Complex modes analysis for powertrain and driveline applications

Transcription

1 Complex modes analysis for powertrain and driveline applications T. Parikyan 1 1 AVL List GmbH, Advanced Simulation Technologies Hans-List-Platz 1, A-8020, Graz, Austria tigran.parikyan@avl.com Abstract It is shown that the modal analyses of rotors and of systems with damping can be considered from a common viewpoint of complex modes analysis. The applications presented in the paper are: 1) damped torsional modal analysis of a 4-cylinder engine cranktrain, 2) damped 3D modal analysis of an engine block suspended in mounts, 3) modal analysis of the rotor of automotive turbocharger. The eigenfrequencies and mode shapes resulting from the real and from the complex modes analyses are compared to each other. Mode type identification based on kinetic energy is used. Critical speed diagrams of rotor with whirl tracking in relative (rotating) and absolute coordinate systems are presented, and the phenomenon of whirl inversion is discussed in connection with stable and unstable resonance points. Mode shapes and node orbits are visualized in 2D and 3D. 1 Introduction Before beginning with nonlinear dynamic forced response, the calculation engineer has a time-saving and quality-assuring possibility to evaluate, verify and eventually modify the dynamic properties of mechanical system or its separate parts by performing modal analysis first. While real-value modal analysis is more widely known and used, it is only the first approximation to the dynamic behavior of a system and its parts. To come up with more realistic values of eigenfrequencies and mode shapes, one has to take some additional properties and effects into account: damping and in case of rotating parts spin effects. Both cases can be treated by the methods of complex modes analysis. There is a number of publications on complex modes analysis: damped modal analysis is dealt with in [1-3], by adding the viscous damping terms to the dynamic equations; modal analysis of rotating structures is considered in [4-12], by taking the spin effects into account. In all these cases the analysis is typically reduced to state-space eigenvalue problem, and some phenomena specific to these types of systems are discussed. By shortly summarizing the existing theory and the methods of solution, the present paper mainly focuses on powertrain and driveline application tasks, which automotive engineers are regularly facing in course of their daily project work. Software tools have been developed for the numerical simulation of the mentioned tasks. 2 Theory The formulation and the methods of solution of complex modes analysis are described in the literature [1-12], and are shortly summarized here. 2761

2 2762 PROCEEDINGS OF ISMA2016 INCLUDING USD2016 In general case, the system of N dynamic equations describing the free vibrations of a system with N degrees of freedom (DOF), can be written as: D Gq ( K C jh q 0 Mq ) (1) where (similar to the definitions used e.g. in [13]): M, D, K are the mass, viscous damping, and stiffness matrices; G, C and H are the gyroscopic/coriolis, centrifugal, and viscoelastic damping 1 matrices; q, q, are the complex-value vectors of generalized displacements, velocities and accelerations. In order to solve N equations in N complex unknowns (i.e. in 2N real unknowns), the system (1) has to be re-formulated in state space: with Az Bz 0 (2) 0 M M 0 q q A ; B M D G ; z 0 K C jh ; z q (3) q and a solution having the form: z θ t ( t) e (4) Here θ is the complex eigenvector and j is the complex eigenvalue of the calculated mode, with being the decay rate and d the damped angular frequency. The ratio damping factor. From (2) and (4) we get the generalized algebraic eigenvalue equation: d is called A Bθ 0 (5) The real and imaginary components of complex eigenvector θ can be transformed into magnitude θ and phase for each of the N degrees of freedom 2 : sin Imθ / θ (6) The general equations shown above can be then easily simplified to cover the two special cases discussed below: 1) Consideration of all types of damping, with no spin effects; 2) Consideration of spin effects with no damping. 1 Also known as hysteretic damping. 2 In fact, only the lower part corresponding to the vector q is needed.

