Sound power radiated from an inverter-driven induction motor. Part 3: statistical energy analysis C. Wang and J.C.S. Lai Abstract: The traditional approach for the prediction of noise radiated from induction motors is usually based on the finite-element and/or boundary-element methods. While these methods have been shown to provide reasonable estimates of the radiated noise at low frequencies (usually well below 2 kHz) and for small motors, the demands on computing memory and time render calculations at high frequencies and for large motors impractical to make as part of the design process. The statistical energy analysis technique, which is especially suited for calculations at high frequencies, has been applied to predict the acoustic noise of electromagnetic origin. Results demonstrate the feasibility of the method and its ability to discern the contribution of various components of a motor to the overall radiated noise.
1
Introduction
Prediction of the acoustic noise radiated from variablespeed inverter-driven induction motors is very useful but difficult. A motor structure, which normally consists of a stator, a rotor, and a casing, is complex for which a thorough analytical analysis is almost impossible. Currently, numerical methods, such as the finite-element method (FEM) and the boundary-element method (BEM), are used for the prediction of motor noise [1, 2]. Normally, one may develop an electromagnetic FEM model to calculate the force acting on the stator, and a structural FEM model to determine the structural vibration behaviour of the motor structure. Then an acoustic BEM model may be used to evaluate the acoustic response due to the electromagentic force. Although the FEM/BEM numerical approach seems to work well, there are quite a number of limitations for it to be applied in practice. This is because, basically, finiteelement/boundary-element methods, by their nature, are limited to low frequencies. If the upper frequency of interest is doubled, the number of elements required for the model would increase by a factor of 8. As a result, if one has to apply FEM/BEM methods to a large motor for frequencies up to 10 kHz, the number of elements and the computing time required may become prohibitive. To overcome the disadvantage of the FEM/BEM method at high frequencies, the statistical energy analysis (SEA) first developed by Lyon [3] may be employed. Although SEA has been applied with success to a number of mechanical systems, such as ship and aircraft structures etc. [4, 5], it has yet to be applied to analyse the noise radiated from electric motors. Recently, Delaere et al. [6] applied SEA to determine the internal
losses in the stator yoke and in the coils as well as the transmission of vibrations from the stator yoke to the motor frame. However, the power input to the stator due to the electromagnetic force excitation and the resulting vibroacoustic responses were not incorporated into the model. In SEA, a complex structure is divided into several simple, identifiable mechanical and/or acoustical subsystems. The behaviour of each subsystem is characterised in a statistical sense. By solving the energy balance equations, which involve expressions for the power flowing from one subsystem to another, the distribution of the vibration energy among these coupled subsystems can be estimated. Since the time and spatial averaged vibration energy of each subsystem is normally the primary variable, SEA cannot describe the vibration details inside each subsystem. Nevertheless, in noise and vibration control engineering, such statistical results are sufficient in most cases to quantify voltage, current
electromagnetic force model force mobility model input power statistical energy model
r IEE, 2005 IEE Proceedings online no. 20045087 doi:10.1049/ip-epa:20045087 Paper first received 4th July and in revised form 26th November 2004. Originally published online: 8th April 2005 The authors are with the Acoustics and Vibration Unit, University College, University of New South Wales, Australian Defence Force Academy, Canberra ACT2600, Australia C. Wang is now with the Milford Proving Ground, General Motors Corp., Milford, MI, 48380, USA E-mail:
[email protected] IEE Proc.-Electr. Power Appl., Vol. 152, No. 3, May 2005
vibration power radiation efficiency
sound power
Fig. 1 Prediction of acoustic power using statistical energy analysis 619
vibro-acoustic effects. Figure 1 shows the scheme for predicting the sound power level from an electrical machine using the statistical approach. The input mechanical power to the motor structure has to be determined from a ‘mobility model’, which requries the calculation of the total electromagnetic force from the input voltage and current based on an electromagnetic force model (analytic or FEM). The vibration power of each subsystem of the motor structure is then calculated using SEA. By using the radiation efficiencies (which could be obtained from an analytic or numerical model) and vibration power of each subsystem, the total sound power level can be obtained. The objective of this study is to employ the SEA technique to pedict the sound power radiated from an induction motor driven by two different inverters. The motor, used in previous investigation [1, 7, 8], is a threephase, four-pole, 415 V, 50 Hz, 2.2 kW induction motor, with 44 rotor slots and 36 stator slots. Two configurations for the speed controllers were considered here: benchmark and PWM inverter. The benchmark drive, consisting of a motor–generator set as described in [8], was used to replicate an almost sinusoid drive to provide an ideal state for which the effects of the harmonics can be neglected. The sinusoidal PWM inverter was type MSC2000 manufactured by Zener Electric with the switching frequency being proportional to the test speed of the motor at a rate of 21Hz/Hz. At the rated speed of 1500 rpm for the motor, the switching frequency was 1050 Hz [8]. In this paper, a foursubsystem SEA model for the motor structure is developed first. The corresponding internal and coupling loss factors were determined experimentally. Then a simple formula for estimating the mechanical power input to the stator subject to electromagnetic force waves was derived. Finally, by using radiation efficiencies of simple structural elements, the SEA predicted sound power is compared with measured results for various operating conditions. 2
SEA model of motor structure
The first step of applying SEA is to formulate the subsystems. For a typical motor structure, there are four basic components: the stator, the rotor, the casing and the endshields. In addition, a motor may have other structural elements because of installation or specific requirements. In this study, the motor was supported by a plate mounted on four isolators, as shown in Fig. 2a. According to SEA principles, a subsystem should represent a group of ‘similar’ energy storage modes. Thus, basically, the determination of the subsystems of a structure should depend on the similarity of the vibration behaviour among the structural elements rather than the physical geometry of structural elements. There are, however, no clear guidelines to identify ‘similar’ energy modes for contiunous structures. Generally, when a structure is excited, vibration modes of three different types (namely, flexural, longitudinal and transverse) may be observed. Normally, the vibration modes of the same type for a simple structure (such as a beam, a plate or a cylindrical shell) would have similar modal energy so that all the vibration modes of the same type for a given simple structure can be grouped together as a subsystem. For the motor structure studied here, the stator was press fitted into the casing. As a result, the vibration of the casing behaves differently in three areas, the part attached to the stator and the other two parts on either side of the stator, as shown by Wang and Lai [7]. Therefore, it is reasonable to divide the casing and the stator into three subsystems: the stator and the part of the casing attached to the stator; and the part of the casing on 620
a
input power
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casing 1
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Fig. 2 2.2 kW induction motor a Test motor b Model of motor structure
each side of the stator. The baseplate is another subsystem. Since the endshields only provide the boundary conditions to the casing, and their vibration and acoustic radiation are not so important, they are neglected in our SEA model. Also, isolators and the rotor are neglected in the model because they are only important at low frequencies [7]. These simplifications may cause some error at low frequencies. Fig. 2b shows the tear-down model of the motor structure. It can be seen that this model only involves the structural subsystems, implying that all the mechanical input power would be dissipated in the structure. Since the acoustic power is usually very small compared with the structural vibration power, this simplification is not expected to cause much error in the final results. Normally, the acoustic power radiated from the motor can be directly calculated by using the sound radiation efficiency and the estimated vibration energy of each sub-structure. Figure 3 illustrates the basic principles of the foursubsystem SEA model for the motor. Here P2 is the timeaveraged power input into the stator due to electromagnetic force; Ei is the time and spatial averaged vibration energy of each subsystem; ni and Zi are the modal density and the internal loss factor of each subsystem, respectively; Pij denotes the time averaged power transferring from subsystem i to j and vice versa for Pij. By conservation of IEE Proc.-Electr. Power Appl., Vol. 152, No. 3, May 2005
