Experimental identification of high-frequency gear mesh vibrations in a planetary gearbox

Transcription

1 Experimental identification of high-frequency gear mesh vibrations in a planetary gearbox D. Plöger 1, P. Zech 1, S. Rinderknecht 1 1 TU Darmstadt, Institut für Mechatronische Systeme im Maschinenbau, Otto-Berndt-Straße 2, D-64287, Darmstadt, Germany ploeger@ims.tu-darmstadt.de Abstract In addition to unbalance excitation, vibrations in gear transmissions are mostly caused by gear meshing. A central challenge is the high frequency range: the meshing frequency is in the range of one to several kilohertz for gearboxes of high power to weight ratio. The first step in the development of an active isolation at gear meshing frequency is the determination of key requirements on the actuation system and the control algorithm. In this work an experimental investigation of the vibration excitation of a planetary gear box is performed. To keep the setup simple, rotational inertia of a rotor of an existing rotor test-rig is used as load in dynamic run-up tests. The test-rig is equipped with force transducers. The vibration behavior as a function of speed and load is measured in multiple test series. Measurement data are visualized and the magnitudes of base band vibration and its harmonics are discussed. A model of the gear mesh excitation is developed. Requirements concerning forces, displacements and arrangement are derived for a projected active vibration isolation system. 1 Introduction In mobile high power applications such as geared turbofan or turboshaft aircraft engines planetary gearing is preferred. In comparison to a parallel axis arrangement it offers a higher power density. Also the epicyclic gearing provides an inherently higher efficiency because of the lower sliding velocity at the tooth contact point. However, even with the benefits of the planetary gearing several difficult trade-offs have to be made. Primary goals are efficiency and weight but there are several constraints which limit the design. Vibration level is one of these constraints. Both efficiency and noise may be influenced by suitable tooth flank corrections. However, the optimal solutions for the respective goals do not necessarily agree. In addition to tooth flank correction other solutions are often employed, such as acoustic liners. These measures also come at a cost. Currently only passive noise control is used in aircraft. The introduction of active vibration control has the potential to overcome the drawbacks of the passive countermeasures. Not only can active vibration control lead to a quieter aircraft engine, but by lifting the design constraint of noise it can lead to a more efficient design. In high power applications the speed of the gear box is usually as high as possible because the transmitted power will be linearly scaled up. The gear meshing frequency will be in the kilohertz dimension. In this frequency range classical feedback control becomes infeasible. Instead adaptive feedforward control offers effective solutions. The aim of our research is the development of a feedfoward active control of vibrations excited by the gear meshing of a fast running high power planetary gear box. In this investigation, the excitation of a planetary gear box will be examined. From this experimental data, the requirements on a feedforward controller will be derived. In feedforward control a reference signal is needed. One of the main questions will be if the rotation of the carrier in the gear box is sufficient as a reference. From previous research it is well known that the vibration of gear boxes can be modeled using Fourier series. 911

