Stability Chart

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International Journal of Non-Linear Mechanics 39 (2004) 1319–1331

Stability chart of parametric vibrating systems usingenergy-rate method

G. Nakhaie Jazar∗

Department of Mechanical Engineering and Applied Mechanics, North Dakota State University, Fargo, ND 58105-5285, USA

Received 5 March 2003; accepted 6 August 2003

Abstract

Based on the integral of energy and numerical integration, we introduce, develop, and apply a general algorithm to predictparameters of a parametric equation to produce a periodic response. Using the new method, called energy-rate, we are ableto 3nd not only stability chart of a parametric equation which indicates the boundaries of stable and unstable regions, butalso periodic responses that are embedded in stable or unstable regions.

There are three main important advantages in energy-rate method. It can be applied not only to linear but also to non-linearparametric equations; most of the perturbation methods cannot. It can be applied to large values of parameters; most ofthe perturbation methods cannot. Depending on the accuracy of numerical integration method, it can also 3nd the value ofparameters for a periodic response more accurate than classical methods, no matter if the periodic response is on the boundaryof stability and instability or it is a periodic response within the stable or unstable region.

In order to introduce the energy-rate method and indicate its advantages we apply the method to the standard Mathieu’sequation,

7x + ax − 2bx cos(2t) = 0

and show how to 3nd its stability chart for the large values of b in a–b plane. The results are compared with McLachlan’sreport (Theory and Application of Mathieu Function, Clarendon, Oxford, 1947).? 2003 Elsevier Ltd. All rights reserved.

Keywords: Mathieu stability chart; Energy-rate method; Parametric vibrations

1. Introduction

In this paper, an applied algorithm based on integralenergy and numerical integration will be introduced,developed, and presented to determine the parame-ters of parametric equations corresponding to periodicsolutions.

∗ Tel.: +1-701-231-8303; fax: +1-701-231-8913.E-mail address: [email protected] (G.N. Jazar).

The algorithm is based on a method calledenergy-rate method that can be applied to the follow-ing general 3rst-order diEerential equation:

7x + f(x) + g(x; x; t) = 0: (1)

There are some important advantages in energy-ratemethod comparing to the perturbation methods andFloquet theory. First, it can be applied to non-linearparametric equations as well as linear equations.Floquet theory, similar to the most of perturbationmethods cannot be applied on non-linear equations.

0020-7462/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved.doi:10.1016/j.ijnonlinmec.2003.08.009

mailto:[email protected]

1320 G.N. Jazar / International Journal of Non-Linear Mechanics 39 (2004) 1319–1331

Second, it can be applied not even to the small val-ues of parameters, but also to large values. Mostof the approximation and perturbation methods arestick to the very small values of parameters. Third,depending on the accuracy of numerical integrationmethod, it can 3nd the value of parameters for a pe-riodic response faster and more accurate than otherclassical methods. The accuracy of most classicalmethods depend on smallness of the value of param-eters, and the number of terms in perturbed solution.Their accuracy cannot sometimes be increased byincreasing the degree of polynomial or the number ofterms in perturbed solution. Fourth, the algorithm ofdetection periodic solution using energy-rate methodcan be adjusted to detect periodic response withina stable or unstable region, as well as periodic so-lutions on the boundary of stability and instabilityregions.I study the well-known Mathieu’s equation as an

example to show the applicability of the energy-ratemethod in identifying the stability chart, and someof the advantages of method. The results are interest-ing because they uncover some hidden facts about theMathieu’s equation.The Mathieu’s equation

7x + ax − 2bx cos(2t) = 0; x =dxdt

(2)

is the simplest and the most widely known paramet-ric diEerential equation, in which a and b are constantparameters. One of the most interesting characteristicsof this linear equation is that depending on the coeI-cients, it may have bounded or unbounded solutions.Fig. 1 depicts the stability chart of Mathieu’s equation(2) based on McLachlan’s [1] results.Parametric equations govern problems of the great-

est diversity in astronomy and theoretical physics andstability of the oscillatory processes in non-linear sys-tems. They have accordingly been the subject of a vastnumber of investigations since the beginning of thelast century [2,3].Parametric equation arises in applied mathematics

in three main groups of problems. The 3rst groupis the transverse vibrations of a taut elastic member.The second group of problems arises typically in themodulation of radio carrier wave [4]. The third groupof problems come from the standard procedure ofinvestigating the stability of a periodic motion in a

Fig. 1. Mathieu stability chart based on the numerical values,generated by McLachlan [1].

non-linear system. A linearization in neighbourhoodof a periodic motion results in a linear diEerentialequation with periodic coeIcients, i.e. Floquet theory.The paradigm example is given by Mathieu’s equa-tion [5]. The Mathieu’s equation is commonly knownas the equation of the elliptic cylinder functions. Thecontribution to the theory of diEerential equations, re-lated to the third group, which has been stimulatedby problems in celestial mechanics, has come chieLyfrom Hill and Poincare [6].Examples of vibrating elastic bodies are: the lateral

vibration of stretched strings and thin rods, which areperhaps the most amenable to theoretical and exper-imental treatment [7]. The linear and various modesof vibration of bars and beams, which was originatedback by Daniel Bernoulli, was investigated completelyby Rayleigh [8] and Love [9]. It is well known thatclassical linearized analysis of the vibrating strings androds could lead to some results related to the Math-ieu’s equation, which are reasonably accurate if onlythe tension and displacements are assumed to be small[10].

