Quenching in a Non-Ideal Mechanical System and the Averaging Method Márcio José Horta Dantas∗, José Manoel Balthazar† and Jorge Luiz Palacios Felix∗∗ ∗ Faculdade de Matemática, UFU ,38400-902 UBERLÂNDIA M.G. , BRAZIL † Departamento de Estatística, Matemática Aplicada e Computação, UNESP,13506-900 RIO CLARO S.P., BRAZIL ∗∗ Departamento de Matemática, Universidade Federal do Pampa, 96413-170, BAGÉ, R.S., BRAZIL Abstract. In this paper, for the first time, a quenching result in a non-ideal system is rigorously obtained. In order to do this a new mechanical hypothesis is assumed, it means that the moment of inertia of the rotating parts of the energy source is big. From this is possible to use the Averaging Method. Keywords: Averaging Theorem, Non-Ideal System, Stability, Quenching PACS: 45.10.Hj 1. INTRODUCTION In the design of structures it is necessary to investi- gate the relevant dynamics in order to predict the struc- tural response due to excitation. In the selection of ro- tating machines for applications in structures, often little thought is given to the effect that the structure has on the machine, i.e., the excitation is considered independent of the system response. Mathematical models of real systems are usually ide- alized by prescribing the forcing term as a known func- tion. In reality, for a great number of structures this is not the case, and such structures are called non-ideal. In general, it is possible that an energy source fixed to a structure may be affected by the structural response. Sys- tems having dynamic coupling between structure and the energy source often exhibit peculiar behavior, especially systems with limited power supply. Non-ideal systems operating in the neighborhood of resonant frequencies are often more expensive and perform poorly as com- pared with ideal counterparts.A number of authors stu- died this class of mechanical systems. We can mention some of them Kononenko [8], Nayfeh and Mook [10] and Balthazar et al. [1], [2], [3]. In applications, for example Mechanical Engineering, a basic question is how to suppress, or at least, how to quench an undesirable vibration, see for example [11]. This work concerns with quenching in a class of non- ideal mechanical systems. In [4] a numerical study is performed with same mechanical system that is inves- tigated here. One of the motivations of [4] is the problem of vibration attenuation in a structure due to a nonlinear energy sink. Such kind of problem has been investigated by several authors such as Felix et all [5], Malatkar et all [9] and Jiang et all [7] in other mechanical systems. In this paper, it is proved that a non ideal system exhibits a stable vibration and it is asked how to quench such oscillation. This is achieved by appending to the original system another one. Quenching of the original vibration has been achieved if there is a reduction of the (x, x′) amplitudes when comparing the initial system