Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 46, 1, pp. 141-156, Warsaw 2008 ANALYSIS OF VIBRATION OF THREE-DEGREE-OF-FREEDOM DYNAMICAL SYSTEM WITH DOUBLE PENDULUM Danuta Sado Warsaw University of Technology, Poland e-mail: dsado@poczta.onet.pl Krzysztof Gajos PEC Legionowo, Poland e-mail: gajosy1@gazeta.pl The nonlinear response of a three-degree-of-freedomvibratory systemwith adouble pendulum in the neighborhoodof internal and external resonances has been examined. Numerical and analytical methods have been applied for these investigations. Analytical solutions have been obtained by using the multiple scales method. This method is used to construct first-order non-linear ordinary differential equations governing themodulation of am- plitudes andphases. Steady state solutionsand their stability are computed for selected values of the system parameters. Key words: nonlinear coupled oscillators, autoparametric vibrations, mul- tiple scale method 1. Introduction In complex three-degree-of-freedom vibrating systems with elements of pen- dulums suspended on a flexible element, the autoparametric excitation as a result of inertial coupling may occur (Sado, 1997). Dynamic systems of this kind with two degrees of freedom were widely discussed in the literature as autoparametric vibration eliminators (Bajaj and Johnson, 1990; Bajaj et al., 1994; Banerjee et al., 1996) or other structural components (Samaranayake and Bajaj, 1993; Sado, 2002; Shoeybi and Ghorashi, 2004). The effect of a parametric or autoparametric excitation on a three-mass system was studied by Tondl and Nabergoj (2004). Numerical simulations of a two mass system 142 D. Sado, K. Gajos with three degrees of freedom with pendulums hanging down from a flexibly suspended bodywas investigated by Sado (2004) for an elastic pendulumand by Sado and Gajos (2003) for a double pendulum. This paper describes the analytical solution of a three-degree-of-freedom system with a double pendulum. As it is a vibrating system with changing values of amplitudes and phases, in the analytical investigation the method of multiple scales was applied (Nayfeh and Mook, 1979). This method was used by several researchers (Ertas andChew, 1990; Ji and Leung, 2003;Moon andKang, 2003; Çevik and Pakdemirli, 2005; Rossikhin and Shitikova, 2006). Eliminating secular terms, we can observe conditions when the phenomenon of internal and external resonances is possible.Next, for the conditions of such resonances, steady-state solutions were investigated. 