2. Historical background and literature review

The 3rst recorded demonstration of parametric be-haviour belongs to Faraday [11] in 1831, in which heproduced wave motion in Luids by vibrating a plate incontact with the Luid [12]. Shortly afterwards, manyscientists discovered parametric phenomena. Among

G.N. Jazar / International Journal of Non-Linear Mechanics 39 (2004) 1319–1331 1321

them are, Melde (1860), Mathieu (1860), Matthiessen(1870), Floquet (1883), Hill (1886), Rayleigh (1887),Stephenson (1906), Meissner (1918), Strutt (1927),van der Pol (1928), and Barrow (1923), in the late19th and the beginning of the 20th century [13].It is known that the 3rst detailed theory relevant

to the study of periodically time varying systems wasgiven by Mathieu [14], Hill [15], and Lord Rayleigh[16] in 1860–1890. One of the most important earlierworks on the behaviour of periodically time-varyingsystems was that by Hill in 1886, in which he laiddown the very mathematical foundations of the stabil-ity theory of parametric systems [15]. The 3rst anal-ysis of parametric resonance of a structural con3g-uration (a pinned perfect column) was presented byBeliaev [17]. Many studies started appearing in theliterature in the late 1940s and 1950s. Reviews on thesubject, including historical sketches, may be found inthe works of Beilin and Dzhanelidze [18]. For morerecent contributions on the subject, the reader is re-ferred to the work done by Yao [19], Hsu [42], Es-mailzadeh and Nakhaie Jazar [20,21], Luo [22], andZounes and Rand [23].Solution of the Mathieu’s equation depends on two

independent parameters, a and b. It is known fromFloquet theory that its solution is, in general, of thefollowing structure:

x(t) = C1et’(t) + C2e−t’(−t) (3)

in which the function ’(t) is periodic. The character-istic exponent, , is a time invariant constant that de-pends in an intricate way upon the parameters a andb. If it is real, the solution becomes exponentially in-3nite, i.e., a so called unstable solution. If the expo-nent is purely imaginary the solutions remain boundedalong the real axis. The intermediate case in which = 0 is of especial importance because it includes asolution known as a Mathieu function which is peri-odic [24]. Broadly speaking, the determination of theMathieu functions is the matter of prime importancein the applications of the equation.The most comprehensive treatment of classical

methods for analyzing the Mathieu’s equation hasbeen given by McLachlan [1]. It is known that theMathieu’s equation (2), could have periodic solutiondepending on parameters a and b, and independent ofthe initial conditions, due to the linearity of the equa-tion. The relationship between a and b for periodic

solution generates a graph which is called a stabilitychart [4]. The relation could be developed analyticallyby Fourier coeIcients or perturbation methods, butall of these methods fail to predict the stability chartwhen b is large.Many authors investigated the stability chart of the

time-varying systems including Mathieu’s equationusing a variety of concepts such as Lyapunov expo-nents [25], PoincareMapping [26], Lyapunov–Floquettransformation, and perturbative Hamiltonian normalforms [27]. None of these analysis could give accurateprediction of the stability boundaries when b is largeand/or some non-linearity is presented. Due to the na-ture of the solutions given by the classical approaches,extensive calculation is generally required to ensuresuIcient convergence to give accurate answers. De-termination of stability, classically, rests upon the useof the Hill determinant procedure [28]. For values ofa and b signi3cantly larger than unity, quite large de-terminant need to be calculated which is also a timeconsuming task. Further, these may diverge initiallyprior to acceptable convergence [13].Taylor and Narendra [29] and later on Gunderson et

al. [30] found the stability boundaries of the dampedMathieu’s equation using Laplace transformation andLyapunov function. Their stability theory was onlycorrect for small multiplier of both damping and pe-riodic terms.Within the limits of numerical accuracy, most of

these procedures have the appeal that they give exactresults since they do not rely upon algebraic ap-proximations to the solutions of the equation. Theirshortcomings, however, lie with their numerical error,especially when solutions near stability boundariesare required. Accuracy is also governed by the num-ber of iteration points chosen for iterative techniquesper period of the periodic coeIcient. Numerical so-lution of the Mathieu’s equation has been extensivelyused in the past to model systems which contain asinusoidal time-varying parameter [31]. Rand andHastings [32] used numerical integration to determineregions of stability for a quasi-periodic Mathieu’sequation. They assumed that the equation is stablefor a given initial condition r(0), if r(t)¡ 106 r(0)for 1000 time units and for all t between 0 and 1000,where r(t) =

√x2(t) + x2(t). Using a numerical inte-

gration, they could develop and present some usefulstability charts.


3. The investigation method

Stability analysis for even a single, second order,linear diEerential equation with periodic coeIcients,such as Mathieu’s equation, is rather cumbersome, butdiEerent methods are available. Among them are themethod of in3nite determinants [33], the perturbationmethods [34], the Galerkin method [35], and the clas-sic Floquet method [36].The method of continued fractions is the most

strong method that McLachlan used to 3nd the sta-bility chart of Mathieu’s equation. This method startsby knowing that the periodic solution of (2) whichadmit the period 2� falls into four classes. Each classdepends on its Fourier series which involve, respec-tively, cosines or sines of even or odd multiples of t.They are de3ned by Ince [37] as

ce2n =∞∑r=0

C(2n)2r cos(2rx) have period �; (4)

se2n+1 =∞∑r=0

S(2n+1)2r+1 sin((2r + 1)x)

have period 2�; (5)

ce2n+1 =∞∑r=0

C(2n+1)2r+1 cos((2r + 1)x)

have period 2�; (6)

se2n+2 =∞∑r=0

S(2n+1)2r+1 sin((2r + 2)x)

have period �; (7)

which are corresponding to characteristic numbersdenoted by a2n; b2n+1; a2n+1, and b2n+2. In everycase, satisfaction of the diEerential equation necessi-tates recurrence formulae connecting three successivecoeIcients. The continued fraction forms the basisof the technique developed by Ince [37] and Gold-stein [38] for the computation of the characteristicsnumbers.If each series of (4)–(7) is substituted in turn in

the Mathieu’s equation (2), and the coeIcients ofcos(2rt); cos((2r+1)t); sin((2r+1)t); sin((2r+2)t)equated to zero for r = 0; 1; 2; : : :, the follow-ing recurrence relations are obtained, respectively

[1,37,38],

r = 0

r = 1

r¿ 2

aC0 + bC2 = 0;

(a− 4)C2 + b(2C0 + C4) = 0;

(a− 4r2)C2r + b(2C2r−2 + C2r+2) = 0;

(8)

r = 0

r¿ 1

(a− 1 + b)C1 + bC3 = 0;