3 Determination of internal loss factors and coupling loss factors
P
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Subsystem 1: casing 1; subsystem 2: stator; subsystem 3: casing 2; subsystem 4: baseplate
energy, one can write the following energy balance equation for each subsystem: Pi ¼ Pid þ
4 X
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This equation indicates that the power input to subsystem i is equal to the power dissipated Pid in subsystem i and the net power transmitted out of subsystem i. The dissipated power can be written in terms of the time and spatial average vibration energy [9]: Pid ¼ oZi Ei
ð2Þ
where o is the circular frequency (rad/s). For Pij, it has been shown [3] that the power transmitted from one subsystem to another is proportional to the difference of the modal vibration energy of the two subsystems: Ei Ej ð3Þ Pij ¼ oZij ni ni nj Here a new SEA parameter Zij, which is called the coupling loss factor between subsystems i and j, is introduced. This parameter is defined as the part of the vibrational energy transmitted in one cycle of the vibration. Therefore, similar to the internal loss factor, it describes the capability of the power transmission between the two subsystems. In SEA, the determination of the coupling loss factors for different coupled structures is the essential point. By using (2) and (3) and the reciprocity relationship Zij ni ¼ Zji nj , the energy balance equations for the system shown in Fig. 3 can be written as, 2 32 3 2 3 0 E1 Z11 Z21 Z31 Z41 6 Z12 Z22 Z32 Z42 76 E2 7 6 P2 7 76 7 6 7 ¼ o6 4 Z13 Z23 Z33 Z43 54 E3 5 ð4Þ 405 0 Z14 Z24 Z34 Z44 E4 where Zii ¼ Zi þ
4 X
Zij ;
i ¼ 1; 2; 3; 4
j¼1 i6¼j
It can be seen that if the input power, the internal loss factor of each subsystem, and the coupling loss factor between each pair of subsystems are known, the vibration energy of each subsystem can be solved from (4). Consequently, by using the radiation efficiency of each subsystem, the corresponding sound power radiated can be derived. IEE Proc.-Electr. Power Appl., Vol. 152, No. 3, May 2005
The determination of internal and coupling loss factors is crucial for establishing the SEA model. Normally, internal loss factors of each subsystem have to be measured and coupling loss factors could be either measured or predicted. In this study, an experimental approach developed by Lalor [10] and described below was employed to determine the internal and coupling loss factors. For a motor structure consisting of four subsystems, there are 16 internal and coupling loss factors that need to be determined. If the external excitation is applied to each of the four subsystems in turn, there would be four sets of equtions similar to (4) in which only the coupling loss factors are the unknowns. By combining these 16 equations, we have 2 3 Z11 Z21 Z31 Z41 6 Z 7 6 12 Z22 Z32 Z42 7 6 7 4 Z13 Z23 Z33 Z43 5 Z14 2 P1 6 16 0 ¼ 6 o4 0 0 2
E11 6E 6 21 6 4 E31 E41
Z24 Z34 Z44 3 0 0 0 P2 0 0 7 7 7 0 P3 0 5 0
0
P4
E12 E22
E13 E23
E32 E42
E33 E43
3 E14 1 E24 7 7 7 E34 5
ð5Þ
E44
where Eij represent the averaged vibrational energy of subsystem i when subsystem j is excited. From (5), all the coupling loss factors can be determined by measuring the input power Pi and the corresponding energy Eij. However, since the inversion of the energy matrix in (5) is generally very sensitive to a small variation in the energy ratios, using (5) directly with the measured energy ratios and the input power may yield a large variation in the loss factors. Lalor [10] found that, by assuming that the coupling between subsystems is weak (i.e., the internal loss factor is greater than the coupling loss factor, Zi4Zij), the coupling loss factors can be directly calculated from the energy ratios as [10]: Pj 1 Eji ð6Þ Zij ¼ o Eii Ejj Correspondingly, the internal loss factors can be determined by ( ) 4 X Pi Eii Zij Eji Zji o j¼1 ðj6¼iÞ ð7Þ Zi ¼ Eii Actually when the coupling between subsystems is weak, i.e. Zi4Zij, it might be acceptable to assume that most of the input power is dissipated within the subsystem that is directly excited, which is Zi
Pi oEii
ð8Þ
Apparently, (8) might overestimate the internal damping loss factors, especially when the coupling betweem subsystems becomes stronger. 621