and the final one. It is worthy to note that all results are rigorously obtained by using the Averaging Theorem, see [6]. We organize this work as the following. In Section 2 the equations of motion of the non-ideal system under in- vestigation are given. By assuming that angular momen- tum of the rotating parts of the motor is big, one has two scales, ε, ε2, in the equations of motion of the system. These scales afford the correct use of Averaging Method in this problem. We believe that this approach is a step beyond of Kononenko’s modelling, see [8]. In Section 3 it is shown that the last foregoing system has two hyper- bolic periodic orbits, one of them is stable. In Section 4 a mass is attached, by springs, to the Main Body in the original system. It is proved that this new system has an asymptotically stable periodic orbit. And in Section 5 it is rigorously proved that the oscillations of the Main Body have been quenched in the new mechanical system. Finally, in Section 6 some comments are given. 2. A NON-IDEAL SYSTEM In this section let us consider a mechanical system com- pound of a damped linear spring linked to a wall and to a principal body. On this principal body there is a DC motor, with limited supply power, which rotates a small mass m0 . Both, the spring and the body-motor set, are at the same height from the ground and such system has 9th International Conference on Mathematical Problems in Engineering, Aerospace and Sciences AIP Conf. Proc. 1493, 274-281 (2012); doi: 10.1063/1.4765501 © 2012 American Institute of Physics 978-0-7354-1105-0/$30.00 274 two degrees of freedom. Such system is given in Figure 1. �� θ r x1 c1 k1 m0 M FIGURE 1. A Non Ideal System In this system, M0 denotes the mass of the fixed part of the mechanism such as its base. In Figure 1 this mass is indicated by the region in gray. The mass of the rotating parts of the motor is denoted by M1 and its moment of inertia by J. Besides M, given by M0 +M1, means the mass of the body-motor set. The constant k1 is the stiffness of the spring. The resistance of the oscillatory motion is a linear force c1x′1. Let r be the distance between the mass m0 and the axis of rotation of the DC motor. It is assumed that −π ≤ θ < ∞. The kinetic and potential are given respectively by T = M 2 ( x′1 (t) )2 + m0 2 (( x′1 (t)− rθ ′ (t)cosθ (t) )2 (1) + ( rθ ′ (t)sinθ (t) )2 ) + J 2 θ ′ (t)2 and V = k1 2 (x1 (t))2 + gm0 r (1+ cosθ (t)) . (2) Let L = T −V be the Lagrangian of this system then the equations of motion are given by⎧⎪⎪⎨⎪⎪⎩ d dt ( ∂L ∂x′ ) − ( ∂L ∂x ) = −c1x′1 (t) , d dt ( ∂L ∂θ ′ ) − ( ∂L ∂θ ) = Γ0 (θ ′ (t)) , (3) where function Γ0 (·) is the difference between the dri- ving torque of the source of energy (motor) and the re- sistive torque applied to the rotor. Such function Γ0 (·) is obtained from experiments. Let us define the following dimensionless variables and parameters τ = √ k1 M+m0 t, y1 (τ) = 1 r x1 (√ M+m0 k1 τ ) , θ1 (τ) = θ (√ M+m0 k1 τ ) , Γ(z) = M+m0 k1 J Γ0 (√ k1 M+m0 z ) . By substituting the above equations in (3) one gets ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ y′′1 (τ) = −y1 (τ)− c1√ k1 √ M+m0 y′1 (τ) + m0 m0 +M ( cos(θ1 (τ)) (θ ′′1 (τ)) −sin(θ1 (τ)) (θ ′1 (τ)) 2 ) , θ ′′1 (τ) = J J+m0 r2 Γ(θ ′1 (τ)) + m0 gr (M+m0) k1 (J+m0 r2) sin (θ1 (τ)) + m0 r2 J+m0 r2 cos(θ1 (τ))y′′1 (τ) . (4) It is assumed that m0 � 1 , c1 � 1, Γ(·) � 1 and J � 1. This last condition means that the moment of inertia of the rotating parts of the motor is big. In terms of adimensional parameters the following conditions are imposed: c1√ k1 √ M+m0 = ε λ1, (5) m0 m0 +M =−ε q1, (6) M0r2 J = ε q4, (7) where λ1, q1 and q4 are adimensional parameters. From (5), (6) and (7) one has m0 r2 J+m0 r2 =−ε2 q2, (8) m0 gr (M+m0) k1 (J+m0 r2) = ε2 q3, (9) where q2 and q3 are dimensionless parameters. In view of (8), (9), it will be assumed that J J+m0 r2 Γ(·) = ε2 Γ1 (·) , (10) where Γ1 (·) is an adimensional function. Using (5), (6),(8), (9) and (10) in (4), the following two scale sys- tem is obtained 275 ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ y′′1 (τ) = −y1 (τ)− ε λ1 y′1 (τ) −ε q1 ( cos(θ1 (τ)) (θ ′′1 (τ)) −sin(θ1 (τ)) (θ ′1 (τ)) 2 ) , θ ′′1 (τ) = ε2 (Γ1 (θ ′1 (τ)) −q2 cos(θ1 (τ))y′′1 (τ) +q3 sin(θ1 (τ))) . (11) 3. STABLE AND UNSTABLE ORBITS From now on the following replacements in the notation will be done, τ → t, y1 → x1, θ1 → θ and Γ1 → Γ. Of course it will be assumed that t, x1 and Γ are dimension- less. By rewritten (11) as a first order system, one gets ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ x′1 (t) = u(t) , u′ (t) = ε ( q1 p(t)2 sin(θ (t)) −λ1u(t) ) −x1 (t)+O ( ε2) θ ′ (t) = p(t) , p′ (t) = ε2 (Γ(p(t))+ q3 sin(θ (t)) +q2 x1 (t) cos(θ (t)))+O ( ε3) . (12) Following [8, pg.38], let us assume the frequency p is close to natural frequency 1. Thus p(t) = 1+ ε p1 (t) . (13) By substituting (13) in (12)3 one gets θ ′ (t)= 1+ε p1 (t). Since 0 < ε ≤ 1 then one can write t in terms of θ and obtains the following reduced system⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ x′1 (θ ) = u(θ )− ε p1 (θ )+ u(θ ) +O ( ε2) , u′ (θ ) = −x1 (θ ) +ε (sin(θ ) q1 +p1 (θ ) x1 (θ )−λ1u(θ )) +O ( ε2) , p′1 (θ ) = ε (sin(θ ) q3 +x1 (θ ) cos(θ ) q2 +Γ(1)) +O ( ε2) . (14) Consider now the following change of variables x1 (θ ) = cos(θ ) x11 (θ )+ sin(θ ) u1 (θ ) , u(θ ) = cos(θ ) u1 (θ )− sin(θ ) x11 (θ ) (15) By using (15) in (14) one obtains a non-autonomous system 2π periodic in the θ variable. In this case the Averaging Theorem can be used. So, performing the average of this system on the interval [0,2π ] one gets the following averaged system⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩ x̂′11 (θ ) = ε −λ1 x̂11− 2 p̂1 û1− q1 2 , û′ (θ ) = ε 2 p1 x̂11− λ1 û1 2 , p̂′1 (θ ) = ε q2 x̂11 + 2Γ(1) 2 . (16) The equilibrium solutions of (16) are given by⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ x̂un11 = −2Γ(1) q2 , ûun1 = − √ 2 √ Γ(1) √ q1 q2− 2Γ(1) λ1√ λ1 q2 , p̂un1 = √ λ1 q1 q2 Γ(1) − 2λ 2 1 2 3 2 , (17) and 276 ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ x̂s11 = −2Γ(1) q2 , ûs1 = √ 2 √ Γ(1) √ q1 q2− 2Γ(1) λ1√ λ1q2 , p̂s1 = − √ λ1 q1 q2 Γ(1) − 2λ 2 1 2 3 2 . (18) Of course the following inequalities must be hold in order (17) and (18) could be valid. λ1 > 0, Γ(1)> 0, q1 q2− 2Γ(1) λ1 > 0. (19) The eigenvalues of the jacobian of the vector field given by the right side of (16) calculated at the equilibrium point (17) are roots of the characteristic polinomial P(λ ) = λ 3 +Aλ 2 +Bλ +C , where A= λ1, B=− 2 5 2 Γ(1) 3 2 √ q1 q2−2Γ(1)λ1−λ 3 2 1 q1 q2 8Γ(1) √ λ1 , C =− √ Γ(1) √ λ1 √ q1 q2−2Γ(1)λ1√ 2 , AB−C= λ 2 1 q1 q2 8Γ(1) . (20) From (20)3 one concludes that C is negative thus it follows from Hurwitz Criterium that P(λ ) has a root which real part is positive. Then one concludes from Averaging Theorem that (14) has an unstable hyperbolic periodic orbit. In the case of the equilibrium point given by (18) one gets A= λ1, B= 2 5 2 Γ(1) 3 2 √ q1 q2−2Γ(1)λ1+λ 3 2 1 q1 q2 8Γ(1) √ λ1 , C = √ Γ(1) √ λ1 √ q1 q2−2Γ(1)λ1√ 2 , AB−C= λ 2 1 q1 q2 8Γ(1) . (21) So all roots of P(λ ) have negative real parts, thus it follows from Averaging Theorem that (14) has an asymptotically stable hyperbolic periodic orbit. As it was assumed in [4], let us take Γ(ϕ ′) = a−bϕ ′, where a and b are adimensional parameters. For physical motivation of this choice see [4, pg.2]. Now, using the parameters given at the following table, the equations (18), (17) and the Averaging Theorem, one can obtain the initial conditions for the cases stable and unstable of (14). The final results are given in the next table. Parameters Initial Conditions ε 0.01 Stable Case Unstable Case q1 −2 x1 (0) = 0.66 x1 (0) = 0.66 q2 −3 u(0) =−1.49 u(0) = 1.49 q3 0.1 p1 (0) =−0.55 p1 (0) = 0.55 γ1 0.5 a 6 b 5 Using this data, one gets the following graphics. -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -2-1.5-1-0.5 0 0.5 1 1.5 2 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 p1 Unstable Orbit Asymptotically Stable Orbit x1 u p1 FIGURE 2. The Stable and Unstable Orbits -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0 50 100 150 200 250 300 350 400 θ p1 Unstable Orbit Asymptotically Stable Orbit FIGURE 3. θ × p1 Graphic 4. ANOTHER NON IDEAL SYSTEM Although the existence of an asymptotically stable peri- odic orbit obtained in Section 3, it should be interesting to suppress or at least to quench this orbit. The way we 277 chose is to append a mass to the principal one, given at Figure 1 in a way given in the following figure. �� θ r x1 c1 k1 m0 M x2 m2 k2 c2 k FIGURE 4. A Non-Ideal System with Three Degree of Free- dom The equations of motion of the mechanical system given in Figure 4, in its dimensionless form and after the introduction of a small parameter ε as in Section 3, are given by ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ x′′1 (t) = −x1 (t) +ε1 (x2 (t)− x1 (t))− ε λ1x′1 (t) +ε q1 ( sin (θ (t))(θ ′ (t))2 −cos(θ (t)) θ ′′ (t) ) , x′′2 (t) = ε2 (x1 (t)− x2 (t))− ε λ2 x′2 (t) −ε γ x2 (t)3 , θ ′′ (t) = ε2 ( Γ(θ ′ (t)) −q2 cos(θ (t)) (x′′1 (t)) +q3 sin(θ (t)) ) . (22) Of course the parameters in (22) depend on the physical constants k1, k2, k,M, m0, m2, r as well as of the moment of inertia of rotating parts and acceleration of gravity. As in the Section 2, the function Γ is the difference between the driving torque of the source energy, in this case a motor, and the resistive torque applied to the rotor. Again this function is given in its dimensionless form. We will write (22) as a pair of perturbed harmonic oscillators coupled with (22)3. Let r1,r2 be positive parameters such that r2 < 1< r1 and ε1 = ( r21− 1 ) ( 1− r22 ) , ε2 = r21 r22. (23) By writing (22) as a first order system, using (23) and the following change of variables ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ x1 (t) u(t) x2 (t) v(t) θ (t) p(t) ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ = ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ x21 (t)+ x11 (t) r1 v1 (t)+ r2 u1 (t) r22 x21 (t) r22− 1 + r12 x11 (t) r12− 1 r1 r22 v1 (t) r22− 1 + r12 r2 u1 (t) r12− 1 θ (t) p(t) ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (24) where u(t) = x′1 (t), v(t) = x′2 (t) and p(t) = θ ′ (t), one obtains the following system x′11 (t) = u1 (t) , u′1 (t) = −r22 x11 (t) +ε r2c3 ( c1 ( q1 p(t)2 sin(θ (t)) −λ1 (v1 (t)+ u1 (t)) ) +c2 ( −γ (c7 x21 (t)+ c8 x11 (t))3 −λ2 (c7 v1 (t)+ c8u1 (t)) )) +O ( ε2) , x′21 (t) = v1 (t) , v′1 (t) = −r21 x21 (t) +ε r1c3 ( c4 ( q1 p(t)2 sin(θ (t)) −λ1 (v1 (t)+ u1 (t)) ) +c5 ( −γ (c7 x21 (t)+ c8 x11 (t))3 −λ2 (c7 v1 (t)+ c8u1 (t)) )) +O ( ε2) , θ ′ (t) = p(t) , p′ (t) = ε2 ( Γ(p(t))+ q3 sin (θ (t)) +(c9 (x21 (t)+ x11 (t)) −c6q2 (c7 x21 (t)+ c8 x11 (t))) ·cos(θ (t)) ) , (25) where c1, . . . ,c9 are adequate rational functions of r1,r2. Assuming now the ansatz p(t) = ω0 + ε p1 (t) , (26) 278 where ω0 is parameter to be chosen later. By following analogous steps to those ones performed in Section 3, see (13), one can obtain a θ dependent reduced system from (25) This new system has five equations and its unknown funtions are given by x11 (θ ), u1 (θ ), x21 (θ ), v1 (θ ) and p1 (θ ). In this new system is applied the following change of variables ⎛⎜⎝x11 u1 x21 v1 ⎞⎟⎠= ⎛⎜⎜⎜⎜⎜⎜⎜⎝ cos ( θ r2 ω0 ) z11 + sin ( θ r2 ω0 ) z1 r2 cos ( θ r2 ω0 ) z1− r2 sin ( θ r2 ω0 ) z11 cos ( θ r1 ω0 ) z21 + sin ( θ r1 ω0 ) z2 r1 cos ( θ r1 ω0 ) z2− r1 sin ( θ r1 ω0 ) z21 ⎞⎟⎟⎟⎟⎟⎟⎟⎠ . (27) By using (27) in the θ dependent system described ear- lier one gets ⎛⎜⎜⎜⎝ z′11 z′1 z′21 z′2 p′1 ⎞⎟⎟⎟⎠= ε ⎛⎜⎜⎜⎝ G1 (z11,z1,z21,z2, p1,θ ) G2 (z11,z1,z21,z2, p1,θ ) G3 (z11,z1,z21,z2, p1,θ ) G4 (z11,z1,z21,z2, p1,θ ) G5 (z11,z1,z21,z2, p1,θ ) ⎞⎟⎟⎟⎠+O ( ε2) , (28) where Gj, j = 1, . . . , 5 areC∞ functions . Their algebraic