43
PEPORTS OF THE FACULTY OF ENGINEERING,
No.7 (1976)
On a Method of Solution for the Equations and Its Application to
Stability of Nonlinear V
NAGASAKI UNIVERSITY
Coupled Hill Type the Study of the
ibrations
by
**
Kazuo TAKAHASHI and Kiyokatsu KAWAHARA
In this paper, a method of the stability analysis for large amplitude steady state response of a nonlinear beam and flat plate under periodic excitation. The nonlinearity is attributed to the membrane tension which is developed when the beam and plate deflections are not small in comparison to their thickness. This problem is analyzed by the application of a Galerkin method, in which the effect of multi‑mode participation isconsidered, and an unspecified function of the time resulting in nonlinear coupled ordinary differential equations of motion is solved by the harrnonic balance method and the Newton Raphson method.
The stability question is investigated by studyingthe behavior of a small perturbation of the steady state response. The perturbation equations of the present method of solution reduce to the coupled Hill type equation. Assuming the solution of the form as a product of characteristic component and a Fourier series which represents the periodicity of motion and application of the harmonic balance method can transform the stability problem into the eigenvalue problem of a nonsymmetric matrix. After a proper transformation, the eigenvalues can be calculated on a digital computer by the QR double step method.
The effectiveness and the accuracy of the proposed method are examined fontr a Mathieu equation whose stability has been worked out in detail and the application to stability analysis of the nonlinear vibrations of beams are presented.
1. Introduction
Nonlinear vibrations of beams and flat plates have been studied by many authorsl‑5) The common approach is to assume some form
for the spatial solution and then solve the ordi‑
nary differential equation that results for time variable. Earlier authors have used a single spatial function in their approach, and there have been very little effort in investi‑
gation of this problem using multiple mode so‑
lutions. This fact is primarily in the difficulty
in solving the coupled nonlinear differential equations. Most coupled problems considered require numerical techniques at some point in the solution. Recently, with the advent of a high speed digital computer, there have been several papers6‑iO) which discuss the multiple mode problem. The methods of analysis to study the coupled nonlinear ordinary differ‑
ential equations for time variable are the har‑
monic balance method6m9) and the method of averaginglO) However, the only and reliable
'Department of Civil Engineering
"Yagi Architect and Associats, Oka‑cho, Himeji, Hyogo
K.TAKAHASHI AND
method that does not loose its usefulness when the amplitude of vibration becomes large is the harmonic balance method. This method involves assuming a solution of summed har‑
monics, substituting this solution into the dif‑
ferential equations, and equating the coef‑
ficients of the harmonics to zero. The resulting set of nonlinear algebraic equations may be solved on a digital computer by the Newton Raphson method. According to the multiple mode participation and the harmonic balance method, nonlinear harmonic, higher harmonic, subharmonic and coupling responses of beams and flat plates are predicted.
Although a solution hasbeen found to the steady state problem, there is neither as‑
surance that these solutions are unique nor that they are stable solutions because of the nonlinearity of the equations. Thus, it will be necessary to check the stability of these so‑
lutions and to investigate the possibility of further steady state solutions. The stability question will be investigated by studying the behavior of a small perturbation of the steady state response. The perturbation equations re‑
duce to the coupled Hill type equations in the case of the stability, problem of the present so‑
lution. The stability has recieved much less attension, although Benett and Eisley6) ana‑
lyzed the coupled Mathieu equation by adi‑
rect numerical application of Floquet theory, and BusbyiO) investigated by means of the method of averaging. However, as for the coupled Hill type equations, it seems that there exists no convenient method.
The present work is concerned with the stability analysis of steady state responseof multiple degree‑of‑freedom systems. Assuming a solution of the form as aproduct of a char‑
acteristic component and a vector which has period components, expanding this vector into a Fourier series and substituting the seriesirrt]o the perturbation equations and using the har‑
monic balance method, we can obtain a system of homogeneous algebraic equations. Then,
K.KAWAHARA
an investigation of stability of the trivial solution now reduces to study the eigenvalue problem of a nonsymmetric matrix for de‑
termin.cr the the characteristic exponents. Then, problem leads to find the condition under which any of the subtraction of the eigenvaluefrom the damping constant have a positive real part or not. This method directly determines the stability of the point under consideration.
