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Let ¯y(t) denote a known equilibrium or periodic solution of period KT (K = 1,2,. .... It is seen from Equation (11) that the nonlinear terms of degree 2 and 3 can ..... It should be noted that the 3T and 4T curves are sets of zero measure similar to ...
Nonlinear Dynamics 23: 35–55, 2000. © 2000 Kluwer Academic Publishers. Printed in the Netherlands.

Normal Forms and the Structure of Resonance Sets in Nonlinear Time-Periodic Systems ERIC A. BUTCHER Department of Mechanical Engineering, University of Alaska Fairbanks, Fairbanks, AK 99775, U.S.A.

S. C. SINHA Nonlinear Systems Research Laboratories, Department of Mechanical Engineering, Auburn University, Auburn, AL 36849, U.S.A. (Received: 16 July 1999; accepted: 6 December 1999) Abstract. The structure of time-dependent resonances arising in the method of time-dependent normal forms (TDNF) for one and two-degrees-of-freedom nonlinear systems with time-periodic coefficients is investigated. For this purpose, the Liapunov–Floquet (L–F) transformation is employed to transform the periodic variational equations into an equivalent form in which the linear system matrix is time-invariant. Both quadratic and cubic nonlinearities are investigated and the associated normal forms are presented. Also, higher-order resonances for the single-degree-of-freedom case are discussed. It is demonstrated that resonances occur when the values of the Floquet multipliers result in MT -periodic (M = 1, 2, . . .) solutions. The discussion is limited to the Hamiltonian case (which encompasses all possible resonances for one-degree-of-freedom). Furthermore, it is also shown how a recent symbolic algorithm for computing stability and bifurcation boundaries for time-periodic systems may also be employed to compute the time-dependent resonance sets of zero measure in the parameter space. Unlike classical asymptotic techniques, this method is free from any small parameter restriction on the time-periodic term in the computation of the resonance sets. Two illustrative examples (one and two-degrees-of-freedom) are included. Keywords: Normal forms, time-periodic systems, Liapunov–Floquet transformation.

1. Introduction In the study of dynamical systems represented by nonlinear ordinary differential equations with periodic coefficients, a set of quasilinear equations generally arise when the system is expanded in a Taylor series about an equilibrium or a periodic motion of the system. The stability and bifurcation conditions are determined from the eigenvalues of the linearized system’s fundamental solution matrix evaluated at the end of the principal period (called the Floquet Transition Matrix or FTM). In order to obtain the stability of critical cases and the nonlinear response in general, the Liapunov–Floquet (L–F) transformation [1–3] may be used to transform the system into an equivalent one in which the linear system matrix is timeinvariant. Subsequently, the method of time-dependent normal forms may be employed as shown by Arnold [4] in order to reduce the equations to their simplest possible forms. The required normal form computations for one and two-degrees-of-freedom time-periodic nonlinear systems were carried out by Pandiyan and Sinha [5] who showed that timeindependent resonances in the corresponding solvability condition translate into the fact that certain time-invariant nonlinear terms must remain in the simplified equations. The case of time-dependent resonances (which imply that corresponding time-varying nonlinearities

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E. A. Butcher and S. C. Sinha

remain in the normal form) were not present in that work. Unlike their time-independent counterparts which are always present for various nonlinearities, these resonances occur only under special conditions which are represented as a set of zero measure in the parameter space similar to the stability boundaries. The general normal form for the case of time-dependent resonances in a single-degree-of-freedom system has previously been presented [6], and normal forms for multi-dimensional systems with weak parametric excitation have also been shown [7]. However, the constraint sets in the parameter space in which the resonances occur have not been explicitly derived to our knowledge. Especially, a constructive technique to obtain the time-dependent normal forms and the corresponding resonance sets in the parameter space for multi-dimensional systems with arbitrarily large parameter values (strong parametric excitation) has been lacking. Since the time-dependent resonance sets in the parameter space are similar in nature to the much-discussed stability boundaries which border the regions of parametric resonance, it is helpful to first examine some of the techniques used to obtain these boundaries in closed form. Two common asymptotic methods that have been used in the past to yield approximate stability boundaries in closed form for small systems include the perturbation method [8] and the averaging technique [9]. It is well known that these methods are limited in application to systems with weak internal excitation since they are based on expanding the solution in terms of a small parameter that multiplies the time-periodic terms. When the excitation is strong, the time-periodic parameter is large and the asymptotic series diverges. Another technique to approximate the parameter-dependent FTM (and hence derive the stability boundaries in closed form), which has been developed recently, involves the use of Picard iteration and Chebyshev polynomial expansion [10]. This method is easily implemented using a symbolic manipulator such as MATHEMATICA since it involves only simple matrix multiplications and additions. The subsequent use of well-known criteria for the local stability and bifurcation conditions of equilibria and periodic solutions enables one to obtain the equations for the bifurcation surfaces in the parameter space. These equations are expressed as homogeneous polynomials of all the system parameters and can be solved for one parameter once values of the others have been chosen. Since this method does not involve expansion in terms of a single small parameter, all parameters are treated equally and the technique can successfully be applied to periodic systems with strong internal excitations. The primary bifurcations in a parametrically excited simple pendulum and a double inverted pendulum subjected to a periodic follower force were subsequently analyzed via this procedure [11]. It was further shown that the secondary bifurcations of a parametrically excited simple pendulum could also be computed [12]. In this paper, the structure of time-dependent resonances and their corresponding parameter sets for one and two-degrees-of-freedom time-periodic systems with both quadratic and cubic nonlinearities are analyzed subsequent to employing the L–F transformation. It is shown that resonances occur when the locations of the Floquet multipliers result in MT -periodic (M = 1, 2, . . .) solutions, and higher-order resonances for the single-degree-of-freedom case are discussed. It is also shown how the parameter sets where these resonances occur may be computed symbolically using the same computational procedure as used previously in the analysis of local stability and bifurcation boundaries [11, 12] as discussed above. The examples of a nonlinear Mathieu equation and a double inverted pendulum subjected to a periodic follower force are employed. For each example, the time-dependent resonance sets are shown along with the standard stability/bifurcation boundaries in the parameter space and the normal forms corresponding to these resonance sets are given explicitly.