power amplifier
B&K2032 analyser
Cremer et al. [9]. For subsystems one to three, two excitation positions, with the normal force applied in the horizontal and vertical directions, respectively, were used. To obtain the vibration energy for each excitation, a B&K 4383 accelerometer was used to measure the vibration levels of each subsystem at a total of 130 points (30 for subsystem one, 40 each for subsystems two and three, and 20 for subsystem four). An HP 3569A dual-channel frequency analyser was used to store and process the data for these 130 points. By transferring the data to a PC, the spatial averaged vibration levels for each subsystem were obtained, and were processed into 1/3 octave bands from 100 Hz to 10 kHz. Figure 5 shows the internal and coupling loss factors obtained from (6) and (8). It can be seen that, at low frequencies, the measured internal loss factors are high, even greater than unity. This is due mainly to the weak coupling assumptions made in the two equations. Generally, the coupling strength is strong at low frequencies, and the energy tends to distribute uniformly over the whole structure. At high frequencies, since the couplings between subsystems are becoming weak as shown in Fig. 5, the four subsystem SEA model appears to be reasonable.
shaker
charge amplifier
impedence head
charge amplifier
accelerometer
charge amplifier
motor
HP-300 computer
HP3569A analyser
Fig. 4 Schematic diagram of experimental setup for measuring loss factors Charge amplifier, B&K2635; power amplifier, B&K2706; shaker, B&K4810; impedance head, B&K8001
The experimental setup for the energy ratio and input power measurements is shown in Fig. 4. The motor was excited by a Bruel & Kjaer (B&K) 4810 shaker driven by a B&K 2706 power amplifier. The B&K 8001 impedance head was inserted between the shaker and the excitation point. The force and acceleration signals, via two B&K 2635 charge amplifiers, were input to a B&K 2032 analyser for cross-spectra processing. The imaginary part of the cross spectrum was then processed on a Hewlett Packard (HP) series 300 microcomputer to obtain the mechanical input power. In the experiment, each of the four subsystems was excited in turn. Note that the vibration distribution over a structure depends on the excitation, as discussed by
4
Input power to stator
To use (4) to predict the vibration response of the motor structure, the mechanical power input to the stator has to be calculated from the specified electromagnetic force. In
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IEE Proc.-Electr. Power Appl., Vol. 152, No. 3, May 2005
for n 2 At high frequencies, the modal overlap is high, that is, the natural frequencies of adjacent vibration modes are close to each other. As a result, the vibration response at an excitation frequency o may be approximated by that of the nearby natural frequency on. By substituting (11) into (10), the real part of the surface mobility is given approximately by ur jo¼on 1 ð12Þ Re M S Mo F where M is the total mass of the stator. Equation (12) indicates that, under the excitation of the electromagnetic force wave, the real part of the equivalent surface mobility IEE Proc.-Electr. Power Appl., Vol. 152, No. 3, May 2005
of a circular cylindrical shell, at high frequencies, is inversely proportional to the mass of the structure, and the frequency. This result actually supports the discussion made in [8] that the random modulation technique may not affect the acoustic power radiated from the motor structure as noticed in [8, 14] because, according to equation (12), a change in the force spectrum content rather than the magnitude would not make any changes to the input power and thus the radiated acoustic power. On the contrary, if any design change (such as rotor skewing) could lead to a change in the magnitude of the total radial force, the corresponding effects should be reflected in the radiated sound power. It should be noted that, although (12) has been obtained from (10), which is only applicable for circumferential modes nZ2, the natural frequencies for a cylindrical shell with both ends free are zero for circumferential modes n ¼ 0 and 1 [13]. In this study, the electromagnetic force for specific operating conditions was calculated by FEM [1, 15], although it may be possible to use the approach of Cho and Kim [16]. The corresponding magnitude of the total radial force acting on the stator inner surface for the benchmark controller and PWM inverter at two operating speeds of 450 and 1500 rpm is shown in Fig. 6. By using (12) and the results shown in Fig. 6, the input power to an electric motor modelled as a cylindrical shell can be calculated from (9).