3 NON-LINEARITIES: IDENTIFICATION AND MODELLING Application cases 3.1 Damped torsional modes In the torsional model (Figure 1) of a 4-cylinder 1.9 L gasoline engine, two variants of the crankshaft are considered: a) with rubber damper (proportional damping coefficient =1.3) and b) with viscous damper (viscous damping constant d=5.0 Nms/rad). The damped frequencies are compared to the natural frequencies (Table 1): while some (lower) eigenfrequencies in both variants of corresponding modes differ, the other (higher) ones are practically the same. Figure 1: Schematic diagram of torsional model of a 4-cylinder engine. Table 1: Undamped and damped eigenfrequencies and damping factors compared in case of rubber and viscous dampers. Note, that in case of prevailing viscoelastic damping, some of the damped modes can have a higher eigenfrequency than the corresponding undamped modes (see the mode #1 in Table 1). In the opposite case when the viscous damping is prevailing the damped modes are lower than the corresponding undamped modes. Together with its damped eigenfrequency, each damped mode is characterized by its damping factor (Table 1). The values in the table show that different modes are differently damped: e.g. the mode 1 is damped stronger than the mode 3 compare also the respective damped sinusoids (Figure 2). To get a deeper insight into the differences between the undamped and damped modal analysis, one has to compare the corresponding mode shapes, too. While the real parts of the modes can practically remain the same, there are the imaginary parts, which make the difference (Figure 3).

4 2764 PROCEEDINGS OF ISMA2016 INCLUDING USD2016 Figure 2: Exponentially decreasing magnitudes of oscillations in damped modes #1 (left) and #3. Figure 3: Real (left) and imaginary parts of the first three damped modes. In parallel to showing the real-imaginary components of the modes, magnitude-phase way of presenting the mode shapes can be used as well. A special attention has to be paid to the phases (Figure 4), which like non-zero imaginary components immediately show a qualitatively different behavior of a damped system. Figure 4: Magnitudes (left) and phases of the first three damped modes. Figure 5: Comparison of the phases of damped and undamped mode shapes (mode #1).

5 NON-LINEARITIES: IDENTIFICATION AND MODELLING 2765 While having only 0 or 180 phase in case of undamped system, other values can be expected in case of damped system (Figure 5). To better imagine the principal difference, one has to associate the mode shapes with the waves: while an undamped mode resembles a standing wave, a damped one can be compared to a propagating wave. 3.2 Damped 3D modes For the same 4-cylinder engine, we perform the modal analysis of the block suspended on the fixed chassis by means of rubber/viscous mounts; both fixed and hinged mounts can be considered (Figure 6). Figure 6: Schematic diagram of a 4-cylinder engine on rubber mounts: two fixed mounts and one hinged mount Preliminary nonlinear static analysis Preliminarily, at each analyzed engine speed, we calculate the nominal position of static equilibrium of the block due to gravity as well as to the mean reaction torque of the driving shaft, which is usually speeddependent (Figure 7). The mounts also have non-linear stiffness and damping coefficients, as a rule Damped modal analysis of pre-loaded system In this pre-loaded nominal position, we calculate the effective stiffness and damping coefficients, and use these values for modal analysis of the linearized system. Comparing the results of the undamped and damped modal analyses (Table 2), one can see some deviations in frequencies and in kinetic energy share of the dominant mode; the dominant DOF in mode #3 even changes. A broader view is given by Figure 8, where the respective eigenfrequencies are compared over the whole speed range: as long as the system is slightly damped, they do not strongly differ (note also, that in both variants of analysis the resulting curves follow the shape of the engine torque curve on Figure 7). Similarly, the damping factor is shown in the whole speed range, too. The magnitudes of free damped vibrations will decrease with time like previously shown on the plots of Figure 2.

6 2766 PROCEEDINGS OF ISMA2016 INCLUDING USD2016 Figure 7: Nonlinear static analysis of a 4-cylinder engine block on rubber mounts. Table 2: Undamped natural frequencies (top) and damped eigenfrequencies and damping factors of suspended engine block at 1000 rpm Figure 8: Critical speeds diagram (left) and damping factors diagram.

7 NON-LINEARITIES: IDENTIFICATION AND MODELLING 2767 Differences can also be seen during 3D animation, especially when watching the trajectories of the moving points on the block (Figure 9): while in undamped case we have only a back-and-forth motion along the same curvilinear segment, in damped case the curves can become spirals starting from a point on the closed-loop curves shown on the right-hand image. Figure 9: Undamped mode with point double trajectory (left) and damped mode with closed-curve where the spiral trajectory starts. Figure 10 shows the 6DOF mode shapes resulting from the both variants of the modal analysis at 3000 rpm, in three different forms: kinetic energy share, motion magnitude and motion phase. Figure 10: Undamped (left) and damped modes by kinetic energy share, magnitude and phase.

