experimental identification of high-frequency gear mesh...

Experimental identification of high-frequency gear meshvibrations in a planetary gearbox

D. Ploger 1, P. Zech 1, S. Rinderknecht 1

1 TU Darmstadt, Institut fur Mechatronische Systeme im Maschinenbau,Otto-Berndt-Straße 2, D-64287, Darmstadt, Germanye-mail: [email protected]

AbstractIn addition to unbalance excitation, vibrations in gear transmissions are mostly caused by gear meshing. Acentral challenge is the high frequency range: the meshing frequency is in the range of one to several kilohertzfor gearboxes of high power to weight ratio. The first step in the development of an active isolation at gearmeshing 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. Tokeep the setup simple, rotational inertia of a rotor of an existing rotor test-rig is used as load in dynamic run-uptests. The test-rig is equipped with force transducers. The vibration behavior as a function of speed and load ismeasured in multiple test series. Measurement data are visualized and the magnitudes of base band vibrationand its harmonics are discussed. A model of the gear mesh excitation is developed. Requirements concerningforces, 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 ispreferred. In comparison to a parallel axis arrangement it offers a higher power density. Also the epicyclicgearing provides an inherently higher efficiency because of the lower sliding velocity at the tooth contactpoint. 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. Vibrationlevel is one of these constraints.

Both efficiency and noise may be influenced by suitable tooth flank corrections. However, the optimalsolutions for the respective goals do not necessarily agree. In addition to tooth flank correction other solutionsare often employed, such as acoustic liners. These measures also come at a cost. Currently only passivenoise control is used in aircraft. The introduction of active vibration control has the potential to overcome thedrawbacks of the passive countermeasures. Not only can active vibration control lead to a quieter aircraftengine, 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 transmittedpower will be linearly scaled up. The gear meshing frequency will be in the kilohertz dimension. In thisfrequency range classical feedback control becomes infeasible. Instead adaptive feedforward control offerseffective solutions. The aim of our research is the development of a feedfoward active control of vibrationsexcited by the gear meshing of a fast running high power planetary gear box. In this investigation, theexcitation of a planetary gear box will be examined. From this experimental data, the requirements on afeedforward controller will be derived. In feedforward control a reference signal is needed. One of the mainquestions 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

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 gearvibration and eigenfrequencies of gears was conducted in [2] and [3]. A general overview of research inplanetary gearbox vibration phenomenons is given in [4]. Control of vibration caused by gear meshing usingactive vibration control systems is an ongoing field of research. Different actuator concepts are comparedin [5–7]. Feedforward control is used in most of the cases as control algorithm and can be seen as state of theart [8–11].

A variety of methods for modeling of gear box vibration have been proposed. However to date quantitativeprediction of gear meshing vibration as function of load and speed for a given setup is impossible. This workuses experiments to identify gear mesh vibration of a chosen planetary gearbox. The dependency on load andspeed is examined. Requirements for an projected test rig for active vibration control of gear mesh vibrationsare derived using the experimental results.

2 Experiments

This section describes the experimental setup that was used for measurements. An existing rotor test rig isused as basis to keep setup simple. Another test rig is projected to investigate solutions for active vibrationcontrol of gear mesh vibration. However in this work the rotor test rig can be used to conduct first fundamentalexperiments. The gearbox is a Neugart PLE-60 type, has a spur toothing and is chosen because it can producemeshing frequency in the range of multiple kilohertz. To integrate the selected gearbox into the test-rig themotor mount is modified. Figure 1 shows the mechanical setup on the whole as well as a close up view of theinstrumented gearbox.

The motor (1) is connected with the input shaft of the gearbox (3) via a metal bellows coupling (2). The outputshaft of the gearbox is connected to the rotor (6) of the rotor test rig via another metal bellow coupling (4). Therotor is mounted with a locating bearing (5) and a double floating bearing (8). A massive disc (7) is mounted tothe rotor to achieve comparable rotordynamic properties as a real engine rotor with a turbine stage. Differencebetween input and output torque at the gearbox is supported by adapters. Two force measuring rings (10) areintegrated into one side of the gearbox mount under the adapters. Furthermore two accelerometers (9) arefixed onto the gearbox.

