(2) ARTICLE IN PRESS M.C. Depassier, V. Haikala / Journal of Sound and Vibration 328 (2009) 338–344. 339. knowledge of the exact solution. While the method is of general validity, we will apply it to two simple examples: a piecewise linear potential and the Dufﬁng nonlinear oscillator which has been extensively studied and therefore allows to compare the results obtained by different methods. In Section 2 we state the variational principles in general form, in Section 3 we apply them to the Dufﬁng nonlinear oscillator and in Section 4 we apply them to a piecewise linear oscillator. 2. Variational principles We consider systems with an equilibrium point x ¼ 0, around which the system oscillates. For the sake of simplicity, we assume that the force is an odd function of x, the results can be generalized in a simple way to a general force term. If the force is odd, the period of oscillation can be evaluated by considering the motion through a quarter of a period. _ Choosing the quarter period in the quadrant ðxo0; x40Þ of phase space, the period may be calculated solving x€ ¼ f ðxÞ; _ ¼ 0 and xðT=4Þ ¼ 0. with xð0Þ ¼ xm , xð0Þ Introducing the new variable t ¼ 4t=T, the problem reduces to ﬁnding the eigenvalue l of x00 ¼ lf ðxÞ;. x0 ð0Þ ¼ 0;. xð0Þ ¼ xm ;. xð1Þ ¼ 0. ð1Þ. with l ¼ T 2 =16. Here, primes denote derivatives with respect to t. Notice that the eigenvalue l depends on the amplitude xm . In terms of l the frequency is then. p o ¼ pﬃﬃﬃ : 2 l. 2.1. Upper bounds for the frequency In previous work we proved that the eigenvalue l of problem (1) is characterized by the variational principle [21,22] Rx 1 ð m g 0 1=3 ðxÞ dxÞ3 ; ð2Þ l½xm ¼ max R0xm g 2 0 f ðxÞgðxÞ dx where the maximum is taken over all positive functions gðxÞ such that gð0Þ ¼ 0, g 0 ðxÞ40. The maximum is achieved for g ¼ g^ which satisﬁes g^ 0 ðxÞ ¼. 1 ðE VðxÞÞ3=2. ;. where VðxÞ is the potential associated to the force f ðxÞ and E ¼ Vðxm Þ is the total energy of the system. A more convenient expression for (2) is obtained deﬁning a new function qðxÞ ¼ ðg 0 ðxÞÞ1=3 : After integrating by parts the denominator in (2), we ﬁnd 1 2. l½xm ¼ max R xm q. 0. where VðxÞ ¼. ð. R xm 0. qðxÞ dxÞ3. q3 ðVðxm Þ VðxÞÞ dx Z. ;. ð3Þ. x. f ðsÞ ds 0. is the potential. From (3) it is direct to verify that for the optimal trial function g^ ðxÞ, or equivalently for q, 1 q^ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ; E VðxÞ the exact formula for the period is recovered. If we evaluate the right side of (3) with an arbitrary qðxÞ40 then we obtain a lower bound for l and an upper bound on o. 2.2. Lower bounds for the frequency A second variational expression for l is given by [23] pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Rx 2 m ð1 2sðsÞÞ ðsðsÞ þ 1Þ=hðsÞ ds 1 ¼ max 0 R xm pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ; s2H l ð 0 ðsðsÞ þ 1Þ=hðsÞ dsÞ3 where H ¼ fsjs 2 Cð0; ym Þ; 0pso12g. To simplify the notation it is convenient to deﬁne the function hðxÞ ¼ E VðxÞ ¼ Vðxm Þ VðxÞ:. ð4Þ.

