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Stability of an equilibrium position of a pendulum with step parameters
International Journal of Non-Linear Mechanics 73 (2015) 12–17
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Stability of an equilibrium position of a pendulum with step parameters Anatoly Markeev Institute for Problems in Mechanics of Russian Academy of Sciences, Pr. Vernadskogo, 101-1, Moscow 119526, Russia
art ic l e i nf o
a b s t r a c t
Article history: Received 25 February 2014 Received in revised form 17 August 2014 Accepted 3 November 2014 Available online 13 November 2014
A mathematical pendulum affected by parametric disturbance with potential energy being periodic step function is considered. Non-linear equation of the pendulum depends on two parameters characterizing the mean value in time of the parametric disturbance and range of its “ripple”. Values of the parameters can be set arbitrarily. The non-linear problem of stability for two particular solutions of the equation corresponding to a hanging and inverse pendulum is solved. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Parametric oscillations Stability Resonance Map
1. Introduction Suppose the motion of a one-degree-of-freedom system is defined by a differential equation of the form 2
d q dt
2
þ ðα þ βφðtÞÞ sin q ¼ 0:
ð1Þ
Here α and β are arbitrary constants, and φðtÞ is a two-step 2π-periodic in t function defined by the expression ( 1 if 0 rt oπ; φðtÞ ¼ ð2Þ 1 if π r t o 2π: A linearized equation 2
d q dt
2
þ ðα þ βφðtÞÞq ¼ 0
ð3Þ
obtained from (1) is called Meissner's equation. It was first examined in the paper [1] in the study of oscillations of an elastic system with variable stiffness. The paper [1] shows the existence of parametric resonance and gives a graphical representation of stability and instability domains. Analysis of Eq. (1) is a subject of many publications [2–6]. A detailed bibliography can be found in the monographs [7,8]. The non-linear equation allows of two equilibrium positions q ¼0 and q ¼ π. The aim of the paper is to summarize results obtained in the study of stability of linear equation (3) and to E-mail address: [email protected] http://dx.doi.org/10.1016/j.ijnonlinmec.2014.11.013 0020-7462/& 2014 Elsevier Ltd. All rights reserved.
make more precise some results in the non-linear analysis of the solutions q ¼0 and q ¼ π of Eq. (1) performed in the paper [5]. Note that Eq. (1) does not change the form if quantities t and β are simultaneously replaced by π þ t and β. Therefore, stability and instability domains are symmetrical relative to the β¼0 axis in the plane of the parameters α and β, and when analyzing the stability we can assume that β Z0. We also note that Eq. (1) does not change the form if we replace simultaneously t by π þ t, q by π þ q, and α by α. So, stability (instability) domains of the equilibrium position q ¼ π are obtained from stability (instability) domains of the equilibrium position q ¼0 by means of specular reflection relative to the α ¼ 0 axis. So, we can confine ourselves to constructing stability and instability domains for the equilibrium position q ¼0. The set G ¼ 1o α o1; β Z0 of considered values of the parameters α and β can be divided into three domains G1, G2 and G3 (see Fig. 1) defined by the conditions G1 ¼ α þ β Z0; α β Z 0; β Z0 ; G2 ¼ α þ β Z0; α β r 0; β Z0 ; ð4Þ G3 ¼ α þ β r0; α β r 0; β Z0 :
2. Algorithm of investigation Eq. (1) can be written as a system of two canonical equation with the Hamiltonian function H ¼ 12 p2 ðα þ βφðtÞÞ cos q:
ð5Þ
A. Markeev / International Journal of Non-Linear Mechanics 73 (2015) 12–17
Calculations show [5] that in the domain G1 π cos xk xk sin xk ðkÞ X ¼ xk ðk ¼ 1; 2Þ; π sin xk cos xk SðkÞ 4 ðqk 1 ; pk 1 Þ ¼
Fig. 1. The domains G1, G2 and G3 of the parameters α and β.
In the neighborhood of the equilibrium position q¼ 0 the function H is represented as a convergent series of the form H ¼ H 2 ðq; p; tÞ þ H 4 ðq; tÞ þ ⋯ þH 2m ðq; tÞ þ⋯;
ð6Þ
where H 2 ¼ 12 p2 þ 12 ðα þ βφðtÞÞq2 ;
H4 ¼
1 24 ðα þ βφðtÞÞq4 :
ð7Þ
The terms in (6) being independent of q and p are omitted. Let q0 and p0 be initial values of q and p at t¼0. The functions q ¼ qðq0 ; p0 ; tÞ and p ¼ pðq0 ; p0 ; tÞ satisfying the differential equations of motion define [9] an univalent canonical transformation q0 ; p0 -q; p (the map T). The right-hand sides of the equations of motion are discontinuous at t ¼ π. However, the functions q ¼ qðq0 ; p0 ; tÞ and p ¼ pðq0 ; p0 ; tÞ are continuous at any t. The map T can be represented within the time interval from t¼0 to t ¼ 2π as a composition of a canonical transformation q0 ; p0 -q1 ; p1 for the time interval from t¼0 to t ¼ π (the map T ð1Þ ) and a canonical transformation q1 ; p1 -q; p for the time interval from t ¼ π to t ¼ 2π (the map T ð2Þ ). Consider the matrixes of fundamental solutions of the linearized equations, with elements xðkÞ ij ðtÞ satisfying the initial conditions xðkÞ ð0Þ ¼ δ , where δ ¼ 1 if i¼j and δij ¼ 0 if i aj. The ij ij ij superscript k corresponds to the map T ðkÞ ðk ¼ 1; 2Þ. We introduce the notations Z π ðkÞ H 4 xðkÞ ðk ¼ 1; 2Þ: SðkÞ 4 ðqk 1 ; pk 1 Þ ¼ 11 ðtÞqk 1 þ x12 ðtÞpk 1 ; t dt 0
ð8Þ The maps T ð1Þ and T ð2Þ are given [10] by the equalities
ð9Þ
and ∂Sð2Þ q q1 ∂p4 þ ⋯ 1 ¼ Xð2Þ ð2Þ p p þ ∂S4 þ ⋯ 1 ∂q1
ð10Þ
x2 ¼ π
pffiffiffiffiffiffiffiffiffiffi α β;
xk ð sin 4xk þ 8 sin 2xk þ12xk Þq4k 1 768π 1 ð cos 4xk þ 4 cos 2xk 5Þq3k 1 pk 1 192 π ð sin 4xk 4xk Þq2k 1 p2k 1 128xk þ
π2 ð cos 4xk 4 cos 2xk þ 3Þqk 1 p3k 1 192x2k
þ
π3 ð sin 4xk 8 sin 2xk þ 12xk Þp4k 1 : 768x3k
ð13Þ
In the domain G2 the matrix Xð1Þ and the function Sð1Þ 4 ðq0 ; p0 Þ are calculated by formulas (11)–(13) and cosh y2 yπ sinh y2 2 ð14Þ Xð2Þ ¼ y2 ; π sinh y2 cosh y2 y2 ðsinh 4y2 þ 8sinh 2y2 þ 12y2 Þq41 768π 1 ðcosh 4y2 þ 4cosh 2y2 5Þq31 p1 192 π ðsinh 4y2 4y2 Þq21 p21 128y2