It is noteworthy that the setup uses no breaking mechanism. In contrast to common approaches where electricdrives or power recirculation systems are applied, in this work the high inertia of the rotor is utilized asload for the gearbox. This has two consequences: On one hand is the maximum load restricted by angularacceleration that can be produced by the motor. On the other hand no defined loads can be adjusted forstationary rotating speeds. In this case only losses in the rotor bearings create minimal load. The investigation

912 PROCEEDINGS OF ISMA2016 INCLUDING USD2016

0 20 40 600

1

2

3

trunup in s

Loa

din

Nm

analyticalexperimental

Figure 2: Load over run-up time from 0 to9000 min–1

BrüelgKjaerI(L9F

KistlerI9bbGA

PTGbbbISensor BuBISensorsITEMODcINC

KistlerI)b7LIChargeIAmplifier

KistlerI)GL(BIPiezotronICoupler

dSPACEIDSGGb(IuIPC

Sensors Signalconditioning DataIaquisition

MeasurementIchainIG

BaumerHübnerI

HGFDNLFbTTLdSPACEIDSGGbLIuIPC

pAccelerationf

pForcef

pTemperaturef

MeasurementIchainIN

pEncoderf

Figure 3: Measurement instrumentation and data acquisitionhardware

concerning correlation between gearbox excitation and load as well as rotating speed has thus to be doneusing 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 ΘTotal describes the inertia of thewhole drive chain related to the rotational degree of freedom of the motor.

Inertia Value

ΘMotor 0.000 661 7 kgm2

ΘGearbox 0.000 006 5 kgm2

ΘRotor 0.064 kgm2

ΘTotal 0.0078 kgm2

Table 1: Inertia of relevant components

The reduced inertia is calculated using

ΘTotal = ΘMotor + ΘGearbox +ΘRotor

i2. (1)

where i = 3 is the transmission ratio of the gearbox.

For a given drive torque MMotor the resulting angular acceleration of the motor ϕMotor can be derived from thefollowing formula when bearing losses are neglected.

ϕMotor =MMotor

ΘReduced

(2)

The run-up time to a given angular speed ϕ max can be calculated using

trunup =ϕ max

ϕMotor

. (3)

The corresponding characteristic for the setup is shown in Figure 2 where analytical and experimental resultsare compared for a run-up to maximum rotational speed of ϕ max = 9000 min–1.

The load was measured during run-up using the force measuring rings in the gearbox support. It can be clearlyseen from Figure 2 that only for run-up times smaller than 10 s the load can be significantly increased. Theexperimental results are in good agreement to the analytically predicted loads. The gearbox is designed for10 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. Onemeasures the rotational speed of the motor and implements motor control. All other sensor signals are

DYNAMICS OF ROTATING MACHINERY 913

0 1500 3000 4500 6000 7500 90000

2000

4000

6000

8000

Motor speed in min–1

Freq

uenc

yin

Hz

Force in dB (relative to 1 N)

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 thesecond system. Consequently signals can be synchronized offline after measurement. The dSPACE DS1104system runs at 30 kHz. The dSPACE DS1103 system runs at 2.5 kHz. Force and measurement signals arefiltered in signal conditioning using a low pass filter with a cut-off frequency of 10 kHz to avoid aliasing. Inaddition the signals are filtered with a 5 Hz high pass filter to remove DC components of the signals. Theprojected 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 0 min–1 to 9000 min–1 gearbox input shaft speed

• Run-down in 15 s, 7 s and 3.5 s from 9000 min–1 to 0 min–1 gearbox input shaft speed

• Stationary gearbox input shaft speed at 9000 min–1

Each measurement was repeated eight times to reduce influence of random disturbances by averaging and todetermine 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. Theresults of run-up and run-down experiments are presented below. Figure 4 shows a Campbell diagram of theforce 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. 4200 Hz formaximum motor speed of 9000 min–1. Second, third and even higher gear mesh orders are visible, howeverthe first gear mesh order has the strongest occurrence. A closer look at the gear mesh orders shows thatthey are split-up. A single dominating order cannot be identified. In fact the first gear mesh order consistsof many sidebands that run very close to each other. Numerous sidebands can even be observed betweenfirst and second gear mesh in the frequency range of 4000 Hz to 6000 Hz. The Campbell diagram revealsseveral eigenfrequencies of the setup that are excited during run-up. They are represented as horizontal lines.