Resonance Sets in Nonlinear Time-Periodic Systems

37

2. Simplification via TDNF Consider a dynamical system described by the set of N nonlinear ordinary differential equations y˙ (t) = f(y(t), t, α) = f(y(t), t + T , α),

y(0) = y0 ,

(1)

where t ∈ R+ denotes time, y ∈ RN is the state vector, α ∈ RL is a vector of control parameters, and f : R+ × RL → RN is analytic in the components of y and α and periodic in t with period T . Let y¯ (t) denote a known equilibrium or periodic solution of period KT (K = 1, 2, . . .) of Equation (1) such that the perturbation x(t) about this position can be defined as y(t) = y¯ (t) + x(t).

(2)

Substituting Equation (2) in Equation (1) and expanding the right-hand side in Taylor series about y¯ (t) yields x˙ = A(t, α)x + f2 (x, t, α) + f3 (x, t, α) + · · · + fk (x, t, α) + O(|x|k+1 ), x(0) = x0 = y0 − y¯ (0),

(3)

where fk ( ) contain homogeneous monomials in xi of order k and A(t, α), and fk (x, t, α) are periodic in t with period KT . From Floquet theory, the fundamental solution matrix 8(t, α) of the linear part of Equation (3) evaluated at the end of the principal period is the FTM H(α) = 8(KT , α) which determines the local stability of Equation (1) in the neighborhood of y¯ (t). The conditions for local bifurcations of y¯ (t) in terms of H(α) are discussed elsewhere [11, 12]. For present purposes, it suffices to state that if complex conjugate Floquet multipliers (eigenvalues of H(α)) satisfy µi , µ∗i = exp(±i2π n/M) for some n, M ∈ N which do not have a common factor, then a bifurcation to a period MKT solution may occur. The conditions on the control parameters for the existence of period MKT solutions may be expressed as det(I − HM (α)) = 0 or alternately as [13] det(I + HM/2 (α)) = 0,

M even, M/2 even;

det(I − H(α) + H2 (α) − H3 (α) . . . + HM/2−1 (α)) = 0, det(I + H(α) + H2 (α) + H3 (α) . . . + HM−1 (α)) = 0,

M even, M/2 odd; M odd.

(4)

As discussed by Pandiyan and Sinha [5], a direct application of normal form theory to Equation (3) is not possible due to the fact that A(t, α) is time-dependent. It was suggested in that paper, however, that for a given parameter set α 0 the KT -periodic L–F transformation x(t) = L(t)z(t)

(5)

be employed to reduce Equation (3) to z˙ = Cz + L−1 (t)F(L(t)z, t)

(6)

in which the (generally complex) linear system matrix C is time-invariant and F( ) = f2 ( ) + f3 ( ). Here the nonlinear terms of order four and higher have been neglected since they have no influence in generic codimension 1 bifurcations [4]. Moreover, it was shown

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that the 2KT -periodic transformation matrix Q(t) produces a real linear system matrix R. The eigenvalues of C (called the characteristic exponents) or R determine the local stability in all hyperbolic (noncritical) cases. Specifically, the eigenvalues λi = (ζi + ξi∗ i)/KT of C are related to the Floquet multipliers µi as ζi = ln |µi |, ξi = Arg(µi ) where ζi < 0, i = 1, 2, . . . , N for asymptotic stability. The eigenvalues of R are similarly found except that ξi = Arctan(Im(µi )/Re(µi )) so that they are equivalent to the characteristic exponents when Re(µi ) > 0. Thus if all of the multipliers are in the right half plane then the two transformations are identical since L(t, α) = Q(t, α) is KT -periodic. The form of Equation (6) is amenable to direct application of the method of time-dependent normal forms (TDNF) for equations with periodic coefficients as shown by Arnold [4]. Techniques for computing the L–F transformation can be found in [1–3]. Equation (6) may be transformed via the modal transformation z = My

(7)

y˙ = 3y + M−1 L−1 (t)F(Q(t)My, t) = 3y + w2 (y, t) + w3 (y, t),

(8)

to

where 3 is a diagonal matrix of the eigenvalues of C (or R) and wk (y, t) are KT - (or 2KT -) periodic functions containing homogeneous monomials of yi of order 2 and 3. Equation (8) can be reduced to its simplest (normal) form using a sequence of near identity transformations of the form y = ν + h(ν, t),