Sð2Þ 4 ðq1 ; p1 Þ ¼
π2 ðcosh 4y2 4cosh 2y2 þ 3Þq1 p31 192y22
π3 ðsinh 4y2 8sinh 2y2 þ12y2 Þp41 : 768y32
ð15Þ
In the domain G3 the matrix Xð2Þ is given by formula (14) and cosh y1 yπ sinh y1 1 ð1Þ ð16Þ X ¼ y1 : π sinh y1 cosh y1 The map T is area-preserving. It has a fixed point q0 ¼ 0, p0 ¼ 0. The stability problem of the equilibrium position q ¼ 0 of the system defined by Eq. (1) is equivalent to the stability problem of the fixed point q0 ¼ 0, p0 ¼ 0 of the map T. We substitute expressions for q1 and p1 from (9) into (10) and rewrite the map T in the form ∂S4 q q0 ∂p þ ⋯ 0 ð17Þ ¼ X ; ∂S4 p p0 þ ∂q þ ⋯ 0
ð2Þ ð1Þ ð1Þ ð1Þ ð1Þ S4 ðq0 ; p0 Þ ¼ Sð1Þ 4 ðq0 ; p0 Þ þ S4 ðx11 ðπÞq0 þ x12 ðπÞp0 ; x21 ðπÞq0 þ x22 ðπÞp0 Þ:
ð18Þ Here the dots denote the sets of terms higher than the third order in q0 and p0. The matrix X in (11) is equal to a product of the matrices Xð2Þ and Xð1Þ , its elements will be denoted by xij. A characteristic equation of the matrix X has the form ϱ2 2aϱ þ 1 ¼ 0
respectively. The matrices XðkÞ ðk ¼ 1; 2Þ on the right-hand sides of equalities (9) and (10) consist of elements xðkÞ ij ðtÞ calculated at t ¼ π. The dots denote the sets of terms higher than the third order in q0 ; p0 and q1 ; p1 respectively. Let pffiffiffiffiffiffiffiffiffiffi α þ β;
ð12Þ
where
∂Sð1Þ q1 q0 ∂p4 þ ⋯ 0 ð1Þ ¼X p1 p þ ∂Sð1Þ 4 þ ⋯ 0 ∂q0
x1 ¼ π
13
y1 ¼ π
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi α β;
y2 ¼ π
pffiffiffiffiffiffiffiffiffiffi β α: ð11Þ
ð2a ¼ x11 þ x22 Þ:
ð19Þ
Its roots (multipliers) depend on two parameters α and β. If the inequality jaj 4 1 is satisfied both multipliers are real and the absolute value of one of them is greater than unity. In this case the equilibrium position in question is unstable not only in the linear approximation but in a rigorous non-linear formulation [11]. In the case jajr 1 the multipliers ϱ can be written in the form ϱ1;2 ¼ expð 7i2πλÞ, where λ is a real number and cos 2πλ ¼ a:
ð20Þ
14
A. Markeev / International Journal of Non-Linear Mechanics 73 (2015) 12–17
If ϱk ¼ 1 for some natural number k, then the k-order resonance is said to occur in the system. If jaj o1, then the multipliers ϱ1;2 are different and complex conjugated numbers with absolute values equal to unity. In this case we have stability in the first (linear) approximation. The case jaj ¼ 1 corresponds to the boundaries of stability and instability domains in the plane of the parameters of the problem. We have the first-order resonance if a ¼1, ϱ1 ¼ ϱ2 ¼ 1 and λ is an integer, and the second-order resonance if a ¼ 1, ϱ1 ¼ ϱ2 ¼ 1 and λ is a half-integer. For these alternatives the equilibrium positions can be stable or unstable in the linear approximation depending on features of the matrix X. Thus, in the case jaj r 1 investigation of non-linear map (17) is necessary to obtain a rigorous solution of the problem.
3. The linear stability problem The coefficient a in characteristic equation (19) is defined as follows: in the domain G1
1 x1 x2 a ¼ cos x1 cos x2 þ ð21Þ sin x1 sin x2 ; 2 x2 x1 in the domain G2
1 x1 y2 sin x1 sinh y2 ; a ¼ cos x1 cosh y2 2 y2 x1
ð22Þ
in the domain G3 a ¼ cosh y1 cosh y2 þ
1 y1 y2 sinh y1 sinh y2 : þ 2 y2 y1
ð23Þ
Instability of the equilibrium position in the domain G3: In the domain G3 the equilibrium under consideration is unstable. This follows from the linear approximation, since quantity (23) is greater than unity. The only point α ¼β¼0 (when a ¼1) is excep2 2 tion. At this point, however, Eq. (1) has the form d q=dt ¼ 0, and instability of the equilibrium position q ¼0 is evident. Stability in the first approximation in the domains G1 and G2: Equalities (11) and (21), (22) define the coefficient a in the domains G1 and G2 as a function of the initial parameters α and β. The sets of
instability and stability domains in the first approximation are countable and these domains can be easily obtained. Fig. 2 illustrates relative location of the domains in the ranges 4:5 o α o 19:5 and 0 o β o24. The stability domains in Fig. 2 are hatched. At the points α; β inside these domains and on the boundaries the inequality jajr 1 satisfies. Inside the unhatched domains we have jaj 4 1, and the equilibrium position in question is unstable in the first approximation (and, therefore, when considering the whole non-linear equations). Now let us describe the stability and instability domains and their boundaries in detail. Case of small values of β: In the limiting case β¼0 we have pffiffiffi a ¼ cos 2π α: ð24Þ Therefore, in the α, β plane the boundary curves of the stability and instability domains on which jaj ¼ 1 issue from the points ðαs ; 0Þ of the β¼0 axis with αs ¼
s2 4
ðs ¼ 0; 1; 2; …Þ:
ð25Þ
On the boundary curves issuing from the points ðαs ; 0Þ at odd values of s we have a ¼ 1 (the second-order resonance), whereas if s is even, a ¼1 (the first-order resonance). Let g s ðs ¼ 0; 1; 2; …Þ be a stability domain with a boundary line segment s2 =4 o α o ðs þ 1Þ2 =4 on the β¼0 axis. Fig. 2 shows fragments of the first nine domains. The left boundary γ0 of the domain g0 issues from the point ð0; 0Þ, for small β it is defined by the equation γ0 :
α¼
π2 2 π6 4 107π 10 6 26711π 14 β þ β β þ β8 þ Oðβ10 Þ: 12 1512 9979200 122594472000
ð26Þ The first term of series (26) is obtained in the paper [2]. Let γ 0s , γ ″s be parts of boundaries of the instability domains in vicinity of the points ðαs ; 0Þ ðs ¼ 1; 2; …Þ of the β¼0 axis at small β. For odd s they are defined by the equations α¼
s2 2 s2 π 2 12 2 3s4 π 4 þ 20s2 π 2 480 3 8 βþ β 7 β þ Oðβ4 Þ: 4 sπ s4 π 2 3s7 π 3
ð27Þ
Here the upper and lower signs correspond to the curves γ 0s and γ ″s . If s ðs a 0Þ is even, then the boundaries of the instability domains are given for small β by
Fig. 2. Stability and instability domains.