(a− (2r + 1)2)C2r+1

+b(C2r−1 + C2r+3) = 0;

(9)

r = 0

r¿ 1

(a− 1− b)S1 + bS3 = 0;

(a− (2r + 1)2)S2r+1

+b(S2r−1 + S2r+3) = 0;

(10)

r = 0

r¿ 1

(a− 4)S2 + bS4 = 0;

(a− (2r + 2)2)S2r+2

+b(S2r + S2r+4) = 0:

(11)

Using the recurrence relations (8)–(11) we can 3nd acontinued fraction equation between a and b such as

a= 2− b2

4

1− a4

14

b2

16

1− a16

116

b2

36

1− a36

× 136

b2

64

1− a64

· · · 14(r − 1)2

b2

4r2

1− a4r2 − 1

4r2 Zr+1

(12)

Zr =− b2

4r2

1− a4r2 − 1

4r2 Zr+1; lim

r→∞Zr = 0

for cos(2rt), to draw the stability chart of Fig. 1.A similar method was used by McLachlan to reportTable 1 for stability chart of the Mathieu’s equa-tion (2). In Table 1, the numerical values related toaci and asi are the stability boundaries for ith co-sine elliptic (cei) and ith sine elliptic (sei) functions,respectively. Fig. 1 is a graphical representation ofTable 1.It can be seen in Table 1 and Fig. 1 that continued

fractions method is strong enough to detect the sta-bility boundaries for large values of parameters. Themethod of continued fractions was used by later in-vestigators, and it is shown that the continued fraction

G.N

.Jazar/International

Journalof

Non-L

inearMechanics

39(2004)

1319–1331

1323Table 1McLachlan’s report for Mathieu stability chartb ac0 as1 ac1 as2 ac2 as2 ac3 as4 ac4 as5 ac5 as6

0, 0, 1, 1, 4, 4, 9, 9, 16, 16, 25, 25, 36,1, −0.4551386, −0.1102488, 1.8591081, 3.9170248, 4.3713010, 9.0477393, 9.0783688, 16.0329701, 16.0338323, 25.0208408, 25.0208543, 36.0142899,2, −1.5139569, −1.3906765, 2.3791999, 3.6722327, 5.1726651, 9.14062277, 9.3703225, 16.1276880, 16.1412038, 25.0833490, 25.0837778, 36.0572070,3, −2.8343919, −2.7853797, 2.5190391, 3.2769220, 6.0451969, 9.2231328, 9.91155063, 16.2727012, 16.3387207, 25.1870798, 25.1902855, 36.12887124, −4.2805188, −4.2591829, 2.3180082, 2.7468810, 6.8290748, 9.2614461, 10.6710271, 16.4520353, 16.6468189, 25.3305449, 25.3437576, 36.229441145, −5.8000460, −5.7900806, 1.8581875, 2.0994604, 7.4491097, 9.2363277, 11.5488320, 16.6482199, 17.0965817, 25.5108160, 25.5499717, 36.35886686, −7.3688308, −7.3639110, 1.2142782, 1.3513812, 7.8700645, 9.1379058, 12.4656007, 16.8446016, 17.6887830, 25.7234107, 25.8172720, 36.51706677, −8.9737425, −8.9712024, 0.4383491, 0.5175454, 8.0866231, 8.9623855, 13.3584213, 17.0266608, 18.4166087, 25.9624472, 26.1561202, 36.70350278, −10.6067292, −10.6053681, −0.4359436, −0.3893618, 8.1152388, 8.7099144, 14.1818804, 17.1825278, 19.2527051, 26.2209995, 26.5777533, 36.91721319, −12.2624142, −12.2616617, −1.3867016, −1.3588101, 7.9828432, 8.3831192, 14.9036797, 17.3030110, 20.1609264, 26.4915472, 27.0918661, 37.156695010, −13.93698, −13.9365525, −2.3991424, −2.3821582, 7.7173698, 7.9860691, 15.5027844, 17.3813807, 21.1046337, 26.7664264, 27.7037687, 37.419858812, −17.3320660, −17.3319184, −4.5701329, −4.5635399, 6.8787369, 7.0005668, 16.3015349, 17.3952497, 22.9721275, 27.3000124, 29.2080550, 38.006008714, −20.7760553, −20.7760004, −6.8934005, −6.8907007, 5.7363123, 5.7926295, 16.5985405, 17.2071153, 24.6505951, 27.7697667, 31.0000508, 38.648471916, −24.2586795, −24.2586578, −9.33523671, −9.3341097, 4.3712326, 0.3978962, 16.4868843, 16.8186837, 26.0086783, 28.136359, 32.9308951, 39.315010818, −27.7728422, −27.7728332, −11.8732425, −11.8727265, 2.8330567, 2.8459917, 16.0619754, 16.2420804, 26.9877664, 28.3738582, 34.8530587, 39.972351120, −31.3133901, −31.3133862, −14.4913014, −14.4910633, 1.15422829, 1.1607057, 15.3958109, 15.4939776, 27.5945782, 28.4682213, 36.6449897, 40.589664124, −38.4589732, −38.4589724, −19.9225956, −19.9225403, −2.5397657, −2.5380779, 13.5228427, 13.5527965, 27.8854408, 28.2153594, 39.5125519, 41.605709928, −45.6733696, −45.6733694, −25.5617471, −25.5617329, −6.5880630, −6.5875850, 11.1110798, 11.1206227, 27.2833082, 27.4057488, 41.2349503, 42.224841532, −52.9422230, −52.9422229, −31.3651544, −31.3651505, −10.9143534, −10.9142090, 8.2914962, 8.2946721, 26.0624482, 26.1083526, 41.9535112, 42.393942836, −60.2555679, −60.2555679, −37.3026391, −37.3026380, −15.4667703, −15.4667243, 5.1456363, 5.1467375, 24.3785094, 24.3960665, 41.9266646, 42.1183561

method

canbe

appliedto

alllinear

parametric

diEer-ential

equations,butitisnot

applicableto

non-linearparam

etricequations

[39].Due

tothis

problem,we

needamore

reliableand

accuratemethod

tobe

ap-plicable

notonly

forlinear

parametric

equationwith

largecoeI

cients,butalsoto

non-linearequations.In

thenext

section,weuse

theprinciple

ofthe

energyintegral

combined

with

numerical

integrationto

de-velop

anapplied

method

todeterm

inethe

transitioncurves.