1000 'ideal' sinusoid, no load
total force, N
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vibro-acoustics, the mobility is a parameter that is often used to describe the ability of a structure in responding to a force excitation. The transmitted power can be easily derived if the mobility and the total force are known [9]. For an arbitrary structure, it has been shown that the mechanical mobility associated with a point excitation, namely the point mobility, is determined by the modal density and the mass of the structure [9]. However, for a surface load, such as the electromagnetic force acting on the stator in induction motors, since the vibration response, which strongly depends on the force distribution, is quite different from that of a point force excitation, the mechanical mobility associated with a surface load, namely the surface mobility, has to be investigated. It should be noted that generally the surface mobility may not have a unique definition because different force distributions on the surface may lead to different behaviour of the surface mobility [11]. However, for a particular case, such as induction motors in which the surface load has common characteristics, it would be meaningful to derive an equivalent surface mobility with a specified equivalent force [9]. As a result, the input power P2 is related to the real part of the surface mobility MS of a structure subjected to an equivalent exciting force F by [9]: 1 ð9Þ P2 ¼ jF j2 Re M S 2 The electromagnetic noise of an electric motor is basically caused by the interaction of electromagnetic forces acting on the rotor and the stator. Since this force generally has a very large radial component (approximately an order of magnitude larger than the tangential component), the vibration of a motor structure is dominated by the flexural vibration [12]. For the vibration response of the stator subject to electromagnetic force wave excitation, Yang [12] argued that axial vibration modes are not as important as circumferential modes, and therefore presented a formula for estimating the amplitude of the radial velocity of a stator core with the circumferential vibration modes nZ2, which is, 12oFa a3 ð10Þ ur jo¼on ¼ 4 n ES h where a is the radius of the stator, h is the thickness, S is the inner surface area of the stator, E is Young’s modulus, and F is the amplitude of the total radial force acting on the inner surface of the stator. The relationship between the circumferential mode number n and the corresponding natural frequencies can be expressed as [13] sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffi 2 Eh2 n2 ðn2 1Þ E h 2 n on ¼ 12r a2 12ra4 n2 þ 1 ð11Þ
10
1
0.1 1500 rpm 0.01 0
2000
4000
6000
8000
10000
frequency, Hz b
Fig. 6 Magnitude of total radial force acting on stator calculated by FEM a 450 rpm b 1500 rpm 623
Prediction of sound power radiated from motor
As described above, the SEA model does not include the acoustic environment. The sound radiation cannot be determined directly from (4). However, with the vibration energy of each subsystem obtained from (4), the acoustic power radiated from each subsystem could be determined if the sound radiation efficiency of each subsystem is known. For the motor structure studied here, the base plate was basically in the piston mode below 2.5 kHz [17]. Based on Silbiger’s derivation [18] for the sound radiation efficiency of an unbaffled circular plate with radius a in the piston mode, 4 2 ð13Þ s ¼ 8ðk0 aÞ =ð27p Þ k0 a 1 1 k0 a 1 where k0 is the acoustic wavenumber, a general expression for an arbitrary plate with the unbaffled acoustic boundary condition can be obtained as: pffiffiffiffiffiffi 2 4 4 4S f =c0 l0 4 p2S ffiffiffiffiffiffi s ð14Þ 1 l0 2S where S is the surface area of one side of the plate, and l0 is the acoustic wavelength. Above 2.5 kHz, the radiation efficiency of this plate is unity simply because the critical frequency of the plate is around 1.3 kHz [17]. The modal averaged sound radiation efficiencies for cylindrical shells also can be estimated analytically [19]. By assuming that the modal energies are uniformly distributed within the frequency band of interest, such as 1/3 octave frequency band, a simplified expression of the modal averaged radiation efficiency can be approximately given as: h i1 N 2 P sn o2n o2 þ4x2n o4n n ð15Þ s N h i1 2 P o2n o2 þ4x2n o4n n
where sn are the modal radiation efficiencies given in [19], on are the natural frequencies of the flexural vibration [13], and xn are the modal damping values of the cylindrical shell. Figure 7 shows the modal averaged radiation efficiencies for each of the four subsystems. It can be seen that, at low frequencies, the stator and baseplate are more efficient in acoustic radiation than the casing. This observation is consistent with the findings by FEM/BEM analysis in [1]. The modelling procedure described above was first examined with a simple case. The sound power level of
