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Vibration Synthesis for Electrical Machines Based on Force Response Superposition Matthias Boesing, Timo Schoenen, Knut A. Kasper, and Rik W. De Doncker, Fellow, IEEE Institute for Power Electronics and Electrical Drives, RWTH Aachen University, 52056 Aachen, Germany A universal and modular approach for synthesizing electromagnetically excited vibrations in electrical machines is presented. The synthesis process uses force responses, i.e., the normalized structural vibration responses for a machine’s generic set of force excitation shapes. The responses are superposed after scaling by the operating point dependent force excitation amplitudes. This leads to a computationally efficient process. It allows to use detailed structural 3-D models and to synthesize the vibrations in the entire operating range of the machine. For validation, measured and synthesized vibration spectrograms of a run-up test of an interior permanent magnet traction motor are presented. Index Terms—Acoustic noise, drives, finite-element methods (FEMs), modeling, structural dynamic analysis, vibrations.
I. INTRODUCTION
II. ELECTROMAGNETIC FORCE EXCITATION
OR electric or hybrid electric vehicles low acoustic emissions of the electric traction motor are required. Furthermore, the acoustic characterization of electric drives needs to be efficiently integrated into the design process. This addresses machine designers (machine configuration, force shapes), control engineers (operating point dependent currents), mechanical design engineers (housing, mounting), structural dynamics specialists (structural dynamics analysis), and acousticians (measurements). The approach presented aims to combine all these fields, to be modular and easily expandable, and to deliver easy-to-interpret results. It can be summarized as follows. In a first step, the force excitation for the entire operating range of a machine is calculated. The force distribution is decomposed in space and time. In a second step, the force responses, i.e., the vibration responses for the machine’s main generic force excitation shapes, are calculated for the entire frequency range by 3-D structural finite-element method (FEM). The operating point dependent machine vibrations are subsequently found by superposing these force responses scaled by the operating point dependent force excitation. The high computational costs that typically limit 3-D models to selected frequencies as pointed out in [1] and seen in [2] can be overcome with the presented approach. Comparisons between simulated and measured run-up spectrograms underline the validity of this approach. The electromagnetic noise components inherent to the machine design [3] are considered. Switching frequency harmonics are currently not included. The machine used for validation is a 15 kW interior permanent magnet synchronous machine (IPMSM) for a hybrid electric vehicle. It is a three-phase machine with 10 pole pairs, 24 stator slots, slots per pole per phase. The outer diameter is 30 and cm and the stack length 7.5 cm. The force decomposition and vibration synthesis is implemented in MATLAB. ANSYS is used for solving the electromagnetic and structural FE models.
F
Manuscript received December 21, 2009; accepted January 25, 2010. Current version published July 21, 2010. Corresponding author: M. Boesing (e-mail:
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMAG.2010.2042291
IPMSM for traction applications are typically current controlled in terms of d- and q-axis current coordinates and [4]. The force excitation is calculated for a set of current coordinates spanning the entire - range within the drive limits. It is mapped to the torque and speed plane as defined by the control strategy. The control strategy calculation takes into , the current limit , and account the inverter voltage , as well as the current-dependent values the speed limit and d- and q-axis inductances and for PM flux linkage 9 . This force calculation process assumes that the machine is operated with sinusoidal currents and that transient effects can be neglected. The electromagnetic simulations here are performed statically using nonlinear 2-D FE models assuming a symmetricurrents, the air-gap cally built machine. For every pair of force is calculated at a number of rotor positions to obtain the quasi time-dependent characteristics. For the given machine, a full period of the force can be concatenated using a span on 60 electrical. It is split in 30 steps of 2 electrical. The force distribution along the air-gap at a time is given where the angle represents a location in the airby gap. The index denotes the direction of the force, i.e., radial (“r”) or tangential (“t”). The force is assumed to have no axial is spatially decomposed and described by component. the superposition of excitation shapes up to the th spatial with component as (1) In (1) every excitation shape is described by two standing and orthogonal force waves and with time-deand which for sympendent amplitude factors metric machines are only phase shifted. and are temporally decomposed into their frequency components [5] —
and
—
(2)
Displaying the amplitudes of the frequency domain forces and over the or plane allows
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BOESING et al.: VIBRATION SYNTHESIS FOR ELECTRICAL MACHINES BASED ON FORCE RESPONSE SUPERPOSITION
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III. STRUCTURAL VIBRATION RESPONSE
Fig. 1. Amplitude of the frequency domain force excitation A . the radial mode 0 at 60 f
f
(60
) of
The structural dynamic simulation is used to find the force responses, i.e., the vibration responses for the machine’s main generic force excitation shapes described by (1). Harmonic analyses based on the modal superposition method using force are performed. represents 1 of excitations force excitations with a unit amplitude of 1 N and a spatial distribution associated with one particular or from (2). We can write (3) is the general structural transfer function of the where the resulting vibration for one machine assembly and spatial force excitation shape with unit amplitude. Both here and in the subsequent vibration synthesis, the contribution of individual structural modes with respect to a certain force excitation shape can be investigated. Due to the modal sucan be written as perposition method used, (4)
Fig. 2. Overview of the force calculation, decomposition, and mapping process.