4.Energy-rate

method

andstability

chartalgorithm

Consider

thatweare

lookingfor

aboundary

ofsta-

bilityfor

thesystem

7x+f(x)

+g(x;x;t)

=0;

(13)

where

g(0;0;t)=0

(14)

andf(x)

isasingle

variablefunction,and

g(x;x;t)is

anon-linear

time-dependent

function.The

functionsfand

gare

dependingon

asetofparam

eters.Writing

Eq.(13)

inthe

following

form:

7x+f(x)

=−g(x;x;t)

(15)

showsthatthe

systemisamodelofa

unitmass

particleattached

toaconservative

spring,acted

uponby

anon-conservative

force−g(x;x;t).T

hefree

motion

ofthe

systemisgoverned

by7x+f(x)=

0.Ingeneral,the

appliedforce

generatesor

absorbsenergy

dependingon

thevalue

ofparam

eters,position

andvelocity

ofthe

particle,andalso

time.If

f(x)

isarestoring

forceand

g(x;x;t)is

aperiodic

functionof

time,

thenwe

shouldexpect

atendency

tooscillate

[40].De3ning

akinetic,

potential,and

mechanical

energyfunction

forthe

systembyV(x)= ∫

f(x)d

x;T(x)=

12x2,and

E=T(x)+

V(x),respectively,w

ecan

write

anintegral

ofenergy

asfollow

s:

E=

ddt (E)=

ddt (12x2+ ∫

(f(x)x)dt )

=−xg(x;x;t):

(16)


This expression represents the rate of energy generatedor absorbed by the term −g(x; x; t).Suppose that for some set of parameters and a

non-zero response x(t); E = −xg(x; x; t) is negativethen, E continuously decreases along the path ofx(t). The eEect of g(x; x; t) resembles damping orresistance; energy is continuously withdrawn fromthe system, and this produces a general decrease inamplitude until the rate of initial amplitude dependedenergy runs out and a new solution make E = 0.On the other hand, if for a set of parameters and anon-zero response x(t); E = −xg(x; x; t) is positivethen, the amplitude increases so long as the path ofx(t) runs away. Evaluating a suitable numerical inte-gration can show if a set of parameters belongs to astable or unstable region by evaluating E.It is known that time derivative of mechanical en-

ergy E must be zero over one period for conservativeand autonomous systems in a steady state periodic cy-cle [41]. We may integrate the Mathieu’s equation (2),to get the following equation:12

ddt(x2 + ax2) = 2bxx cos(2t) = E; (17)

where E is the mechanical energy of the system. Inorder to 3nd a set of a and b to indicate a steady-stateperiodic response, we may choose a pair of parametersa and b and integrate Eq. (17) numerically.We may evaluate the averaged energy over a period

Eav =1T

∫ T

0E dt =

1T

∫ T

0(2bxx cos(2t)) dt;

T = 2� (18)

to compare with zero. If Eav is greater than zero, (a; b)belongs to a region that energy being inserted to thesystem and then Eq. (2) is unstable. However, if itis less than zero, then (a; b) belongs to a region thatenergy being extracted from the system and Eq. (2) isstable. On the common boundary of these two regions,Eav =0.Assume b is 3xed and we are looking for a to be

on a boundary that its left-hand side is stable and itsright-hand side is unstable then, if the chosen param-eters show that Eav is less than zero, increasing a in-creases Eav. On the other hand, if Eav is greater thanzero, decreasing a decreases Eav. Using this strategywe may 3nd the appropriate value of a such that Eavequates zero. In this way, we will 3nd a point on the

boundary that has a stable region on its left-hand sideand unstable region on its right-hand side. This groupof points constitutes a branch of �-periodic boundarycorresponding to ce2n or se2n+2.

By changing the strategy and looking for a to beon a boundary that its right-hand side is stable andits left-hand side is unstable then, we can generatethe 2�-periodic branches corresponding to ce2n+1 orse2n+1. Now we may change b by some incrementand repeat the procedure. This procedure could be ar-ranged in an algorithm called “stability chart algo-rithm”.

4.1. Stability chart algorithm

1. set a, equal to one of its special values;2. set b, equal to some arbitrary small value;3. solve the diEerential equation numerically;4. evaluate Eav;5. decrease (increase) a, if Eav¿ 0 (Eav¡ 0) by

some small increment;6. the increment of a must be decreased if Eav(ai) ·Eav(ai−1)¡ 0;

7. save a and b when Eav�1;8. while b¡b3nal, increase b and go to step 3;9. set a, equal to another special value and go to

step 2;10. reverse the decision in step 5 and go to step 1.

It is known that the boundary between stable and un-stable regions in the stability chart starts from somespecial values of a. So, it is better to start the algo-rithm from one of these characteristic values.

5. Applying the stability chart algorithm

In order to 3nd more accurate values than thosepresented in Table 1, we have used the elementsof McLachlan’s table as an initial value. Then,the Mathieu’s equation was solved numerically for0¡t¡T; T=2�, in order to include both �-periodicand 2�-periodic solutions. Two solutions of identicalperiod, even � or 2�, bound the region of instability,while two solutions of diEerent periods bound theregion of stability.