2 912 PROCEEDINGS OF ISMA216 INCLUDING USD216 Figure 1: Left: Sectional view of the setup Right: Gearbox with measurement instrumentation This approach was applied to epicyclic gearing in [1]. Experimental research concerning high frequency gear vibration and eigenfrequencies of gears was conducted in [2] and [3]. A general overview of research in planetary gearbox vibration phenomenons is given in [4]. Control of vibration caused by gear meshing using active vibration control systems is an ongoing field of research. Different actuator concepts are compared in [5 7]. Feedforward control is used in most of the cases as control algorithm and can be seen as state of the art [8 11]. A variety of methods for modeling of gear box vibration have been proposed. However to date quantitative prediction of gear meshing vibration as function of load and speed for a given setup is impossible. This work uses experiments to identify gear mesh vibration of a chosen planetary gearbox. The dependency on load and speed is examined. Requirements for an projected test rig for active vibration control of gear mesh vibrations are derived using the experimental results. 2 Experiments This section describes the experimental setup that was used for measurements. An existing rotor test rig is used as basis to keep setup simple. Another test rig is projected to investigate solutions for active vibration control of gear mesh vibration. However in this work the rotor test rig can be used to conduct first fundamental experiments. The gearbox is a Neugart PLE-6 type, has a spur toothing and is chosen because it can produce meshing frequency in the range of multiple kilohertz. To integrate the selected gearbox into the test-rig the motor mount is modified. Figure 1 shows the mechanical setup on the whole as well as a close up view of the instrumented gearbox. The motor (1) is connected with the input shaft of the gearbox (3) via a metal bellows coupling (2). The output shaft of the gearbox is connected to the rotor (6) of the rotor test rig via another metal bellow coupling (4). The rotor is mounted with a locating bearing (5) and a double floating bearing (8). A massive disc (7) is mounted to the rotor to achieve comparable rotordynamic properties as a real engine rotor with a turbine stage. Difference between input and output torque at the gearbox is supported by adapters. Two force measuring rings (1) are integrated into one side of the gearbox mount under the adapters. Furthermore two accelerometers (9) are fixed onto the gearbox. It is noteworthy that the setup uses no breaking mechanism. In contrast to common approaches where electric drives or power recirculation systems are applied, in this work the high inertia of the rotor is utilized as load for the gearbox. This has two consequences: On one hand is the maximum load restricted by angular acceleration that can be produced by the motor. On the other hand no defined loads can be adjusted for stationary rotating speeds. In this case only losses in the rotor bearings create minimal load. The investigation

3 DYNAMICS OF ROTATING MACHINERY 913 Load in Nm analytical experimental Sensors Signalconditioning DataIaquisition BrüelgKjaerI(L9F paccelerationf KistlerI)GL(BIPiezotronICoupler KistlerI9bbGA pforcef PTGbbbISensor ptemperaturef KistlerI)b7LIChargeIAmplifier BuBISensorsITEMODcINC dspaceidsggb(iuipc MeasurementIchainIG t runup in s BaumerHübnerI HGFDNLFbTTL pencoderf dspaceidsggbliuipc MeasurementIchainIN Figure 2: Load over run-up time from to 9 min 1 Figure 3: Measurement instrumentation and data acquisition hardware concerning correlation between gearbox excitation and load as well as rotating speed has thus to be done using run-up experiments. The projected test rig will use an eddy current brake to produce load. The inertias of the relevant components are listed in Table 1. The total inertia Θ Total describes the inertia of the whole drive chain related to the rotational degree of freedom of the motor. The reduced inertia is calculated using Inertia where i = 3 is the transmission ratio of the gearbox. Value Θ Motor kgm 2 Θ Gearbox. 6 5 kgm 2 Θ Rotor.64 kgm 2 Θ Total.78 kgm 2 Table 1: Inertia of relevant components Θ Total = Θ Motor + Θ Gearbox + Θ Rotor i 2. (1) For a given drive torque M Motor the resulting angular acceleration of the motor ϕ Motor can be derived from the following formula when bearing losses are neglected. The run-up time to a given angular speed ϕ max can be calculated using ϕ Motor = M Motor Θ Reduced (2) t runup = ϕ max ϕ Motor. (3) The corresponding characteristic for the setup is shown in Figure 2 where analytical and experimental results are compared for a run-up to maximum rotational speed of ϕ max = 9 min 1. The load was measured during run-up using the force measuring rings in the gearbox support. It can be clearly seen from Figure 2 that only for run-up times smaller than 1 s the load can be significantly increased. The experimental results are in good agreement to the analytically predicted loads. The gearbox is designed for 1 Nm load at the input shaft. In this work input loads up to 1.7 Nm were applicable. The measurement chains are shown in Figure 3. Two data aquisition systems are used in parallel. One measures the rotational speed of the motor and implements motor control. All other sensor signals are

