Helmholtz salınıcısının rezonans durumlarında dallanma analizi
Bifurcation analysis of helmholtz oscilator in resonances
- Tez No: 46163
- Danışmanlar: DOÇ.DR. GAZANFER ÜNAL
- Tez Türü: Yüksek Lisans
- Konular: Matematik, Mathematics
- Anahtar Kelimeler: Belirtilmemiş.
- Yıl: 1995
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: 100
Özet
ÖZET Bu çalışmada, birçok fiziksel ve mühendislik problemlerini modelleyen, katastrof teorisinde önemli rol oynayan ve ilk olarak Helmholtz tarafindan kulak zarının titreşimini açıklamak için önerilen [4,7], tek serbestlik dereceli, sönümlü, kuadratik nonlineer terim içeren ve zorlamalı Helmholtz salınıcısının analitik yöntemlerle statik ve dinamik dallanması incelenmiştir. Birinci bölümde, incelemelerde kullanılan ve pertürbasyon yöntemlerinden olan çoklu ölçekler yöntemi, ortalama alma yöntemi ile normal form analizi kısaca tamtilmışür. İkinci bölümde, Helmholtz salınıcı denklemine (0,0) denge noktası civarında yukarıda sözü edilen yöntemler uygulanmış ve elde edilen sonuçlar karşılaştınlmıştır. Üç farklı yöntemin de aynı yapıda sonuçlar verdiği, fakat uygulama açısından her yöntemin diğerlerine göre avantajı veya dezavantajı olduğu görülmüş ve bunlar sonuçlar kısmında tartışılmıştır. Bu bölümün son aşamasında elde edilen denklemlerin denge noktalan ve bunların kararlılık durumları incelenerek dallanma analizi yapılmıştır. Burada denge noktasının kararlılığını kaybetmesi sonucu periyodik yörüngelere dallandığı gözlenmiştir. Üçüncü bölümde, [5] referansmdaki çalışmalarda ^ periyodundaki periyodik yörüngelerin periyodik çiftleme dallanmalanyla kararlılığını kaybedip periyodu ^, - olan yörüngelerin sayısal ve analog çalışmalarla gözlenmesinden yola çıkarak ikinci mertebe ortalama alma yöntemi ile bu dallanmalar incelenmiş ve dallanma diyagramları elde edilmiştir. v
Özet (Çeviri)
BIFURCATION ANALYSIS OF HELMHOLTZ OSCILLATOR IN RESONANCES SUMMARY In this study, static and dynamic bifurcations of Helmholtz oscillator are studied by using some analytical methods. The equation of Helmholtz oscillator which was first proposed by Helmholtz to explain the vibration of pre-stressed membrane models many physical and engineering problems. It also plays a significant role in catastrophe theory [7]. Helmholtz oscillator is a driven damped, continuously excited and single-degree of freedom system with quadratic nonlinearity. In the first section, methods of multiple scales, normal form analysis and method of second order averaging are introduced. We have used reference [3] and [1] to introduce methods of multiple scales. In this method, x, which is assumed to be an approximate solution of original equation depends on the various new scales, instead of t and can be represented by an expansion having the form M x(t,s)=x(T0,T1,...,Tm,8)= E8aıxm(To,Tı,...,Tm) + 0(8Tm) (1) m=0 where Tm = s^t. If the equation (1) is differentiated with respect to t and substituted into the original equation and if the coefficients of all s which have the same degree in the obtained equation, we arrived at ordinary differential equations. In the solutions of these differential equations we meet secular terms which makes the rest of the equations unsolvable. By eliminating secular terms we obtain model equations which must be satisfied by approximate solution of the original differential equation. The examination of these model equations gives us some information about behaviour of original system. We have used reference [4] to introduce normal form analysis. Normal form analysis plays an important role in modern bifurcation theory. This analysis involves smooth transformation of non-linear co-ordinates so that it simplifies the original dynamical system. Since we have used fourth order normal form analysis in our study, the coefficients of fourth order normal form equations are given here. We assume that the dynamical systems is autonomous, the linear part of vector field have Jordan form and non-linear terms of the system