2. Equations of motion The investigated system is shown in Fig.1. The system consist of a double pendulumandabodyofmass m1 suspendedonaflexible element of rigidity k, thus S(y) = ky. The pendulum of length l1 and mass m2 hangs down from Fig. 1. Schematic diagram of the considered system the body of mass m1. The pendulum of length l2 andmass m3 is suspended on the body of mass m2. It is assumed that a linear viscous damping force acts upon the body m1 (R(ẏ) = c1ẏ), and a linear damping momentum acts upon the pendulum of mass m2 (M1(ϕ̇1) = c2ϕ̇1), and a linear damping Analysis of vibration of three-degree-of-freedom... 143 momentum applied in the hinge opposesmotion of the pendulumof mass m3 (M2(ϕ̇1, ϕ̇2)= c3(ϕ̇2− ϕ̇1)). The body ofmass m1 is subjected to a harmonic vertical excitation F(t)=P0cosνt. This systemhas three degrees of freedom. Asgeneralized coordinates, thevertical displacement y of thebodyofmass m1 measured fromthe equilibriumposition and the angles ϕ1 and ϕ2 of deflection of the pendulumsmeasured from the vertical lines are assumed. The equations of motion are derived as Lagrange’s equations (m1+m2+m3)ÿ− l1(m2+m3)ϕ̈1 sinϕ1−m3l2ϕ̈2 sinϕ2+ −(m2+m3)l1ϕ̇21cosϕ1−m3l2ϕ̇22cosϕ2+ky+ c1ẏ=P0cosνt −(m2+m3)ÿ sinϕ1+(m2+m3)l1ϕ̈1+m3l2ϕ̈2cos(ϕ2−ϕ1)+ (2.1) −m3l2ϕ̇22 sin(ϕ2−ϕ1)+(m2+m3)g sinϕ1+c2ϕ̇1− c3(ϕ̇2− ϕ̇1)= 0 −ÿ sinϕ2+ l1ϕ̈1cos(ϕ2−ϕ1)+ l2ϕ̈2+ l1ϕ̇21 sin(ϕ2−ϕ1)+ +gsinϕ2+ c3(ϕ̇2− ϕ̇1)=0 Next, we introduce the dimensionless time τ = ω1t and the following defini- tions y1 = y l1 y1st = yst l1 d1 = m2 m1 d2 = m3 m1 d3 = d1 1+d1+d2 d4 = d2 1+d1+d2 d5 = d3+d4 d6 = d4 d3 d7 =1+d6 ω21 = k m1+m2+m3 ω22 = g l1 ω23 = g l2 c= l2 l1 β1 = ω2 ω1 γ1 = c1 m2ω1 γ2 = c2 m2l 2 1ω1 γ3 = c3 m2l 2 1ω1 µ= ν ω1 p= P0 m2l1ω 2 1 (2.2) 3. The method of multiple scales In order to find approximate solutions to equations of motion we use the method of multiple scales (Nayfeh and Mook, 1979). Partially, this problem 144 D. Sado, K. Gajos for a systemwithadouble pendulumwas presentedbySadoandGajos (2005). For small oscillations, after transformations the equations of motion can be written down in the form ÿ1+y1−d5 ( ϕ1+ ϕ31 6 ) ϕ̈1−d4c ( ϕ2+ ϕ32 6 ) ϕ̈2− ϕ̇21d5 ( 1−ϕ 2 1 4 ) + −d4cϕ̇22 ( 1− ϕ 2 2 4 ) =−d3γ1ẏ1+d3pcos(µτ) d5ϕ̈1−d5 ( ϕ1+ ϕ31 6 ) ÿ1+d4c ( ϕ1ϕ2+1− ϕ2 4 − ϕ1 4 ) ϕ̈2+ +d4cϕ̇ 2 2 ( ϕ1+ ϕ31 6 −ϕ1 ϕ22 4 −ϕ2− ϕ32 6 −ϕ2 ϕ21 4 ) +d5β 2 1 ( ϕ1+ ϕ31 6 ) + +d3[γ2ϕ̇1−γ3(ϕ̇2− ϕ̇1)] = 0 (3.1) cϕ̈2− ( ϕ2+ ϕ32 6 ) ÿ1+ ( ϕ1ϕ2+1− ϕ22 4 − ϕ21 4 ) ϕ̈1− ϕ̇21 ( ϕ1+ ϕ31 6 −ϕ1 ϕ22 4 + −ϕ2− ϕ32 6 +ϕ2 ϕ21 4 ) +β21 ( ϕ2+ ϕ32 6 − d2c d4 γ2(ϕ̇2− ϕ̇1)= 0 We introduce independent variables {T0,T1,T2, . . . ,Tn}= {τ,ετ,ε2τ, . . . ,εnτ} (3.2) and parameters p1 = ε 2p1 γ1 = εγ1 γ2 = εγ2 γ3 = εγ3 (3.3) Solutions to the dimensionless equations can be represented by y1 = εy10+ε 2y11+ . . . ϕ1 = εϕ10+ε 2ϕ11+ . . . (3.4) ϕ2 = εϕ20+ε 2ϕ21+ . . . It follows that the derivatives with respect to τ become expansions in terms of partial derivatives with respect to Tn as d dτ = ∂ ∂T0 +ε ∂ ∂T1 +ε2 ∂ ∂T2 + . . .=D0+εD1+ε 2D2+ . . . (3.5) d2 dτ2 = ∂2 ∂T20 +ε ∂2 ∂T0∂T1 +ε2 ∂2 ∂T0∂T2 +ε ∂2 ∂T1∂T0 +ε2 ∂2 ∂T21 +ε2 ∂2 ∂T2∂T0 +. . .= = ∂2 ∂T20 +2ε ∂2 ∂T0∂T1 +ε2 ( 2 ∂2 ∂T0∂T2 + ∂2 ∂T21 ) + . . .= =D20 +2εD0D1+ε 2(2D0D2+D 2 1)+ . . . Analysis of vibration of three-degree-of-freedom... 145 Substituting (3.3) and (3.4) into dimensionless equations (3.1) and equating the coefficients standing at ε1 and ε2 on both sides, we obtain: — for ε1 D20y10+y10 =0 D20ϕ10−d6β21ϕ20+d7β21ϕ10 =0 (3.6) D20ϕ20−d7β22ϕ10+d7β22ϕ20 =0 — for ε2 D20y11+y11 =−2D0D1y10+d5(D0ϕ10)2+d4c(D0ϕ20)2+d3pcos(µτ)+ −d3γ1D0ϕ10+2d6d5β21ϕ10ϕ20−d6d5β21ϕ220− d25 d3 β21ϕ 2 10 D20ϕ11−d6β21ϕ21+d7β21ϕ11 =−2D0D1ϕ10−d7y10ϕ10+d6y10ϕ20+ (3.7) − (γ3 c +γ2+γ3 ) D0ϕ10+ ( γ3+ γ3 c ) D0 D20ϕ21− 1 c (d7β 2 1ϕ11−d7β21ϕ21)=−2D0D1ϕ20+ 1 c (d7y10ϕ10−d7y10ϕ20)+ − d7γ3 c2 (D0ϕ20−D0ϕ10)+γ2D0ϕ10−γ3(D0ϕ20−D0ϕ10) General solutions to equations (3.6) can be represented by y10(T0,T1,T2)=A1(T1,T2)e iω1T0 +A1(T1,T2)e −iω1T0 ϕ10(T0,T1,T2)=A2(T1,T2)e iω2T0 +A2(T1,T2)e −iω2T0 + +A3(T1,T2)e iω3T0 +A3(T1,T2)e −iω3T0 (3.8) ϕ20(T0,T1,T2)=Λ2A2(T1,T2)e iω2T0 +Λ2A2(T1,T2)e −iω2T0 + +Λ3A3(T1,T2)e iω3T0 +Λ3A3(T1,T2)e −iω3T0 We find natural frequencies of system (3.6) by substituting y1 =A1e iωT0 + cc ϕ1 =A2e iωT0 + cc ϕ2 =ΛA2e iωT0 + cc (3.9) where cc represents the complex conjugate, and using the condition that the determinant of the matrix of coefficients is zero. In this case ω1 =1 and ω22,3 = 1 2 [ −d7β21 ( 1+ 1 c ) ±β21 √ d27 ( 1+ 1 c )2 − 4d5d6 c ] 146 D. Sado, K. Gajos Λ2,3 = −ω22,3+d7β21 d6β 2 1 Amplitudesandphases canbe foundbysubstituting (3.8) into (3.7).Weobtain a system of equations D20y11+y11 =−2iA′1eiT0 +d5(−ω22A22e2iω2T0 −ω23A23e2iω3T0 + +2ω2ω3A2A3e i(ω3−ω2)T0−2ω2ω3A2A3ei(ω2+ω3)T0+2ω22A2A2+2ω23A3A3)+ +d4c(−ω22Λ22A22e2iω2T0 −ω23Λ23A23e2iω3T0 +2ω2ω3Λ2A2Λ3A3ei(ω3−ω2)T0 + −2ω2ω3Λ2A2Λ3A3ei(ω2+ω3)T0 +2ω22Λ2A2Λ2A2+2ω23Λ3A3Λ3A3)+ + 1 