4 914 PROCEEDINGS OF ISMA216 INCLUDING USD216 8 Force in db (relative to 1 N) 6 Frequency in Hz Motor speed in min 1 Figure 4: Campbell diagram of force signal of force measuring ring measured using the second data aquisition system. A trigger signal from the first system is recorded by the second system. Consequently signals can be synchronized offline after measurement. The dspace DS114 system runs at 3 khz. The dspace DS113 system runs at 2.5 khz. Force and measurement signals are filtered in signal conditioning using a low pass filter with a cut-off frequency of 1 khz to avoid aliasing. In addition the signals are filtered with a 5 Hz high pass filter to remove DC components of the signals. The projected test rig will concentrate all data acquisition on one real time system. The following measurements are carried out: Run-up in 15 s, 7 s and 3.5 s from min 1 to 9 min 1 gearbox input shaft speed Run-down in 15 s, 7 s and 3.5 s from 9 min 1 to min 1 gearbox input shaft speed Stationary gearbox input shaft speed at 9 min 1 Each measurement was repeated eight times to reduce influence of random disturbances by averaging and to determine repeatability. 3 Discussion of experimental results One goal of this paper is to identify forces caused by gear meshing as a function of load and speed. The results of run-up and run-down experiments are presented below. Figure 4 shows a Campbell diagram of the force sensor that is mounted on the rotor side for a run-up in 15 s. Gear mesh orders are clearly visible. The first gear mesh order has a frequency of approx. 42 Hz for maximum motor speed of 9 min 1. Second, third and even higher gear mesh orders are visible, however the first gear mesh order has the strongest occurrence. A closer look at the gear mesh orders shows that they are split-up. A single dominating order cannot be identified. In fact the first gear mesh order consists of many sidebands that run very close to each other. Numerous sidebands can even be observed between first and second gear mesh in the frequency range of 4 Hz to 6 Hz. The Campbell diagram reveals several eigenfrequencies of the setup that are excited during run-up. They are represented as horizontal lines.

5 DYNAMICS OF ROTATING MACHINERY Force in db (relative to 1 N) 8 Frequency in Hz Motor speed in min 1 Figure 5: Campbell diagram of force signal of force measuring ring CloseUp-View on Frequencies below 1 khz Figure 5 shows a closeup view on the lower frequency range of the Campbell diagram. Eigenfrequencies at 1 Hz, 6 Hz and 8 Hz can be observed. Especially when an excitation crosses an eigenfrequency they can be seen. Figure 5 shows also that rotor and motor synchronous forces have the highest amplitude of all excitations. These forces are caused by unbalance as well as misalignment. In addition a strong occurrence of subharmonics of the first gear mesh can be observed. Subharmonics can be produced by meshing of sun gear and planet gears. Figure 4 shows two different motor speeds for which vertical lines in the spectrum occur: 4 min 1 and 55 min 1. During run-up the setup was temporarily very loud at these speeds. A vertical line shows that almost all frequencies are excited at this particular motor speeds. It is interesting to note that especially the second phenomenon is slowly fading in. Starting at a speed of 5 min 1 all frequencies are continuously increased in amplitude up to 55 min 1 where an abrupt reduction of excitation can be observed. It is assumed that the teeth of the gearbox strike against each other in these situations. This pulsed excitations would explain vertical lines in the spectrum. Calculations as well as experimental identifications show that torsional eigenfrequencies of the setup occur at approx. 18 Hz and 25 Hz. As consequence vertical lines can be explained as situations where first gear mesh crosses torsional eigenfrequencies. At 55 min 1 motor speed a special situation occurs: Two resonance phenomenons are superposed: First gear mesh excites a torsional eigenfrequency and as can be seen in figure 5 motor synchronous forces excite another eigenfrequency of the setup: the first bending mode of the rotor at 1 Hz. This explains the high force amplitudes at this particular motor speed. While in the lower frequency range resonances are distinct in the Campbell diagram for higher frequency range this is not the case. This could be caused by higher modal damping for high frequency modes. Figure 6 shows the spectral components of the gear mesh force for a stationary motor speed of 9 min 1. As observed in the Campbell diagram several sidebands are visible. The main peaks have uniform distances between each other. Their amplitudes are not symmetrically distributed. Sidebands can be explained by amplitude and phase modulation effects [12] as explained in Section 4 of this work. This property of the gear mesh force has consequences for the active vibration isolation system that is projected to reduce transmission of gear mesh forces. The spectrum in Figure 6 shows that the gear mesh force is a superposition of several spectral components. The active vibration isolation system needs to be able to deal with this spectral diversity.