Numerical results are presented at first for
a undamped and damped Mathieu equation whose stability has been worked out in detail for checking the accuracy and compared with various methods of solution. In addition, the stability of nolinear steady state response of the hinged‑hinged beam for single mode ap‑
proach and two mode approach is performed by the present method. The stabilities of the amplitude of harmonic, higher harmonic and subharmonic response are investigated.
2. Equations of Motion (1) A Straight Beam
The equations of motioni) describing the transverese vibration of a straight beam,which axially restrained in moving and large de‑
flection is permitted as follows
p== ‑!I3i/! %2([.llii )2 clx, ' (i) 04y
02y
02y 0y
L(y,I]b=:EI
(2) + pA
+P +c
0x2 0x4
0t2 0t ‑Pcos9t = 0,
where E denotes Young's modulus, I the
moment inertia of the cross section, p the mass density, A the cross sectional area, e the beam length, y the transverse deflection, t the time, x the axial coordinate, c the linear viscous damping coeflficient, P the load intensity of the external force, and 9 the circular frequen‑
cy of the external force, and P the only nonlinear term due to the effect of the trans‑
verese displacement on the axial force caused by immovable supports. The beam vibrates with moderately large amplitude but the curvatures are assumed small and Hooke's law applied at all times.
THE COUPLED HILL TYPE EQUATIONS
(2) A Circular Plate(with Radial Symmetry) The nonlinear equations5)governing large amplitude motions of a circular plate with . radial symmetry are taken to be
E 0w 02w
V̀F== ‑7or or2' (3)
o2 w 0w
L(w,n=Dv̀w+ph ot2 +C ot ‑ (4)
‑ll‑ EI}‑(0,lil gge)‑pcosgt‑ o,
where Fdenotes Airy's stress function defined
10F o2F
bY ar =7or, ae= or2 ar the radial stress, ae the circumierential stress, r theradial co‑
ordinate, w the transverese displacement, tthe time, D=Eh3/ I12(1‑ v2)l the flexural rigidity, h the plate thickness, v Poisson's ratio.
3. Minltiple Degree‑of‑Freedom Approach A normal mode solution is assumed as follows
co
y= r Z X} (x) Z' ( t) for the case of a i=1
beam,
oo (5)
w=hZ Ri(r) Z(t) for the case of a i=1
circular plate, where r is the radius of the gyration of the beam, Z(t) an unknown function of the time, ahd Xi (x), Ri (r) space variables satisfyingthe geometric boundary conditions of the beam or the circular plate, which denote the as‑
sociated linear problem. Substituting the second equation of Eq.(5)into Eq.(3), the desired representation of the stress function F can be given by9)
F= ‑ {l ,S ,S, n 7) ,Z=co, ,Z.co., afr. ai{ g2(i+j‑2)‑
M,j g2 >, (6)
where N'J'=(i+i‑2)(2i+27'‑5‑v)/(1‑v) for the immovable edge, N'J'=i+1' ‑2forthe movable edge Mij = (i ‑ 1)(2i ‑ 3) (7' m 1)/
l(i+ ]' ‑ 2) 2(i+ 7' ‑ 3) 2l, g= r/a.
Applying a Galerkin method to Eq.(2) or (4), the following set of nonlinear ordinary differential equation's is obtained
45
AND STABILITY OF NONLINEAR VIBRATIONS
.‑ oo co co
d
7‑;i +2hn an 7H)i +aZ 7‑;t + Z Z Z B:i. Tk k=1 l=lm=1 (7)
T} 7;rt = ,(9n Pcos co T,
where an = (.A. / Ai)2, a. == Y[li x:] dg, ff, =: Yg' x. dg/ a. Af,
Bzi. = ‑ ygi fl:ggkddig¥) dg ./g' diee, x},dg /(2Af s.)
g= x/l, IZJ7= pl̀/ETr for the beam, 0n=Y[i?R#dg,
Bzi‑ = 6(oi. ‑A f"2) ,tOO, ,tco, af. a,k･ MiJ'
( o,,･R,M･j" ‑ ks,m･jn ‑ A71･f vvmn), Bn = Y[i' Rn gdg/ AS 6n,
R;j" = y[ii g2(i+j‑3) !illglMRndg,
s,m.jn = yCi g2(i+j‑3) dd2gR,m R.gdg,
vvmn = yg'( fl71glm + g dii5,M) Rn dg,
Oij ‑‑ (i+7'‑2)(i+1' ‑3), Pij=i+7' ‑2, T=coit for the circular plate.