(9)

where h(ν, t) = h2 (ν, t) + h3 (ν, t) is a formal power series in ν of degree r (r = 2, 3) with KT - (or 2KT -) periodic coefficients. (Since we are restricting consideration to codimension one bifurcations, non-semisimple eigenvalues are not allowed so that the diagonal Jacobian matrix in Equation (8) decouples the bases of the above nonlinear transformation. However, for a general singular vector field with non-semisimple eigenvalues, the resulting coupling of the bases makes the computation of the normal forms more complicated [14].) Applying the above change of coordinates, Equation (8) takes the form !  2  ∂h ∂h ∂h ν˙ = I − − ··· 3ν + 3h − + + W(ν + h(ν, t), t) , (10) ∂ν ∂ν ∂t where the inversion of [I + ∂hr /∂ν] in a neighborhood of ν = 0 has been used and W(y, t) = w2 (y, t) + w3 (y, t). Expanding Equation (10) and collecting like powers of |ν| results in     ∂h2 ∂h3 ν˙ = 3ν − LA (h2 ) + + g2 (ν, t) − LA (h3 ) + + g3 (ν, t) + O(|ν|4 ), (11) ∂t ∂t where LA (hr ) =

∂hr 3ν − 3hr , ∂ν

r = 2, 3

(12)

is the Lie operator which carries vector polynomials over to vector polynomials of the same degree and g2 (ν, t) = w2 (ν, t)

Resonance Sets in Nonlinear Time-Periodic Systems g3 (ν, t) = w3 (ν, t) +  +

∂h2 ∂ν

39

∂h2 ∂h2 ∂h2 ∂h2 − w2 (ν, t) − 3h2 ∂ν ∂t ∂ν ∂ν

2 3ν +

∂w2 (ν, t)h2 (ν, t) ∂y

(13)

in which the additional cubic terms above depend on the quadratic part h2 of the transformation in Equation (9). (Note that the quadratic term in the inversion of [I + ∂hr /∂ν] is present in one of these additional cubic terms.) It is seen from Equation (11) that the nonlinear terms of degree 2 and 3 can be eliminated if the homological equations LA (hr ) +

∂hr = g2 (ν, t), ∂t

r = 2, 3

(14)

are satisfied. These equations may thus be solved sequentially for hr , r = 2, 3 if the set of eigenvalues of (LA + ∂/∂t) does not contain any zeros (i.e. LA + ∂/∂t is invertible). For this purpose, the known wr (y, t) and unknown hr (y, t) monomials are expressed in a Fourier series as gr (ν, t) =

∞ N X X X l=1

hr (ν, t) =

m

n=−∞

∞ N X X X l=1

m

gl,m,n eint |ν|m el

hl,m,n eint |ν|m el

(15)

n=−∞

where N X

m = (m1 , m2 , . . . , mN );  =

π KT

 or

2π KT

mi = r;

r = 2, 3;

i=1

 ;

|ν|m = ν1m1 ν2m2 . . . νNmN ;

and el is the lth member of the natural basis. In practice the higher harmonics are neglected so that n runs from −q to q for some positive integer q. Substituting Equation (15) into Equation (14) yields the solvability condition hl,m,n =

gl,m,n , in + m · λ − λl

(16)

where λ = (λ1 , λ2 , . . . , λN ). Since the number of different possible vectors m for a given nonlinear degree r may be expressed as   (N + r − 1)! N +r −1 = , r r! (N − 1)! the number of unknowns hl,m,n which must be solved for in this way is (2q + 1)N(N + r − 1)! . r! (N − 1)!

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It is clear that when the denominator in Equation (16) is nonzero for all l, m, n that hr (ν, t), r = 2, 3 may be obtained so that Equation (8) can be reduced to a linear form. On the other hand, if in + m · λ − λl = 0

(17)

for some l, m, n then the corresponding resonant term must remain and Equation (8) takes the simplest possible nonlinear normal form. It was shown by Pandiyan and Sinha [5] that time-independent resonances (corresponding to Equation (17) with n = 0) always occur for certain types of eigenvalue arrangements such as a purely imaginary pair. In this study it is further shown that time-dependent resonances (corresponding to Equation (17) with nonzero n) also occur in these situations for certain sets of the system parameters which may be obtained analytically once the parameter-dependent FTM H(α) is known. Hence, these timedependent resonance sets can be symbolically evaluated using procedures similar to those used previously [11, 12] in determining the local stability and bifurcation boundaries. 3. TDNF for One-Degree-of-Freedom Systems Following the procedure in Section 2, the KT- or 2KT -periodic (henceforth called T - or 2T periodic) L–F transformation is applied to the two-dimensional state space form of a singledegree-of-freedom system. After using the modal transformation of Equation (7), the resulting system is of the form of Equation (8), namely !      w1,(2,0)(t)y12 + w1,(1,1)(t)y1 y2 + w1,(0,2)(t)y22 y˙1 λ1 0 y1 = + 0 λ2 y˙2 y2 w2,(2,0)(t)y12 + w2,(1,1)(t)y1 y2 + w2,(0,2)(t)y22 +

w1,(3,0)(t)y13 + w1,(2,1)(t)y12 y2 + w1,(1,2)(t)y1 y22 + w1,(0,3)(t)y23 w2(3,0)(t)y13 + w2,(2,1)(t)y12 y2 + w2,(1,2)(t)y1 y22 + w2,(0,3)(t)y23