γ 0s :
α¼
γ ″s :
α¼
s2 1 2 s2 π 2 36 4 s4 π 4 400s2 π 2 þ 14400 6 β þ β β þ Oðβ8 Þ; 4 s2 3s6 60s10 ð28Þ s2 3 2 s2 π 2 þ84 4 s4 π 4 þ 720s2 π 2 þ31680 6 þ β β þ β þ Oðβ8 Þ: 4 s2 3s6 60s10 ð29Þ
Series expansions (27)–(29) in terms up to the second order in β inclusive are previously obtained in [4]. Thus, for odd s the boundary curves γ 0s and γ ″s are close to the straight-line segments intersecting each other at the point ðαs ; 0Þ. But for even s they are close to segments of the parabolas osculating at the point ðαs ; 0Þ. Double points on the boundaries of stability and instability domains: With growth of β the parametric resonance boundaries issuing from the point ðαs ; 0Þ can intersect one another at one or more points. This leads to “kinking” of the instability domains (see Fig. 2). Analysis showed [5] that all double points lie inside the domain G1 and the set of these points is countable. The coordinates of the
A. Markeev / International Journal of Non-Linear Mechanics 73 (2015) 12–17
double points P i ðα0 ; β0 Þ (i ¼ 1; 2; 3; …) are defined by ( ðm þ 2n 1Þ2 if a ¼ 1ðs is oddÞ α 0 β 0 ¼ m2 ; α 0 þ β 0 ¼ if a ¼ 1ðs is evenÞ ðm þ 2nÞ2
ð30Þ
where m, n are natural numbers. There are no double points on the boundary curves issuing from the points (0,0), (1/4,0) and (1,0). In the domain shown in Fig. 2 there are ten double points: P 1 ð5=2; 3=2Þ; P 6 ð13; 12Þ;
P 2 ð5; 4Þ;
P 3 ð13=2; 5=2Þ;
P 7 ð25=2; 7=2Þ;
P 4 ð17=2; 15=2Þ;
P 8 ð29=2; 21=2Þ;
P 5 ð10; 6Þ;
P 9 ð37=2; 35=2Þ;
P 10 ð17; 8Þ:
They are denoted by numbers 1; 2; …; 10 in Fig. 2. In vicinity of the double point P i ðα0 ; β0 Þ the equations of the boundaries σ 0i , σ ″i of the instability domains are defined by series expansions of the form σ 0i :
α ¼ α0 þ
3ðα20 β20 Þ β0 ðβ β0 Þ þ ðβ β0 Þ2 þ O ðβ β0 Þ3 ; α0 4α30
σ ″i :
α ¼ α0
α0 ðβ β0 Þ2 þ O 4ðα20 β20 Þ
ðβ β0 Þ3 :
ð31Þ ð32Þ
In a small neighborhood of the double point P i ðα0 ; β0 Þ the curve σ 0i is close to the straight-line segment passing through this point and making the angle arctanðβ0 =α0 Þ with the horizontal axis. However, the curve σ ″i represents a piece of the parabola lying to the left of the point P i ðα0 ; β0 Þ, with a vertical tangent at this point. Stability intervals in the special cases α ¼ β and α ¼0: The stability domains g s ðs ¼ 0; 1; 2…Þ are simply connected. They originate on the β ¼0 axis but do not end anywhere inside the domains G1 and G2 with growth of β. For large values of β the stability domains become very narrow, and their boundaries lead asymptotically to the curves with the tangent inclination equal to 1 to the horizontal axis. Each of the domains gs is once intersected by the straight line α ¼ β separating the domains G1 and G2, and once by the vertical straight line α ¼0. In the case α¼ β the quantity a is equal to pffiffiffiffiffiffi a ¼ cos 2x x sin 2x ðx ¼ π 2α=2Þ: ð33Þ The straight line α ¼ β separates the intervals ðrÞ αðlÞ s o α o αs
ðs ¼ 0; 1; 2…Þ
ð34Þ αðlÞ s
αsðrÞ
¼ s =2 and is the nearest to in the stability domain gs. Here αðlÞ s (and larger than it) root of the equation x tan x ¼ 1 in the case of even s and the equation tan x ¼ x in the case of odd s. Fig. 2 shows the first seven stability intervals on the straight line α¼β defined by 2
ð0; 0:15Þ; ð1=2; 0:834Þ; ð2; 2:378Þ; ð9=2; 4:8917Þ; ð8; 8:3973Þ; ð25=2; 12:9Þ; ð18; 18:4016Þ: At s-1 width of the stability intervals leads to the limiting quantity 4=π 2 . On the straight line α ¼ 0 we have pffiffiffi pffiffiffi a ¼ cosh ðπ βÞ cos ðπ βÞ; ð35Þ and the stability intervals are defined by the inequalities ðrÞ βðlÞ s oβ o βs
ðs ¼ 0; 1; 2…Þ:
ð36Þ
The left boundary βðlÞ s of interval (3.16) is a root of the equation a ¼1 for even s and the equation a ¼ 1 for odd s. The right boundary βsðrÞ , to the contrary, is a root of the equations a ¼1 and a ¼ 1 for odd and even s respectively. The first five stability intervals on the straight line α ¼ 0 are defined by ð0; 0:3562Þ; ð2:2326; 2:2669Þ; ð6:2488; 6:2512Þ; ð12:2499; 12:25Þ; ð20:2499; 20:25Þ:
15
Width of the stability intervals decreases as s increases. For sufficiently large s it is exponentially small and close to the quantity 8 expð π=2Þ expð πsÞ: π Linear normalization of map (17): We make the linear change of variables of the form q ¼ n11 Q þ n12 P;
p ¼ n21 Q þ n22 P
ðd ¼ n11 n22 n12 n21 a 0Þ
ð37Þ
As a result the matrix X of the linearized map takes a real normal form J. Replacement of variables (37) is a canonical transformation 1 with the valency c ¼ d . The quantities nij and the matrix J are given in the paper [5]. In the new variables the map conserves square and can be written in the form similar to (17) ∂F 4 Q Q 0 ∂P þ ⋯ 0 ð38Þ ¼ J : ∂F 4 P P 0 þ ∂Q þ⋯ 0 Here the dots denote the sets of terms higher than the third order in Q 0 ; P 0 , and F4 is the fourth-order form in Q 0 ; P 0 defined by F 4 ðQ 0 ; P 0 Þ ¼ cS4 ðn11 Q 0 þ n12 P 0 ; n21 Q 0 þ n22 P 0 Þ
∑ f νμ Q ν0 P μ0 :
νþμ ¼ 4
ð39Þ Consider now several cases differing from one another. (1) If the point ðα; βÞ lies inside one of the domains g s ðs ¼ 0; 1; 2; …Þ of stability in the first approximation, then the normal form of the linearized map defines turn at the angle 2πλ as cos 2πλ sin 2πλ J¼ ð40Þ ; sin 2πλ cos 2πλ 8 1 s > > arccos a þ < 2π 2 λ¼ 1 sþ1 > > arccos a þ : 2π 2
if s ¼ 0; 2; 4; …; if s ¼ 1; 3; 5; …:
ð41Þ
(2) Consider the points ðα; βÞ on the boundaries of stability and instability domains when a ¼ 1 or 1. We will get the results on stability in the first approximation for these points. We first consider the double points P i ðα0 ; β0 Þ ði ¼ 1; 2; 3; …Þ at we have 1 0 J¼ ð42Þ ; 0 1 on the curves a ¼ 1 and 1 0 J ¼ : 0 1
ð43Þ
on the curves a ¼1. Since matrices (42) and (43) are diagonal, the equilibrium position in question is stable in the first (linear) approximation for the sets of parameters α0 ; β0 corresponding to any of the double points P i ðα0 ; β0 Þ ði ¼ 1; 2; 3; …Þ. Off the double points on the boundary curves a ¼ 1 the matrix J of the transformed map (38) has the form of a Jordan box 1 1 J ¼ ð44Þ ; 0 1 and in the boundaries a¼ 1, 1 1 J¼ : 0 1
ð45Þ
Therefore, for the points ðα; βÞ lying on the boundaries a ¼ 7 1 of the stability domains the equilibrium position under consideration is
16
A. Markeev / International Journal of Non-Linear Mechanics 73 (2015) 12–17
unstable in the first approximation, except for the double points P i ðα0 ; β0 Þ ði ¼ 1; 2; 3; …Þ at which it is stable in the first approximation.