0
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−2
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base plate casing -1 stator casing -2
−3
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Fig. 7 motor 624
Radiation efficiencies of various structural components of
the motor subjected to random mechanical excitation applied to a point on the casing was measured by the sound intensity technique and calculated using the SEA model and the sound radiation efficiencies shown in Fig. 7 for 1/3 octave centre frequencies. It can be seen from Fig. 8 that there is reasonable agreement between the measurements and SEA calculations for frequencies above 1000 Hz. The discrepancies at low frequencies are primarily due to the simplifications made in estimating the sound radiation efficiencies and more importantly the omission of lowfrequency interference among the sound waves radiated from each subsystem. In SEA, sound waves radiated from each subsystem are assumed to be incoherent.
60 sound power level, dB
5
50 40 30 20 SEA 10 0 100
measurement 1000
10000
1/3 octave centre frequency, Hz
Fig. 8 Comparison of acoustic power calculated with sound radiation efficiencies to experimental results
The sound power level predicted by SEA for the motor driven by the PWM inverter with no load at 450 rpm, for the motor driven by the benchmark controller with no load and with full load at 450 rpm and with no load at 1500 rpm, together with the corresponding experimental results, are compared in Figs. 9a, b, c and d, respectively. To facilitate the comparison, the measurement data were synthesised into the same bandwidth as that of the force data, as given in Fig. 6. Since the experimental results include the electromagnetic noise and the aerodynamic and mechanical noise, the SEA results shown in Fig. 9 also include the electromagnetic, the aerodynamic and the mechanical noise. It should be noted that the aerodynamic and mechanical noise spectra at 450 and 1500 rpm have already been given in [1]. As already discussed in [1, 8], at low speeds, such as 450 rpm, the electromagnetic noise dominates in the total sound power level, whereas at high speeds, such as 1500 rpm, the aerodynamic noise is dominant. Figure 9 shows that the total sound power level obtained from the statistical model agree fairly well with measured results in general. At 450 rpm, since the force calculation was done only up to 3 kHz [15], the SEA calculations were also made up to 3 kHz. For the benchmark controller, the agreement at high frequencies is quite encouraging (Figs. 9b, c, and d). At low frequencies, the variation in the SEA sound power level prediction is directly related to that in the total force. Note that the FEM electromagnetic force model is twodimensional [1, 15]. A better correlation might be expected with improved electromagnetic force simulations. However, since the primary concern in SEA is the averaged response, specific peaks caused by prominent structural resonances, particularly at low frequencies, cannot be detected by this statistical approach, for example, the peak around 2.5 kHz in Fig. 9a. Fortunately, FEM and BEM can be applied in this frequency range if the behaviour of a specific vibration mode is of interest. IEE Proc.-Electr. Power Appl., Vol. 152, No. 3, May 2005
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Fig. 10 Increment of sound power level due to PWM inverter compared to benchmark controller at no load and 450 rpm
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Fig. 11 Increment of sound power level with full load compared to no load at 450 rpm for benchmark controller
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the PWM inverter compare fairly well with the measurements. The effect of load on the radiated sound power level can be examined by the increment of the sound power level due to full load relative to that at no load for the benchmark controller at 450 rpm, as shown in Fig. 11. Overall, the radiated sound power level has been increased by operating the motor at full load and the SEA predictions agree quite well with the measurements, especially at high frequencies. By comparing the SEA results presented here with those of FEM/BEM given in [1], they are quite comparable in terms of their predictive ability, but SEA calculations can be done at a fraction of the cost of FEM/ BEM calculations.