! (18)

in which the constant linear system matrix contains the two eigenvalues of C or R on the diagonal and the quadratic and cubic wl,m (t) terms are Fourier-expanded T - or 2T -periodic coefficients. The near identity transformation !     h1,(2,0)(t)ν12 + h1,(1,1)(t)ν1 ν2 + h1,(0,2)(t)ν22 y1 ν1 = + y2 ν2 h2,(2,0)(t)ν12 + h2,(1,1)(t)ν1 ν2 + h2,(0,2)(t)ν22 ! h1,(3,0)(t)ν13 + h1,(2,1)(t)ν12 ν2 + h1,(1,2)(t)ν1 ν22 + h1,(0,3)(t)ν23 × (19) h2,(3,0)(t)ν13 + h2,(2,1)(t)ν12 ν2 + h2,(1,2)(t)ν1 ν22 + h2,(0,3)(t)ν23 to remove the cubic terms in Equation (18) results in the homological equations   ∂ LA + (h2 ) = g2 (ν, t), ∂t   ∂ LA + (h3 ) = g3 (ν, t), ∂t

(20)

where gi (ν, t) are given in Equation (13) and gi and hi are Fourier-expanded as in Equation (15). Quadratic and cubic terms in Equation (18) may be removed as long as Equation (20)

Resonance Sets in Nonlinear Time-Periodic Systems

41

Table 1. Resonances for cubic and quadratic nonlinearities using time-dependent normal forms with a stable Hamiltonian one-degree-of-freedom problem. Time-independent resonances are indicated where n = 0 while time-dependent resonances result when n = ±1.

can be solved for the 12q + 6 (for r = 2) or 16q + 8 (for r = 3) unknown Fourier coefficients hl,m,n where n runs from −q to q. However, terms of the form wl,m,n eint νlm1 ν2m2 el remain when the corresponding resonance condition in + m1 λ1 + m2 λ2 − λl = 0

(21)

is satisfied where m1 + m2 = 3,  = π/T (or 2π/T ), and el is the lth member of the natural basis. As discussed in Section 2, Pandiyan and Sinha [5] showed that time-independent (n = 0) resonances always occur for certain types of eigenvalue arrangements such as a purely imaginary pair. The problem considered here, however, is what eigenvalue pairs satisfy Equation (21) for n 6 = 0. It can be seen that Equation (21) has no n 6 = 0 solutions for real λ1,2 . In addition, there are no n 6 = 0 solutions for complex conjugate eigenvalues (with nonzero real part) since the real part of Equation (21) results in m1 + m2 = 1. Therefore, the two eigenvalues are restricted to be purely imaginary pairs of the form λ1,2 = ±iν which implies that only the (marginally) stable Hamiltonian case is relevant. Equation (21) thus becomes n = pν;

p = m2 − m1 ± 1,

(22)

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where (+1) corresponds to l = 1 and (−1) corresponds to l = 2. Table 1 lists all possible (l, m1 , m2 ) combinations and the resulting p value from Equation (22) for both the quadratic (r = 2) and cubic (r = 3) cases. It can be seen in this table that there are exactly six (for r = 2) and eight (for r = 3) different combinations which yield different values of p from -4 to 4. It should be noted that only odd values of p occur for r = 2 while only even ones exist for r = 3. The two cubic cases which result in p = 0 correspond to the time-independent (n = 0) resonances which are parameter-independent since λ1 + λ2 = 0 is always satisfied for the Hamiltonian case. Without loss of generality the frequency is set to  = 2 from which the magnitude ν of the characteristic exponents shall be determined. This corresponds to using a π -periodic L–F transformation. Two of the (l, m1 , m2 ) combinations for r = 2 are repeated at the bottom where the frequency has been changed to  = 1. This is done because the corresponding normal forms depend on whether the T - or 2T -periodic L–F transformation is used. For the case of cubic terms, however, there are no independent 2T -periodic transformations which result in a time-dependent resonance. The magnitude ν of the imaginary characteristic exponents λ1,2 is restricted to satisfy T ν < π (see Section 2), and this can be rearranged as ν < /2. Multiplying this inequality by |p| and using Equation (22) results in |n|
3) nonlinearities in a single-degree-of-freedom problem is considered. Essentially, for each nonlinear degree r the resonance condition (Equation (21)) is satisfied for nonzero n when the solution is (r + 1)T -periodic. The structure of these resonance sets becomes increasingly more complicated, however. Figure 2 shows the locations of the Floquet multipliers µi on the unit circle in the complex plane for MT -periodic solutions where M = 1 (tangent bifurcation), 2 (period-doubling bifurcation), and 3, . . . , 8 corresponding to time-dependent resonances for nonlinearities of degree r up to 7. The angles for these locations are given in Table 2 by Arg(µi ) for both T- and 2T -periodic (if distinct) L–F transformations along with the resulting linear eigenvalues of C or R which correspond to time-dependent resonances (where T = π ). It should be remembered that the 2T transformation is distinct if Re(µi ) < 0 or |Arg(µi )| > π/2. It can be seen in Figure 2 and Table 2 that, beginning with M = 5, there is more than one possible location for the Floquet multipliers for a given M and r. In addition, some locations for higher M values are identical to those for lower M values so that resonance sets for multiple monomial degrees occur simultaneously for a given parameter set. In the last column of Table 2, the exact b = 0 crossings on the a-axis of the Mathieu equation stability chart for these periodic solutions are given up to a = 9 (the second period-