Since there is no known exact solution of Eq.(7), it will be neccesary to get the solution approximately. The harmonic balance method is used here. For Duffing type of Eq.(7), a so‑
lution of the foum is assumed as oo
n= : (asn coss‑aJ‑T + bs. sins‑E[IT), (8) s=1
where a; and ba are amplitude components and ca the frequency ratio defined by 9/cai.
Substituting Eq.(8) into Eq.(7) andapply‑
ing the harmonic balance method, a set of nonlinear algebraic equations will be obtained and solved by the Newton Raphson method on a high speed digitital computer for a proper initial guess.
4. Stability Analysis
The stability question wil1 be investigated by studying the behavior for a small pertur‑
bation of the steady state responseli)Let T;, : n+8n where Tn is the steady steady solution for Eq.(7) and 8n isasmall perturbation of the n‑th mode. Substituting this equation into Eq.(7), and retaining only first order terms,the
K.TAKAHASHI AND
following is obtained
co co co 6n +2hn an an +an2 Sn + Z Z Z (9) k=1 l=1 m=1 BZi. ( Tk Ti am + Tk Tm Si + Ti Tm 0k)=O.
Substititing Eq. (8) into Eq.(9),theperturbation equations will take the form of the matrix notatlon
‑ oo
E6+ 2Ho" + A6 + Z (Bh cos2kiJHT + h =1
(10) Sksin2kZ[FT)6 = O,
where E is the consistent mass matrix which reduces to unit matrix in this case. H,the damping matrix, A the linear stiffness matrix, Bk and Ck the geometric stiffness matrix,6 the vector of generalized coordinates, S the generalized velocity vector and '6' the general‑
ized acceleration vector. Eq.(10) is thecoupled Hill type equation. Let introduce the matrix transformation 6= e‑fHdr 6 where S is avec‑
tor with component which must be determined.
Taking into account that e‑f"dT is a nonsingu‑
lar matrix and assuming that the damping matrix is the scalar matrix H = hE where is the damping constant which is the same for all forms of vibration, Eq.(10) can be written in the form
!a" +{ A ‑ H2 + k2.)i (Bkcos2k Z[JT +
(11)
Ch sin2k 'E[JT) }S = o.
'We seek the solution of Eq.(11) in the formi2) ‑S‑= eAr {‑ll)o + kS.1 (ahsin2le 'aJ‑T +
(12)
bhcos2k EJ‑T) } ,
where ak and bh are some vectors not de‑
pendent on time.
Substituting the series Eq.(12) in the dif‑
ferential equation (11) and applying the har‑
monic balance method, we obtain a set of homogeneous algebraic equations
GX=O, (13)where G is the coefficient matrix, and X vec‑
tor of generalized coordinates.
In oder for this system to have solutions other than the trivial ones, the deteminant
K.KAWAHARA
of the coefficient must be equal to zero. Thus, we obtain the equation for determining the characteristic components as
det(G.)‑O.i (14)
Fromthe," proP'erty of the matrix G, Eq.(14) may be decomposed into the following three matrlces as
det (d) = det(Mo ‑ A Mi ‑ A2 M2) =O, (15) where Mo, Mi and M2 are the coefficient matrices of the constant, first and second power of A, respectively･
To obtain the value of A in Eq.(15), we employ the method which obtains the eigen‑
value of the following double size matrix [ M,‑PM , ‑ M,‑PM,] (Xy) == A (¥), (16)
where Y= AX.
As the matrix of Eq.(16) is a generally nonsymmetric matrix with real element, the eigenvalges consist of pairs of complex number.