6 916 PROCEEDINGS OF ISMA216 INCLUDING USD Force in N Frequency in Hz Figure 6: Frequency components of first gear mesh in force signal for stationary motor speed 9 min 1 To analyze the influence of motor speed and load on the forces in first gear mesh ordercuts are used as a tool as shown in Figure 7. To generate ordercuts the force signal is filtered with an adaptive bandpass filter that cuts out only forces produced by first gear mesh out of the entire signal. The center frequency is adapted using a multiple of the motor speed. After cutting out the forces of the first gear mesh the envelope of the remaining signal is formed to produce the curves shown in Figure 7. They are the projection of the gear mesh force onto frequency axis of the Campbell diagram. Each curve represents the mean of eight conducted experiments. During run-up gear mesh forces are qualitatively very similar at both force sensors. Forces on the rotor side sensor are about one newton less in amplitude for the highest peaks. Maxima during run-up are visible at 8 Hz, 18 Hz, 26 Hz and 35 Hz. The peaks at 18 Hz and 26 Hz correspond to the torsional eigenfrequencies of the setup that were mentioned before. In frequency ranges without resonances e.g. in the range from 1 Hz to 17 Hz a clear increase of the gear mesh force as function of the load can be observed. This is not the case in the resonances. On one hand resonances are stronger excited with higher loads. On the other hand higher loads can only be realized with faster run-up times, which decreases the excitation. It is difficult to analyze the dependency of gear mesh forces as function of speed due to the high number of resonances. Run-up and run-down ordercuts are different. In the run-down situation one can see a clear increase of gear mesh forces with higher loads for all frequencies, even in resonances. Furthermore in run-down some resonances are not visible at all. This indicates the presence of nonlinearities. Displacements of the gearbox support due to gear mesh were measured using accelerometers and were found to be smaller than 1 µm in the measured frequency range. In summary it can be stated that forces of the first gear mesh order are measurable in a frequency range up to 42 Hz. An increase of gear mesh forces with the load was observed outside of resonance frequencies during run-up and for all frequencies in run-down. Gear mesh forces are by trend higher for higher speeds. However an exact analysis of speed dependency is difficult due the to many resonances in the operating range. 4 Excitation model From the frequency domain representation of the force signal in Figure 6 we can observe that there are several excitation frequencies. In the following section we will show that the force signal can be derived from the carrier angle ϕ(t) by means of a Fourier series. The carrier is connected to the rotor. If the components of the