expressions are very complicated and we will not give them here. Now, assuming that ω0 = r1, (29) let us adopt the following resonance condition r1 = 3 2 r2. (30) Substituting (29),(30) in (28) one obtains another θ de- pendent system with period equal to 6π . Computing the averaged system, one has⎛⎜⎜⎜⎝ ẑ′11 ẑ′1 ẑ′21 ẑ′2 p̂′1 ⎞⎟⎟⎟⎠= ε ⎛⎜⎜⎜⎝ F1 (ẑ11, ẑ1, ẑ21, ẑ2, p̂1) F2 (ẑ11, ẑ1, ẑ21, ẑ2, p̂1) F3 (ẑ11, ẑ1, ẑ21, ẑ2, p̂1) F4 (ẑ11, ẑ1, ẑ21, ẑ2, p̂1) F5 (ẑ11, ẑ1, ẑ21, ẑ2, p̂1) ⎞⎟⎟⎟⎠ (31) where F1 = ⎛⎜⎜⎜⎜⎝ ( 18π c2 c7 2 c8 γ r22 ẑ1 ẑ221 +8π c2 c7 2 c8 γ ẑ1 ẑ22 +9π c2 c8 3 γ r22 ẑ1 ẑ211 +(−12π λ2 c2 c8− 12π λ1 c1) r24 ẑ11 +9π c2 c8 3 γ ẑ31− 16π c3 p̂1 r22 ẑ1 ) ⎞⎟⎟⎟⎟⎠ 36π c3 r24 F2 =− ⎛⎜⎜⎜⎜⎝ ( 18π c2 c7 2 c8 γ r22 ẑ11 ẑ221 +8π c2 c7 2 c8 γ ẑ11 ẑ22 +9π c2 c8 3 γ r22 ẑ311 + ( 9π c2 c8 3 γ ẑ21− 16π c3 p̂1 r22) ẑ11 +(12π λ2 c2 c8 + 12π λ1 c1) r22 ẑ1 ) ⎞⎟⎟⎟⎟⎠ 36π c3 r22 F3 = ⎛⎜⎜⎜⎜⎜⎜⎝ ( 18π c5 c7 3 γ r22 ẑ2 ẑ221 +(−54π λ2 c5 c7− 54π λ1 c4) r24 ẑ21 +8π c5 c7 3 γ ẑ32 + ( 36π c5 c7 c8 2 γ r22 ẑ211 +36π c5 c7 c8 2 γ ẑ21− 48π c3 p̂1 r22) ẑ2 −81π c4q1 r25 ) ⎞⎟⎟⎟⎟⎟⎟⎠ 108π c3 r24 F4 =− ⎛⎜⎜⎜⎜⎜⎜⎝ ( 9π c5 c7 3 γ r22 ẑ321 + ( 4π c5 c7 3 γ ẑ22 +18π c5 c7 c8 2 γ r22 ẑ211 +18π c5 c7 c8 2 γ ẑ21 −24π c3 p̂1 r22) ẑ21 +(12π λ2 c5 c7 + 12π λ1 c4) r22 ẑ2 ) ⎞⎟⎟⎟⎟⎟⎟⎠ 24π c3 r22 F5 = (18π c9− 18π c6 c7 q2) r2 ẑ21 + 36π r2 Γ ( 3 r2 2 ) 54π r22 An equilibrium point of (31) is given by ẑs11 = 0, ẑs1 = 0, ẑs21 =− 8Γ ( 3r2 2 ) 9q2 r22 , ẑs2 = 1 3q2 r2 √ R1 R2 Γ ( 3r2 2 ) , p̂s1 =− ⎛⎜⎜⎜⎜⎝ 24γ q1 r22 ( 9r22− 4 )√ R2 Γ ( 3 r2 2 )2 + √ R1q2 ( r22− 1 ) R2 2 √ Γ ( 3 r2 2 ) ⎞⎟⎟⎟⎟⎠ 40q2(r22−1)R 3 2 2 Γ ( 3 r2 2 ) . (32) Of course the parameters ci were substituted by the ori- ginal ones. It is assumed that the following inequalities hold: Γ ( 3r2 2 ) > 0, (33) R1 = 243q1q2 ( 1− r22 ) r23− 16R2 Γ ( 3r2 2 ) > 0, (34) R2 = (9λ2− 9λ1) r22− 4λ2 + 9λ1 > 0. (35) Hence, computing the jacobian of (31) at the equilib- rium point (32) one gets the matrixM1⊕M2 where M1 = ( A B −r22 B A ) , (36) 279 A= (9λ2− 9λ1) r22− 9λ2 + 4λ1 15r2 and B= ⎛⎜⎝ 12γ q1 r22 ( 99r22− 89 ) Γ ( 3 r2 2 )2 + √ R1 q2 ( r22− 1 ) R 3 2 2 √ Γ ( 3 r2 2 ) ⎞⎟⎠ 90q2 r22 ( r22− 1 ) R2 Γ ( 3 r2 2 ) Besides M2 is given by M2 = ⎛⎝ M211 M212 M213 M221 M222 M223 M231 M232 M233 ⎞⎠ (37) where M211 =− ⎛⎝ √ R1 ( 144γ r22− 64γ ) Γ ( 3 r2 2 ) 3 2 +81R 3 2 2 q2 2 (r22− 1 )2 r2 ⎞⎠ 1215 √ R2 q22 ( r22− 1 )2 r22 , M221 =− ⎛⎜⎝ ( 2304γ r22− 1024γ ) Γ ( 3 r2 2 )3 +81 √ R1R2 q2 2 (r22− 1 )2 r2 √ Γ ( 3 r2 2 ) ⎞⎟⎠ 3240q22 ( r22− 1 )2 r2 Γ ( 3 r2 2 ) , M231 = 3q2 r2 4 , M212 = ⎛⎜⎜⎜⎜⎜⎜⎝ − ( 9r22− 4 )( 256R2 γ Γ ( 3 r2 2 )3 +3888γ q1 q2 ( r22− 1 ) r23 Γ ( 3 r2 2 )2 ) +81 √ R1R 3 2 2 q2 2 (r22− 1 )2 r2 √ Γ ( 3 r2 2 ) ⎞⎟⎟⎟⎟⎟⎟⎠ 7290R2q22 ( r22− 1 )2 r23 Γ ( 3 r2 2 ) , M222 = ⎛⎝ √ R1 ( 144γ r22− 64γ ) Γ ( 3 r2 2 ) 3 2 −81R 3 2 2 q2 2 (r22− 1 )2 r2 ⎞⎠ 1215 √ R2 q22 ( r22− 1 )2 r22 , M232 = 0, M213 =− 4 √ R1 √ Γ ( 3 r2 2 ) 27 √ R2 q2 r23 , M223 =− 8Γ ( 3 r2 2 ) 9q2 r22 , M233 = 0. Since we are interested in stable periodic