(3) ARTICLE IN PRESS 340. M.C. Depassier, V. Haikala / Journal of Sound and Vibration 328 (2009) 338–344. From this variational expression upper bounds on l, hence lower bounds for the frequency, can be obtained. The maximum is attained at s ¼ 0 and it is given precisely by !2 Z ym 1 1 p ﬃﬃﬃﬃﬃﬃﬃﬃ ds ¼2 : ð5Þ l hðsÞ 0 Before any application we will give an elementary proof of this variational expression. We need to show that for any 0pso 12, R x pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Z xm sﬃﬃﬃﬃﬃﬃﬃﬃ !2 ð 0 m ðsðsÞ þ 1Þ=hðsÞ dsÞ3 1 1 p ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ ð6Þ ds : Z Rx hðsÞ 2 0 m ð1 2sðsÞÞ ðsðsÞ þ 1Þ=hðsÞ ds 2 0 First, it is evident that when s ¼ 0 equality holds and the exact value for l is recovered. If sZ0, the numerator on the left side of (6) satisﬁes Z xm sﬃﬃﬃﬃﬃﬃﬃﬃ Z xm sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sðsÞ þ 1 1 ds ds: ð7Þ hðsÞ hðsÞ 0 0 In order to estimate the denominator consider the integrand as a function of s and deﬁne rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sþ1 mðsÞ ¼ ð1 2sÞ : h This is a decreasing function of s, effectively dm 3 2s þ 1 ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ds 2 ð1 þ sÞh is always negative for s 0. Therefore mðsÞpmð0Þ. It follows then that the denominator on the left side of (6) satisﬁes Z xm sﬃﬃﬃﬃﬃﬃﬃﬃ Z xm qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 ð1 2 sðsÞÞ ðsðsÞ þ 1Þ=hðsÞ dsp ds: ð8Þ hðsÞ 0 0 From (7) and (8) the desired result, Eq. (6) follows. 3. Application to the Dufﬁng oscillator We will apply the variational principles to obtain accurate analytic expressions for the frequency of the Dufﬁng oscillator x€ ¼ lðx þ dx3 Þ. ð9Þ. _ with xð0Þ ¼ 0 and xð1Þ ¼ 0. The amplitude of the oscillations xm ¼ xð0Þ. In order to obtain analytical expressions for the frequency of oscillations we must construct suitable trial functions to be used in the variational principles. Upper bounds for the frequency (lower bounds for l) follow from the variational expression equation (3). In previous work we showed that a simple closed analytic approximation can be obtained by taking as a trial function qðxÞ the exact trial function for the linear problem, qL , which is given by pﬃﬃﬃ 2 ð10Þ qL ðxÞ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ : x2m x2 Performing the integrals in Eq. (3) we obtain. opoU0 ¼. qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 þ 34dx2m :. ð11Þ. This approximation for the frequency gives a close estimate for all amplitudes, and has been obtained by other methods as well. Here, however, we establish that it constitutes an upper bound to the exact frequency. This ﬁrst expression oU0 was obtained by using the simplest possible trial function. In order to construct a closer approximation we make use of the fact that the optimal trial function q^ is known. For our example it is given by pﬃﬃﬃ 1 2 q^ exact ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ : x2m x2 d 2 1 þ ðxm þ x2 Þ 2 If we use the above function we will obtain the exact value of the frequency since this is an exactly solvable example. A simple way to obtain analytic upper bounds is to approximate the second factor in the right side by a series expansion. We.

(4) ARTICLE IN PRESS M.C. Depassier, V. Haikala / Journal of Sound and Vibration 328 (2009) 338–344. 341. ﬁnd it convenient to deﬁne 1 pðxÞ ¼ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ :. d. 1 þ ðx2m þ x2 Þ 2 As a ﬁrst trial we may approximate pðxÞ by a series expansion of degree 2, centered at xm =2, that is pﬃﬃﬃ 2 pﬃﬃﬃ x xm 2 dxm x m 8ð 2d x2m þ 4 2dÞ x 1 2 2 : pðxÞ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3=2 ð5dx2m þ 8Þ5=2 5dx2m 5dx2m þ1 4 þ1 8 8 Then, with the trial function q1 ðxÞ ¼ qL ðxÞpðxÞ we perform the integrals in (3) to obtain. opoU1 ¼. A1 ; B1. rﬃﬃﬃﬃﬃﬃ 2 2 ð9175040p þ 39567360dpx2m þ 107520d ð48 þ 695pÞx4m 35. where. A1 ¼ 2p. 3. 4. þ11200d ð1440 þ 7157pÞx6m þ 4032d ð5157 þ 13115pÞx8m 5. 6. 12 þ120d ð116064 þ 179795pÞ x10 m þ 9d ð541276 þ 565145pÞxm 7. 