7 DYNAMICS OF ROTATING MACHINERY Sensor motor - run-up.42 Nm.84 Nm 1.67 Nm 1 8 Sensor motor - run-down.42 Nm.84 Nm 1.67 Nm Force in N 6 4 Force in N Frequency in Hz Frequency in Hz Figure 7: Ordercuts of first gear mesh for motor side force sensor with different loads as well as run-up and run-down gearbox are assumed rigid, the carrier angle can be calculated from the motor angle using the gear ratio. The ratio of the gearbox used in the experiments is 3. ϕ = ϕ Motor 3 (4) The Fourier series constitutes a sum of harmonic functions at different frequencies. Alternatively this can also be understood as a periodic modulation of a single frequency. Instead of directly modeling the force signal as a Fourier series we will choose a more graphic approach. At first glance one would expect the force signal F to be a harmonic function of the carrier angle ϕ(t) multiplied by the number of teeth Z on the ring gear. F e izϕ (5) Observation however rejects this overly simple model. This can be explained by the motion of the planetary gears [1]. The transmission path of the vibration changes depending on the location of the mesh contact points. Therefore both amplitude and phase of the vibration will be modulated. From this observation we come to a more sophisticated model: The complex amplitude A(ϕ) is not constant but a periodic function of ϕ with a periodic length of 2π. F = 1 2 A(ϕ)eiZϕ A(ϕ)e izϕ (6) The amplitude function is chosen complex so it can also encode the phase of the resulting vibration. Equation (6) guaranties a real force signal F. In the next step the complex amplitude A(ϕ) will be expressed as Fourier series. A(ϕ) = l= c A,l e ilϕ (7) Here, c A,l represent the Fourier coefficients of the Amplitude function A. Inserting (7) into (6) yields

8 918 PROCEEDINGS OF ISMA216 INCLUDING USD216 F = 1 2 [ ] c A,l e i(z+l)ϕ + c A,l e i(z+l)ϕ l= l=. (8) By substituting the index l by k = Z + l the more common representation [ F = 1 ] c 2 k e ikϕ + c k e ikϕ k= k= (9) follows. From this derivation we can learn that a modulated base frequency can be expressed as a Fourier series. Also from (8) we can expect that the most significant Fourier coefficients c k will be found in the vicinity of the number of teeth in the ring gear Z. This will be shown in detail in the next section (see Fig. 9). The coefficients c k cannot be calculated from the fast Fourier transform (FFT) of the time series representation of the force signal F. The FFT connects time domain and frequency domain, but (9) uses the carrier angle ϕ(t) instead of the time. Only for the case that ϕ = const. the coefficients c k can be derived from the FFT. Even if the FFT could be used to determine the Fourier coefficients of the proposed model it would not be useful in a practical implementation. The FFT needs to be calculated block-wise. This introduces an undesirable time delay if used by an active noise cancellation algorithm. Therefore, the numerical computation of the Fourier coefficients needs to be derived. First the formulation (9) has to be written using only real symbols. F a N 2 + (a k cos kϕ + b k sin kϕ) (1) k=1 Also, the Fourier series is truncated to the order N. The complex coefficients c k and the real coefficients a k and b k can be translated using a for k = c k = 1 2 (a k ib k ) for k > 1 2 (a k + ib k ) for k <. (11) As demonstrated by Figure 8, this model reproduces the experimental data quite well. Here N = 3.5 Z = 294 is chosen. Because (1) is linear with respect to a k and b k a least squares (LS) approach may be employed. The drawback of the LS approach is the high computational complexity. Alternatively an integral transform my be used. During the experiment the signals are uniformly sampled at discrete points with time step t. The underlying continuous signals will be denoted by F(t) and ϕ(t), where t denotes the continuous time. The discrete signals will be written F[l] and ϕ[l], where l is an integer and t = t l. The Fourier coefficients need to change over time for a correct reproduction of the experimental data. Therefore, the calculation will be derived for a finite segment beginning with the index m and ending with n. In order to prevent spectral leakage, the coefficients are evaluated on complete turns of the carrier, so that ϕ[n] ϕ[m] = 2π. (12) On a complete turn of the carrier the coefficient a k could be calculated from a k = 1 2π F(ϕ) cos kϕ dϕ (13) π