0 1500 3000 4500 6000 7500 90000

200

400

600

800

1000

Motor speed in min–1

Freq

uenc

yin

Hz

Force in dB (relative to 1 N)

Figure 5: Campbell diagram of force signal of force measuring ring – CloseUp-View on Frequencies below1 kHz

Figure 5 shows a closeup view on the lower frequency range of the Campbell diagram. Eigenfrequencies at100 Hz, 600 Hz and 800 Hz can be observed. Especially when an excitation crosses an eigenfrequency theycan be seen. Figure 5 shows also that rotor and motor synchronous forces have the highest amplitude of allexcitations. These forces are caused by unbalance as well as misalignment. In addition a strong occurrence ofsubharmonics of the first gear mesh can be observed. Subharmonics can be produced by meshing of sun gearand planet gears.

Figure 4 shows two different motor speeds for which vertical lines in the spectrum occur: 4000 min–1 and5500 min–1. During run-up the setup was temporarily very loud at these speeds. A vertical line shows thatalmost all frequencies are excited at this particular motor speeds. It is interesting to note that especially thesecond phenomenon is slowly fading in. Starting at a speed of 5000 min–1 all frequencies are continuouslyincreased in amplitude up to 5500 min–1 where an abrupt reduction of excitation can be observed. It isassumed that the teeth of the gearbox strike against each other in these situations. This pulsed excitationswould explain vertical lines in the spectrum. Calculations as well as experimental identifications show thattorsional eigenfrequencies of the setup occur at approx. 1800 Hz and 2500 Hz. As consequence vertical linescan be explained as situations where first gear mesh crosses torsional eigenfrequencies.

At 5500 min–1 motor speed a special situation occurs: Two resonance phenomenons are superposed: Firstgear mesh excites a torsional eigenfrequency and as can be seen in figure 5 motor synchronous forces exciteanother eigenfrequency of the setup: the first bending mode of the rotor at 100 Hz. This explains the highforce amplitudes at this particular motor speed. While in the lower frequency range resonances are distinct inthe Campbell diagram for higher frequency range this is not the case. This could be caused by higher modaldamping for high frequency modes.

Figure 6 shows the spectral components of the gear mesh force for a stationary motor speed of 9000 min–1.As observed in the Campbell diagram several sidebands are visible. The main peaks have uniform distancesbetween each other. Their amplitudes are not symmetrically distributed. Sidebands can be explained byamplitude and phase modulation effects [12] as explained in Section 4 of this work. This property of the gearmesh force has consequences for the active vibration isolation system that is projected to reduce transmissionof gear mesh forces. The spectrum in Figure 6 shows that the gear mesh force is a superposition of severalspectral components. The active vibration isolation system needs to be able to deal with this spectral diversity.


3600 3800 4000 4200 4400 4600 4800 50000

1

2

3

4

5

Frequency in Hz

Forc

ein

N

Figure 6: Frequency components of first gear mesh in force signal for stationary motor speed 9000 min–1

To analyze the influence of motor speed and load on the forces in first gear mesh ordercuts are used as a toolas shown in Figure 7. To generate ordercuts the force signal is filtered with an adaptive bandpass filter thatcuts out only forces produced by first gear mesh out of the entire signal. The center frequency is adapted usinga multiple of the motor speed. After cutting out the forces of the first gear mesh the envelope of the remainingsignal is formed to produce the curves shown in Figure 7. They are the projection of the gear mesh force ontofrequency 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 sidesensor are about one newton less in amplitude for the highest peaks.