orbits, we have to find out sufficient conditions on the parameters such that all eigenvalues of the jacobian M1⊕M2 have negative real parts. For M1, in view of (36) and since det(M1) = A2 +B2r22 > 0 the following inequality (9λ2− 9λ1) r22− 9λ2 + 4λ1 < 0 (38) is enough. The characteristic polynomial of (37) is given by p(s) = s3 +A1 s2 +B1 s+C1. From Hurwitz Criterium, all roots of p(s) have negative real parts, if, and only if, A1 > 0, B1 > 0, C1 > 0 and A1B1−C1 > 0. After a long computation one has A1 = 2R2 15r2 > 0, (39) B1 = ⎛⎜⎜⎜⎜⎝ 16 √ R1R2 Γ ( 3 r2 2 ) 3 2 · ( 3γ q1 r22 ( 9r22− 4 ) +25q2 ( 1− r22 ) ) +243q1q2 2 ( 1− r22 )2r32R 2 2 ⎞⎟⎟⎟⎟⎠ 3600q2 ( 1− r22 ) r22 R2 Γ ( 3 r2 2 ) > 0 (40) C1 = 2 √ R1 √ R2 √ Γ ( 3 r2 2 ) 135r32 > 0 (41) A1B1−C1 = ⎛⎜⎜⎝ 16γ q1 √ R1R2 Γ ( 3 r2 2 ) 3 2 · ( 9r22− 4 ) +81q1q2 2 ( 1− r22 )2 r2R2 2 ⎞⎟⎟⎠ 9000q2 ( 1− r22 ) r2 Γ ( 3 r2 2 ) > 0 (42) Since is assumed that ε > 0, it follows from (6), (8) that q1 < 0, q2 < 0, but this does not change the following conclusions. From (34) and (35) it follows that (39), (41) are true. And (40), (42) hold if, for example, 3r2− 2 ≥ 0. This last condition is compatible, by (30), with the condition r1 > 1. From this we have that (31) has an asymptotically stable equilibrium point. Thus, it follows from Averaging Theorem that (28) has an asymptotically stable periodic orbit near the point (32). Particularly, the first two coordinates of such orbit are near zero. And from (27) one gets that x11,u1 are near zero. 5. QUENCHING In this section the body with mass M in Figure 1 and Figure 4 will be referred as Main Body and the body with mass m2 in Figure 4 as Secondary Body. 280 From (18) and (15) one gets that the two first compo- nents of the stable periodic orbit of (14) obtained Section 3 can be written as x1 (θ ) = x̂s11 cos(θ ) + ûs1 (θ ) sin(θ )+O(ε) , u(θ ) = −x̂s11 sin(θ )+ ûs1 cos(θ )+O(ε) . (43) So, the amplitude of (x1 (t) ,u(t)) of (12) is given by√ x2 1 + u2 = √ 2Γ(1) q1 λ1 q2 +O(ε) . (44) This last result implies that, even for a periodic stable orbit, the Main Body of the non-ideal system given in Section 2 can suffer great oscillations in position and velocity when λ1 � 1, and this can be undesirable. For the system given in Section 4 and assuming the stability conditions obtained in that section, one has from (32) that z11 (θ ) = O(ε) , z1 (θ ) = O(ε) , z21 (θ ) = ẑs21 +O(ε) , z2 (θ ) = ẑs2 +O(ε) , p1 (θ ) = p̂s1 +O(ε) . (45) Note that this means, in the coordinate system in which (28) is written, the Main Body is "almost" stopped. Anyway by using (45), (27), (24) and (26) one has the following estimate √ x2 1 + u2 ≤ √√√√√16R2 Γ ( 3 r2 2 )2 +R1 4q2 2R2 +O(ε) . (46) Comparing (46) with (44) and using (35) one has that the Main Body, in the mechanical system given in Section 4, has an uniformly bounded amplitude for all λ1 � 1. Thus the movement of the Main Body has been quenched due to the attachment of the Secondary