9 DYNAMICS OF ROTATING MACHINERY measured reconstructed Force F in N Time t in s 1 3 Figure 8: Reconstruction of experimental data by a truncated Fourier series but for a numerical implementation it is necessary to substitute the variable ϕ of the integral by the time t. a k = 1 t(ϕ=2π) F(ϕ(t)) cos kϕ(t) ϕ(t)dt (14) π t(ϕ=) The integral can now be approximated using the discrete signals. a k t π n F[l] ϕ[l] cos kϕ[l] (15) l=m The angular velocity of the carrier angle ϕ also needs to be approximated. For this the central difference method is chosen. ϕ[l] ϕ[l + 1] ϕ[l 1] 2 t From these steps the following numerical approximation for the Fourier coefficients may be obtained: (16) a k 1 2π b k 1 2π n F[l] (ϕ[l + 1] ϕ[l 1]) cos kϕ[l] (17) l=m n F[l] (ϕ[l + 1] ϕ[l 1]) sin kϕ[l] (18) l=m The numerical complexity depends heavily on the order N of the truncated Fourier series. For big N these calculations can be more complex than the FFT approach, needing O(N(n m)) operations where a FFT based approach would need O((n m) log(n m)) operations. Depending on the implementation, a least squares approach can easily require O((n m) 3 ) operations because it is closely linked to the computation of a (n m) (n m) pseudo inverse. It has to be noted that a least squares based calculation will be more accurate because it is not disturbed by imperfections in the numerical derivation of ϕ. Additionally it is completely immune to leakage errors. These will always be present in the integral transform as (12) cannot be fulfilled exactly using a uniformly sampled signal ϕ. However, these effects are small. As visible in Figure 8 the integral transform method yields sufficient accuracy. Therefore the much faster integral transform method will be used.

10 92 PROCEEDINGS OF ISMA216 INCLUDING USD216 mean ( c k ) in N k first order vibrations second order vibrations mean ( c k ) in N k mean ( c k ) in N k Figure 9: Mean absolute values of Fourier coefficients 5 Analysis of the model parameters Figure 9 gives an overview over the distribution of the coefficients of the experimental data. From stand-still the rotor is accelerated to 9 min 1 within 3.5 s. This is repeated eight times. For every full revolution of the carrier a set of 294 coefficients is calculated. Finally average absolute values for each complex coefficient c n are calculated and plotted. The highest magnitude can be observed at n =. This coefficient does not correspond to a true vibration as it describes a static offset. The piezoelectric force sensor inherently acts as a high pass filter. Therefore, the coefficient c must be regarded as arbitrary. The next six coefficients also exhibit a large magnitude. This results from the misalignment of both couplings. The preliminary experiment offers insufficient facilities to align the rotating parts precisely. This will be corrected in the projected test rig. These first six coefficients cannot be regarded as significant with regard to the objective of this investigation. With increasing n a drop off can be observed, from which a peak emerges. The peak centers around n = 84 where the main excitation from ring gear meshing is expected. From the detail we can observe that the mean magnitude of the coefficient c 84 is not the largest. Instead the neighboring coefficients c 83 and c 85 exhibit more than double the magnitude. This results from the modulation of the fundamental excitation frequency. In addition to the first peak, a second and a third can be observed. The second peak is very similar to the first peak, but of smaller magnitude. Gear meshing is a nonlinear phenomenon and therefore introduces harmonics of the fundamental frequencies. For a large part of experimental data, the third peak is already beyond the cut off frequency of the anti aliasing low pass filter. This obscures its true magnitude. In the planned test rig a faster A/D converter will be used so that the higher harmonics can be observed undistorted. The mean absolute values of coefficients can only give a simplified view on the excitation mechanism. Comparing Figure 6 to Figure 9 exposes significant differences between the spectral composition of the force signal at fixed carrier speed and the mean composition for a full run-up. From the previously computed 294 coefficients the most significant coefficient c 83 is selected for further analysis. Figure 1 shows that both magnitude and phase vary during a run-up of the test rig. For each revolution of the carrier both magnitude and phase are plotted separately. The frequency f denotes the mean frequency the

11 DYNAMICS OF ROTATING MACHINERY Magnitude π Phase c 83 in N 2 1 arg c f in Hz π f in Hz Figure 1: Coefficient c 83 from eight run-up experiments and mean (black line). Each color corresponds to one run-up experiment. n Coefficients for eight run-ups and run-downs Detail view 168 n Revolutions of the planet carrier Brightness: Magnitude; Hue: Phase. Figure 11: Fourier coefficients for the continuously acquired data.