4. Results on non-linear investigation To obtain a rigorous solution of the stability problem in the case jajr 1 non-linear analysis is necessary. This problem is equivalent to one of the immovable point Q 0 ¼ 0; P 0 ¼ 0 of map (38). Conditions for stability and instability can be expressed in terms of the coefficients f νμ in the fourthorder form (39) [10,12]. The points ðα; βÞ lie inside the stability domains g s ðs ¼ 0; 1; 2; …Þ in the first approximation. In this case we must distinguish between resonance and non-resonance values of the parameters. Since there are no the second-order terms in Q 0 ; P 0 on the right-hand side of equality (38), the third-order resonance is unimportant. In each of the domains gs one of the fourth-order resonances occurs for which 4λ ¼ 2s þ 1 ðs ¼ 0; 1; 2; …Þ. For small β the fourth-order resonance curves are defined by α¼
ð2s þ 1Þ2 4 þ 16 ð2s þ 1Þ2
4ð 1Þs 2 β þ Oðβ4 Þ: 1 πð2s þ1Þ
ð46Þ
the equilibrium position is stable [10,12]. If the parameters α, β belong to one of the fourth-order resonance curves, then the sufficient condition for stability is the inequality ð3f 40 þf 22 þ 3f 04 Þ2 ðf 40 f 22 þ f 04 Þ2 ðf 31 f 13 Þ2 4 0:
ð47Þ
Checking stability conditions (46), (47) was carried out numerically for ranges of the parameters α and β corresponding to Fig. 2. Analytical results were obtained for the cases α ¼ β and 0 oβ 5 1. It was found that in all domains g s ðs ¼ 0; 1; 2; …Þ of stability in the first approximation conditions (46), (47) are satisfied. Therefore, the equilibrium position under consideration is also stable when considering the rigorous non-linear statement of the problem. The parameters α and β correspond to the double points on the boundaries of the domains g s ðs ¼ 0; 1; 2; …Þ. Calculations showed [5] that at the double points P i ðα0 ; β0 Þ ði ¼ 1; 2; 3; …Þ on the boundaries of the stability domains in the first approximation the function S4 has the form ! π α0 4 2 2 4 α0 q0 þ2q0 p0 þ 2 p0 : S4 ðq0 ; p0 Þ ¼ 32 α0 β20 Since the function S4 ðq0 ; p0 Þ is of fixed sign, the equation S4 ð sin ψ; cos ψ Þ does not have real roots. According to [10,12], it follows that at all double points the equilibrium position considered is stable in Lyapunov's sense. Segments of the boundaries of the domains g s ðs ¼ 0; 1; 2; …Þ outside the double points. If the point ðα; βÞ lies on the boundaries of the domains g s ðs ¼ 0; 1; 2; …Þ and does not coincide with one of the double points, then a sign of the coefficient f40 in (39) is required to obtain a conclusion on stability in the non-linear problem [5,10,12]. If a ¼1 and the inequality f 40 o 0
f 40 ¼
π6 2 β þ Oðβ4 Þ: 6
ð48Þ
ð49Þ
For sufficiently small β the quantity f40 is positive and, therefore, instability occurs. Consider now left and right boundaries γ 0s and γ ″s of instability domains issuing from the points ðs2 =4; 0Þ of the β ¼0 axis. For small values of β they are defined by Eq. (27) if s is odd, and (28), (29) if s is even. For odd s the corresponding coefficient f40 has the form f 40 ¼ 7
16π 3 β þ Oðβ4 Þ s7
ð50Þ
Here the upper and lower signs are related to the boundaries γ 0s and γ ″s respectively. In the case of even s we get f 40 ¼
The fourth-order resonant curves are not shown in Fig. 2. If the parameters α, β lie inside the stability domain in the first approximation and do not belong to the fourth-order resonance curves, then in the case 3f 40 þ f 22 þ 3f 04 a 0
is satisfied, then the equilibrium position is stable in Lyapunov's sense, whereas in the case of the opposite sign in inequality (48), unstable. In the case a ¼ 1, to the contrary, the equilibrium position is unstable if inequality (48) is satisfied and stable at the opposite sign in it. Calculations show that on the boundary γ0 of the domain g0 defined by (26) the coefficient f40 has for small β the form
32π 4 6 β þ Oðβ7 Þ s10
ð51Þ
on the boundary γ 0s and f 40 ¼
8π 6 8 β þ Oðβ9 Þ s12
ð52Þ
on the boundary γ ″s . From (50)–(52), it follows that stability occurs on the left boundaries γ 0s and instability on the right boundaries γ ″s . Suppose the parameters α and β correspond to the boundaries of the stability and instability domains in vicinity of the double points P i ði ¼ 1; 2; 3; …Þ. On the boundaries σ 0i defined by (31) we get f 40 ¼ 8
π 6 β30 ðβ β0 Þ5 þO ðβ β0 Þ6 ; 2 4 2 512α0 ðα0 β0 Þ
ð53Þ
and on the boundaries σ ″i defined by (3.12), f 40 ¼ 7
π 4 α0 β30 32ðα20 β20 Þ3
ðβ β0 Þ3 þ O ðβ β0 Þ4 :
ð54Þ
In equalities (53), (54) the upper signs correspond to the cases when s is odd and the double point P i ðα0 ; β0 Þ is intersection of the instability domain boundaries issuing from the point ðs2 =4; 0Þ of the β¼0 axis. The lower signs are related to the similar points in the case of even s. From expressions (53), (54) and the aforementioned conditions for stability and instability, we get the following results. If s is odd, then in a small vicinity of the point P i ðα0 ; β0 Þ on the curve σ 0i the equilibrium position is unstable for β 4β0 and stable for β o β0 , whereas on the curve σ ″i it is stable for β 4 β0 and unstable for β oβ0 . Similarly, if s is even, then on the curve σ 0i the equilibrium position is unstable for β 4 β0 and stable for β o β0 , and on the curve σ ″i it is stable for β 4 β0 and unstable for β oβ0 . At arbitrary points ðα; βÞ on the boundaries of the stability domains g s ðs ¼ 0; 1; 2; …Þ which do not coincide with any of the double points P i ðα0 ; β0 Þ ði ¼ 1; 2; 3…Þ the conditions for stability were checked numerically. The equilibrium position in question was found to be unstable at points on the left boundaries of the domains gs in Fig. 2 and stable at points on the right boundaries.
A. Markeev / International Journal of Non-Linear Mechanics 73 (2015) 12–17
5. Conclusions Let us summarize the results obtained (see Fig. 2). For any values of the parameters α and β lying inside the domains g s ðs ¼ 0; 1; 2; …Þ or on their right boundaries the equilibrium position q ¼0 of the system defined by Eq. (1) is stable in Lyapunov's sense. If the parameters α and β lie on the left boundaries, then the equilibrium position is unstable, except for the double points P i ði ¼ 1; 2; 3; …Þ at which it is stable in Lyapunov's sense. Acknowledgment The work is carried out at the cost of the grant of the Russian Scientific Foundation (Project 14-21-00068) at the Moscow Aviation Institute (National Research University). References [1] E. Meissner, Ueber Schüttelerscheinungen in Systemen mit periodisch veränderlicher Elastizität, Schweiz. Bauztg. 72 (11) (1918) 95–98.
17
[2] B. Van der Pol, M.J.O. Strutt, On the stability of solutions of Mathieu's equation, Philos. Mag. Ser. 7 (5) (1928) 18–38. [3] M.J.O. Strutt, Lamèsche, Matchieusche und verwandte Funktionen in Physik und Technik, vol. 1. Ergeh. Math. Grenzgeb., 1932, pp. 199–323. [4] V.I. Arnold, Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics, vol. 60, 2nd ed., Springer, September 23, 1997. [5] A.P. Markeev, On Meissner's nonlinear equation, Nonlinear Dyn. 7 (3) (2011) 531–547 (in Russian). [6] J.M. Almira, P.J. Torres, Invariance of the stability of Meissner's equation under a permutation of its intervals, Ann. Mat. Pura Appl. 180 (2) (2001) 245–253. [7] S. Timoshenko, Vibration Problems in Engineering, John Wiley & Sons, 1974 521 p. [8] Oscillations in nonlinear systems, Hand-book, vol. 1, in: V.V. Bolotin (Ed.), Vibrations in Techniques, 6 vols., Mashinostroyeniye, Moscow, 1999 (in Russian). [9] A.P. Markeev, Theoretical Mechanics, Regular and Chaotic Dynamics, Moscow – Izhevsk, 2007, 592 p. (in Russian). [10] A.P. Markeev, Stability of equilibrium states of Hamiltonian systems: a method of investigation, Mech. Solids. J. Russ. Acad. Sci. 39 (6) (2004) 1–8. [11] I.G. Malkin, Theory of stability of motion, United States Atomic Energy Commission, 1952, 456 p. [12] A.P. Markeev, On area-preserving mappings and their applications in dynamics of systems with collisions, Mech. Solids J. Russ. Acad. Sci. 2 (1996) 37–54.