The nonsymmetric matrix is converted to the general Hessenberg form and the eigen‑
value is evaluated using the QR double step method. Therefore , the stability isdetermined point by point for both increasing and decreas‑
ing frequency. If any of the subtraction of the eigenvalue from the damping constant have
a positive real part and the corresponding basic solution is unbounded asT‑> oo andthe so‑
lution is unstable. On the other hand, if all of the subtraction of the eigenvalue from the damping constant h have negative real part, then:"e(a'h)r.O as T.oo and the solution is stable.
5. Stability of the Solution of the.Mathieu Equation
For the purpose of examining the accuracy of the present solution, in this section, we carry out such a discussion for the special case of the damped Mathieu equation
'a'+2PO+ (a+ ecos T) 6= O, (17) where B= hra･
Introducing transformation 6= e‑BT 6 gives
"5r+ (a‑ B2 + ecos t)‑Sr=O. (18)
47
THE COUPLED HILL TYPE EQUATIONS AND STABILITY OF NONLINEAR VIBRATIONS
Eq.(12) may be rewritten in this case asfollows
‑IS "= e"T(ml}‑ bo + kE.), (ah sink T+ bkcosk rt)g.) (bAki,‑l‑,ll ‑o,'(i?2 ‑k2) bk +2Akak + ‑S‑( bh"i +(2o)
Substituting Eq.(19) into Eq.(18) and equating (A2+a‑B2 ‑k2)ak ‑2Ak bk+S( ak+i+
coeflicients of identical sinle T, coskT, we ob‑
ak‑1) :O, tain the following set of recurrence relations
where k=1, 2,''' ', ao == O.
for the ak and bk If we take the first three terms of the series,
we find that
‑ll‑(A2 +a‑ B2) bo +'li‑ ebi= O, Gx =o, (21)
where
A2+,‑Kl?2 s o o o o o
S/2 A2+ cr‑B2‑l 8/2 O 2A O O O S/2 A2+a‑ xg2‑4 0/2 O 4A O
G= O O 0/2 A2+a‑ B2‑9 O O 6A
O ‑2A O O A2+ cr‑/92‑1 6･/2 O O O ‑4A O 6t/2 A2+ cr‑B2‑4 6)l2
O O O ‑6A O 8/2 12+a‑ ,e2‑9.
Consequently, Eq.(16) has the form byusing Eq.(21) as
AZ=AZ, (22)
where A=[M2M, ‑M.i PM,], Z= {xy)i･
Y=AX,
Mo=
Mi=
M2=
a‑B2
8/2 o
o o o o o o o o o o o
‑1
o o o o o o
6
cr ‑32 ‑1 ･6/2 .
o o o o o o o o 2 o o o ‑1 o o o o o
o 6/2
cr ‑B2 ‑4
0/2 o o o o o o o o 4 o o o
‑‑ 1
o o o o
o o S/2
cr‑B2‑9
o o o o o o o o o 6 o o o
‑1
o o o
o o o o
cr‑B2‑1
0/2 o o
‑2
o o o o o o o o o
‑1
o o
o o o o 6/2
a‑B2 ‑4
0/2 o o
‑4
o o o o o o o o o
‑1
o
o o o o o 0/2 a‑32‑9
o o o
‑6
o o o o o o o o o
‑1 ‑
'
'
,
K.TAKAHASHI
The stability regions of the Mathieu equa‑
tion will be determined by using Eq.(22). We show the undamped and damped (h=O.1) sta‑
bility region for a Mathieu equation in Figs.1
2
,1
1
Sl '2
C1 2 gegS1
M
E E ti)f
EFEA E
f Eiiil!F5 ,,.. ,.,<t <"̀
,e2Ef ,,<'N
62i .
present solution (2 term) present solution (:3 term) ‑‑‑･exact solution ‑ ‑ ‑ perterbation solution KZZiK stable side of boundary
2
,1
1 o
Fig.1 Undamped taj
E
i cr Mathieu stability
'i)5
t,>,l4)
6 k
k E
2
diagram
x
present solution (2 term) present solution (3 term)
‑‑‑ exact solution Z7T7 stablo side of boundary )E>>>5
cr
Fig. 2 Damped Mathieu stability diagram and 2, respectively, in which the shaded region are the stable regions. In these figures, the thicker solid lines show the two term solution (bo,bi, ai) and thinner lines show the three term solution(bo,bi,b2,ai,a2). In Fig.l, dotted lines show the stability boundaries which are de‑
termined from the condition under which the differential equation has periodic systems with 2rr or 4n according to the Floquet theory]i) Boundaries of the principal region of the sta‑
bility marked by Sii2 and Cii2 have periodic
AND K.KAWAHARA
solutions with period 4rr and the boundaries of the second region of the stability marked by Si and Ci have periodic solution with peri‑
od 2n. As these results obtained by the Floquet theory are converged values, we can estimate these results as the exact solution for comparison of the present solution with the establish6d solution.