8 2768 PROCEEDINGS OF ISMA2016 INCLUDING USD2016 While the mode shapes in the first two forms are rather close to each other, the phase-based shapes can differ considerably (see the comparison on Figure 11). Figure 11: Comparing phases of undamped and damped modes. 3.3 Rotor modes A bar-mass model of a typical automotive turbocharger rotor is based on lumped mass representation of its constituent parts and on Timoshenko beam theory [14] for the elastic elements connecting these mass points. With this approach, one needs to specify only geometric and material parameters of the elementary parts to analytically calculate the stiffness and mass properties of the system, without a need to involve any method based on volumetric FE-mesh. As a result, one obtains assembled dynamic model of the rotor featuring mass and stiffness matrices (Figure 12) Bending mode shapes with whirl Due to spin effects, bending modes acquire additional property forward or backward whirl because of non-zero phase. Here, the modes shown on Figure 13 have a forward whirl and those on Figure 14 backward whirl. The conical mode shape can be qualified a suspended rigid-body mode: this is usually observed in the systems with compliant bearings; in all other cases the rotor behaves like a flexible body.

9 NON-LINEARITIES: IDENTIFICATION AND MODELLING 2769 Figure 12: Automotive turbocharger rotor, its simplified definition and resulting bar-mass model. Figure 13: Typical mode shapes: forward whirl bending modes conical (left) and spindle. Figure 14: Typical mode shapes: backward whirl bending modes hyperboloid (left) and double spindle.

10 2770 PROCEEDINGS OF ISMA2016 INCLUDING USD Mode and whirl tracking The complex modes analysis of the rotor is performed in rotating coordinate system, and after that additionally transformed into absolute coordinate system. In coordinate system rotating with the rotor, considering the full solution (positive and negative frequencies vs speed) is necessary in order to track the modes. Eigenfrequencies referring to bending modes vary with the spin speed, while those of the torsion/tension modes remain constant. Figure 15 shows such smooth mode tracks in pairs, symmetric about horizontal axis. At the same time, each bending mode track indicates a definite whirl. The paired (symmetric) tracks possess opposite whirl sign. Note that in rotating coordinate system there can be mode tracks crossing the zero-frequency line. Figure 15: Mode and whirl tracking using the positive and negative frequency range (in relative coordinate system). Figure 16: Mode and whirl tracks transformed into absolute coordinate system.

11 NON-LINEARITIES: IDENTIFICATION AND MODELLING Transforming the results into absolute coordinate system With the help of order 1 line, the bending mode tracks can be transformed into absolute coordinate system: the order 1 line has to be added to the forward-whirl tracks and subtracted from the backward-whirl ones. Figure 16 shows the result of that transformation. Note that there are no tracks crossing the zero-frequency line in the absolute coordinate system Resonances and instability In what follows, only the positive frequency range will be considered. The intersections of order 1 line with each of the mode tracks (Figure 17) designates a resonance point. On the critical speeds diagram in absolute coordinate system, it is difficult to say something more about the types of the resonances. Figure 17: Critical speeds diagram with resonance points (absolute coordinate system). Figure 18: Critical speeds diagram with stable and unstable resonances (relative coordinate system).

12 2772 PROCEEDINGS OF ISMA2016 INCLUDING USD2016 The critical speeds diagram in rotating coordinate system can help in this case (Figure 18). First, due to the different whirl sign, the backward-whirl tracks have to be intersected by order 2 line, while forward-whirl tracks by order 0 line (i.e. by zero-frequency line). As long as the backward-whirl tracks do not intersect the zero-frequency line, one can consider the corresponding resonance points stable. The situation is quite different in the second case: here we have a singular point with 0 frequency, at which the track breaks and the whirl sign inverts. Eigenvalue analysis of the modified stiffness matrix ( K C) shows additional zero eigenvalues at these points, and at higher speeds some eigenvalues become negative; all this can become a source of instability. Therefore, such resonance points, as well as the complementary backward-whirl parts of the tracks have to be considered unstable Node orbits The whirl information can be further detailed by considering node orbits, both in relative and in absolute coordinate systems. Figure 19 shows the projections of the orbits of the compressor wheel node on the plane perpendicular to the rotation axis of the rotor for the first four non-trivial modes; the positive rotation is counter-clockwise. According to Figure 18, rpm belongs to the region of stable speeds. Figure 19: Node orbits in relative (top row) and absolute (bottom row) coordinate systems at rpm. The Figure 20 shows the orbits of the same node at rpm, i.e. in the region beyond the first unstable speed. As expected, the sequence of 2 nd and 3 rd mode tracks appearing as frequencies has interchanged. Note that for the mode #3, the whirl signs in relative and absolute coordinate systems are opposite. The explanation of this apparent contradiction is: the orbits belong to different tracks to the main (stable) in absolute system, and to the complementary (unstable) in relative system, according to Figure 17 and Figure 18.