12 922 PROCEEDINGS OF ISMA216 INCLUDING USD216 coefficient excites during each turn of carrier. While there is some variance between the different experiments, a clear characteristic can be observed. The vibration excited by the gear meshing cannot be explained by constant Fourier coefficients. It can be concluded that the run-up experiments exhibit mostly deterministic behavior because repetition yields similar results. Figure 1 exposes that the contribution of coefficient c 83 to the overall Force level almost vanishes at different rotor speeds. A comparison to Figure 7 (left plot, 1.67 Nm) does not reveal much similarity. The peak between 18 Hz and 19 Hz in Figure 7 is not visible from coefficient c 83. In reverse, the peak at 21 Hz in Figure 1 appears as part of a plateau in Figure 7. In general, the maximum magnitudes of the Fourier coefficients cannot be identified with the natural frequencies observed in the Campbell plot in Figure 4. While Figure 9 is restricted by the lack of time information Figure 1 is restricted because it only displays one coefficient. Figure 11 eliminates these restriction by encoding the amplitude information as the brightness and the phase as the hue. Figure 11 displays all eight run-ups at maximum load from the previous figures including the run-downs and the stationary phases. For visibility reasons the brightness does not correspond linearly to the magnitude of the coefficients. The excitation mechanism appears as horizontal lines. In a Campbell diagram (see Figure 4) it appears as straight lines crossing the origin. This difference in appearance results from the respective choice of the ordinate. The Campbell diagram uses absolute frequency, while Figure 11 normalizes frequency to the angular velocity of the carrier separately for every full turn. For the same reason, natural frequencies result in curved lines resembling a valley. In the Campbell diagram they appear as horizontal lines. Overall good agreement between Figures 4 and 11 can be found. 6 Active vibration control This section presents some considerations concerning an active vibration control system for gear mesh vibration of the chosen gearbox. Table 2 shows some basic requirements that were defined using the findings of the conducted experiments. The required actuator force amplitude is rated 3 N to have enough reserves for Requirement Value Actuator force - Amplitude at least 3 N Actuator force - Frequency up to 4.2 khz Actuator displacement 1 µm-1 µm Actuator weight at most.2 kg Sampling Frequency at least 5 khz Table 2: Requirements for AVC-system higher loads. The needed sampling frequency of the projected AVC-system is specified with 5 khz to be more than ten times higher then the highest mechanical frequency that has to be controlled. The actuation concept is planned to be based on piezoelectric inertial mass actuators that are fixed onto the gearbox mount. In contrast to active mounts the solution using inertial actuators offers more flexibility regarding positioning and construction of the actuators. Conventional gearbox support can remain. The AVC-system can be seen as an add-on system which is advantageous. Inertial actuators are tuned to have their first eigenfrequency below the frequency to be controlled. In the working range the force that acts onto the inertial mass m can be transferred into the structure F act = m act ẍ (19) where ẍ is the acceleration of the inertial mass. In reality the transferred forces will of course be reduced by flexibility of the connection area. If sinusoidal movements of the mass are supposed the following formula describes the actuator force F act = m act ω 2 ˆx sin(ω t) (2)