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Fig. 9 Comparison of sound power level calculated by SEA with experimental results a PWM inverter, no load, 450 rpm b Benchmark controller, no load, 450 rpm c Benchmark controller, full load, 450 rpm d Benchmark controller, no load, 1500 rpm
The effect of the PWM inverter on the radiated sound power level can be examined by the difference in the sound power level between the motor driven by the PWM inverter and that driven by the benchmark controller. This difference, referred to here as the increment of the sound power level due to the PWM inverter and determined from the experimental measurements at 450 rpm for no load, is compared with that predicted by SEA in Fig. 10. It can be seen that, overall, the radiated sound power level has been increased quite substantially owing to the effects of the time harmonics introduced by the PWM inverter. The SEA predictions of the increment of the sound power level due to IEE Proc.-Electr. Power Appl., Vol. 152, No. 3, May 2005
6
Conclusions
The feasibility of applying SEA to the vibro-acoustic analysis of a motor structure has been discussed. Based on an experimental method of determining SEA parameters, a four subsystem SEA model for the motor structure has been developed. The results show that the SEA method provides a good estimate of the vibro-acoustic characteristics of the motor structure, especially for high frequencies. Since SEA is analytically based, it is more efficient and much easier to employ compared with FEM and BEM. It has to be emphasised that the work presented is preliminary. There are a lot of challenges that remain to be tackled. Specifically, a general guideline for creating a reasonable SEA model for a motor structure should be developed. The development of a reasonable SEA model for a motor structure is crucial not only for experimental SEA study, but also for the corresponding analytical analysis. In this study, the rotor and the endshields were not included in the SEA model, and the stator and the casing were divided into three subsytems. 625
Analytical expressions for estimating the coupling loss factors should be developed. In this study, the coupling loss factors were determined by experiments. Obviously, these results do not apply to other motor structures, and cannot be used for discussing the relationship between the structural parameters and the vibration power flow in the structure. This would require a detailed analysis of the vibration power transfer between two coupled acoustically thick cylindrical shells, and an analytical expression for the coupling loss factor. Moreover, to improve the low frequency SEA results, a detailed theoretical analysis of coupling loss factors for various junctions, including strong coupling and non-conservative coupling, should be made. An effective way to calculate the electromagnetic force waves in the motor analytically should be developed. The electromagnetic force is the source of the vibration and the acoustic noise, and thus is the input to the SEA model. Obviously the accuracy of the estimated force would affect the results from SEA. In this study, the electromagnetic force was calculated by the finite-element method, which is basically suitable for low-frequency analysis, and the FEM model was just two-dimensional. Since SEA is analytically based, and is good at high frequencies, an analytical method of calculating the electromagnetic force waves in the motor would be beneficial.
7
Acknowledgments
The work presented was conducted at the Australian Defence Force Academy campus of the University of New South Wales. The project was supported by the Australian Research Council under the large grant scheme. The motor and the inverter were provided by Fasco, Australia, and Zener Electric, Australia, respectively. C. Wang acknowledges receipt of an Overseas Postgraduate Research Scholarship for the pursuit of this study.
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References
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IEE Proc.-Electr. Power Appl., Vol. 152, No. 3, May 2005