Approximate regionii) of the stability of the Mathieu equation for small E obtained by the use of the perturbation method are also shown in Fig.1.
As can be seen in Fig.1 in the case of the undamped case, the difference between the two term solution and the three term solution of the present solution does not noted for not too large values of E and a certain difference is coming out with increase of E. However, more than four term solution almost coincides with the three term solution. Therefore, the three term solution will be assumed to be the converged value.
The twoterm solution of the stability boundary Si is equal to unity independently of the magnitude of a. But,as the three term solution agrees with the exact solution, the stability boundary Si of the present method is omitted from Fig.1. If we compare the
present solution with the exact solution, it will be seen that the present solution agrees well with the exact solution as to the stability boundaries Sii2 and Si and do not agree well as to the stability boundaries Cii2 and Ci in the range where E is not small. The result of this method overestimates the area of the sta‑
bility range. However, the present solution gives a satisfactory results compared with the first term solution of the perturbation method which is usually used in the stability analysis of a nonlinear system
As to the damped Mathieu equation as shown in Fig.2, the present solution isassumed to give a similar result as well as undamped case. In this case, the stability boundary cor‑
responding to Cii2 is pot affected by the
THE COUPLED HILL TYPE EQUATIONS AND
number of the term.
6. Stability of the Hamionic, Higher Har‑
monic and Subharinonic Response of the Nonlinear System
(1) Single Degree‑of‑Freedom System For the case of the symmetric vibration about the center of a hinged‑hinged beam,the equation for the first mode can be obtained from Eq.(7) in the following form
'T+ 2h 7+ T+ 0. 25 T3= ::,ri5cos to‑T. (23) Employing the first two harmonics as T= ai cos co T + a3 cos3co T + bi sin co T +
b3 sin3co T. (24)
Substituting Eq.(24) into Eq.(23) and using the
5 Al 4
3
2
1
o
12
Fig.3 Amplitude frequency curves of a damped case)
Fig.3 shows theundamped overall solution (h =O) of Ekl.(25) near the first natural frequen‑
cy range in the rectangular coordinategraph, where the abscissa indicates the frequency rati‑
o di and the ordinate indicates the absolute value of the amplitude, i.e., A= lai+a3l
which occures at tu‑T=n rr (n = O, 1, 2, ･ ･ ･). In Fig.3, the solid lines correspond to amplitude in phase with the external force and the broken lineS correspond to amplitude out of phase with external force. Theamplitudes
49
STABILITY OF NONLINEAR VIBRATIONS
harmonic balance method, the following set of four nonlinear algebraic equations is ob‑
tained as
(1‑ di2)ai + 2h te‑bi + Iiil6T(a? + a? a3 + 2ai a?
+ ai b? ‑ a3 bi + 2ai di + 2ai bi be)= .4s i,
(1‑ 9ca‑2 ) a3 + 6h ca‑ be +.f6( ai + 3ag + 6a? as
‑ 3ai bl + 3a3 bZ + 6as bi) : 0,
( 1‑ tod2 ) bi ‑ 2h to‑ai + Iil6T( b? ‑ b? be + 2bi di
+ 2a: bi + a? bo + a?bi ‑ 2ai as bi) == 0, (25) (1‑9j 2) be ‑ 6h w‑as +f6( ‑ b? +3rd + 6bi b3 + 3a? bi + 6tzi be + 3aZ be ) = 0.
q7 e7
7/
7/oo
7/oo
nyny/
9/
1/
//xxk
inphase
‑‑‑ outofphase
oo
o‑,=ok‑p‑×3)bkp=
7o
x
JoxJ ostablexunstable
o.N..N
eO‑o 3 4t.5o
hinged‑hingedbeam (single degree‑of‑freedom, un‑
marked by O correspond to stable points and the amplitude marked by × correspond to unstable points.