G.N

.Jazar/International

Journalof

Non-L

inearMechanics

39(2004)

1319–1331

1325Table 2Stability chart for Mathieu’s equation generated by the algorithmb ac0 as1 ac1 as2 ac2 as2 ac3 as4 ac4 as5 ac5 as6

0, 0, 1, 1, 4, 4, 9, 9, 16, 16, 25, 25, 361, −0.455138604, −0.110249001, 1.8591084, 3.9170245, 4.3713010, 9.04773900, 9.0783688, 16.0329698, 16.0338323, 25.0208405, 25.0208543, 36.0142899,2, −1.5139568851, −1.3906767, 2.3791999, 3.6722324, 5.1726648, 9.14062247, 9.3703225, 16.1276877, 16.1412038, 25.0833487, 25.0837778, 36.0572070,3, −2.834391889959, −2.7853799, 2.519039087, 3.2769217, 6.0451972, 9.2231325, 9.91155033, 16.2727009, 16.3387207, 25.1870795, 25.1902855, 36.12887094, −4.280518818368, −4.2591831, 2.31800817, 2.7468807, 6.8290745, 9.2614458, 10.6710271, 16.452035, 16.64681817, 25.3305446, 25.3437576, 36.229440845, −5.8000460207, −5.7900808, 1.8581875414, 2.0994601, 7.4491094, 9.2363274, 11.5488317, 16.6482196, 17.0965817, 25.5108157, 25.5499717, 36.35886656, −7.36883083214, −7.3639112, 1.2142781642, 1.3513809, 7.8700648, 9.1379055, 12.465601, 16.8446013, 17.688783, 25.7234104, 25.8172720, 36.51706647, −8.97374250574, −8.9712026, 0.43834908996, 0.5175451, 8.086623145, 8.9623852, 13.358421, 17.0266605, 18.416609, 25.9624469, 26.1561202, 36.70350248, −10.60672923566, −10.6053683, −0.43594360159, −0.3893621, 8.11523883, 8.7099141, 14.1818807, 17.1825275, 19.2527054, 26.2209992, 26.5777533, 36.91721289, −12.2624142182, −12.2616619, −1.386701566696, −1.3588104, 7.982843163, 8.3831189, 14.90368, 17.3030107, 20.1609267, 26.4915469, 27.0918661, 37.156694710, −13.93697995675, −13.9365527, −2.399142400351, −2.3821585, 7.7173698494, 7.9860688, 15.5027847, 17.3813804, 21.1046334, 26.7664261, 27.7037687, 37.419858512, −17.33206603507, −17.3319186, −4.570132851088, −4.5635402, 6.87873685481, 7.0005665, 16.301534946, 17.3952494, 22.9721278, 27.3000121, 29.2080550, 38.006008414, −20.77605531238, −20.7760006, −6.8934005,3343 −6.890701, 5.736312349375, 5.7926292, 16.598540469, 17.207115, 24.6505954, 27.7697664, 31.0000505, 38.648471616, −24.25867947476, −24.258658, −9.33523701, −9.33411, 4.371232605296, 0.3978959, 16.4868842565, 16.8186834, 26.008678, 28.1363587, 32.9308948, 39.315010518, −27.77284216366, −27.7728334, −11.87324251702, −11.8727268, 2.833056732297, 2.8459914, 16.06197536045, 16.2420801, 26.98776644, 28.3738579, 34.8530584, 39.972350820, −31.31339007051, −31.3133864, −14.49130142564, −14.4910636, 1.1542285900001, 1.1607054, 15.39581091191, 15.4939773, 27.594578154, 28.468221, 36.6449894, 40.589663824, −38.4589731694, −38.4589726, −19.92259564555, −19.9225406, −2.539765704347, −2.5380782, 13.52284271453, 13.5527962, 27.8854407976, 28.2153591, 39.5125519, 41.605709628, −45.673369663942, −45.673369545, −25.561747709501, −25.5617332, −6.588062973934, −6.5875853, 11.1110798375, 11.1206224, 27.28330817057, 27.4057485, 41.234950267, 42.224841232, −52.94222296412, −52.9422229395, −31.36515444857, −31.3651508, −10.91435338774, −10.9142093, 8.291496150361, 8.2946718, 26.062448443, 26.1083523, 41.9535111621, 42.393942536, −60.25556789136, −60.2555681, −37.30263912327, −37.3026383, −15.46677033703, −15.4667246, 5.145636255447, 5.1467372, 24.37850942577, 24.3960662, 41.92666456905, 42.1183558

Fig.2.

PlotofTE

avas

afunction

ofafor

b=

1.

The

averagedenergy

Eavisevaluated

bythe

simple

trapezoidalintegrationmethod

∫T

0Edt= ∫

T

0(2bxx

cos(2t))dt

=n−

1∑k=

0

h2[E(kh)

+E((k

+1)h)];

h=Tn;n=1000;

(19)

using1000

segments

overtheperiod

T.Follow

ingthe

algorithmpresented

inthe

previoussection,the

valueofamust

bedecreased

(orincreased)

ifEav¿

0(or

Eav¡

0).IfEavatstep

iwas

greaterthanzero

andEav

atstep

i−1was

lessthan

zero,thenthe

valueof

in-crem

entwhich

mustbe

addedtoa(orsubtracted

froma)

would

bedecreased

atstepi+

1in

ordertoprevent

cyclicrepetition.

Wehave

appliedthe

algorithmand

foundthe

resultstabulated

inTable

2,which

couldbe

compared

with

Table

1.Distribution

ofregions

ofinstability

will

bemore

clearifweletb→

0then,using

relations(8)–(11),w

e3nd

thatthesolutions

with

a2�-period

lieinpairs

nearthe

characteristicvalues

a=(2k

+1)

2;k=0;1;2;3;:::,

andthe

solutionswith

a�-period

liein

pairsnear

thecharacteristic

valuesa=

(2k)2;k=

0;1;2;3;:::.Both

casescan

becom

binedin

oneform

ulato

indi-cate

characteristicvalues,

a=k2;k=

0;1;2;3;:::.

Indeed,2�-periodic

transitioncurves

startfrom

a=

1;32;5

2;:::,and�-periodic

transitioncurves

startfroma=0;2

2;42;:::.