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International Journal of Non-Linear Mechanics journal homepage: www.elsevier.com/locate/nlm
Stability of an equilibrium position of a pendulum with step parameters Anatoly Markeev Institute for Problems in Mechanics of Russian Academy of Sciences, Pr. Vernadskogo, 101-1, Moscow 119526, Russia
art ic l e i nf o
a b s t r a c t
Article history: Received 25 February 2014 Received in revised form 17 August 2014 Accepted 3 November 2014 Available online 13 November 2014
A mathematical pendulum affected by parametric disturbance with potential energy being periodic step function is considered. Non-linear equation of the pendulum depends on two parameters characterizing the mean value in time of the parametric disturbance and range of its “ripple”. Values of the parameters can be set arbitrarily. The non-linear problem of stability for two particular solutions of the equation corresponding to a hanging and inverse pendulum is solved. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Parametric oscillations Stability Resonance Map
1. Introduction Suppose the motion of a one-degree-of-freedom system is defined by a differential equation of the form 2
d q dt
2
þ ðα þ βφðtÞÞ sin q ¼ 0:
ð1Þ
Here α and β are arbitrary constants, and φðtÞ is a two-step 2π-periodic in t function defined by the expression ( 1 if 0 rt oπ; φðtÞ ¼ ð2Þ 1 if π r t o 2π: A linearized equation 2
d q dt
2
þ ðα þ βφðtÞÞq ¼ 0
ð3Þ
obtained from (1) is called Meissner's equation. It was first examined in the paper [1] in the study of oscillations of an elastic system with variable stiffness. The paper [1] shows the existence of parametric resonance and gives a graphical representation of stability and instability domains. Analysis of Eq. (1) is a subject of many publications [2–6]. A detailed bibliography can be found in the monographs [7,8]. The non-linear equation allows of two equilibrium positions q ¼0 and q ¼ π. The aim of the paper is to summarize results obtained in the study of stability of linear equation (3) and to E-mail address: [email protected] http://dx.doi.org/10.1016/j.ijnonlinmec.2014.11.013 0020-7462/& 2014 Elsevier Ltd. All rights reserved.
make more precise some results in the non-linear analysis of the solutions q ¼0 and q ¼ π of Eq. (1) performed in the paper [5]. Note that Eq. (1) does not change the form if quantities t and β are simultaneously replaced by π þ t and β. Therefore, stability and instability domains are symmetrical relative to the β¼0 axis in the plane of the parameters α and β, and when analyzing the stability we can assume that β Z0. We also note that Eq. (1) does not change the form if we replace simultaneously t by π þ t, q by π þ q, and α by α. So, stability (instability) domains of the equilibrium position q ¼ π are obtained from stability (instability) domains of the equilibrium position q ¼0 by means of specular reflection relative to the α ¼ 0 axis. So, we can confine ourselves to constructing stability and instability domains for the equilibrium position q ¼0. The set G ¼ 1o α o1; β Z0 of considered values of the parameters α and β can be divided into three domains G1, G2 and G3 (see Fig. 1) defined by the conditions G1 ¼ α þ β Z0; α β Z 0; β Z0 ; G2 ¼ α þ β Z0; α β r 0; β Z0 ; ð4Þ G3 ¼ α þ β r0; α β r 0; β Z0 :
2. Algorithm of investigation Eq. (1) can be written as a system of two canonical equation with the Hamiltonian function H ¼ 12 p2 ðα þ βφðtÞÞ cos q:
ð5Þ
A. Markeev / International Journal of Non-Linear Mechanics 73 (2015) 12–17
Calculations show [5] that in the domain G1 π cos xk xk sin xk ðkÞ X ¼ xk ðk ¼ 1; 2Þ; π sin xk cos xk SðkÞ 4 ðqk 1 ; pk 1 Þ ¼
Fig. 1. The domains G1, G2 and G3 of the parameters α and β.
In the neighborhood of the equilibrium position q¼ 0 the function H is represented as a convergent series of the form H ¼ H 2 ðq; p; tÞ þ H 4 ðq; tÞ þ ⋯ þH 2m ðq; tÞ þ⋯;
ð6Þ
where H 2 ¼ 12 p2 þ 12 ðα þ βφðtÞÞq2 ;
H4 ¼
1 24 ðα þ βφðtÞÞq4 :
ð7Þ
The terms in (6) being independent of q and p are omitted. Let q0 and p0 be initial values of q and p at t¼0. The functions q ¼ qðq0 ; p0 ; tÞ and p ¼ pðq0 ; p0 ; tÞ satisfying the differential equations of motion define [9] an univalent canonical transformation q0 ; p0 -q; p (the map T). The right-hand sides of the equations of motion are discontinuous at t ¼ π. However, the functions q ¼ qðq0 ; p0 ; tÞ and p ¼ pðq0 ; p0 ; tÞ are continuous at any t. The map T can be represented within the time interval from t¼0 to t ¼ 2π as a composition of a canonical transformation q0 ; p0 -q1 ; p1 for the time interval from t¼0 to t ¼ π (the map T ð1Þ ) and a canonical transformation q1 ; p1 -q; p for the time interval from t ¼ π to t ¼ 2π (the map T ð2Þ ). Consider the matrixes of fundamental solutions of the linearized equations, with elements xðkÞ ij ðtÞ satisfying the initial conditions xðkÞ ð0Þ ¼ δ , where δ ¼ 1 if i¼j and δij ¼ 0 if i aj. The ij ij ij superscript k corresponds to the map T ðkÞ ðk ¼ 1; 2Þ. We introduce the notations Z π ðkÞ H 4 xðkÞ ðk ¼ 1; 2Þ: SðkÞ 4 ðqk 1 ; pk 1 Þ ¼ 11 ðtÞqk 1 þ x12 ðtÞpk 1 ; t dt 0
ð8Þ The maps T ð1Þ and T ð2Þ are given [10] by the equalities
ð9Þ
and ∂Sð2Þ q q1 ∂p4 þ ⋯ 1 ¼ Xð2Þ ð2Þ p p þ ∂S4 þ ⋯ 1 ∂q1
ð10Þ
x2 ¼ π
pffiffiffiffiffiffiffiffiffiffi α β;
xk ð sin 4xk þ 8 sin 2xk þ12xk Þq4k 1 768π 1 ð cos 4xk þ 4 cos 2xk 5Þq3k 1 pk 1 192 π ð sin 4xk 4xk Þq2k 1 p2k 1 128xk þ
π2 ð cos 4xk 4 cos 2xk þ 3Þqk 1 p3k 1 192x2k
þ
π3 ð sin 4xk 8 sin 2xk þ 12xk Þp4k 1 : 768x3k
ð13Þ
In the domain G2 the matrix Xð1Þ and the function Sð1Þ 4 ðq0 ; p0 Þ are calculated by formulas (11)–(13) and cosh y2 yπ sinh y2 2 ð14Þ Xð2Þ ¼ y2 ; π sinh y2 cosh y2 y2 ðsinh 4y2 þ 8sinh 2y2 þ 12y2 Þq41 768π 1 ðcosh 4y2 þ 4cosh 2y2 5Þq31 p1 192 π ðsinh 4y2 4y2 Þq21 p21 128y2