1=2 þ6d ð121119 þ 90755pÞx14 ; m Þ 2. B1 ¼ ½128p þ 152d px2m þ 6d ð4 þ 9pÞx4m 3=2 : A simpler expression is obtained approximating pðxÞ by a Pade polynomial of ﬁrst degree in the numerator, that is, 1 dxm ðx xm Þ pðxÞ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ dx2m þ 1 2ðdx2m þ 1Þ3=2 with which the trial function becomes. ! pﬃﬃﬃ 1 dxm ðx xm Þ 2 : q2 ðxÞ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ dx2m þ 1 2ðdx2m þ 1Þ3=2 x2m x2. Performing the integrals in Eq. (3) we obtain a second analytical upper bound,. opoU2 ¼ where. A2 ; B2. 2. ð12Þ. 3. 4. 1 3 ð1791p 2768Þd x6m þ 80 ð645p 1232Þd x8m Þ1=2 ; A2 ¼ pð8p þ 6ð7p 4Þdx2m þ 4ð21p 23Þd x4m þ 24. 3=2 pﬃﬃﬃ3p B2 ¼ 2 2 1 dx2m þ p : 2 Better accuracy is obtained approximating pðxÞ by a series expansion of degree 3, centered at xm =2, we obtain. p A opoU3 ¼ pﬃﬃﬃ 3 2 2 B3. with 10. 9. 25ð2688256 þ 2683065pÞd x20 5ð237126448 þ 243489015pÞd x18 m m þ 4928 5544. A3 ¼. 5 1 8 7 ð180808736 þ 198264297pÞd x16 ð1328298320 þ 1631028861pÞd x14 m þ m 616 231 8 16 6 5 þ ð183586828 þ 269824695pÞd x12 ð46827208 þ 90738165pÞd x10 m þ m 105 35 128 4 3 þ ð5429638 þ 15998115pÞd x8m þ 2560ð3952 þ 22075pÞd x6m 35 1=2 2 þ24576ð88 þ 1443pÞd x4m þ 12976128pdx2m þ 2097152p ; þ.

(5) ARTICLE IN PRESS 342. M.C. Depassier, V. Haikala / Journal of Sound and Vibration 328 (2009) 338–344 3. 2. B3 ¼ ð56ð8 þ 15pÞd x6m þ 2ð11 þ 32pÞd x4m þ 116pdx2m þ 64pÞ3=2 : We shall now use the dual variational principle (4) to obtain lower bounds on o. The exact solution is obtained when s ¼ 0. Our purpose is to illustrate the method, therefore we need to ﬁnd a good trial function sðxÞ. It is convenient to deﬁne ˜pðxÞ ¼. rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ h0 ðs þ 1Þ ; h. where h0 ¼ ðx2m x2 Þ=2 in terms of which. s ¼ ð˜pÞ2. h 1: h0. The exact value from the dual principle is obtained when s ¼ 0, that is, when ˜p ¼ p. We construct a trial function s replacing ˜p with a suitable approximation for p as was done to construct the upper bounds. A simple way to obtain an analytic approximate value for the frequency is to use a Pade approximant of degree 2 in the numerator and denominator for pðxÞ. The trial function constructed in this way is. sðxÞ ¼. d3 x6 ; ðdx2m þ 2Þð3dx2 þ 4dx2m þ 8Þ2. which satisﬁes the condition s 0. The lower bound calculated performing the integrations in (4) is given by pﬃﬃﬃﬃ N ; oL1 ¼ D where 928 512 1152 196 49 N ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ þ þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ þ ; 7dx2m þ 8 ð7dx2m þ 8Þ3=2 ð7dx2m þ 8Þ5=2 dx2m þ 2 ðdx2m þ 2Þ3=2 !3=2 4 1 ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ p p þ D¼7 : 7dx2m þ 8 dx2m þ 2 This lower bound gives quite accurate values for the frequency as we show in the table below. If higher precision is desired we may choose a high order polynomial trial function, such as 8 1 2 2 sðxÞ ¼ 1 þ d ðx þ x Þ 1 þ m 2 ð8 þ 5dx2m Þ1 5 ðdð2x þ xm Þ2 ð4 þ dx2m Þð8 þ 5dx2m Þ5 þ dxm ð2x xm Þð8 þ 5dx2m Þ6 2. ð8 þ 5dx2m Þ7 þ d xm ð2x xm Þ3 ð8 þ 5dx2m Þ4 ð12 þ 5dx2m Þ 2. 2. þd ð2x xm Þ4 ð8 þ 5dx2m Þ3 ð24 þ 5d x4m Þ 3. 2. þd xm ð2x xm Þ5 ð8 þ 5dx2m Þ2 ð120 þ 80dx2m þ 11d x4m Þ 4. 2. 3. þd xm ð2x xm Þ7 ð1120 840dx2m 84d x4m þ 29d x6m Þ 3. 2. 3. d ð2x xm Þ6 ð8 þ 5dx2m Þð160 þ 120dx2m þ 180d x4m þ 41d x6m ÞÞ2 : The integrals can be performed and a high order rational polynomial approximation for the frequency, which we call oL2 is found. We do not give the analytical expression, only its numerical value at different values of the amplitude. Numerical values for the upper and lower bounds for the frequency are given in the following table: xm 1 10 100. oL1. oL2. otrue. oU0. oU1. oU2. oU3. 1.31686 8.4551 83.9146. 1.31777 8.53225 84.714. 1.31778 8.53359 84.7275. 1.32288 8.7178 86.6083. 1.3178 8.53759 84.769. 1.31873 8.53661 84.5672. 1.31778 8.53376 84.7294. From the table we see that with adequate trial functions we obtain analytic bounds which can be as accurate as desired. In the example, at the amplitude xm ¼ 100 we have 84:714pop84:7249, a degree of certainty which cannot be achieved by other methods..