Maxima during run-up are visible at 800 Hz, 1800 Hz, 2600 Hz and 3500 Hz. The peaks at 1800 Hz and2600 Hz correspond to the torsional eigenfrequencies of the setup that were mentioned before. In frequencyranges without resonances e.g. in the range from 1000 Hz to 1700 Hz a clear increase of the gear mesh forceas function of the load can be observed. This is not the case in the resonances. On one hand resonances arestronger excited with higher loads. On the other hand higher loads can only be realized with faster run-uptimes, which decreases the excitation. It is difficult to analyze the dependency of gear mesh forces as functionof speed due to the high number of resonances. Run-up and run-down ordercuts are different. In the run-downsituation one can see a clear increase of gear mesh forces with higher loads for all frequencies, even inresonances. Furthermore in run-down some resonances are not visible at all. This indicates the presence ofnonlinearities.

Displacements of the gearbox support due to gear mesh were measured using accelerometers and were foundto be smaller than 1 µm in the measured frequency range. In summary it can be stated that forces of the firstgear mesh order are measurable in a frequency range up to 4200 Hz. An increase of gear mesh forces with theload was observed outside of resonance frequencies during run-up and for all frequencies in run-down. Gearmesh forces are by trend higher for higher speeds. However an exact analysis of speed dependency is difficultdue 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 severalexcitation frequencies. In the following section we will show that the force signal can be derived from thecarrier angle ϕ(t) by means of a Fourier series. The carrier is connected to the rotor. If the components of the


0 1000 2000 3000 40000

2

4

6

8

10

Frequency in Hz

Forc

ein

N

Sensor motor - run-up

0.42 Nm0.84 Nm1.67 Nm

0 1000 2000 3000 40000

2

4

6

8

10

Frequency in Hz

Forc

ein

N

Sensor motor - run-down

0.42 Nm0.84 Nm1.67 Nm

Figure 7: Ordercuts of first gear mesh for motor side force sensor with different loads as well as run-up andrun-down

gearbox are assumed rigid, the carrier angle can be calculated from the motor angle using the gear ratio. Theratio of the gearbox used in the experiments is 3.

ϕ =ϕMotor

3(4)

The Fourier series constitutes a sum of harmonic functions at different frequencies. Alternatively this can alsobe understood as a periodic modulation of a single frequency. Instead of directly modeling the force signal asa Fourier series we will choose a more graphic approach. At first glance one would expect the force signal Fto be a harmonic function of the carrier angle ϕ(t) multiplied by the number of teeth Z on the ring gear.

F ∝ eiZϕ (5)

Observation however rejects this overly simple model. This can be explained by the motion of the planetarygears [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 amore sophisticated model: The complex amplitude A(ϕ) is not constant but a periodic function of ϕ with aperiodic length of 2π.

F =12

A(ϕ)eiZϕ +12

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 Fourierseries.

A(ϕ) =∞∑

l=–∞cA,le

ilϕ (7)

Here, cA,l represent the Fourier coefficients of the Amplitude function A. Inserting (7) into (6) yields


F =12

[ ∞∑l=–∞

cA,lei(Z+l)ϕ +

∞∑l=–∞

cA,le–i(Z+l)ϕ

]. (8)

By substituting the index l by k = Z + l the more common representation

F =12

[ ∞∑k=–∞

ckeikϕ +∞∑

k=–∞cke–ikϕ

](9)

follows. From this derivation we can learn that a modulated base frequency can be expressed as a Fourierseries. Also from (8) we can expect that the most significant Fourier coefficients ck will be found in thevicinity 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 ck cannot be calculated from the fast Fourier transform (FFT) of the time series representationof 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 ck can be derived from the FFT. Even ifthe FFT could be used to determine the Fourier coefficients of the proposed model it would not be useful in apractical implementation. The FFT needs to be calculated block-wise. This introduces an undesirable timedelay 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 ≈ a02

+N∑

k=1

(ak cos kϕ + bk sin kϕ) (10)

Also, the Fourier series is truncated to the order N. The complex coefficients ck and the real coefficients akand bk can be translated using

ck =

a0 for k = 012 (ak – ibk) for k > 012 (a–k + ib–k) for k < 0 .