13 NON-LINEARITIES: IDENTIFICATION AND MODELLING 2773 Figure 20: Node orbits in relative (top row) and absolute (bottom row) coordinate systems at rpm. 4 Conclusion and outlook Compared to the real modal analysis, the complex one results in more realistic eigenfrequencies and mode shapes. This provides a deeper understanding of the dynamic properties of engine parts, and lays a sound basis for a better prediction of their behavior within dynamic system. With the software tools developed, engineers get a possibility to easily and quickly perform such kind of analysis of shafts, mounts and other parts of powertrain / driveline and, consequently, to adjust their mass, stiffness and damping properties in the early concept phase of the development project to match the design requirements. While currently performing the 3D complex modes analysis of a single suspended (rotating or nonrotating) body, the method can be easily re-formulated for the multi-body case, i.e. for the whole vehicle. The other extensions and improvements being considered are: More general types of shafts: non-symmetric rotors, as well as crankshafts, camshafts, etc.; Rotors with residual unbalance; Rotors in presence of damping; Mounted rotor housing; Breaking down the dissipation power of the damped modes into the damping shares of the separate elements/nodes and revealing those with dominant damping action; Speed-dependent stiffness and damping in bearings.

14 2774 PROCEEDINGS OF ISMA2016 INCLUDING USD2016 Acknowledgements I greatly appreciate the contribution of Paul Bodenbenner (AVL Graz), in writing the GUI and 2D/3D post-processing of Mount Layout Tool [15]. I also like to thank Krešimir Hećimović (AVL Croatia) for 2D post-processing and Josip Jurić (AVL Graz) for extending the GUI and 3D animation of gyroscopic modal analysis in Shaft Modeler [16]. My special thanks to Saša Bukovnik (AVL Graz) for fruitful discussions and to Bronislav Plemenitaš (AVL Slovenia) for verification of results of rotor modal analysis using commercial FEA software packages. References [1] T. K. Caughey, M. E. J. O'Kelly, Effect of Damping on the Natural Frequencies of Linear Dynamic Systems, Journal of the Acoustical Society of America, Vol. 33, No. 11 (1961), pp [2] R. R. Craig, A. J. Kurdila, Fundamentals of Structural Dynamics, 2 nd Edition, John Wiley & Sons, Inc. (2006). [3] L. San Andrés, Modal Analysis of MDOF Systems with Viscous Damping, Mechanical Vibrations Course MEEN 617, Handouts HD 11 (2013). [4] O. A. Bauchau, A solution of the eigenproblem for undamped gyroscopic systems with the Lanczos algorithm, International Journal for Numerical Methods in Engineering, Vol. 23 (1980), pp [5] C. Kessler, J. Kim, Complex Modal Analysis and Interpretation for Rotating Machinery, IMAC XVI - 16th International Modal Analysis Conference (1998), pp [6] M. L. Adams, Rotating Machinery Vibration, Marcel Dekker, Inc. (2001). [7] E. S. Gutierrez-Wing, Modal Analysis of Rotating Machinery Structures, Ph. D. Thesis, Univ. of London (2003). [8] D. B. Stringer, P. N. Sheth, P. E. Allaire, Modal Synthesis of a Non-Proportionally Damped, Gyroscopically Influenced, Geared Rotor System via the State-Space, Army Research Laboratory, Report ARL-TR-4582 (2008), Adelphi (MD) [9] M. I. Friswell, et al., Dynamics of Rotating Machines, Cambridge Univ. Press (2010). [10] NX Nastran 8.5, Rotor Dynamics User s Guide, Siemens PLM Software Inc. (2012). [11] S. Y. Yoon, Z. Lin, P. E. Allaire, Control of Surge in Centrifugal Compressors by Active Magnetic Bearings, Advances in Industrial Control, Springer (2013). [12] H. Nguyen-Schäfer, Rotordynamics of Automotive Turbochargers, 2nd Edition, Springer (2015). [13] T. Parikyan, T. Resch, Statically indeterminate main bearing load calculation in frequency domain for usage in early concept phase, ASME Paper ICEF (2012). [14] K.-J. Bathe, Finite Element Procedures in Engineering Analysis, Prentice Hall (1982). [15] EXCITE Designer Theory Manual, AVL (2016). [16] Shaft Modeler with AutoSHAFT Users Guide, AVL (2016).