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Hardening Softening Behavior of Antiresonance for Non Linear Torsional Vibration Absorbers

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Hardening Softening Behavior of Antiresonance for Non Linear Torsional Vibration Absorbers XXIV ICTAM, 21-26 August 2016, Montreal, Canada HARDENING SOFTENING BEHAVIOR OF ANTIRESONANCE FOR NON LINEAR TORSIONAL VIBRATION ABSORBERS Alexandre Renault1,2, Olivier Thomas ∗1, Herve´ Mahe´2, and Yannick Lefebvre2 1Arts et Metiers´ ParisTech, LSIS UMR CNRS 7296, 8 bd. Louis XIV 59046 Lille, France 2 Valeo Transmissions, Centre d’Etude´ des Produits Nouveaux Espace Industriel Nord, Route de Poulainville, 80009 Amiens Cedex 1, France Summary We address non linear torsional vibration absorbers (TVA), used in rotating machinery to counteract irregularities of rotation at a some order of the engine speed of rotation. The TVA is analogous to a tuned mass damper (TMD), tuned on the desired order. It exhibits non-linearities of various natures which affect resonance and antiresonance frequencies at large amplitude of motion, which consequently causes the detuning of the system from the targetted order. This study focuses on some non linear systems (several TVA designs and a more general Duffing like system) to study the impact of non-linearities on the hardening / softening behavior of antiresonances. Non linear solutions are obtained by a numerical continuation procedure coupled with the harmonic balance method to follow periodic solutions in forced steady-state. Moreover, we propose an original direct antiresonance continuation method for undamped systems. INTRODUCTION Non linear torsional vibration absorber (TVA) is used in rotating machinery T (t) P rimary to counteract irregularities of rotation, called “acyclisms”, at a some order of the inertia engine speed of rotation. It is compose of a primary and secondary inertia and Secondary acts as the classical tuned mass damper (TMD). It is tuned on the desired engine inertia order. In practice, the secondary inertia represents a mass subjected to move in a particular path on the primary inertia which is linked to the rotating assembly. However the TVA includes strong non-linearities. Geometric non-linearities and those due to Coriolis effect, intrinsic to rotating articulated systems. They cause the detuning of the TVAand the shifting of the operating order of TVA(an Non linear antiresonance of the whole system) from the targeted engine order. Knowledge interactions of operating point behavior of the device is therefore essential to ensure optimal acyclisms filtering. This study focuses on tow points. First, several non linear Figure 1: TVA scheme systems, including torsional and translational systems, are studied to highlight how the hardening / softening behavior is modified by non-linearities. Non linear frequency responses in steady state are obtained by numerical continuation procedure, the harmonic balance method (HBM) [3] coupled with the asymptotic numerical method (ANM) [1], which leads to a fast and robust algorithm. Secondly, we propose an original direct antiresonance continuation method for undamped systems. MODELLING The equations of motion of the TVA can be written in general following form : M(x)x¨ + fin(x; x_ ) + Cx_ + fint(x) = F cos (!t) ; (1) where x is the vector of unknowns. M(x) is the mass matrix and depends on x. C is the damping matrix. fin(x; x_ ) is the inertial forces vector, including Coriolis terms, and depends on x and x_ . fint(x) is the internal forces vector and depends on x. Here, because only one oscillator is forced, the external forces vector is F = [0 ··· f ··· 0] where f is the amplitude of excitation of the forced oscillator. Assuming a periodic solution of (1), the vector of unknowns is expanded in truncated Fourier series : H X x(t) = x0 + xci cos(i!t) + xsi sin(i!t): (2) i=0 Then substituting (2) into (1) and applying HBM, we obtain a system of algebraic equations relating x0, xci, xsi, ! and f. The accuracy of the periodic solution depends on H, the number of harmonics retains on (2). The final system to solve can be written : R(U; w; f) = 0: (3) ∗Corresponding author. Email: [email protected] Where U = [x0 xci xsi ::: xcH xsH ]. Finally, (3) is solved by an asymptotic numerical method, for which a quadratic recast of R(U; !; f) is convenient [2]. In practice, the software Manlab 2.0 is used [4]. Unlike to predictor-corrector algorithms, where the solution is computed point by point, ANM adopts a piecewise continuous representation of the solution using a power series expansion of the pseudo arc length along the branch of solution. ANTIRESONANCE CONTINUATION Standard continuation procedure consist in considering ! and f as continuation parameters and compute the solutions with respect to one of them in forced vibration [2]. In free vibration (f = 0), the oscillation frequency can also be computed as a function of the amplitude, obtaining so-called backbone curves. In this case, the system has to be conservative so that periodic solutions are obtained and damping must be cancelled (C = 0)[5]. Here, we propose an original method to perform non linear antiresonance continuation, i.e. the computation of the antiresonance frequency as a function of the amplitude of forcing. Unlike free oscillations continuation, the system has to be forced. For a linear system, an antiresonance occurs only for undamped systems, for which the response of one of the unknowns is strictly zero, xi(t) = 0 8t at a given frequency, xi being the i-th component of x. If the system shows non-linearities, the antiresonance is not strict and only some harmonics of xi(t) can be set to zero, whereas the others are non zero. The antiresonance continuation is obtained by solving (3) with ! and f left free and by adding an additional condition: here the first harmonic of a given unknown is set to zero. RESULTS F (t) (c) primary secondary mass mass f x1(t) x2(t) ! f Amplitude of the primary mass Amplitude of the primary mass ! (a) ! (b) Figure 2: Frequency responses of the non linear TMD for several amplitudes of excitation (a) and antiresonance continuation (black bold line) (b) Here, we consider a free-free non linear TMD. The equations of motion can be written : ( x¨1 (m1 + m2) +x ¨2m2 = f cos (!t) ; (4a) 3 x¨1m2 +x ¨2m2 + kx2 + γx2 = 0; (4b) where γ and k are the non linear and linear stiffness constants, respectively. Figure 2(a) shows the frequency responses of the first harmonic of the primary mass, subjected to an harmonic forcing, for several amplitudes of excitation f. The system exhibits a hardening behavior due to the positive non linear stiffness constant. The backbone curve (red bold line), represents the oscillation frequency in free vibration. The antiresonance continuation can be viewed on the three dimensions representa- tion, on the figure 2(b), and also on the plane (!,f), on the figure 2(c). This procedure is very useful to accurately predict the detuning with respect to the amplitude of excitation. Moreover, it avoids redundant frequency responses simulations. References [1] Cochelin B. A path-following technique via an asymptotic-numerical method. Computers & Structures, 53(5):1181 – 1192, 1994. [2] Cochelin B. and Vergez C. A high order purely frequency-based harmonic balance formulation for continuation of periodic solutions. Journal of Sound and Vibration, 324(12):243 – 262, 2009. [3] Ali H Nayfeh and Dean T Mook. Nonlinear oscillations. John Wiley & Sons, 1979. [4] Arquier R., Karkar S., Lazarus A., Thomas O., Vergez C., and Cochelin B. Manlab2.0 : an interactive path-following and bifurcation analysis software. Tech. rep., Laboratoire de Mecanique´ et d’Acoustique, CNRS. [5] Karkar S., Cochelin B., and Vergez C. A comparative study of the harmonic balance method and the orthogonal collocation method on stiff nonlinear systems. Journal of Sound and Vibration, 333(12):2554 – 2567, 2014..
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