(11)

As demonstrated by Figure 8, this model reproduces the experimental data quite well. Here N = 3.5 · Z = 294is chosen. Because (10) is linear with respect to ak and bk a least squares (LS) approach may be employed.The drawback of the LS approach is the high computational complexity. Alternatively an integral transformmy be used.

During the experiment the signals are uniformly sampled at discrete points with time step ∆t. The underlyingcontinuous signals will be denoted by F(t) and ϕ(t), where t denotes the continuous time. The discrete signalswill be written F[l] and ϕ[l], where l is an integer and t = ∆t · l. The Fourier coefficients need to change overtime for a correct reproduction of the experimental data. Therefore, the calculation will be derived for a finitesegment beginning with the index m and ending with n. In order to prevent spectral leakage, the coefficientsare evaluated on complete turns of the carrier, so that

ϕ[n] – ϕ[m] = 2π . (12)

On a complete turn of the carrier the coefficient ak could be calculated from

ak =1π

∫ 2π

0F(ϕ) cos kϕ dϕ (13)


0 0.5 1 1.5 2 2.5 3

·10–3

80

90

100

110

Time t in s

Forc

eF

inN

measuredreconstructed

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.

ak =1π

∫ t(ϕ=2π)

t(ϕ=0)F(ϕ(t)) cos kϕ(t) ϕ(t)dt (14)

The integral can now be approximated using the discrete signals.

ak ≈∆tπ

n∑l=m

F[l]ϕ[l] cos kϕ[l] (15)

The angular velocity of the carrier angle ϕ also needs to be approximated. For this the central differencemethod is chosen.

ϕ[l] ≈ ϕ[l + 1] – ϕ[l – 1]2∆t

(16)

From these steps the following numerical approximation for the Fourier coefficients may be obtained:

ak ≈1

2π

n∑l=m

F[l] (ϕ[l + 1] – ϕ[l – 1]) cos kϕ[l] (17)

bk ≈1

2π

n∑l=m

F[l] (ϕ[l + 1] – ϕ[l – 1]) sin kϕ[l] (18)

The numerical complexity depends heavily on the order N of the truncated Fourier series. For big N thesecalculations can be more complex than the FFT approach, needing O(N(n – m)) operations where a FFTbased approach would need O((n – m) log(n – m)) operations. Depending on the implementation, a leastsquares approach can easily require O((n – m)3) operations because it is closely linked to the computation ofa (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 disturbedby 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 uniformlysampled signal ϕ. However, these effects are small. As visible in Figure 8 the integral transform methodyields sufficient accuracy. Therefore the much faster integral transform method will be used.


0 84 168 252

101

100

10–1

10–2

k

mea

n( |c k|

) inN

74 84 940

0.25

0.5

0.75

1

k

mea

n( |c k|

) inN

first order vibrations

158 168 1780

0.25

0.5

0.75

1

k

mea

n( |c k|

) inN

second order vibrations

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-stillthe rotor is accelerated to 9000 min–1 within 3.5 s. This is repeated eight times. For every full revolution ofthe carrier a set of 294 coefficients is calculated. Finally average absolute values for each complex coefficientcn are calculated and plotted.

The highest magnitude can be observed at n = 0. This coefficient does not correspond to a true vibration as itdescribes a static offset. The piezoelectric force sensor inherently acts as a high pass filter. Therefore, thecoefficient c0 must be regarded as arbitrary. The next six coefficients also exhibit a large magnitude. Thisresults from the misalignment of both couplings. The preliminary experiment offers insufficient facilities toalign the rotating parts precisely. This will be corrected in the projected test rig. These first six coefficientscannot 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 = 84where the main excitation from ring gear meshing is expected. From the detail we can observe that the meanmagnitude of the coefficient c84 is not the largest. Instead the neighboring coefficients c83 and c85 exhibitmore 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 firstpeak, but of smaller magnitude. Gear meshing is a nonlinear phenomenon and therefore introduces harmonicsof the fundamental frequencies. For a large part of experimental data, the third peak is already beyond the cutoff frequency of the anti aliasing low pass filter. This obscures its true magnitude. In the planned test rig afaster 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 forcesignal at fixed carrier speed and the mean composition for a full run-up. From the previously computed 294coefficients the most significant coefficient c83 is selected for further analysis.