2 d3pe iµT0 −d3γ1iω1A1eiT0 +2d5d6β21[Λ2A22e2iω2T0 +Λ3A23e2iω3T0 +(3.10) +(Λ2+Λ2)A2A2+(Λ3+Λ3)A3A3+(Λ2+Λ3)A2A3e i(ω2+ω3)T0 + +(Λ2+Λ3)A2A3e i(ω3−ω2)T0]−d5d6β21(Λ22A22e2iω2T0 +Λ23A23e2iω3T0 + +2Λ2A2Λ3A3e i(ω3−ω2)T0 +2Λ2A2Λ3A3e i(ω2+ω3)T0 +2Λ2A2Λ2A2+ +2Λ3A3Λ3A3)− d25β 2 1 d3 (A22e 2iω2T0 +A23e 2iω3T0 +2A2A2+ +2A3A3+2A2A3e i(ω2+ω3)T0 +2A2A3e i(ω3−ω2)T0) D20ϕ11−d6β21ϕ21+d7β21ϕ11 =−2iω2A′2eiω2T0 −2iω3A′3eiω3T0 + −d7(A1A2ei(1+ω2)T0+A1A2ei(1−ω2)T0+A1A3ei(1+ω3)T0+A1A3ei(−1+ω3)T0)+ +d6(Λ2A1A2e i(1+ω2)T0 +Λ2A1A2e i(1−ω2)T0 +Λ3A1A3e i(1+ω3)T0 + (3.11) +Λ3A1A3e i(−1+ω3)T0)− (γ3 c +γ2+γ3 ) (iω2A2e iω2T0 +iω3A3e iω3T0)+ + ( γ3+ γ3 c ) (iω2Λ2A2e iω2T0 +iω3Λ3A3e iω3T0) D20ϕ21− d7β 2 1 c ϕ11+ d7β 2 1 c ϕ21 =−2iω2Λ2A′2eiω2T0 −2iω3Λ3A′3eiω3T0 + −d7 c (A1A2e i(1+ω2)T0+A1A2e i(1−ω2)T0+A1A3e i(1+ω3)T0+A1A3e i(−1+ω3)T0)+ − d7 c (Λ2A1A2e i(1+ω2)T0 +Λ2A1A2e i(1−ω2)T0 +Λ3A1A3e i(1+ω3)T0 + (3.12) +Λ3A1A3e i(−1+ω3)T0)− (d7γ3 c2 + γ2 c + γ3 c ) (iω2A2e iω2T0 +iω3A3e iω3T0)+ − (d7γ3 c2 + γ3 c ) (iω2Λ2A2e iω2T0 +iω3Λ3A3e iω3T0) Analysis of vibration of three-degree-of-freedom... 147 In thiswork,we analyze one combination of internal resonances and the exter- nal resonance µ=1 2ω2 =1 ω3 =3ω2 We introduce detuning parameters σ1, σ2, σ3 defined by 2ω2+εσ1 =1 ω3 =3ω2+εσ2 µ=1+εσ3 (3.13) Substituting (3.13) into equation (3.10) and eliminating terms that produce secular terms, we obtain −2iA′1+2 [ d5+d4cΛ2Λ3+d5d6β 2 1(Λ2+Λ3)−d5d6β21Λ2Λ3+ − d25 d3 β21 ] ω2ω3A2A3e iT1(−σ1+σ2)− ( d5+d4cΛ 2 2−2d5d6β21Λ2+ (3.14) +d5d6β 2 1Λ 2 2+ d25 d3 β21 ) ω22A 2 2e −iT1σ1 + 1 2 d3pe iσ3T1 −d3γ1iω1A1 =0 By introducing A1 = 1 2 a1e iα1 A2 = 1 2 a2e iα2 A3 = 1 2 a3e iα3 (3.15) and θ1 =2α2−α1−T1σ1 θ2 =α3−α2−α1−T1σ1+T1σ2 (3.16) θ3 =−α1+T1σ3 into (3.14), we obtain the first modulation equation −ia′1+a1α′1+ 1 4 f1a2a3e −iθ2 + 1 4 f2a 2 2e iθ1 + 1 2 d3pe iθ3 − 1 2 id3γ1a1 =0 (3.17) where f1 =2d5ω2ω3+2d4cω2ω3Λ2Λ3+2d6d5β 2 1(Λ2+Λ3−Λ2Λ3)− 2d25β 2 1 d3 f2 =−d5ω22−d4cω22Λ22+d5d6β21(2Λ2−Λ22)− d25β 2 1 d3 To determine the solvability conditions of (3.11) and (3.12), we seek for par- ticular solutions in the form ϕ11 =P11e iω2T0 +P12e iω3T0 ϕ21 =P21e iω2T0 +P22e iω3T0 (3.18) 148 D. Sado, K. Gajos Substituting particular solutions (3.18) into equations (3.11), (3.12) and using resonant conditions (3.13) and equaling the coefficients of exp(iω2T0) and exp(iω3T0) on both sides, we obtain system of four equations −ω22P11−d6β21P21+d7β21P11 =R11 (3.19) −ω22P21− d7β 2 1 c (P11+P21)=R21 and −ω23P12−d6β21P22+d7β21P12 =R12 (3.20) −ω23P22− d7β 2 1 c (P12+P22)=R22 where R11 =−2iω2A′2−d7(A1A2eiT1σ1 +A1A3eiT1(σ2−σ1))+ +d6(A1A2Λ2e iT1σ1 +A1A3Λ3e iT1(σ2−σ1))+ − (γ3 c +γ2+γ3 ) (iω2A2+ (γ3 c +γ3 ) (iω2A2Λ2 R21 =−2iω2A′2δΛ2−d7(A1A2eiT1σ1 +A1A3eiT1(σ2−σ1))+ + d7 c (A1A2Λ2e iT1σ1 +A1A3Λ3e iT1(σ2−σ1))+ + (d7γ3 c2 + γ2 c + γ3 c ) (iω2A2− (d7γ3 c2 + γ3 c ) (iω2A2Λ2 R12 =−2iω3A′3−d7A1A2eiT1(−σ2+σ1)+d6A1A2Λ2eiT1(−σ2+σ1))+ + (γ3 c +γ2+γ3 ) (iω3A3+ (γ3 c +γ3 ) (iω3A3Λ3 R22 =−2iω3A′3Λ3+d7A1A2eiT1(−σ2+σ1)− d7 c A1A2Λ2e iT1(−σ2+σ1))+ + (d7γ3 c2 + γ2 c + γ3 c ) (iω3A3− (d7γ3 c2 + γ3 c ) (iω3A3Λ3 We reduce the problem of determination of the solvability conditions of equ- ations (3.11), (3.12) to finding solvability conditions of equations (3.19) and (3.20).The determinants of the coefficient matrices of equations (3.19) and (3.20) are the same and equal 0 according to conditions on the natural frequ- encies of system (3.6). Analysis of vibration of three-degree-of-freedom... 149 Then the solvability conditions are ∣ ∣ ∣ ∣ ∣ ∣ ∣ R11 −d6β21 R21 −ω22+ d7β 2 1 c ∣ ∣ ∣ ∣ ∣ ∣ ∣ =0 (3.21) for equations (3.19) and ∣ ∣ ∣ ∣ ∣ ∣ ∣ R12 −d6β21 R22 −ω23+ d7β 2 1 c ∣ ∣ ∣ ∣ ∣ ∣ ∣ =0 (3.22) for equations (3.20). Substituting (3.15) and (3.16) and after some transformations, we obtain two modulation equations −ia′2+a2α′2+ f4 4f3ω2 a1a2e −iθ1 + f5 4f3ω2 a1a3e iθ2 + f6 2f3 ia2 =0 (3.23) −ia′3+a3α′3+ f8 4f7ω3 a1a2e −iθ2 + f9 2f7 ia3 =0 where f3 =−ω22+ d7β 2 1 c +d6Λ2β 2 1 f4 = ( −ω22+ d7β 2 1 c ) (−d7+d6Λ2)+ d6d7β 2 1 c (1−Λ2) f5 = ( −ω22+ d7β 2 1 c ) (−d7+d6Λ3)+ d6d7β 2 1 c (1−Λ3) f6 = ( −ω22+ d7β 2 1 c )[ − γ3 c −γ3−γ2+Λ2 (γ3 c +γ3 )] + + d6β 2 1 c [d7γ3 c +γ3+γ2−Λ2 (d7γ3 c +γ3 )] (3.24) f7 =−ω23+ d7β 2 1 c +d6Λ3β 2 1 f8 = ( −ω23+ d7β 2 1 c ) (−d7+d6Λ2)+ d6d7β 2 1 c (1−Λ2) f9 = ( −ω23+ d7β 2 1 c )[ − γ3 c −γ3−γ2+Λ3 (γ3 c +γ3 )] + + d6β 2 1 c [d7γ3 c +γ3+γ2−Λ3 (d7γ3 c +γ3 )] 150 D. Sado, K. Gajos To separate the real and imaginary parts of modulation equations (3.17), (3.23) and (3.24), we have to transform exp(iθ) into a complex form exp(iθ)= cosθ+isinθ.We obtain six modulation equations a′1 = a2a3f1 sinθ2+a 2 2f2 sinθ1+ 1 2 d3psinθ3− 1 2 d3γ1a1 a1α ′ 1 =−a2a3f1cosθ2−a22f2cosθ1+ 1 2 d3pcosθ3 a′2 =− f4 4f3ω2 a1a2 sinθ1+ f5 4f3ω2 a1a3 sinθ2+ f6 2f3 a2 (3.25) a2α ′ 2 =− f4 4f3ω2 a1a2cosθ1− f5 4f3ω2 a1a3cosθ2 a′3 =− f8 4f7ω3 a1a2 sinθ2+ f9 2f7 a3 a3α ′ 3 =− f8 4f7ω3 a1a2cosθ2 From these equations, we look for steady-state motion. In this case, we have a′1 =0 a ′ 2 =0 a ′ 