Sð2Þ 4 ðq1 ; p1 Þ ¼
π2 ðcosh 4y2 4cosh 2y2 þ 3Þq1 p31 192y22
π3 ðsinh 4y2 8sinh 2y2 þ12y2 Þp41 : 768y32
ð15Þ
In the domain G3 the matrix Xð2Þ is given by formula (14) and cosh y1 yπ sinh y1 1 ð1Þ ð16Þ X ¼ y1 : π sinh y1 cosh y1 The map T is area-preserving. It has a fixed point q0 ¼ 0, p0 ¼ 0. The stability problem of the equilibrium position q ¼ 0 of the system defined by Eq. (1) is equivalent to the stability problem of the fixed point q0 ¼ 0, p0 ¼ 0 of the map T. We substitute expressions for q1 and p1 from (9) into (10) and rewrite the map T in the form ∂S4 q q0 ∂p þ ⋯ 0 ð17Þ ¼ X ; ∂S4 p p0 þ ∂q þ ⋯ 0
ð2Þ ð1Þ ð1Þ ð1Þ ð1Þ S4 ðq0 ; p0 Þ ¼ Sð1Þ 4 ðq0 ; p0 Þ þ S4 ðx11 ðπÞq0 þ x12 ðπÞp0 ; x21 ðπÞq0 þ x22 ðπÞp0 Þ:
ð18Þ Here the dots denote the sets of terms higher than the third order in q0 and p0. The matrix X in (11) is equal to a product of the matrices Xð2Þ and Xð1Þ , its elements will be denoted by xij. A characteristic equation of the matrix X has the form ϱ2 2aϱ þ 1 ¼ 0
respectively. The matrices XðkÞ ðk ¼ 1; 2Þ on the right-hand sides of equalities (9) and (10) consist of elements xðkÞ ij ðtÞ calculated at t ¼ π. The dots denote the sets of terms higher than the third order in q0 ; p0 and q1 ; p1 respectively. Let pffiffiffiffiffiffiffiffiffiffi α þ β;
ð12Þ
where
∂Sð1Þ q1 q0 ∂p4 þ ⋯ 0 ð1Þ ¼X p1 p þ ∂Sð1Þ 4 þ ⋯ 0 ∂q0
x1 ¼ π
13
y1 ¼ π
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi α β;
y2 ¼ π
pffiffiffiffiffiffiffiffiffiffi β α: ð11Þ
ð2a ¼ x11 þ x22 Þ:
ð19Þ
Its roots (multipliers) depend on two parameters α and β. If the inequality jaj 4 1 is satisfied both multipliers are real and the absolute value of one of them is greater than unity. In this case the equilibrium position in question is unstable not only in the linear approximation but in a rigorous non-linear formulation [11]. In the case jajr 1 the multipliers ϱ can be written in the form ϱ1;2 ¼ expð 7i2πλÞ, where λ is a real number and cos 2πλ ¼ a:
ð20Þ
14
A. Markeev / International Journal of Non-Linear Mechanics 73 (2015) 12–17
If ϱk ¼ 1 for some natural number k, then the k-order resonance is said to occur in the system. If jaj o1, then the multipliers ϱ1;2 are different and complex conjugated numbers with absolute values equal to unity. In this case we have stability in the first (linear) approximation. The case jaj ¼ 1 corresponds to the boundaries of stability and instability domains in the plane of the parameters of the problem. We have the first-order resonance if a ¼1, ϱ1 ¼ ϱ2 ¼ 1 and λ is an integer, and the second-order resonance if a ¼ 1, ϱ1 ¼ ϱ2 ¼ 1 and λ is a half-integer. For these alternatives the equilibrium positions can be stable or unstable in the linear approximation depending on features of the matrix X. Thus, in the case jaj r 1 investigation of non-linear map (17) is necessary to obtain a rigorous solution of the problem.
3. The linear stability problem The coefficient a in characteristic equation (19) is defined as follows: in the domain G1
1 x1 x2 a ¼ cos x1 cos x2 þ ð21Þ sin x1 sin x2 ; 2 x2 x1 in the domain G2
1 x1 y2 sin x1 sinh y2 ; a ¼ cos x1 cosh y2 2 y2 x1
ð22Þ
in the domain G3 a ¼ cosh y1 cosh y2 þ
1 y1 y2 sinh y1 sinh y2 : þ 2 y2 y1
ð23Þ
Instability of the equilibrium position in the domain G3: In the domain G3 the equilibrium under consideration is unstable. This follows from the linear approximation, since quantity (23) is greater than unity. The only point α ¼β¼0 (when a ¼1) is excep2 2 tion. At this point, however, Eq. (1) has the form d q=dt ¼ 0, and instability of the equilibrium position q ¼0 is evident. Stability in the first approximation in the domains G1 and G2: Equalities (11) and (21), (22) define the coefficient a in the domains G1 and G2 as a function of the initial parameters α and β. The sets of
instability and stability domains in the first approximation are countable and these domains can be easily obtained. Fig. 2 illustrates relative location of the domains in the ranges 4:5 o α o 19:5 and 0 o β o24. The stability domains in Fig. 2 are hatched. At the points α; β inside these domains and on the boundaries the inequality jajr 1 satisfies. Inside the unhatched domains we have jaj 4 1, and the equilibrium position in question is unstable in the first approximation (and, therefore, when considering the whole non-linear equations). Now let us describe the stability and instability domains and their boundaries in detail. Case of small values of β: In the limiting case β¼0 we have pffiffiffi a ¼ cos 2π α: ð24Þ Therefore, in the α, β plane the boundary curves of the stability and instability domains on which jaj ¼ 1 issue from the points ðαs ; 0Þ of the β¼0 axis with αs ¼
s2 4
ðs ¼ 0; 1; 2; …Þ:
ð25Þ
On the boundary curves issuing from the points ðαs ; 0Þ at odd values of s we have a ¼ 1 (the second-order resonance), whereas if s is even, a ¼1 (the first-order resonance). Let g s ðs ¼ 0; 1; 2; …Þ be a stability domain with a boundary line segment s2 =4 o α o ðs þ 1Þ2 =4 on the β¼0 axis. Fig. 2 shows fragments of the first nine domains. The left boundary γ0 of the domain g0 issues from the point ð0; 0Þ, for small β it is defined by the equation γ0 :
α¼
π2 2 π6 4 107π 10 6 26711π 14 β þ β β þ β8 þ Oðβ10 Þ: 12 1512 9979200 122594472000
ð26Þ The first term of series (26) is obtained in the paper [2]. Let γ 0s , γ ″s be parts of boundaries of the instability domains in vicinity of the points ðαs ; 0Þ ðs ¼ 1; 2; …Þ of the β¼0 axis at small β. For odd s they are defined by the equations α¼
s2 2 s2 π 2 12 2 3s4 π 4 þ 20s2 π 2 480 3 8 βþ β 7 β þ Oðβ4 Þ: 4 sπ s4 π 2 3s7 π 3
ð27Þ
Here the upper and lower signs correspond to the curves γ 0s and γ ″s . If s ðs a 0Þ is even, then the boundaries of the instability domains are given for small β by
Fig. 2. Stability and instability domains.