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Figure 2. Locations of the Floquet multipliers on the unit circle for MT -periodic solutions.

doubling intersection). If all of these crossing locations are arranged in a single list ordered from lowest to highest, then the resulting M-sequence formed by writing the corresponding M values next to the intersection locations is identical to that obtained from the ordered list of MT locations that the Floquet multipliers encounter as they traverse the unit circle in Figure 2. As this circle is initially traversed from +1 to −1 the M sequence is seen to be 1, 8, 7, 6, 5, 4(8), 7, 3(6), 8, 5, 7, 2 while this sequence is reversed as the multipliers retrace their path back to +1. This pattern continues indefinitely as a is increased. The entire curves in the (a, b) parameter space through these crossings are similar to those for 3T and 4T solutions and can be computed from the FTM using the appropriate relation in Equation (4). It can thus be seen that, although each resonance set has zero measure in the parameter space, when all (infinite) monomial degrees are simultaneously considered, the stable region of the parameter plane is foliated with such sets with any random location occurring arbitrarily close to some resonance set. Similar to the structure of the rational numbers, in any given interval (say a ∈ [0, 1]) there is a countable infinity of such time-dependent resonance sets. Also, it should be noted that the perturbation method [8] could also be used to compute the period MT curves in this problem since the a-axis crossings are easily determined analytically; however, the results would not be accurate for large b. 4. TDNF for Two-Degrees-of-Freedom Systems As before, the T-periodic or 2T -periodic L–F transformation is performed on the fourdimensional state space form. After applying the modal transformation of Equation (7) and expanding the periodic coefficients of the nonlinear terms in Fourier series, the resulting

Resonance Sets in Nonlinear Time-Periodic Systems

47

Table 2. Locations of period MT Floquet multipliers on the unit circle and the corresponding constant eigenvalues for time-dependent resonances. The b = 0 crossings on the Mathieu equation stability chart are also given out to a = 9.

system is of the form    y˙1 λ1 0  y˙2   0 λ2   =   y˙3  0 0 0 0 y˙4

0 0 λ3 0

 P3 r=2

 y1 0  y2 0   0   y3 λ4 y4 P

   

P∞ m

int m1 m2 m3 m4 y1 y2 y3 y4 n=−∞ w1,m,n e

  P3 P P∞ int m1 m2 m3 m4  r=2 y1 y2 y3 y4 m n=−∞ w2,m,n e  +P  3 P P∞ int m1 m2 m3 m4 y1 y2 y3 y4  r=2 m n=−∞ w3,m,n e  P3 P P∞ int m1 m2 m3 m4 y1 y2 y3 y4 m r=2 n=−∞ w4,m,n e

        

(34)

in which the diagonalized constant linear system matrix contains the four eigenvalues of C or R. The above notation for the nonlinear monomials is used for convenience (instead of listing

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E. A. Butcher and S. C. Sinha

all 10 quadratic and 20 cubic terms as in Equation (18)). The corresponding near identity transformation   P 3 P P∞ int m1 m2 m3 m4 ν1 ν2 ν3 ν4 m r=2 n=−∞ h1,m,n e       y1 ν1  P3 P P∞ m1 m2 m3 m4  int ν1 ν2 ν3 ν4   y2   ν2   m r=2 n=−∞ h2,m,n e   =  + (35)    y3   ν3   P3 P P∞ int m1 m2 m3 m4  h e ν ν ν ν   r=2 3,m,n m 1 2 3 4 n=−∞ y4 ν4   P3 P P∞ int m1 m2 m3 m4 ν1 ν2 ν3 ν4 m r=2 n=−∞ h4,m,n e to remove the quadratic and cubic terms in Equation (34) results in the homological equations in Equation (20). Quadratic and cubic terms in Equation (34) may be removed as long as Equation (20) can be solved for the 80q + 40 (r = 2) or 160q + 80 (r = 3) unknown Fourier coefficients hl,m,n where n runs from −q to q. However, in this case terms of the form wl,m,n eint ν1m1 ν2m2 ν3m3 ν4m4 el remain when the corresponding resonance condition in + m1 λ1 + m2 λ2 + m3 λ3 + m4 λ4 − λl = 0