3 =0 θ′1 =0 θ ′ 2 =0 θ ′ 3 =0 (3.26) We obtain a system of equations a2a3f1 sinθ2+a 2 2f2 sinθ1+ 1 2 d3psinθ3− 1 2 d3γ1a1 =0 −a2a3f1cosθ2−a22f2cosθ1+ 1 2 d3pcosθ3−a1σ3 =0 − f4 4f3ω2 a1a2 sinθ1+ f5 4f3ω2 a1a3 sinθ2+ f6 2f3 a2 =0 (3.27) − f4 4f3ω2 a1a2cosθ1− f5 4f3ω2 a1a3cosθ2−a2 σ3+σ1 2 =0 − f8 4f7ω3 a1a2 sinθ2+ f9 2f7 a3 =0 − f8 4f7ω3 a1a2cosθ2−a3 3σ3−2σ2+3σ1 2 =0 Analysis of vibration of three-degree-of-freedom... 151 After transformations, we get amplitude equations f28a 2 1a 2 2−4[f29 +f27(3σ3−2σ2+3σ1)]ω23a23 =0 f24f 2 8a 2 1a 4 2− (2f5f9ω3a23+2f6f8ω2a22)2+ −[2f5f7ω3(3σ3−2σ2+3σ1)a23−2f3f8ω2(σ3+σ1)a22]2 =0 (3.28) f24f 2 8d 2 3p 2a21− [4f9ω3(f4f1+f5f2)a23+4f6f8f9ω2a22−f4f8d3γ1a21]2+ +[4f7ω3(f4f1−f5f2)(3σ3−2σ2+3σ1)a23+4f3f8f2ω2(σ3+σ1)a22+ −f4f8σ3a21]2 =0 From equations (3.28), we obtain a42 [ −h 2 1h4 h23 a41+ ( h3− h6h1 h3 ) a21−h5 ] =0 (3.29) We have two types of solutions, and these possibilities are examined in turn: — case I – one-frequency solution a2 =0 then a3 =0 and a 2 1(h10a 2 1−h7)= 0 so a1 =0 or a1 = √ h7 h10 (3.30) — case II – multi-frequency solution h21h4 h23 a41− ( h3− h6h1 h3 ) a21+h5 =0 (3.31) so a1 = √ √ √ √ √ h3− h6h1h3 ± √ ∆1 2h2 1 h4 h2 3 (3.32) where ∆1 = ( h3− h6h1 h3 )2 −4h4h5h 2 1 h23 and from (3.28) a2 = √ √ √ √ √ √ − ( h13h1 h3 a41+h11a 2 1 ) ± √ ∆2 2 ( h2 1 h8 h2 3 a41+ h12h1 h3 a21+h9 ) a3 = √ h1 h3 a1a2 (3.33) 152 D. Sado, K. Gajos where ∆2 = (h13h1 h3 a41+h11a 2 1 )2 −4 (h21h8 h23 a41+ h12h1 h3 a21+h9 ) (h10a 4 1−h7a21) and h1 = f 2 8 h2 =4ω 2 3(f 2 9 +f 2 7)(3σ3−2σ2+3σ1)2 h3 = f 2 4f 2 8 h4 =4f 2 5ω 2 3[f 2 9 +f 2 7(3σ3−2σ2+3σ1)2] h5 =4f 2 8ω 2 2[f 2 6 +f 2 3(σ3+σ1) 2] h6 =8f5f8ω2ω3[f6f9−f3f7(3σ3−2σ2+3σ1)(σ3+σ1) h7 = f 2 4f 2 8d 2 3p 2 h10 = f 2 4f 2 8(d 2 3γ 2 1 +σ 2 3) h8 =16f 2 9ω 2 3(f4f1+f5f2) 2+16f27ω 2 3(f4f1−f5f2)2(3σ3−2σ2+3σ1)2 h9 =16f 2 6f 2 8f 2 9ω 2 2+16f 2 2f 2 3f 2 8ω 2 2(σ3+σ1) 2 h11 =−8f4f28ω2[f6f9d3γ1+f2f3σ3(σ3+σ1)] h12 =32f8ω2ω3[f6f 2 9(f4f1+f5f2)+ +f1f3f7(f4f1−f5f2)(3σ3−2σ2+3σ1)(σ3+σ1)] h13 =−8f4f8ω3[f9d3γ1(f4f1+f5f2)+f7σ3(f4f1−f5f2)(3σ3−2σ2+3σ1)] Both cases of solutions (one-frequency andmulti-frequency) are presented in Figs.2-5. In Fig.2 and Fig.3 amplitudes a1, a2, a3 are plotted as functions of the amplitude of excitation p. We can see the jump phenomenon associa- ted with the varying amplitude p. We have regions where two of the three solutions are stable. The initial conditions determine which of these solutions gives the response. We can clearly see the saturation phenomenon, when the amplitude a1 assumes its maximum value for stable solutions. In Fig.4 and Fig.5, these amplitudes are presented versus the detuning parameter σ1. We can see the jump phenomenon associated with the varying frequency ω1 according with the amplitude a1. 