Figure 10 shows that both magnitude and phase vary during a run-up of the test rig. For each revolution of thecarrier both magnitude and phase are plotted separately. The frequency f denotes the mean frequency the


1000 2000 3000 40000

1

2

3

f in Hz

|c83|

inN

Magnitude

1000 2000 3000 4000

–π

0

π

f in Hz

arg

c 83

Phase

Figure 10: Coefficient c83 from eight run-up experiments and mean (black line). Each color corresponds toone run-up experiment.

0 200 400 600 800 1000 1200 1400 1600 18000

84

168

252

n

Coefficients for eight run-ups and run-downs

300 350 400 450 500 550 600 650 700 7500

84

168

Revolutions of the planet carrier

n

Detail view

Brightness: Magnitude; Hue: Phase.

Figure 11: Fourier coefficients for the continuously acquired data.


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 byconstant Fourier coefficients.

It can be concluded that the run-up experiments exhibit mostly deterministic behavior because repetitionyields similar results. Figure 10 exposes that the contribution of coefficient c83 to the overall Force levelalmost vanishes at different rotor speeds. A comparison to Figure 7 (left plot, 1.67 Nm) does not reveal muchsimilarity. The peak between 1800 Hz and 1900 Hz in Figure 7 is not visible from coefficient c83. In reverse,the peak at 2100 Hz in Figure 10 appears as part of a plateau in Figure 7. In general, the maximum magnitudesof the Fourier coefficients cannot be identified with the natural frequencies observed in the Campbell plot inFigure 4.

While Figure 9 is restricted by the lack of time information Figure 10 is restricted because it only displays onecoefficient. Figure 11 eliminates these restriction by encoding the amplitude information as the brightnessand the phase as the hue. Figure 11 displays all eight run-ups at maximum load from the previous figuresincluding the run-downs and the stationary phases. For visibility reasons the brightness does not correspondlinearly to the magnitude of the coefficients. The excitation mechanism appears as horizontal lines. In aCampbell diagram (see Figure 4) it appears as straight lines crossing the origin. This difference in appearanceresults from the respective choice of the ordinate. The Campbell diagram uses absolute frequency, whileFigure 11 normalizes frequency to the angular velocity of the carrier separately for every full turn. For thesame reason, natural frequencies result in curved lines resembling a valley. In the Campbell diagram theyappear 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 meshvibration of the chosen gearbox. Table 2 shows some basic requirements that were defined using the findingsof the conducted experiments. The required actuator force amplitude is rated 30 N to have enough reserves for

Requirement ValueActuator force - Amplitude at least 30 NActuator force - Frequency up to 4.2 kHzActuator displacement 1 µm-10 µmActuator weight at most 0.2 kgSampling Frequency at least 50 kHz

Table 2: Requirements for AVC-system

higher loads. The needed sampling frequency of the projected AVC-system is specified with 50 kHz to bemore 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 ontothe gearbox mount. In contrast to active mounts the solution using inertial actuators offers more flexibilityregarding positioning and construction of the actuators. Conventional gearbox support can remain. TheAVC-system can be seen as an add-on system which is advantageous. Inertial actuators are tuned to have theirfirst eigenfrequency below the frequency to be controlled. In the working range the force that acts onto theinertial mass m can be transferred into the structure

Fact = mact · x (19)

where x is the acceleration of the inertial mass. In reality the transferred forces will of course be reduced byflexibility of the connection area. If sinusoidal movements of the mass are supposed the following formuladescribes the actuator force