Inorder

toinvestigate

theapplicability

ofthe

method,w

ehave

plottedTE

avversus

ain

Fig.2,for


the line b = 1 shown in Fig. 1. It can be seen thatthe curve TEav cuts the a axis exactly at the samepoints where line b = 1 cuts the transition curves.Therefore, Fig. 1 could be assumed as intersection ofa three-dimensional surface

q(a; b) = TEav =∫ T

0(2bxx cos(2t)) dt

and the plane TEav = 0.Due to physical consideration, the 3rst instability

region, started at a = 1, is the most dangerous andhas therefore the greatest practical importance. Bolotin[33] calls this region the “principal region of dynamicinstability”. In the next section, we compare Tables 1and 2, and show the advantages of the stability chartalgorithm.

6. Comparison, application, and modi%edalgorithm

How important the determined transient curve of aparametric equation is, depends on the physical sys-tem, real period of free oscillation, closeness of designparameters to the transient curves, mechanism and rateof dissipation system. If mistakenly design parame-ters belong to an unstable region instead of periodic orstable region, then parametric resonance might occur.Table 3 shows the value of TEav for the 3rst ten dig-

its of �-periodic branch ac0 of Tables 1 and 2. The lastthree rows indicate that how important is the accuracyof 3nding the transition lines. On the 10th row, de-creasing 0.00000004325 unit could reduce TEav from

Table 3The values of TEav for 3rst 10 digits of ac0 for a and b from Tables 1 and 2

b a, Table 1 a, Table 2 TEav for Table 1 TEav for Table 2

1 −0.4551386 −0.455138604 −0.41036782785e − 6 −0.1213741795e − 102 −1.5139569 −1.5139568851 0.000033303989524 0.147415342e − 73 −2.8343919 −2.83439189959 0.000269337200715 0.58511601e − 74 −4.2805188 −4.280518818368 −0.004127213067066 −0.382606859e − 65 −5.800046 −5.8000460207 −0.03087386648124 −0.0004839133576346 −7.3688308 −7.36883083214 −0.2604694332158 0.0003061688201717 −8.9737425 −8.97374250574 −0.2235618874348 0.0002899104008728 −10.6067292 −10.60672923566 −5.978866028905 0.0027716387763759 −12.2624142 −12.2624142182 −12.03902889408 −0.0061065122282

10 −13.93698 −13.93697995675 104.6426231398 −0.07985525604

Fig. 3. Time response of the diEerence of amplitude.

104.6426231398 to −0:07985525604 signifying thatpoint (−13:93697995675; 10) is much more closer tothe transient curve than point (−13:93698; 10).We compare Tables 1 and 2 by study the Math-

ieu’s equation (2) for two pairs of a and b equal to(−0:4551386; 1) and (−0:455138604; 1); denotingby x1 and x2, respectively. The pairs of a and b belongto two identical elements of Tables 1 and 2, andTable 3 says that TEav¡ 0 for both of them. Althoughwe cannot recognize any diEerence between the plotsof time responses of the Mathieu’s equation for x1 andx2, the function y= |x1−x2| for 0¡t¡ 10� shown inFig. 3 indicates that the response of Mathieu’s equa-tion for x1 and x2 are not identical. It can be seen thatthe diEerence of the solutions is growing up, whichshows that the solution of Mathieu’s equation for pointx1 deviates from periodic response faster than itssolution for x2.


Fig. 4. Time response for a point on the boundary of stability.

Fig. 5. TEav as a function of time for a point on the boundary ofstability.

The time response shown in Fig. 4 depicts that theperiod of oscillation is �. It is also indicated in theplots of time history of TEav shown in Fig. 5.

The plot of TEav in Fig. 5 has an interestingbehaviour. The time integral of TEav is zero for0¡t¡�, but the value of time integral of TEav for0¡t¡�=2 is minus of the time integral of TEav for�=2¡t¡�. If time integral of Eav for 0¡t¡T isminus of the time integral of Eav for T ¡ t¡ 2T wecall that Eav has skew-symmetric property. When-ever Eav has skew-symmetric property then, thesystem is 2T -periodic instead of T -periodic. Hence,searching for responses having period T = n�, and�=n; n = 1; 2; 3; : : :, also includes responses havingperiod T = 2n� and 2�=n; n= 1; 2; 3; : : : .Now another advantage of energy-rate method will

be revealed that are not easily available by traditional

Fig. 6. Plot of TEav as a function of a for b = 5 integrated fordiEerent period of time.

perturbations and approximation methods. We ana-lyze super and sub-harmonic oscillation, and open newdoors of research to discover periodic responses of theparametric equations.Although mathematical theories can provide some

analytic basis, we may use a cross section of thethree-dimensional energy plots, similar to Fig. 2, to3nd the possibility of super and sub-harmonic oscil-lations. Integration of the Mathieu’s equation for aconstant b, and plotting Eav versus a indicates whereEav is negative (stability) and positive (instability).If the plot of Eav versus a, integrated for T = n� or�=n; n=1; 2; 3; : : :, crosses the axis Eav =0, then thereis a transition or periodicity curve separating stableand unstable regions.Fig. 6 depicts a plot of TEav for the Mathieu’s equa-

tion as a function of a. The value of b is set to 5. Theline b = 5 is plotted in Fig. 1 to give a view of thedomain of stability and instability. The period of in-tegration of TEav in Fig. 6 is diEerent for each curve.It is seen that the curve for T = �=3 and �=2 has justone zero for positive a, close to a = 9:1 and 7.4, re-spectively. Time response of the Mathieu’s equation atthose points would be periodic, although both of thembelong to instability regions of the Mathieu’s stabil-ity chart. Fig. 7 shows time response of the Mathieu’sequation for (a= 7:45; b= 5).The curve for T=� in Fig. 6 intersects the horizontal

axis exactly at points where the line b = 5 intersectsthe transition curves in Fig. 2, but the curve for T=2�not only intersects at the �-periodic and 2�-periodictransition curves of the Mathieu’s equation, but also


Fig. 7. Time history of �-periodic response of Mathieu’s equationfor (a = 7:45; b = 5), a periodic response in instability region.