4. Conclusions Themultiple scales method can be used to find an approximate solution for a systemwith three degrees of freedomwith variable amplitudes andphases.We can find resonance conditions (sometimes the resonance area is very narrow Analysis of vibration of three-degree-of-freedom... 153 Fig. 2. Amplitudes of the response as functions of the amplitude of the excitation; d1 =0.9, d2 =1.6, c=1, β1 =0.67082,µ=1, γ1 =0.0001, γ2 =0.00001, γ3 =0.00001, σ1 =σ2 =σ3 =0 Fig. 3. Amplitudes of the response as functions of the amplitude of the excitation; d1 =0.9, d2 =1.6, c=1, β1 =0.67082,µ=1, γ1 =0.0001, γ2 =0.00001, γ3 =0.00001, σ1 =σ2 =σ3 =1 154 D. Sado, K. Gajos Fig. 4. Frequency-response curves; d1 =0.9, d2 =1.6, c=1, β1 =0.67082,µ=1, p=4.4, γ1 =0.0001, γ2 =0.00001, γ3 =0.00001, σ2 =0, σ3 =1 Fig. 5. Frequency-response curves; d1 =0.9, d2 =1.6, c=1, β1 =0.67082,µ=1, p=4.4, γ1 =0.0001, γ2 =0.00001, γ3 =0.00001, σ2 =0, σ3 =−1 and difficult to find numerically). It is possible to investigate steady state solutions for different combinations of external and internal resonances. We can observe regions where the solutions are stable or unstable, and can clearly see the saturation phenomenon. Analysis of vibration of three-degree-of-freedom... 155 References 1. Bajaj A.K., Chang S.I., Johnson J.M., 1994, Amplitude modulated dy- namics of a resonantly excited autoparametric two degree-of-freedom system, Nonlinear Dynamics, 5, 433-357 2. 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Non-Linear Mechanics, 39, 821-832 Analiza drgań dynamicznego układu z podwójnym wahadłem o trzech stopniach swobody Streszczenie W pracy przebadano drgania nieliniowego układu o trzech stopniach swobody z podwójnymwahadłemw otoczeniu rezonansówwewnętrznych i zewnętrznych. Ba- dania przeprowadzono analitycznie i numerycznie. Rozwiązanie analityczne uzyskano przy użyciumetodywielu skali czasowych.Metoda posłużyła do zbudowania nielinio- wych równań różniczkowych pierwszego rzędu opisującychmodulację amplitud i faz. Rozwiązanie ustalone i jego stabilność zostały przedstawione dla wybranychwartości parametrów układu. Manuscript received February 21, 2007; accepted for print April 4, 2007