to judge and compare the force excitation characteristics of a machine. Due to the magnetic configuration of the machine in this study, the frequency components of all excitation modes are with being the shaft speed in Hz multiples of [5]. The circumferential orders of the three dominant excitation modes are 0, 4, and 8. Mode 0 has frequency components at while those for modes 4 and 8 are at . , i.e., the amplitude of the The amplitude of , is shown in Fig. 1 as radial mode 0 force excitation at 60 is displayed over an example. In the left figure, the plane while in the right it is displayed over the plane. The left type of display is suited to compare different machine designs, the right helps to compare different control strategies. Fig. 2 gives a summarizing overview of the implementation of the force component calculation for IPMSM. Force distribution calculation, force decomposition, and control map calculation are performed offline. Depending on settings and model size, this task takes between 6 and 18 h of computation time (mainly for the FE calculations) on a 2 GHz dual-core PC but is typically parallelized. The remapping is performed online when the vibrations for a particular operating point are synthesized.
where is 1 of eigenshapes and is the frequency dependent modal coordinate. This reduces the size of the result data and even allows to reduce the structural model to the most important eigenshapes. Excitation force modes 0, 4, and 8 with their radial and tangential force components are considered here. With increasing order their influence typically decreases and it was found that orders higher than 8 can be neglected here. From (1) we see that harmonic analyses (for mode 0, the sine this requires term is negligible because ). The 3-D structural model is built directly from the machine assembly’s CAD drawing. Noncritical structural details are eliminated to decrease the model size (see Fig. 6). In this study, the model consists of the machine stator in a test bench housing which is flange mounted to a test bench. The windings are modeled as solids with approximate material parameters. The rotor is not included for the results presented here. The computational costs for the 3-D structural simulations are approximately 6 h on a 2 GHz dual-core PC for a 44 405 second-order element model and 345 frequency points. IV. VIBRATION SYNTHESIS at one freFor synthesizing the machine vibrations quency for a particular operating point at torque and speed , the unit force velocity responses from (3) are of the force superposed with the complex amplitudes excitation modes as weighting factors for this particular operating point and frequency (5) represents an or from (2) operating point. This process is perfor the respective frequency components. They are calformed for
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Fig. 3. Overview of the vibration response calculation and synthesis process.
culated from (5) with with spectrum
leading to the operating vibration
(6) This superposition can be performed within seconds. If the operational deflection shape is to be animated, comprises the velocity on the surface of the machine at one particular operating point and frequency. For synthesizing machine run-up spectrograms, is determined for all frequency components at all operating points along a torque-speed trajectory. For the spectrograms that are compared to measurements, the normal component of is evaluated at the same positions where the accelerometers are placed in the measurements. Fig. 3 sketches the vibration response calculation process and how in the synthesis process these results are combined with the results from the electromagnetic force calculation. V. RESULTS Using the results from force excitation calculation and structural dynamic response calculations, the spectrogram of a machine run-up at full load is readily synthesized as shown in Fig. 4(c). Run-ups are well suited for validation as covering the entire speed range is essential for variable-speed drives. The torque dependency of the noise is typically less pronounced and omitted here for conciseness. The force excitation uses mode 0 and modes 4 and 8 at 20, 40, 80, 100, and at 60 and 120 . The simulated spectrogram is compared to a mea140 sured spectrogram of this run-up in Fig. 4(b) which for the sake of comparability is reduced to the same seven frequency components by filtering. Both figures use the same decibel-scale on the color-axis and the same function for display. As reference, the original measured spectrogram is shown in Fig. 4(a). The vibration was measured at 4 sensor positions around the circumference of the housing, evaluated as the power sum of these signals, and synthesized accordingly. This confirms that for the given machine, the analyzed noise components are the most prominent electromagnetic noise components. Synthesized and measured spectrograms match well. The low-frequency noise components 300 Hz) in Fig. 4(a) are mechanical noise and not within (
Fig. 4. Spectrograms of normal surface velocity for run-up at full load from 0 to 6000 r/min. (a) Original measurement data. (b) Filtered measurement data. (c) Synthesized result.