(36)

is satisfied where m1 + m2 + m3 + m4 = r,  = π/T or 2π/T , l = 1, . . . 4, and el is the lth member of the natural basis. As with the single-degree-of-freedom system, time-independent (n = 0) cubic resonances occur for certain types of eigenvalue arrangements such as a purely imaginary pair. Obviously, at least one complex or imaginary pair is needed for the existence of a time-dependent (n 6 = 0) resonance; hence, the case of all real λ1,2,3,4 can be discarded. In order to utilize previous results, we only consider the Hamiltonian case which results in resonances that are similar in structure to those obtained in Section 3. Specifically, we restrict the problem to the case of at least one imaginary pair λ1,2 = ±iν while the remaining pair must, due to the Hamiltonian property, satisfy λ3 + λ4 = 0 or λ3,4 = ±κ where κ is either real or imaginary. Multiplying by −i and rearranging, Equation (36) thus becomes   q1,2 = m4 − m3   p1 = m2 − m1 + 1; p2 = m2 − m1 − 1; q3 = m4 − m3 + 1 . n = pl ν − ql κi; (37)   p3,4 = m2 − m1 ; q4 = m4 − m3 − 1 It can be seen that if ql = 0 for l = 1 or 2, then Equation (37) reduces to Equation (22). Tables 3 and 4 list in an abbreviated form all possible combinations of l, p, and q for cubic and quadratic terms, respectively. As before, the T -periodic L–F transformation has been assumed in all cases except the last two rows of Table 4 in which the distinct 2T -periodic transformation is again applied to the two quadratic time-dependent resonance combinations. The eight arrangements which yield zero values of both p and q (indicated by two bold zeros in the appropriate columns of Table 3) correspond to the time-independent (n = 0) resonances, for which there are two for each given l in the cubic case and none for the quadratic case. In the other cases with zero q, Equations (23) and (24) from Section 3 may again be used to obtain the values of ν and M corresponding to the time-dependent resonances at these points. Consequently, the four resonances which are shown (two for cubic and two for quadratic – again indicated by two bold numbers in the appropriate columns) have the same values of ν and M as in Section 3. The locations in the parameter plane where the time-dependent resonances occur are thus given in terms of the parameter-dependent FTM by Equation (25).

Resonance Sets in Nonlinear Time-Periodic Systems

49

Table 3. Resonances for cubic nonlinearities using time-dependent normal forms with a Hamiltonian two-degrees-of-freedom problem. Time-independent resonances are indicated where n = 0 while time-dependent resonances result when n = ±1. The appropriate values of p and q are in bold print.

The resulting normal forms can be seen from Tables 3 and 4 as ν˙ 1 =

1 iν1 + w1,(2,1,0,0),0ν12 ν2 + w1,(1,0,1,1),0ν1 ν3 ν4 + w1,(0,3,0,0),1 ei2t ν23 , 2

1 ν˙ 2 = − iν2 + w2,(1,2,0,0),0ν1 ν22 + w2,(0,1,1,1),0ν2 ν3 ν4 + w2,(3,0,0,0),−1 e−i2t ν13 , 2 ν˙ 3 = µν3 + w3,(0,0,2,1),0ν32 ν4 + w3,(1,1,1,0),0ν1 ν2 ν3 , ν˙ 4 = −µν4 + w4,(0,0,1,2),0ν3 ν42 + w4,(1,1,0,1),0ν1 ν2 ν4 ,

(38)

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E. A. Butcher and S. C. Sinha

Table 4. Resonances for quadratic nonlinearities using time-dependent normal forms with a Hamiltonian two-degrees-of-freedom problem. Time-independent resonances are indicated where n = 0 while time-dependent resonances result when n = ±1. The appropriate values of p and q are in bold print.

for the cubic case (in which the pair of Floquet multipliers corresponding to the resonance is µ1,2 = ±i) and ν˙ 1 =

2 iν1 + w1,(0,2,0,0),1 ei2t ν22 , 3

2 ν˙ 2 = − iν2 + w2,(2,0,0,0),−1 e−i2t ν12 , 3 ν˙ 3 = µν3 , ν˙ 4 = −µν4 ,

(39)

for the quadratic case using the T -periodic transformation (in which the resonant multipliers are µ1,2 = −0.5 ± 0.866i). As with the single-degree-of-freedom problem in Section 3, an independent 2T transformation and corresponding normal form ν˙ 1 =

1 iν1 + w˜ 1,(0,2,0,0),1 eit ν22 , 3

1 ν˙ 2 = − iν2 + w˜ 2,(2,0,0,0),−1 e−it ν12 , 3 ν˙ 3 = µν3 , ν˙ 4 = −µν4 ,

(40)

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51

Figure 3. A double inverted pendulum subjected to a periodic follower force.

exists for the quadratic case, but again this is equivalent to Equation (39) by a change in timescale in the first two coupled equations and become linear when the solution is not 3T periodic because of the absence of time-independent resonances. 4.1. E XAMPLE : A D OUBLE I NVERTED P ENDULUM F ORCE

WITH A

P ERIODIC F OLLOWER

As an example of a two-degrees-of-freedom system, consider the double inverted pendulum of Figure 3 subjected to a follower force with both constant and periodically-varying components. The time-periodic equations of motion for this system may be expressed as [11] 3ϕ˙1 + cos(ϕ2 − ϕ1 )ϕ¨2 − sin(ϕ2 − ϕ1 )ϕ˙ 22 + (B1 + B2 )ϕ˙ 1 ¯ 1 − kϕ ¯ 2 − p(t) − B2 ϕ˙ 2 + 2kϕ ¯ sin(ϕ1 − γ ϕ2 ) = 0, cos(ϕ2 − ϕ1 )ϕ¨1 + ϕ¨2 + sin(ϕ2 − ϕ1 )ϕ˙ 12 − B2 ϕ˙ 1 ¯ 1 + kϕ ¯ 2 − p(t) + B2 ϕ˙ 2 − kϕ ¯ sin((1 − γ )ϕ2 ) = 0,