Body. 6. CONCLUSIONS In this paper, for the first time, a quenching result in a non-ideal system is rigorously obtained by using the Av- eraging Method. The main reason behind this result is a hypothesis on the angular momentum of the rotating parts of the energy source (motor) of the system. Esti- mates of the amplitude of the Main Body in the original system, see Figure 1 as well as the Main Body in the sys- tem given in the Section 4, Figure 4, are obtained proving the quenching. The method used here can be applied in other inter- esting mechanical systems and it will be the subject of future research. ACKNOWLEDGMENTS The first author acknowledges the support given by Fun- dação de Amparo à Pesquisa do Estado de Minas Gerais- FAPEMIG REFERENCES 1. Balthazar, J.M., Mook, D.T., Brasil, R.M.L.R.F., Weber.H.I., Fenili, A., Belato, D. and Felix, J.L.P., ‘Recent results on vibrating problems with limited power supply’, in: Awrejcewicz, J., Brabski, J. and Nowakowski, J (eds), Sixth Conference on Dynamical Systems Theory and Applications, Lodz, Poland, December 10-12, 2001. 2. Balthazar, J.M., Mook, D.T., Brasil, R.M.L.R.F., Weber.H.I., Fenili, A., Belato, D. and Felix, J.L.P.,‘An overview on Non-Ideal Vibrations’, Meccanica 38, 2003, 613-621. 3. Balthazar, J.M., Brasil, R.M.L.R.F., Weber.H.I., Fenili, A., Belato, D., Felix, J.L.P. and Garzelli F.J. ‘ A Review of New Vibration Issues due to Non-Ideal Energy Sources’, in Udwadia, F.E., Webwer, H.I. and Leitmann, G. (eds), Dynamical Systems and Control, Stability and Control: Theory, Methods and Applications 22, Chapman & Hall/ CRC, Boca Raton, 2004. 4. Felix, J. L. P. , Balthazar, J. M. , Dantas, M. J. H. On energy pumping, synchronization and beat phenomenon in a nonideal structure coupled to an essentially nonlinear oscillator. Nonlinear Dyn. 56, 1-11 (2009). 5. Felix, J. L. P. , Balthazar, J. M. , Dantas, M. J. H. On a Nonideal (MRD) Damper-Electro-Mechanical Absorber Dynamics. International Journal of Bifurcation and Chaos, Vol.21, No. 10, 2871âĂŞ2882, (2011). 6. Guckenheimer,J., Holmes, P. Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer-Verlag, New York Inc. (1983). 7. Jiang, X. ,McFarland, D.M., Bergman, L.A., Vakakis, A.F. Steady state passive nonlinear energy pumping in coupled oscillators: theoretical and experimental results. Nonlinear Dyn. 33, 87-102 (2003). 8. Kononenko, V.O.Vibrating Systems with a Limited Power Supply Power, Iliffe Books Ltd (1969). 9. Malatkar,P. , Nayfeh,A.H. Steady-state dynamics of a linear structure weakly coupled to an essentially nonlinear oscillator. Nonlinear Dyn. 47, 167-179 (2007). 10. Nayfeh, A.H. and Mook, D.T., Nonlinear Oscillations, John Wiley and Sons, New York, 1979. 11. Tondl, A., Quenching of Self-Excited Vibrations, Elsevier, Amsterdan, (1991) 281 Copyright of AIP Conference Proceedings is the property of American Institute of Physics and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use.