the scope of this investigation. However, they can be easily integrated into the synthesis process. While the spectrograms in Fig. 4 give a good overview, they make it hard to trace the amplitudes (and they do not reproduce well in black-and-white prints). Fig. 5 therefore compares the amplitudes of the individual frequency components in separate graphs. It can be seen that measured and synthesized amplitudes generally match, in particular the two mode 0 force excited freand 120 . A notable deviquency components at 60 between 3500 and 4500 r/min where ation is seen for 20 the simulation does not include a presumably resonance related amplitude increase. Given that the structural model is not optimized but uses general material parameters and damping coefficient, this deviation is considered to be acceptable. Structural tests (i.e., an experimental modal analysis [6]) or a detailed analysis of material parameters (e.g., for laminations [7] and windings) have not been performed. This work here focuses on the general synthesis process but one of its features is that the structural model can now be improved independently by the respective experts. Fig. 7 shows as an example a still-frame of the operational deflection of the outside of the housing and the stator at
BOESING et al.: VIBRATION SYNTHESIS FOR ELECTRICAL MACHINES BASED ON FORCE RESPONSE SUPERPOSITION
Fig. 7. Operational deflection at n f at 3822 Hz.
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= 5733 r/min and T = 20 N m for 40 1
harmonics, skewed synchronous machines, switched reluctance machines, and eccentricity are planned for the future. Auralzation of airbourne sound can be achieved building on the approach proposed in [9]. VII. CONCLUSION
Fig. 5. Comparison of measured and synthesized amplitudes of the frequency to 140 f during run-up at full load from 0 to components from 20 f 6000 r/min.
A modular and universal approach for simulating electrical machine vibrations is presented and validated. The given results show that the vibration characteristics of an electric drive can be synthesized a priori without measurement input, only based on machine configuration, operating limits, and engineering drawings. The process is computationally efficient, integrates well into the drive design process, and delivers easy-to-interpret results of high quality. The approach presented can be easily applied to other electrical machines or vibrating structures in general. REFERENCES
Fig. 6. Simplified CAD model of machine assembly (back-cover not shown).
5733 r/min, 20 N m for the frequency component of 40 which then is at 3822 Hz. Animations of the operational deflections allow for interactively investigating the vibration behavior. VI. MODULARITY The interface between electromagnetic force excitation and structural dynamic response on the basis of force excitation shapes is very flexible. Any type of model (FE, analytical, etc.) and simulation (2-D or 3-D, static or transient, etc.) can be used. For example, an analytical force calculation may be used as in [8] for an induction machine. Extensions for switching
[1] M. van der Giet, C. Schlensok, B. Schmülling, and K. Hameyer, “Comparison of 2-D and 3-D coupled electromagnetic and structure-dynamic simulation of electrical machines,” IEEE Trans. Magn., vol. 44, no. 6, pp. 1594–1597, Jun. 2008. [2] M. Furlan, A. Cernigoj, and M. Boltezar, “A coupled electromagneticmechanical-aoucstic model of a dc electric motor,” COMPEL, vol. 22, no. 4, pp. 1155–1165, 2003. [3] J. Gieras, C. Wang, and J. Lai, Noise of Polyphase Electric Motors. Boca Raton, FL: Taylor and Francis Group, 2006. [4] B.-H. Bae, N. Patel, S. Schulz, and S.-K. Sul, “New field weakening technique for high saliency interior permanent magnet motor,” in Proc. 38th IAS Annu. Meet. Conf. Rec. Ind. Appl. Conf., Oct. 2003, vol. 2, pp. 898–905. [5] M. Boesing, K. Kasper, and R. De Doncker, “Vibration excitation in an electric traction motor for a hybrid electric vehicle,” in Proc. 37th Int. Congr. Expo. Noise Control Eng. (INTER-NOISE), Shanghai, China, Nov. 2008. [6] D. J. Ewins, “Basic and state-of-the-art modal testing,” in Proc. Eng. Sci., Indian Academy of Science, 2000, vol. 25, pp. 287–370. [7] H. Wang and K. Williams, “Effects of laminations of the vibrationals behaviour of electrical machine stators,” J. Sound Vibr., vol. 202, no. 2, pp. 703–715, 1997. [8] J. Le Besnerais, V. Lanfranchi, M. Hecquet, G. Lemaire, E. Augis, and P. Brochet, “Characterization and reduction of magnetic noise due to saturation in induction machines,” IEEE Trans. Magn., vol. 45, no. 4, pp. 2003–2008, Apr. 2009. [9] S. Fingerhuth, P. Dietrich, M. Pollow, M. Vorländer, D. Franck, M. van der Giet, K. Hameyer, M. Boesing, K. Kasper, and R. D. Doncker, “Towards the auralization of electrical machines in complex virtual scenarios,” presented at the 40th Nat. Acoust. Congr., Tecniacústica, Cádiz, Spain, Sep. 2009.