(41)

where k¯ = k/ml 2 is the normalized stiffness, B1 = b1 /ml 2 and B2 = b2 /ml 2 are the normalized damping constants, p(t) ¯ = (Pˆ1 + Pˆ2 cos ωt)/ml = P1 + P2 cos ωt is the normalized

52

E. A. Butcher and S. C. Sinha

Figure 4. Stability diagram for the double inverted pendulum with ω = 2, k¯ = 1, B1 = B2 = 0 and γ = 0 showing the 3T and 4T curves.

applied load, γ is the load direction parameter, and ω is the driving frequency of the applied load. Denoting xT = (x1 x2 x3 x4 ) = (ϕ1 ϕ2 ϕ˙ 1 ϕ˙ 2 ) as the state vector and expanding Equation (41) in a Taylor series about the vertical φ1 = φ2 = 0 equilibrium position, the above equation may be written in state space form as     0 0 1 0 x˙1  x˙2   0 0 0 1    =   ¯  x˙3   (−3k¯ + p(t))/2 ¯ k − p(t)/2 ¯ −B1 /2 − B2 B2  x˙4 (5k¯ − p(t))/2 ¯ −2k¯ + (3/2 − γ )p(t) ¯ B1 /2 + 2B2 −2B2     x1 0  x2    0    ×  (42)  x3  +  f33 (x, t)  , f34 (x, t) x4 where f33 (x, t) and f34 (x, t) contain terms up to the cubic order only. The following analysis is applied to the parameter set B1 = B2 = 0, k¯ = 1, and ω = 2 for two separate values of the follower angle: γ = 0.0 (vertical load) and 1.0 (tangential load). The resulting resonance sets in the (P1 , P2 ) plane are investigated. After computing the parameter-dependent FTM H(P1 , P2 ) in which each matrix element j consists of various powers of P1i P2 up to degree P = 17 such that i + j ≤ P , i =

Resonance Sets in Nonlinear Time-Periodic Systems

53

Figure 5. Stability diagram for the double inverted pendulum with ω = 2, k¯ = 1, B1 = B2 = 0 and γ = 1 showing the 3T and 4T curves.

0, . . . , P , j = 0, . . . , P , the locations of the period 3T and 4T curves were symbolically computed for the two follower angle cases using Equation (25). As was indicated in [11], the cumulative cpu time was approximately 14 hours of running MATHEMATICA on a SUN SPARC 20. The resulting parameter sets are seen in Figures 4 and 5 along with the standard stability boundaries due to tangent, period-doubling, and Krein collision (Hamiltonian Hopf) bifurcations. (See [11] for an explanation of the stability diagram including which regions are stable/unstable.) Since only cubic terms are present in Equation (42), time-dependent resonances occur only on the 4T (not 3T ) curves with the normal form of Equation (38). Furthermore, if the Taylor series expansion were carried out to fifth order, resonances would occur on curves corresponding to 6T -periodic solutions (not shown), and so forth for higher-order expansions. It can be observed that, in contrast to those computed for the single-degree-offreedom problem, the period 3T and 4T lines here may cross the stability boundaries which involve only two eigenvalues (i.e. tangent and period-doubling). For this reason, κ in Equation (37) may be either real or purely imaginary. However, it can also be seen that these lines may never cross the Krein collision stability boundary since all four eigenvalues are required to participate in this particular destabilization route. This does not exclude the possibility of time-dependent resonances within the Krein collision regions, however. In fact, for a quadruplet of complex eigenvalues λ1−4 = ±κ ± iν, Equation (37) holds where

54

E. A. Butcher and S. C. Sinha p1,3 = m2 + m4 − m1 − m3 + 1;

q1,2 = m3 + m4 − m1 − m2 + 1,

p2,4 = m2 + m4 − m1 − m3 − 1;

q3,4 = m3 + m4 − m1 − m2 − 1.

(43)