(6) ARTICLE IN PRESS M.C. Depassier, V. Haikala / Journal of Sound and Vibration 328 (2009) 338–344. 343. 4. Application to a piecewise linear oscillator As a second example consider the piecewise linear oscillator y€ þ f ðyÞ ¼ 0 with. ( f ðyÞ ¼. y. for yp1;. y þ mðy 1Þ. for y41. _ with initial conditions yð0Þ ¼ 1, yð0Þ ¼ y_ 0 40. The potential associated to the force f is given by ( 2 for yo1; y =2 VðyÞ ¼ y2 =2 þ mðy 1Þ2 =2 for y41: This is a linear problem, easily solvable, the solution of which is given by pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 4 T ¼ 2p þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ arctanðy_ 0 1 þ mÞ 4arctanðy_ 0 Þ: 1þm. ð13Þ. The amplitude of the motion A41 can be expressed in terms of the initial conditions solving E¼. 1 y_ 20 A2 mðA 1Þ2 ¼ þ : þ 2 2 2 2. Replacing y_ 0 in terms of the amplitude A in Eq. (13) we obtain the exact expression for the period in terms of the amplitude: pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃqﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 4 TE ¼ 2p þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ arctanð 1 þ m A2 þ mðA 1Þ2 1Þ ð14Þ 1þm 4 arctanð. qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ A2 þ mðA 1Þ2 1Þ:. ð15Þ. We will show here that an accurate approximation for the period can be obtained from the variational principles. We shall use the ﬁrst variational principle to construct a simple analytic approximation to the exact solution, and the second variational principle to establish that the exact solution constitutes an upper bound. To obtain a lower bound for the period choose the trial function 1 qðyÞ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ A2 y2 and evaluate (3). The integrals are not difﬁcult to perform and we obtain pﬃﬃﬃﬃﬃﬃ p 2p T TL h : m pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃi1=2 arcsinA1 þ ð1 þ mÞarccosA1 2 A2 1 A. ð16Þ. With the trial function s ¼ 0 in the variational principle (4) we obtain ToTE , the exact value. Let us now compare the numerical accuracy of the lower bound (16) with the exact value, which we have established as an upper bound. m ¼ 0:5. A. 2 10 100. m¼2. m ¼ 10. TL. TE. TL. TE. TL. TE. 5.74652 5.24246 5.14112. 5.74972 5.24264 5.14112. 4.7068 3.79182 3.64309. 4.7262 3.79238 3.6431. 2.83556 2.01441 1.90551. 2.87603 2.01501 1.90552. From the table we see that the lower bound TL gives a very accurate analytical approximation to the exact value of the period particularly at large amplitudes. Since this is a linear problem the exact solution is already simple, the lower bound is also simple and clearly it cannot be obtained by a series expansion of the exact solution. In summary, we have shown that from variational principles the frequency of second-order nonlinear oscillators can be determined accurately at low and large amplitudes. We illustrated the method through an application to the Dufﬁng equation to allow comparison with the results obtained by other methods, and to a piecewise linear oscillator. For any.

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