Resonance which occures in the neighbor‑
hood of toM‑‑ 1.0 presents a harmonic solution.
As shown in Fig.3, the amplitude transmits from a stable point to a unstable point where points at which the resonance curves have a vertical tangent( di= 1.42 when 5= 70 and ca‑‑‑
1.28 when P= 30) in the A, to‑ plane. Thelocus of the vertical tangents of the response curves
50
KTAKAHASHI
in the A, di plane maps on the boundary Cii2 between a stable and unstable region of the a, E plane as shown in Fig.1. These results coincide with the foregoingconclusion based on physical ground and the accuracy of the present method seems to be satisfactory.
The resonance which occures in the
neighborhood of di=O.3 shows the higher har‑
monic whose frequency is three times asmany as the frequency of the external force. This re‑
sponse occures continuously with the increase ordecrease of the frequency.The amplitudes in phase with the exte'rnal iorce transmits from the stable point to the unstable point where the response curves have vertical tangents.
The study of the stability of this higher har‑
monic is obtained by using the threeterm Solution 'of the harmonic balance method.
According to the two terni solution, all the amplitudes are stable. This means that the variational equation which is calculated by
5 Al 4
3
2
1
AND K.KAWAHARA
the amplitude components requires to present precisely the variational equation of the non‑
linear differential equation.
The unstable amplitude of the harmonic response in the neighborhood of te"=O.6 ?/
corresponds to the second instability region of the Mathieu equation. As the unstable region is considerably narrow, this ampli‑
tude becomes stable when damping term is taken into account.
The resonance which results through bi‑
furcation from the harmonic response in the neighborhood of di=3.0 is the subharmonicre sponse of order 1/3. The amplitude is defined
by A=lay3+ailwhere we assume the so‑
lution T as T= aicos c'o T +ai/3 cos c'o r/3. The amplitude out of phase with the external force is stable and the amplitude in phase with the external force is unstable.
h=O.05
‑h=O.O05
‑p=70
'p=O
ostable xunstable･
Fig.4 Amplitude frequency curves of a damped case)
Fig.4 shows the frequency response curve when the viscous damping term is included.We use the instantaneous maximum value of the amplitude as the definiglon of the amplitude in this case. The dampirig constant h is taken.
to 5% in the case of the harmonic and higher
3 hinged‑hinged beam
4 ‑‑.‑ 5
to (single degree‑ of‑freedom,
harmonic response, and O.5% in the case of the subharmonic response.
Thewidth of the higher harmonic response near to‑=O.3 is narrow as shown in Fig.4, the amplitude becomes considerably small. As damping is added, the response curve ofthe
THE COUPLED HILL TYPE EQUATIONS AND
harmonic motion in the neighborhood of di = 1.0 is round off in the vicinity of the curve for
p‑=Oat A==4.3, and the response curve is bented to the right, there are two vertical tapgents as shown in Fig.4. The curve which is surrounded by these teq,o vertical tanengts seems to be unstable. The present solution for the lower transmitting point from stable amplitude to unstable one agrees with the lower point of the vertical tangent. However the result of the stability analysis in the neighbor hood of the upper vertical tangent does not coincide with the point of the vertical tangent when the damping constant is smaller than 10%. The accuracy of the stability analysis is not improved even if we adopt further terms of the harmonic balance method. This reason may be caused by the fact that the stability boundary corresponding to Cy2 isnotaffected by the numbers of the term as shown in Fig.2.
The subharmonic response with damping is also presented in Fig.4. In this case, the response curves have the upper and lower round off points. The region where the sub‑
harmonic response occures is considerably li‑
mited.
(2) Two Degree‑of‑Freedom System
If more than one mode is considered in the analysis, then a coupled set of nonlinear dif‑
ferential equations results. For a example,the follwing set of equations isgivenfor two sym‑
metric modes of the hinged‑hinged beam 'Ti + 2h lz7i + Ti + o.2s( Ti3+ gTi T;)
=ll!ii‑ [ZJcos cffoT, (26)
rr
'7b + 2h T3 + 81 T3 + 2.25( 7g 7b +9Tg)=
‑4‑ ff
3 ns PCos to T.
