf or Neimark Sacker bifurcation. The expression for the corresponding chatter frequencies is given as follows: f = { ± γ + n2π T }, n =, 1, where γ = Imλ b, T = 2π R p / Ω, and R p and Ω are as defined previously. λ b = 1: This situation is known as a period-one bifurcation, where the eigenvalue escapes the unit circle at 1, as shown in Fig. 5b. In this case, the chatter frequencies are given as follows: f = n2π, n =, 1, T λ b = 1: In this case, the eigenvalue escapes the unit circle at 1. This situation is known as a period-doubling or flip bifurcation, and is shown in Fig. 5c. The chatter frequencies are calculated as follows: f = { π + n2π T }, n =, 1, The chatter frequencies f are generally compared with the modulation frequencies f m : f m = n2π, n =, 1, T Thus, in the case of a period-one bifurcation, the chatter frequencies 46 are exactly equal to the modulation frequencies 48. The bifurcation and chatter frequency diagrams of 4 are shown in Figs. 6 and 7, respectively. In Fig. 6, we use the following parameter values: ζ =.2, R P = 5, and R A =.1. In Fig. 6a, we observe that Ω =.7 results in a secondary Hopf or Neimark Sacker bifurcation because the dominant eigenvalues cross the unit circle in the form of a complex pair. In Fig. 6b, we find that Ω =.675 results in a periodone bifurcation because the dominant eigenvalues cross the unit circle at 1. In Fig. 6c, we find that we obtain a period-doubling or flip bifurcation when Ω = because, in this case, the dominant eigenvalues cross

11 Galerkin Arnoldi algorithm for stability analysis 193 a b c Fig. 6 Bifurcation diagrams for the single-degree-of-freedom turning model 4 obtained using the Galerkin Arnoldi method: a Ω =.7; b Ω =.675; and c Ω = The unit circle is shown in blue; the dominant eigenvalues are indicated with red dots. Note that the eigenvalues at λ = 1ina andc are spurious. Color figure online Fig. 7 Chatter frequency diagrams for the single-degree-of-freedom turning model 4 obtained using the Galerkin Arnoldi method: a ζ =.2, R P = 2, and R A =.1; and b ζ =.2, R P = 5, and R A =.1. Dashed lines indicate the modulation frequencies 48 a f b f Ω Ω the unit circle at 1. Here, the motion is periodic with period 2 T. Finally, the chatter frequency diagrams of 4 are plotted in Fig. 7, where the spindle speed Ω is varied. We find that these figures are in very close agreement with the results reported by Insperger and Stépán [1]. Based on the results of Fis. 6 and 7, we can be confident that the Galerkin Arnoldi method is able to accurately capture bifurcations at the stability boundaries. 4 Conclusions We have established the utility of applying the Galerkin method with the Arnoldi algorithm for studying the stability of time-periodic DDEs. The original timeperiodic DDEs are first expressed as PDEs and are then converted into a system of ODEs using the Galerkin method. Shifted Legendre polynomials are used as the basis functions in the Galerkin approximation, and the boundary condition is incorporated using the tau method. We then use a modified version of the Arnoldi algoritho determine the dominant eigenvalues of the Floquet transition matrix. Five numerical examples demonstrate that the Galerkin Arnoldi algorithm is capable of establishing the stability of time-periodic DDEs, and does so substantially more efficiently than classical Floquet theory. The Galerkin Arnoldi method is particularly useful when many terms are required in the Galerkin approximation to achieve convergence, in which case evaluating the entire Floquet transition matrix can be prohibitively expensive. Acknowledgments C.P.V. gratefully acknowledges the Department of Science and Technology for funding this research through the Fast Track Scheme for Young Scientists Ref:SB/ FTP/ETA-462/212. Funding This research was funded by the Department of Science and Technology through the Fast Track Scheme for Young Scientists Ref:SB/FTP/ETA-462/212.

12 194 Z. Ahsan et al. Compliance with ethical standards The authors declare that they have no con- Conflict of interest flict of interest. References 1. Arnoldi, W.E.: The principle of minimized iterations in the solution of the matrix eigenvalue problem. Q. Appl. Math. 9, Balachandran, B.: Nonlinear dynamics of milling processes. Philos. Trans. R. Soc. A , Bauchau, O.A., Nikishkov, Y.G.: An implicit Floquet analysis for rotorcraft stability evaluation. J. Am. Helicopter Soc. 463, Bocharov, G.A., Rihan, F.A.: Numerical modelling in biosciences using delay differential equations. J. Comput. Appl. Math , Golub, G.H., Van Loan, C.F.: Matrix Computations, 4th edn. Johns Hopkins University Press, Baltimore Insperger, T., Stépán, G.: Stability of the milling process. Period. Polytech. Mech. Eng. 441, Insperger, T., Stépán, G.: Semi-discretization method for delayed systems. Int. J. Numer. Methods Eng. 555, Insperger, T., Stépán, G.: Stability chart for the delayed Mathieu equation. Proc. R. Soc. A , Insperger, T., Stépán, G.: Updated semi-discretization method for periodic delay-differential equations with discrete delay. Int. J. Numer. Methods Eng. 611, Insperger, T., Stépán, G.: Stability analysis of turning with periodic spindle speed modulation via semidiscretization. J. Vib. Control 112, Insperger, T., Stépán, G.: Semi-Discretization for Time- Delay Systems. Springer, Berlin Koto, T.: Method of lines approximations of delay differential equations. Comput. Math. Appl , Kuang, Y.: Delay Differential Equations: With Applications in Population Dynamics. Academic Press, San Diego Long, X., Insperger, T., Balachandran, B.: Systems with periodic coefficients and periodically varying delays: semidiscretization-based stability analysis. In: Balachandran, B., Kalmár-Nagy, T., Gilsinn, D.E. eds. Delay Differential Equations, pp Springer, Berlin Maset, S.: Numerical solution of retarded functional differential equations as abstract Cauchy problems. J. Comput. Appl. Math. 1612, Richard, J.P.: Time-delay systems: an overview of some recent advances and open problems. Automatica 391, Saad, Y.: Krylov subspace methods for solving large unsymmetric linear systems. Math. Comput , Sadath, A., Vyasarayani, C.P.: Galerkin approximations for stability of delay differential equations with time periodic coefficients. ASME J. Comput. Nonlinear Dyn. 12, Sadath, A., Vyasarayani, C.P.: Galerkin approximations for stability of delay differential equations with time periodic delays. ASME J. Comput. Nonlinear Dyn. 16, Safonov, L.A., Tomer, E., Strygin, V.V., Ashkenazy, Y., Havlin, S.: Multifractal chaotic attractors in a system of delay-differential equations modeling road traffic. Chaos 124, Segalman, D.J., Butcher, E.A.: Suppression of regenerative chatter via impedance modulation. J. Vib. Control 62, Vyasarayani, C.P.: Galerkin approximations for higher order delay differential equations. ASME J. Comput. Nonlinear Dyn. 73, Wahi, P., Chatterjee, A.: Galerkin projections for delay differential equations. ASME J. Dyn. Syst. Meas. Control 1271,