Fig. 8. Time history of 4�-periodic response of Mathieu’s equationfor (a = 8:13; b = 5) and (a = 13:47; b = 5).

touches the horizontal axis at some other points. Forinstance, the 3rst touching point is around a = 8:13which is a point in stable region of Mathieu’s stabilitychart. Fig. 8 depicts two 4�-periodic time responsesusing the 3rst two touching points (a = 8:13; b = 5)and (a= 13:47; b= 5) of the curve T = 2� in Fig. 6.These two periodic solutions were found when wewere looking for 2�-periodic responses due to skewsymmetric characteristic of Eav. A 3�-periodic and a6�-periodic responses are shown in Fig. 9 for twotouching points (a=12:57; b=5) and (a=14:54; b=5) of the curve T = 3� in Fig. 6. These two superharmonic responses belong to the stability region ofthe Mathieu’s equation.Is it possible to have other periodicity lines in the

stability or instability is regions of Mathieu’s equa-

Fig. 9. Time history of 3�- and 6�-periodic response of Mathieu’sequation for (a = 12:57; b = 5) and (a = 14:54; b = 5).

tion? Figs. 6, 8 and 9 indicate that it is possible. Nowwe may have for instance, a 2�-periodic curve, whereboth left- and right-hand sides of that curve are stable.It is a periodic line “splitting” a stable region. We mayalso have a 2�-periodic curve, where both left- andright-hand sides are unstable; a periodic line “split-ting” an unstable region. The same situation could beseen for other sub and super harmonics.We may now modify the stability chart algorithm

to be able to detect splitting lines. The following algo-rithm is prepared to detect splitting lines of the Math-ieu’s equation.

6.1. Splitting chart algorithm

0. set period of integration T equal to a multipleof �;

1. set a, equal to a1, and a2, two arbitrary valuesin a stability region;

2. set b, equal to some arbitrary small value;3. solve the diEerential equation numerically for

both pairs of a, and b;4. evaluate (Eav)1 and (Eav)2;5. set a1 = a2;6. increase a2, by some small increment if

|(Eav)1|¿ |(Eav)2| and (a2 − a1)¿ 0or|(Eav)1|¡ |(Eav)2| and (a2 − a1)¡ 0else decrease a2, if|(Eav)1|¿ |(Eav)2| and (a2 − a1)¡ 0or|(Eav)1|¡ |(Eav)2| and (a2 − a1)¿ 0;


0 2 4 6 8 10 12

12

10

8

6

4

2

b

a0 2 4 6 8 10 12

12

10

8

6

4

2

b

a

0 2 4 6 8 10 12

12

10

8

6

4

2

b

a0 2 4 6 8 10 12

12

10

8

6

4

2

b

a

0 2 4 6 8 10 12

12

10

8

6

4

2

b

a0 2 4 6 8 10 12

12

10

8

6

4

2

b

a

(a) (b)

(c) (d)

(f)(e)

Fig. 10. (a) Mathieu’s �-periodicity chart and splitting curves, based on the “splitting chart algorithm”; (b) Mathieu’s 2�-periodicitychart and splitting curves, based on the “splitting chart algorithm”; (c) Mathieu’s 3�-periodicity chart and splitting curves, based on the“splitting chart algorithm”; (d) Mathieu’s 4�-periodicity chart and splitting curves, based on the “splitting chart algorithm”; (e) Mathieu’s5�-periodicity chart and splitting curves, based on the “splitting chart algorithm”; and (f) Mathieu’s periodicity chart and splitting curves,for �; 2�; 3�; 4�, and 5�-periodic solutions.


7. the increment of a must be decreased if a2repeats twice;

8. save a and b when (a2 − a1)�0;9. while b¡b3nal, increase b and go to step 1;10. set a, equal to a1, and a2, two arbitrary values

in another stability region and go to step 2.

It should be mentioned that the algorithms can beextended to cover the other parametric diEerentialequations whose stability chart is two dimensional orcould be reduced to two dimensional. Applying thealgorithms on parametric equation with more thantwo parameters produces a set of stability charts byvarying only two parameters and 3xing the others.We have applied the “splitting chart algorithm” to

3nd the splitting curves of Mathieu’s equation, andFig. 10(a)–(e) illustrates the results for �; 2�; 3�; 4�,and 5�-periodic solutions on the second and third sta-ble regions. Fig. 10(f) depicts all splitting curves re-lated to �; 2�; 3�; 4�, and 5�-periodic solutions. Thealgorithm can easily be applied to a wider range, andalso to the other periodic splitting curves.

7. Closure

We have developed a stability chart algorithm basedon energy-rate method, to 3nd the stability bound-aries of the parametric diEerential equations. The al-gorithm provides more informative results than per-turbation and approximation methods. The algorithmhas been applied to the Mathieu’s equation, and its sta-bility chart has been found, comparable to previouslyreported results. Although the procedure is sometimestime consuming, it generates a more eIcient and moreaccurate stability chart.The idea may be used to detect other periodicity

curves not necessarily dividing stability and instabilityregions. It may be applied to detect splitting curves,a periodicity curve embedded in a stable or unstableregion. The algorithm can be applied to any linearparametric diEerential equation as well as non-lineartime-varying equations.

Acknowledgements

The author acknowledges M. Rastgaar Aagaah asan assistant who developed the computer program toapply the energy-rate Algorithm.

References

[1] N.W. McLachlan, Theory and Application of MathieuFunctions, Clarendon, Oxford, 1947, U.P., Reprinted byDover, New York, 1964.

[2] E.T. Whittaker, G.N. Watson, A Course of Modern Analysis,4th Edition, Cambridge University Press, Cambridge, 1927.

[3] B. Van der Pol, M.J.O. Strutt, On the stability of the solutionof Mathieu’s equation, Philos. Mag. 5 (1928) 18.

[4] W.G. Bickley, The tabulation of Mathieu equations, Math.Tables Other Aids to Comput. 1 (11) (1945) 409–419.

[5] C. Hayashi, Nonlinear Oscillations in Physical Systems,Princeton University Press, Princeton, 1964.

[6] F.R. Moulton, W.D. MacMillan, On the solutions of certaintypes of linear diEerential equations with periodic coeIcients,Amer. J. Math. 33 (10) (1911) 65–96.