13 DYNAMICS OF ROTATING MACHINERY 923 where ω denotes the angular frequency. It can be seen clearly that actuator force depends linearly on mass and amplitude and quadratically on the angular frequency. In reverse this means that for a desired actuator force and displacement the mass of the actuator can be chosen much smaller for higher frequencies. The development of high frequency inertial mass actuators is subject of further work. Concerning the control algorithm it can be stated that using the described excitation model it is possible to create an artificial reference signal using encoder signals. This reference signal represents the forces produced by gear meshing. It is planned to use an adaptive feedforward control algorithm that controls several narrow band disturbances in parallel. 7 Conclusion The vibration of a planetary gear box can be explained by a Fourier series with respect to the angle of the carrier. The Fourier coefficients exhibit the largest magnitudes in the vicinity of number of teeth of the annular gear. This has been previously known from literature. Beyond the state of the art it could be found that the Fourier coefficients are not constant but a function of speed and load. It could be shown that gear meshing in planetary gear boxes is a very complex process and the Fourier coefficients cannot be easily derived from structural parameters of the components. Instead the Fourier coefficients have to be learned from the operation of the gear box. The proposed model of the force signal excited by gear meshing does not need any block wise calculations. Instead it can accurately predict the vibration signal form the instantaneous angle of the carrier. Therefore the model is well suited for active vibration control purposes. In future investigations an active vibration control system will be developed. The test rig used in this investigation is not well suited to this goal. Therefore, an improved test rig will be built. From this investigations several requirement on the new test rig could be derived: The rotating parts need to be aligned precisely. Also instead of a rotor an eddy current brake will be used to load the gear box. Inertial mass actuators will be developed using piezoelectric technology. The feedforward controller will be a central objective of future investigations, once the projected test rig is operational. References [1] P. D. McFadden, J. D. Smith, An Explanation for the Asymmetry of the Modulation Sidebands about the Tooth Meshing Frequency in Epicyclic Gear Vibration, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, vol. 199, no. 1, (1985), pp [2] F. M. Agemi, M. Ognjanovic, Gear Vibration in Supercritical Mesh-Frequency Range, FME Transactions, vol. 32, (24), pp [3] T. M. Ericson, R. G. Parker, Natural Frequency Clusters in Planetary Gear Vibration, Journal of Vibration and Acoustics, vol. 135, no. 6, (213), p [4] C. G. Cooley, R. G. Parker, A Review of Planetary and Epicyclic Gear Dynamics and Vibrations Research, Applied Mechanics Reviews, vol. 66, no. 4, (214), p [5] M. Li, T. C. Lim, Y. H. Guan, W. S. S. Jr, Actuator design and experimental validation for active gearbox vibration control, Smart Materials and Structures, vol. 15, no. 1, (26), pp. N1 N6. [6] Y. H. Guan, M. Li, T. C. Lim, W. Shepard, Comparative analysis of actuator concepts for active gear pair vibration control, Journal of Sound and Vibration, vol. 269, no. 1-2, (24), pp [7] M. H. Chen, M. J. Brennan, Active control of gear vibration using specially configured sensors and actuators, Smart Materials and Structures, vol. 9, no. 3, (2), p. 342.

14 924 PROCEEDINGS OF ISMA216 INCLUDING USD216 [8] T. Sutton, S. Elliott, M. Brennan, K. Heron, D. Jessop, Active isolation of multiple structural waves on a helicopter gearbox support strut, Journal of Sound and Vibration, vol. 25, no. 1, (1997), pp [9] M. Li, T. C. Lim, W. S. S. Jr, Y. H. Guan, Experimental active vibration control of gear mesh harmonics in a power recirculation gearbox system using a piezoelectric stack actuator, Smart Materials and Structures, vol. 14, no. 5, (25), p [1] Y. H. Guan, T. C. Lim, W. Steve Shepard, Experimental study on active vibration control of a gearbox system, Journal of Sound and Vibration, vol. 282, no. 35, (25), pp [11] G. T. Montague, A. F. Kascak, A. Palazzolo, D. Manchala, E. Thomas, Feedforward Control of Gear Mesh Vibration Using Piezoelectric Actuators, Shock and Vibration, vol. 1, no. 5, (1994), pp [12] M. Inalpolat, A. Kahraman, A theoretical and experimental investigation of modulation sidebands of planetary gear sets, Journal of Sound and Vibration, vol. 323, no. 35, (29), pp