γ 0s :
α¼
γ ″s :
α¼
s2 1 2 s2 π 2 36 4 s4 π 4 400s2 π 2 þ 14400 6 β þ β β þ Oðβ8 Þ; 4 s2 3s6 60s10 ð28Þ s2 3 2 s2 π 2 þ84 4 s4 π 4 þ 720s2 π 2 þ31680 6 þ β β þ β þ Oðβ8 Þ: 4 s2 3s6 60s10 ð29Þ
Series expansions (27)–(29) in terms up to the second order in β inclusive are previously obtained in [4]. Thus, for odd s the boundary curves γ 0s and γ ″s are close to the straight-line segments intersecting each other at the point ðαs ; 0Þ. But for even s they are close to segments of the parabolas osculating at the point ðαs ; 0Þ. Double points on the boundaries of stability and instability domains: With growth of β the parametric resonance boundaries issuing from the point ðαs ; 0Þ can intersect one another at one or more points. This leads to “kinking” of the instability domains (see Fig. 2). Analysis showed [5] that all double points lie inside the domain G1 and the set of these points is countable. The coordinates of the
A. Markeev / International Journal of Non-Linear Mechanics 73 (2015) 12–17
double points P i ðα0 ; β0 Þ (i ¼ 1; 2; 3; …) are defined by ( ðm þ 2n 1Þ2 if a ¼ 1ðs is oddÞ α 0 β 0 ¼ m2 ; α 0 þ β 0 ¼ if a ¼ 1ðs is evenÞ ðm þ 2nÞ2
ð30Þ
where m, n are natural numbers. There are no double points on the boundary curves issuing from the points (0,0), (1/4,0) and (1,0). In the domain shown in Fig. 2 there are ten double points: P 1 ð5=2; 3=2Þ; P 6 ð13; 12Þ;
P 2 ð5; 4Þ;
P 3 ð13=2; 5=2Þ;
P 7 ð25=2; 7=2Þ;
P 4 ð17=2; 15=2Þ;
P 8 ð29=2; 21=2Þ;
P 5 ð10; 6Þ;
P 9 ð37=2; 35=2Þ;
P 10 ð17; 8Þ:
They are denoted by numbers 1; 2; …; 10 in Fig. 2. In vicinity of the double point P i ðα0 ; β0 Þ the equations of the boundaries σ 0i , σ ″i of the instability domains are defined by series expansions of the form σ 0i :
α ¼ α0 þ
3ðα20 β20 Þ β0 ðβ β0 Þ þ ðβ β0 Þ2 þ O ðβ β0 Þ3 ; α0 4α30
σ ″i :
α ¼ α0
α0 ðβ β0 Þ2 þ O 4ðα20 β20 Þ
ðβ β0 Þ3 :
ð31Þ ð32Þ
In a small neighborhood of the double point P i ðα0 ; β0 Þ the curve σ 0i is close to the straight-line segment passing through this point and making the angle arctanðβ0 =α0 Þ with the horizontal axis. However, the curve σ ″i represents a piece of the parabola lying to the left of the point P i ðα0 ; β0 Þ, with a vertical tangent at this point. Stability intervals in the special cases α ¼ β and α ¼0: The stability domains g s ðs ¼ 0; 1; 2…Þ are simply connected. They originate on the β ¼0 axis but do not end anywhere inside the domains G1 and G2 with growth of β. For large values of β the stability domains become very narrow, and their boundaries lead asymptotically to the curves with the tangent inclination equal to 1 to the horizontal axis. Each of the domains gs is once intersected by the straight line α ¼ β separating the domains G1 and G2, and once by the vertical straight line α ¼0. In the case α¼ β the quantity a is equal to pffiffiffiffiffiffi a ¼ cos 2x x sin 2x ðx ¼ π 2α=2Þ: ð33Þ The straight line α ¼ β separates the intervals ðrÞ αðlÞ s o α o αs
ðs ¼ 0; 1; 2…Þ
ð34Þ αðlÞ s
αsðrÞ
¼ s =2 and is the nearest to in the stability domain gs. Here αðlÞ s (and larger than it) root of the equation x tan x ¼ 1 in the case of even s and the equation tan x ¼ x in the case of odd s. Fig. 2 shows the first seven stability intervals on the straight line α¼β defined by 2
ð0; 0:15Þ; ð1=2; 0:834Þ; ð2; 2:378Þ; ð9=2; 4:8917Þ; ð8; 8:3973Þ; ð25=2; 12:9Þ; ð18; 18:4016Þ: At s-1 width of the stability intervals leads to the limiting quantity 4=π 2 . On the straight line α ¼ 0 we have pffiffiffi pffiffiffi a ¼ cosh ðπ βÞ cos ðπ βÞ; ð35Þ and the stability intervals are defined by the inequalities ðrÞ βðlÞ s oβ o βs
ðs ¼ 0; 1; 2…Þ:
ð36Þ
The left boundary βðlÞ s of interval (3.16) is a root of the equation a ¼1 for even s and the equation a ¼ 1 for odd s. The right boundary βsðrÞ , to the contrary, is a root of the equations a ¼1 and a ¼ 1 for odd and even s respectively. The first five stability intervals on the straight line α ¼ 0 are defined by ð0; 0:3562Þ; ð2:2326; 2:2669Þ; ð6:2488; 6:2512Þ; ð12:2499; 12:25Þ; ð20:2499; 20:25Þ:
15
Width of the stability intervals decreases as s increases. For sufficiently large s it is exponentially small and close to the quantity 8 expð π=2Þ expð πsÞ: π Linear normalization of map (17): We make the linear change of variables of the form q ¼ n11 Q þ n12 P;
p ¼ n21 Q þ n22 P
ðd ¼ n11 n22 n12 n21 a 0Þ
ð37Þ
As a result the matrix X of the linearized map takes a real normal form J. Replacement of variables (37) is a canonical transformation 1 with the valency c ¼ d . The quantities nij and the matrix J are given in the paper [5]. In the new variables the map conserves square and can be written in the form similar to (17) ∂F 4 Q Q 0 ∂P þ ⋯ 0 ð38Þ ¼ J : ∂F 4 P P 0 þ ∂Q þ⋯ 0 Here the dots denote the sets of terms higher than the third order in Q 0 ; P 0 , and F4 is the fourth-order form in Q 0 ; P 0 defined by F 4 ðQ 0 ; P 0 Þ ¼ cS4 ðn11 Q 0 þ n12 P 0 ; n21 Q 0 þ n22 P 0 Þ
∑ f νμ Q ν0 P μ0 :
νþμ ¼ 4
ð39Þ Consider now several cases differing from one another. (1) If the point ðα; βÞ lies inside one of the domains g s ðs ¼ 0; 1; 2; …Þ of stability in the first approximation, then the normal form of the linearized map defines turn at the angle 2πλ as cos 2πλ sin 2πλ J¼ ð40Þ ; sin 2πλ cos 2πλ 8 1 s > > arccos a þ < 2π 2 λ¼ 1 sþ1 > > arccos a þ : 2π 2
if s ¼ 0; 2; 4; …; if s ¼ 1; 3; 5; …:
ð41Þ
(2) Consider the points ðα; βÞ on the boundaries of stability and instability domains when a ¼ 1 or 1. We will get the results on stability in the first approximation for these points. We first consider the double points P i ðα0 ; β0 Þ ði ¼ 1; 2; 3; …Þ at we have 1 0 J¼ ð42Þ ; 0 1 on the curves a ¼ 1 and 1 0 J ¼ : 0 1
ð43Þ
on the curves a ¼1. Since matrices (42) and (43) are diagonal, the equilibrium position in question is stable in the first (linear) approximation for the sets of parameters α0 ; β0 corresponding to any of the double points P i ðα0 ; β0 Þ ði ¼ 1; 2; 3; …Þ. Off the double points on the boundary curves a ¼ 1 the matrix J of the transformed map (38) has the form of a Jordan box 1 1 J ¼ ð44Þ ; 0 1 and in the boundaries a¼ 1, 1 1 J¼ : 0 1
ð45Þ
Therefore, for the points ðα; βÞ lying on the boundaries a ¼ 7 1 of the stability domains the equilibrium position under consideration is
16
A. Markeev / International Journal of Non-Linear Mechanics 73 (2015) 12–17
unstable in the first approximation, except for the double points P i ðα0 ; β0 Þ ði ¼ 1; 2; 3; …Þ at which it is stable in the first approximation.