begin with quadratic terms. According to this assumption, standard form to which normal form can be applied is xv =XvXv+EaJi1XjXh + 2balkXjXhXk (2) where Xv are the eigenvalues of the linear part. The coefficients ajj, and bj^ are assumed to be symmetric with respect to their subscripts without loss of generality vi(v,j,h,k= l,2,...,n). According to fundamental theorem of Brjuno there exists a normalizing transformation xv = yv + S oCyıym + 2 PLpYiymYp + 2 YLpryiymyPyr (3) which generally has complex coefficients, in equation (3), a^ and P^, are assumed to be symmetric with respect to their subscripts. Normalizing transformation (3) reduces the original system (2) to normal form yv =Xvyv + S(ptoyiym + 2K^yiymyp+2n^pryiymypyr (4) Substituting the transformation equation (3) into the original equation (2), and using (4), we obtain the following equation Jmp + PjLJpl -ajcpinp-aycpin]} (7e) Now, we assume that the original dynamical system contains a linear term related to bifurcation parameter. We will examine contribution of this term to normal form equations. In this case, dynamical system can be written as x=L0x+L|Oc+f(x) (8) where, L0 is the matrix which consists of the coefficient of linear terms, Lş is the matrix which consists of the coefficient of linear terms related to bifurcation parameter and f(x) represents the non-linear terms. Since the normal form analysis of the non-linear terms is already studied, transformation sought should affect only the matrix L§. Moreover, it is also desired that this transformation should give terms in the main diagonal of normal form equations in order to make the further analysis easier. Hence the normalizing transformation takes the form x=y+9(y)+G£)y (9) where o + ea and Q = 2coo + eo have been used for primary resonance, second order superharmonic resonance and second order subharmonic resonance respectively. For primary resonance, the coefficients of equation (19) are taken as ğ = s2g, S = sa, A = s2 A and we expressed the approximate solution in the following form x(t,8) = xo(To,Tı,T2)+8x1(To5Tı,T2)+e2x2(To,Tı,T2) +... (20) where Tm = smt. If the equation (20) is differentiated and substituted into the equation (19), and if the coefficients of 8°, e1, s2 on both sides of the obtained equation is equated, we obtain three ordinary differential equations. Eliminating secular terms in the solution of this equations and introducing integration constant as K = |aeip, we obtain a^-fa + ^sin^Ta-p) (21a)./- 5a2 a3 A 12coS 2®° a0' = -^-a3 - r^- cos(oT2 - p) (21b) Making the transformation y = oT2 - p in the equations (21), we get al'' = aO + ^ + ^C0Sr (22b) Secondary resonances are examined for two cases; hard excitation (A = A) and weak excitation (A = s A). An approximate solution of original equation can be given as x(8,t) = xo(To,T1)+8Xi(T0,T1) +... (23) For second order superharmonic resonance, taking the coefficient of equation (19) as ğ = sg, â = sa, doing some simple calculations as introduced above, eliminating secular terms and introducing integration constant as K = |aeip, we find thata/ =“I a”, Ka“2 sin(aTx - P) (24a) 2 4©0(a>5-n2) *' =, A2a^cos ^24b> 4(oo(ü)o-n2) where excitation is hard, and a' = -f (25a) ap' = 0 (25b) where excitation is weak. In like manner, for second order subharmonic resonance, we obtain a/ = -fa”, /°^^sin(aTl - P) o(a>o-&) ^ =,,°? r^cos 4©o((ûo-ft2) where excitation is hard, and zJ = -|a + t^- sin(n - coo - P)T0 (27a) 2 2(0 o ap/ = -^cos(Q-©o-P)To (27b) 2©o where excitation is weak. For normal form analysis, the coefficients of equation (19) are taken as ğ = g, 53 = a, A = A. Using xi = x, x2 =x, X3 = Acosflit and X4 = Asin Qt, we can write the equation (19) as autonomous dynamical system xi =x2 (28a) x2 =-gx2-ffloXi+x3 (28b) x3 =-Qx4 (28c) x4 =nx3 (28d) In case of primary resonance, having used the equality Q = ®a+o and making a suitable transformation, the standard form to which normal form analysis can be applied is obtained. Using the code given