[7] G.F. Carrier, On the Nonlinear vibration problem of the elasticstring, Quart. Appl. Math. 3 (2) (1945) 157–165.

[8] J.W.S. Rayleigh, The Theory of Sound, Vol. 1, DoverPublications, New York, 1945 (Macmillan Company, 1894).

[9] A.E.H. Love, A Treatise on the Mathematical Theory ofElasticity, Dover, New York, 1927.

[10] L. Meirovitch, Analytical Methods in Vibrations, Macmillan,New York, 1967.

[11] M. Faraday, On a particular class of acoustical 3gures; and oncertain forms assumed by a group of particles upon vibratingelastic surfaces, Philos. Trans. R. Soc. London 121 (1831)299–318.

[12] E.R.M. Iwanowski, On the parametric response of structures,Appl. Mech. Rev. 18 (9) (1965) 699–702.

[13] J.A. Richards, Analysis Periodically Time Varying Systems,Springer, New York, 1983.

[14] E. Mathieu, Memoire sur le mouvement vibratoire d’unemembrane de forme elliptique, J. Math. Pure Appl. 13 (1868)137.

[15] G.W. Hill, On the part of the moon’s motion which is afunction of the mean motions of the sun and the moon, ActaMath. 8 (1886) 1–36.

[16] J.W. Strutt, (Lord Rayleigh), On the crispations of Luidresting upon a vibrating support, Philos. Mat. 16 (1883)50–53.

[17] N.M. Beliaev, Stability of prismatic rods subject tovariable longitudinal forces, Collection of Papers: EngineeringConstructions and Structural Mechanics, Leningrad, 1924, pp.149–167.

[18] E.A. Beilin, G.H. Dzhanelidze, Survey of work on thedynamic stability of elastic systems, Prikl. Math. I Mekh. 16(5) (1952) 635–648.

[19] J.C. Yao, Dynamic stability of cylindrical shells under staticand periodic axial and radial loads, AIAA J. 1 (6) (1963)1391–1396.

[20] E. Esmailzadeh, G. Nakhaie Jazar, Periodic solution of aMathieu–DuIng type equation, Internat. J. Nonlinear Mech.32 (5) (1997) 905–912.

[21] E. Esmailzadeh, G. Nakhaie Jazar, Periodic behaviour ofa cantilever with end mass subjected to harmonic baseexcitation, Internat. J. Nonlinear Mech. 33 (4) (1998)567–577.


[22] A.C.J. Luo, Chaotic motion in the generic separatrix bandof a Mathieu–DuIng oscillator with a twin-well potential, J.Sound Vib. 248 (3) (2001) 521–532.

[23] R.S. Zounes, R.H. Rand, Global behaviour of a nonlinearquasiperiodic Mathieu equation, Proceedings of DETC01,ASME 2001 Design Engineering Technical Conference,Pittsburgh, PA, September 2001, pp. 1–11.

[24] R.E. Langer, The solution of the Mathieu equation with acomplex variable and at least one parameter large, Trans.Amer. Math. Soc. 36 (3) (1934) 637–695.

[25] F. Colonius, W. Kliemann, Stability of time varying systems,ASME Design Engineering Technical Conferences DE. 84-1,Vol. 3, Part A, 1995, pp. 365–373.

[26] R.S. Guttalu, H. Flashner, An analytical study of stability ofperiodic systems by Poincare mappings, ASME Des. Eng.Tech. Conf. DE 84-1 Part A 3 (1995) 387–398.

[27] E.A. Butcher, S.C. Sinha, On the analysis of time-periodicnonlinear hamiltonian dynamical systems, ASME Des.Eng. Tech. Conf. DE 84-1 Part A 3 (1995)375–386.

[28] D.W. Jordan, P. Smith, Nonlinear Ordinary DiEerentialEquations, Oxford University Press, Oxford, 1999.

[29] J. Taylor, K. Narendra, Stability regions for the dampedMathieu equation, SIAM J. Appl. Math. 17 (1969)343–352.

[30] H. Gunderson, H. Rigas, F.S. VanVleck, A techniquefor determining stability regions for the dampedMathieu equation, SIAM J. Appl. Math. 26 (2) (1974)345–349.

[31] P.H. Dawson, N.R. Whetten, Ion storage in three dimensional,rotationally symmetric, quadrupole 3eld. I. Theoreticaltreatment, J. Vac. Sci. Technol. 5 (1968) 1–6.

[32] R. Rand, R. Hastings, A quasiperiodic Mathieu equation,ASME Des. Eng. Tech. Conf. DE 84-1 Part A 3 (1995)747–758.

[33] V.V. Bolotin, The Dynamic Stability of Elastic Systems,Holden-Day, San-Francisco, USA, 1964.

[34] A.H. Nayfeh, D.T. Mook, Nonlinear Oscillation, Wiley, NewYork, USA, 1979.

[35] P. Pederson, On stability diagrams for damped Hill equations,Quart. Appl. Math. 42 (1985) 477–495.

[36] L. Cesari, Asymptotic Behaviour and Stability Problems inOrdinary DiEerential Equations, 2nd Edition, Academic Press,New York, USA, 1964.

[37] E.L. Ince, Tables of the elliptic-cylinder functions, R. Soc.Edinburgh Proc. 52 (1932) 355–423.

[38] S. Goldstein, Mathieu functions, Trans. Cambridge Philos.Soc. 23 (1927) 303–336.

[39] G. Nakhaie Jazar, Analysis of nonlinear parametric vibratingsystems, Ph.D. Thesis, Mechanical Engineering Department,Sharif University of Technology, 1997.

[40] E. Esmailzadeh, B. Mehri, G. Nakhaie Jazar, Periodic solutionof a second order, autonomous, nonlinear system, NonlinearDyn. 10 (1996) 307–316.

[41] W. Szemplinska-Stupnicka, The Behaviour of NonlinearVibrating Systems, Kluwer Publishers, Dordrecht, 1990.

[42] C.S. Hsu, Impulsive parametric excitation: Theory, Journalof Applied Mechanics 39 (1972) 551–559.

Stability Chart

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