For ql = 0, a cubic time-dependent resonance occurs when ν = 1/2 and the corresponding normal form can be shown to be   1 ν˙ 1 = κ + i ν1 + w1,(2,0,0,1),0ν12 ν4 + w1,(1,1,1,0),0ν1 ν2 ν3 + w1,(0,2,0,1),1 ei2t ν22 ν4 , 2   1 ν˙ 2 = κ − i ν2 + w2,(0,2,1,0),0ν22 ν3 + w2,(1,1,0,1),0ν1 ν2 ν4 + w2,(2,0,1,0),−1 e−i2t ν12 ν3 , 2   1 ν˙ 3 = −κ + i ν3 + w3,(0,1,2,0),0ν2 ν32 + w3,(1,0,1,1),0ν1 ν3 ν4 + w3,(0,1,0,2),1 ei2t ν2 ν42 , 2   1 ν˙ 4 = −κ − i ν4 + w4,(1,0,0,2),0ν1 ν42 + w4,(0,1,1,1),0ν2 ν3 ν4 + w4,(1,0,2,0),−1 e−i2t ν1 ν32(.44) 2 Since the value of κ is irrelevant, it is apparent that only the phase (not the magnitude) of the Floquet multipliers determines if time-dependent resonances occur in this case. However, since these do not correspond to MT -periodic solutions, the present computational method cannot specify the corresponding parameter sets from the FTM. It should also be noted that additional resonances for the Hamiltonian case are possible. For instance, if the magnitudes of two pairs of imaginary eigenvalues are rationally related, time-dependent resonances with |n| > 1 may occur. Obviously, these and other combinations must be investigated on a caseby-case basis. A complete list of all possibilities for two-degrees-of-freedom (including timedependent resonances for higher-order nonlinearities and/or non-semisimple eigenvalues) is not pursued here, but would be a topic for further investigation. 5. Conclusions The structure of time-dependent resonances present in the method of time-dependent normal forms for one and two-degrees-of-freedom nonlinear systems with time-periodic coefficients has been investigated. First, the Liapunov–Floquet (L–F) transformation was employed to transform the periodic variational equation into an equivalent form in which the linear system matrix is constant. Both quadratic and cubic nonlinearities were investigated and the associated normal forms were explicitly obtained. Higher-order resonances for the singledegree-of-freedom case were briefly discussed. Since resonances occur when the Floquet multipliers result in MT -periodic solutions, the discussion was limited to the Hamiltonian case (which encompassed all possible resonances for one-degree-of-freedom). The symbolic algorithm for computing stability and bifurcation boundaries for time-periodic systems was also employed to compute the associated parameter sets of zero measure in which the resonances occur. Unlike classical asymptotic techniques, this method has been shown to be free from any small parameter restriction on the time-periodic term and thus found to be accurate for large excitations. The examples of a nonlinear Mathieu equation and a double inverted pendulum subjected to a periodic follower force were employed. In each case, the timedependent resonance sets were shown along with the standard stability/bifurcation boundaries

Resonance Sets in Nonlinear Time-Periodic Systems

55

in the parameter space and the normal forms corresponding to these resonance sets were given explicitly. Acknowledgment Partial financial support for one author (EB) from the Arctic Region Supercomputing Center (ARSC) is gratefully acknowledged. References 1.

Sinha, S. C., Pandiyan, R., and Bibb, J. S., ‘Liapunov–Floquet transformation: Computation and applications to periodic systems’, Journal of Vibration and Acoustics 118, 1996, 209–219. 2. Sinha, S. C. and Pandiyan, R., ‘Analysis of quasilinear dynamical systems with periodic coefficients via Liapunov–Floquet transformation’, International Journal of Non-Linear Mechanics 29, 1994, 687–702. 3. Sinha, S. C. and Joseph, P., ‘Control of general dynamic systems with periodically varying parameters via Liapunov–Floquet transformation’, ASME Journal of Dynamic Systems, Measurement, and Control 116, 1994, 650–658. 4. Arnold, V. I., Geometrical Methods in the Theory of Ordinary Differential Equations, Springer-Verlag, New York, 1988. 5. Pandiyan, R. and Sinha, S. C., ‘Analysis of time-periodic nonlinear dynamical systems undergoing bifurcations’, Nonlinear Dynamics 8, 1995, 21–43. 6. Chow, S.-N., Li, C., and Wang, D., Normal Forms and Bifurcation of Planar Vector Fields, Cambridge University Press, Cambridge, 1994. 7. Nayfeh, A. H., Method of Normal Forms, Wiley, New York, 1993. 8. Nayfeh, A. H., Perturbation Methods, Wiley, New York, 1973. 9. Sanders, J. A. and Verhulst, F., Averaging Methods in Nonlinear Dynamical Systems, Springer-Verlag, New York, 1985. 10. Sinha, S. C. and Butcher, E. A., ‘Symbolic computation of fundamental solution matrices for linear timeperiodic dynamical systems’, Journal of Sound and Vibration 206, 1997, 61–85. 11. Butcher, E. A. and Sinha, S. C., ‘Symbolic computation of local stability and bifurcation surfaces for nonlinear time-periodic systems’, Nonlinear Dynamics 17, 1998, 1–21. 12. Butcher, E. A. and Sinha, S. C., ‘Symbolic computation of secondary bifurcations in a parametrically excited simple pendulum’, International Journal of Bifurcation and Chaos 8, 1998, 627–637. 13. Guttalu, R. S. and Flashner, H., ‘Stability analysis of periodic systems by truncated point mappings’, Journal of Sound and Vibration 189, 1996, 33–54. 14. Bi, Q. and Yu, P., ‘Computation of normal forms of differential equations associated with non-semisimple zero eigenvalues’, International Journal of Bifurcation and Chaos 8, 1998, 2279–2319. 15. Pandiyan, R., ‘Analysis and control of nonlinear dynamic systems with periodically varying parameters’, Ph.D. Dissertation, Auburn University, Auburn, AL, 1994. 16. Bruno, A. D., Local Methods in Nonlinear Differential Equations, Springer-Verlag, New York, 1989.