in Appendix- 1 which is written for Mathematica, we find the coefficient of normal form equations. Thus, the normal form equations can be written as Yx =©0iyi+oiyi (29a) y2 = yi +o0iy2 - f y2 - ^tjIya (29b) y3 =-©oiy3-oiy3 (29c) y4 = y3 - cD0iy4 - fy4 + Şryayî (29d) xiHaving found the solutions yi, V3, substituting them into y2, y4 and assuming that the solution of the last two equation have the form y2 = reie, y4 = re-ie (30) and using y = [(©o + a)t - 9], we obtain the following equations ^-fr + 4İSİnY (31a) ry=ar + ^4r3+-A-cosy (31b) 3©*, 4oo As it is seen, the equations obtained by using normal form analysis and the methods of multiple scales are the same except the coefficient of cubic term in (31b). In case of second order subharmonic resonance, the relation fi = 2©0 + o is used. After bringing (28) to standard form to which normal form analysis can be applied, we use the programme given in Appendix- 1 for Mathematica to obtain the coefficients of normal form equations. Thus, the normal form equations are found as follows, y ! = -ooiy 1 - f y 1 - £y2y3 + fry2y2 + yry iy3y4 (32a) 4©o 36©o 720(5)5 y2 = ©oiy2 - |y2 + ^yiy4 - ^p©oyiy2 - ^WW (32b>, a3i 5a3i 2, 221a3i, xr 2 4oq 36©q 720©o y3 =-2ffioiy3-criy3 (32c) y4 = 2©0iy4 + aiy4 (32d) If y3, y4 are found from (32c,d), substituted them into (32a,b) and if we assumed that yu y2 have the form yi=reie, y2 = re-ie (33) and if y = [(2©0 + o)t + 20] is used, we obtain r= -|r + [cır - c4r3 - c5r3 + cerjsiny (34a) y= a + 2c2r2 + 2c3 + [2ci + 2c4r2 - 2c5r2 + 2c6]cosy (34b) where - _ A r _5cc2“ _ a2 A2 ”a3A“ 5a2A ”221a3 A3 6©^' 3©?' 16©2' 24©^' 216©^' 155520ÖJ1 xnIn case of second order superharmonic resonance we use ^2. instead of Q. After bringing (28) to standard form, we obtain the coefficients of normal form equation by using the programme given in Appendix-1 which is made for Mathematica. Thus, we can construct normal form equations as. g ai 2. 5a2i 2, 46a2i 1 2 2tDo 3©;* 15©oyiy3 2“ ”z©o“ ”3©o i5©oyiy3y4 y2=ffl0iy2_fy2+^_^yiyi_^wy4, 1003a3i“2,.2, 107a3i w 2, 1081a3i v3 +- - T”v2y3 + 1A 5 yiY2y4 +... 5 y^ 180©5 10©5 450©5 i ai (Ool y3 = - i en 2-y3-Ty3 ©oi, ai y4 =^-y4+jy4 (35a) (35b) (35c) (35d) Here, if y3, y4 are found and substituted into (35a,b), if the solutions are sought in the following form yi = re£8 y2 = re-'9 (36) and if y = [(©0 + o)t + 9] is used, we obtain r= -fr-[cı -C4r2 +C5r2 +Cö]siny ry=or+C2r3+C3r-[cı+C4r2 + C5r2+C6]cosy (37a) (37b) where 2A2a Cl = c4 = 9©Ü 1003a3A2 405©o C2 = 5a2 3©o c3- 184a2 A2 1352) wnerea- z'0-!^ 0, (l/a)(-d±Vc2+b2-a2)>0 (44a,b) must be met. If one such solution for r does exist, then examination of equation (42) shows that there are two distinct solutions, (r, 8) = (n, 9i), fa, 9 + %). The bifurcation sets on which these pair of equilibrium are created and annihilated are given by equations (44a,b), with equalities replacing the inequalities. Using this bifurcation sets and taking A = 0.4, we draw bifurcation diagram (r, co), and then taking o = 1.9 we draw bifurcation diagram (r, A). Next, applying the method of second order averaging for k = j, we obtain ü= au + bv + 2cuv + av(u2 + v2) (45a) v= -bu + av + c(u2 - v2) - ccu(u2 + v2) (45b) or, in polar co-ordinates r=r(a+crsin39) (46a) 0= -(b + or2) + cr cos 39 (46b) wherea--! b (9- 62)2 ' ^ 4o3(1-û>2)' U ~ 4m3 The averaged equation (45) always have the trivial equilibrium solution (u,v)=(0>0). Non-trivial solution of equation (46) are r = ^(l/2a2X(c2 - 2ab) ± J(?- 2ab)2 - 4a2(b2 - a2) ) (47) and occur in sets of three with 0 = 9i, 9i + y, 9i + y. Here, we also create bifurcation sets and then using these sets, and taking some value of oo, we draw bifurcation diagram (r,A), finally taking some value of A, we draw bifurcation diagram (r,o). We can not find any equations which give bifurcation for other subharmonics with method of second order averaging. We used Mathematica to control stability analysis of all equilibrium points. xv
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