4. Results on non-linear investigation To obtain a rigorous solution of the stability problem in the case jajr 1 non-linear analysis is necessary. This problem is equivalent to one of the immovable point Q 0 ¼ 0; P 0 ¼ 0 of map (38). Conditions for stability and instability can be expressed in terms of the coefficients f νμ in the fourthorder form (39) [10,12]. The points ðα; βÞ lie inside the stability domains g s ðs ¼ 0; 1; 2; …Þ in the first approximation. In this case we must distinguish between resonance and non-resonance values of the parameters. Since there are no the second-order terms in Q 0 ; P 0 on the right-hand side of equality (38), the third-order resonance is unimportant. In each of the domains gs one of the fourth-order resonances occurs for which 4λ ¼ 2s þ 1 ðs ¼ 0; 1; 2; …Þ. For small β the fourth-order resonance curves are defined by α¼
ð2s þ 1Þ2 4 þ 16 ð2s þ 1Þ2
4ð 1Þs 2 β þ Oðβ4 Þ: 1 πð2s þ1Þ
ð46Þ
the equilibrium position is stable [10,12]. If the parameters α, β belong to one of the fourth-order resonance curves, then the sufficient condition for stability is the inequality ð3f 40 þf 22 þ 3f 04 Þ2 ðf 40 f 22 þ f 04 Þ2 ðf 31 f 13 Þ2 4 0:
ð47Þ
Checking stability conditions (46), (47) was carried out numerically for ranges of the parameters α and β corresponding to Fig. 2. Analytical results were obtained for the cases α ¼ β and 0 oβ 5 1. It was found that in all domains g s ðs ¼ 0; 1; 2; …Þ of stability in the first approximation conditions (46), (47) are satisfied. Therefore, the equilibrium position under consideration is also stable when considering the rigorous non-linear statement of the problem. The parameters α and β correspond to the double points on the boundaries of the domains g s ðs ¼ 0; 1; 2; …Þ. Calculations showed [5] that at the double points P i ðα0 ; β0 Þ ði ¼ 1; 2; 3; …Þ on the boundaries of the stability domains in the first approximation the function S4 has the form ! π α0 4 2 2 4 α0 q0 þ2q0 p0 þ 2 p0 : S4 ðq0 ; p0 Þ ¼ 32 α0 β20 Since the function S4 ðq0 ; p0 Þ is of fixed sign, the equation S4 ð sin ψ; cos ψ Þ does not have real roots. According to [10,12], it follows that at all double points the equilibrium position considered is stable in Lyapunov's sense. Segments of the boundaries of the domains g s ðs ¼ 0; 1; 2; …Þ outside the double points. If the point ðα; βÞ lies on the boundaries of the domains g s ðs ¼ 0; 1; 2; …Þ and does not coincide with one of the double points, then a sign of the coefficient f40 in (39) is required to obtain a conclusion on stability in the non-linear problem [5,10,12]. If a ¼1 and the inequality f 40 o 0
f 40 ¼
π6 2 β þ Oðβ4 Þ: 6
ð48Þ
ð49Þ
For sufficiently small β the quantity f40 is positive and, therefore, instability occurs. Consider now left and right boundaries γ 0s and γ ″s of instability domains issuing from the points ðs2 =4; 0Þ of the β ¼0 axis. For small values of β they are defined by Eq. (27) if s is odd, and (28), (29) if s is even. For odd s the corresponding coefficient f40 has the form f 40 ¼ 7
16π 3 β þ Oðβ4 Þ s7
ð50Þ
Here the upper and lower signs are related to the boundaries γ 0s and γ ″s respectively. In the case of even s we get f 40 ¼
The fourth-order resonant curves are not shown in Fig. 2. If the parameters α, β lie inside the stability domain in the first approximation and do not belong to the fourth-order resonance curves, then in the case 3f 40 þ f 22 þ 3f 04 a 0
is satisfied, then the equilibrium position is stable in Lyapunov's sense, whereas in the case of the opposite sign in inequality (48), unstable. In the case a ¼ 1, to the contrary, the equilibrium position is unstable if inequality (48) is satisfied and stable at the opposite sign in it. Calculations show that on the boundary γ0 of the domain g0 defined by (26) the coefficient f40 has for small β the form
32π 4 6 β þ Oðβ7 Þ s10
ð51Þ
on the boundary γ 0s and f 40 ¼
8π 6 8 β þ Oðβ9 Þ s12
ð52Þ
on the boundary γ ″s . From (50)–(52), it follows that stability occurs on the left boundaries γ 0s and instability on the right boundaries γ ″s . Suppose the parameters α and β correspond to the boundaries of the stability and instability domains in vicinity of the double points P i ði ¼ 1; 2; 3; …Þ. On the boundaries σ 0i defined by (31) we get f 40 ¼ 8
π 6 β30 ðβ β0 Þ5 þO ðβ β0 Þ6 ; 2 4 2 512α0 ðα0 β0 Þ
ð53Þ
and on the boundaries σ ″i defined by (3.12), f 40 ¼ 7
π 4 α0 β30 32ðα20 β20 Þ3
ðβ β0 Þ3 þ O ðβ β0 Þ4 :
ð54Þ
In equalities (53), (54) the upper signs correspond to the cases when s is odd and the double point P i ðα0 ; β0 Þ is intersection of the instability domain boundaries issuing from the point ðs2 =4; 0Þ of the β¼0 axis. The lower signs are related to the similar points in the case of even s. From expressions (53), (54) and the aforementioned conditions for stability and instability, we get the following results. If s is odd, then in a small vicinity of the point P i ðα0 ; β0 Þ on the curve σ 0i the equilibrium position is unstable for β 4β0 and stable for β o β0 , whereas on the curve σ ″i it is stable for β 4 β0 and unstable for β oβ0 . Similarly, if s is even, then on the curve σ 0i the equilibrium position is unstable for β 4 β0 and stable for β o β0 , and on the curve σ ″i it is stable for β 4 β0 and unstable for β oβ0 . At arbitrary points ðα; βÞ on the boundaries of the stability domains g s ðs ¼ 0; 1; 2; …Þ which do not coincide with any of the double points P i ðα0 ; β0 Þ ði ¼ 1; 2; 3…Þ the conditions for stability were checked numerically. The equilibrium position in question was found to be unstable at points on the left boundaries of the domains gs in Fig. 2 and stable at points on the right boundaries.
A. Markeev / International Journal of Non-Linear Mechanics 73 (2015) 12–17
5. Conclusions Let us summarize the results obtained (see Fig. 2). For any values of the parameters α and β lying inside the domains g s ðs ¼ 0; 1; 2; …Þ or on their right boundaries the equilibrium position q ¼0 of the system defined by Eq. (1) is stable in Lyapunov's sense. If the parameters α and β lie on the left boundaries, then the equilibrium position is unstable, except for the double points P i ði ¼ 1; 2; 3; …Þ at which it is stable in Lyapunov's sense. Acknowledgment The work is carried out at the cost of the grant of the Russian Scientific Foundation (Project 14-21-00068) at the Moscow Aviation Institute (National Research University). References [1] E. Meissner, Ueber Schüttelerscheinungen in Systemen mit periodisch veränderlicher Elastizität, Schweiz. Bauztg. 72 (11) (1918) 95–98.
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[2] B. Van der Pol, M.J.O. Strutt, On the stability of solutions of Mathieu's equation, Philos. Mag. Ser. 7 (5) (1928) 18–38. [3] M.J.O. Strutt, Lamèsche, Matchieusche und verwandte Funktionen in Physik und Technik, vol. 1. Ergeh. Math. Grenzgeb., 1932, pp. 199–323. [4] V.I. Arnold, Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics, vol. 60, 2nd ed., Springer, September 23, 1997. [5] A.P. Markeev, On Meissner's nonlinear equation, Nonlinear Dyn. 7 (3) (2011) 531–547 (in Russian). [6] J.M. Almira, P.J. Torres, Invariance of the stability of Meissner's equation under a permutation of its intervals, Ann. Mat. Pura Appl. 180 (2) (2001) 245–253. [7] S. Timoshenko, Vibration Problems in Engineering, John Wiley & Sons, 1974 521 p. [8] Oscillations in nonlinear systems, Hand-book, vol. 1, in: V.V. Bolotin (Ed.), Vibrations in Techniques, 6 vols., Mashinostroyeniye, Moscow, 1999 (in Russian). [9] A.P. Markeev, Theoretical Mechanics, Regular and Chaotic Dynamics, Moscow – Izhevsk, 2007, 592 p. (in Russian). [10] A.P. Markeev, Stability of equilibrium states of Hamiltonian systems: a method of investigation, Mech. Solids. J. Russ. Acad. Sci. 39 (6) (2004) 1–8. [11] I.G. Malkin, Theory of stability of motion, United States Atomic Energy Commission, 1952, 456 p. [12] A.P. Markeev, On area-preserving mappings and their applications in dynamics of systems with collisions, Mech. Solids J. Russ. Acad. Sci. 2 (1996) 37–54.