To apply the harmonic balance method, let Ti =aicos c‑oT+ bisin c'oT, (27)
Tb = a3 cos c‑oT+ b3 sin c‑o T .
Substituting Eq.(27) into Eq.(26) and taking ･//･‑
.the only first terms of the perturbation yields
51
STABILITY OF NONLINEAR VIBRATIONS
.
E'S' +2H6+ A6 + (B + Ccos2 w‑T+
Dsin2 E[iT)6 :0, (28)
where .,.[3 9], H‑[2 :], A=[S sg), B‑[zg z:], c==[gi gi], D‑[gl gg], s‑([il:l, o ={o o}T,
qo = 0.375{ ai2 + bi2 + 3( a32 + b32) }, qi = 0.375{ai2 ‑ bi2 + 3( a32 ‑ b32 ) }, q2 = 0. Z5( ai bi + 3a3 b3),
q3 = 2. 25(ai a3 + bi b3), q, == 2.25(ai a3 ‑ bi Q3), qs= 2.25(ai b3 +a3 bi).
q, = 1. 125{ai2 + bi2 + 27( a32 + b32 ) }, q, == 1. L25{ai2 ‑ b? +27(a32 ‑ b32)}, qs :1.125(arbi+27tz3b3), Let introduce 6 :eMHT6' , then we obtain EL6'+(A+B‑ H2 +Ccos2 to‑T+
Dsin2 c‑o T)S=O. (29)
We assume
S= eAT { ‑li‑bo + ktle ,3( ah sin2 kto‑T+
bk cos2 k cMo T) }, (30)
where bo ={bo' bo3}', bh={bh' bk3}', ah={ak' ah3}T.
Substituting Ekl.(30) intoEq.(29) and equating the coefficients of cos2 cneoT and sin2 c‑oT, we obtain a set of homogeneous algebraic e‑
quatlons
'
[A+B‑cDH2+ A2E2(A+B‑‑Hs2 E:;fiigE‑4 ib2 E)
D 8 E[JAE 2(A+B‑ H2 + A2E‑4 aj2 This results in an
determinant is required to E)
ili‑o
problem
(31)
eigenvalue whose
vanish.
Amplitude frequency curves of the hinged‑
hinged beam for the first and the second modes are shown in Fig.5. These response curves are obtained by using damping constants equal to
O% and 5%.
5 Al 4
‑3
2
1
K. TAKAHASHI
qh‑=o h=O.05 7i
i
fe
‑p=70
p=o×ex‑.
N‑o‑
o .tu(a) first natural frequency region 3
3 .1
"i 1 ri
p ==70
oE==e p‑‑o
8.5 9.0 9.5 10.0
w
(b) second natural frequency region
Fig.5 Amplitude frequency curves of a hinged‑hinged beam (two degree‑of‑
freedom, undamped and damped cases) From the stability analysis, it will be seen
that a similar result is obtained in the cases of the first and second natural frequency region as well as the result of the single de‑
gree‑of‑freedom system. Consequently, the present method of stability analysis can be easily applied to multiple degree‑of‑freedom systems and can be treated as well as single degree‑of‑freedom system.
7. Conclusions
The results of the numerical examples indicate that the present method of solution for the stability question gives a excellent ap‑
proximate solution for a Mathieu equation and can be applied to investigate the stability analysis of steady state response of nonlinear vibrations.
The method only 2nalyzes one point on the reponse curve at a time so it isnecessary
AND K.KAWAHARA
to determine the stability point of several re‑
sponse in order to construct regions of insta‑
bility. Therefore, in the problem in which the nonlinear coupling is week, it is possible to use the instability region in the Mathieu dia‑
gram maps or to obtain the position og the vertical tangent when the amplitude frequen‑
cy curve is obtained. However, in the case when the nonlinear coupling appears or when the amplitude frequency curve can be not ob‑
tained easily, the proposed method of solution will be useful and play a significant role.
Acknowledgement
The authors wish to thank Pro. Togawa of Kyoto Industrial University for hishelpful advice.
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