Fact = mact · ω2 · x · sin(ω · t) (20)


where ω denotes the angular frequency. It can be seen clearly that actuator force depends linearly on massand amplitude and quadratically on the angular frequency. In reverse this means that for a desired actuatorforce and displacement the mass of the actuator can be chosen much smaller for higher frequencies. Thedevelopment 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 tocreate an artificial reference signal using encoder signals. This reference signal represents the forces producedby gear meshing. It is planned to use an adaptive feedforward control algorithm that controls several narrowband 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 thecarrier. The Fourier coefficients exhibit the largest magnitudes in the vicinity of number of teeth of the annulargear. This has been previously known from literature. Beyond the state of the art it could be found that theFourier coefficients are not constant but a function of speed and load. It could be shown that gear meshing inplanetary gear boxes is a very complex process and the Fourier coefficients cannot be easily derived fromstructural parameters of the components. Instead the Fourier coefficients have to be learned from the operationof the gear box. The proposed model of the force signal excited by gear meshing does not need any blockwise calculations. Instead it can accurately predict the vibration signal form the instantaneous angle of thecarrier. 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 thisinvestigation is not well suited to this goal. Therefore, an improved test rig will be built. From thisinvestigations several requirement on the new test rig could be derived: The rotating parts need to be alignedprecisely. Also instead of a rotor an eddy current brake will be used to load the gear box. Inertial massactuators will be developed using piezoelectric technology. The feedforward controller will be a centralobjective 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 aboutthe Tooth Meshing Frequency in Epicyclic Gear Vibration, Proceedings of the Institution of MechanicalEngineers, Part C: Journal of Mechanical Engineering Science, vol. 199, no. 1, (1985), pp. 65–70.

[2] F. M. Agemi, M. Ognjanovic, Gear Vibration in Supercritical Mesh-Frequency Range, FME Transac-tions, vol. 32, (2004), pp. 87–94.

[3] T. M. Ericson, R. G. Parker, Natural Frequency Clusters in Planetary Gear Vibration, Journal ofVibration and Acoustics, vol. 135, no. 6, (2013), p. 061002.

[4] C. G. Cooley, R. G. Parker, A Review of Planetary and Epicyclic Gear Dynamics and VibrationsResearch, Applied Mechanics Reviews, vol. 66, no. 4, (2014), p. 040804.

[5] M. Li, T. C. Lim, Y. H. Guan, W. S. S. Jr, Actuator design and experimental validation for active gearboxvibration control, Smart Materials and Structures, vol. 15, no. 1, (2006), pp. N1–N6.

[6] Y. H. Guan, M. Li, T. C. Lim, W. Shepard, Comparative analysis of actuator concepts for active gearpair vibration control, Journal of Sound and Vibration, vol. 269, no. 1-2, (2004), pp. 273–294.

[7] M. H. Chen, M. J. Brennan, Active control of gear vibration using specially configured sensors andactuators, Smart Materials and Structures, vol. 9, no. 3, (2000), p. 342.


[8] T. Sutton, S. Elliott, M. Brennan, K. Heron, D. Jessop, Active isolation of multiple structural waves on ahelicopter gearbox support strut, Journal of Sound and Vibration, vol. 205, no. 1, (1997), pp. 81–101.

[9] M. Li, T. C. Lim, W. S. S. Jr, Y. H. Guan, Experimental active vibration control of gear mesh harmonicsin a power recirculation gearbox system using a piezoelectric stack actuator, Smart Materials andStructures, vol. 14, no. 5, (2005), p. 917.

[10] Y. H. Guan, T. C. Lim, W. Steve Shepard, Experimental study on active vibration control of a gearboxsystem, Journal of Sound and Vibration, vol. 282, no. 35, (2005), pp. 713–733.

[11] G. T. Montague, A. F. Kascak, A. Palazzolo, D. Manchala, E. Thomas, Feedforward Control of GearMesh Vibration Using Piezoelectric Actuators, Shock and Vibration, vol. 1, no. 5, (1994), pp. 473–484.

[12] M. Inalpolat, A. Kahraman, A theoretical and experimental investigation of modulation sidebands ofplanetary gear sets, Journal of Sound and Vibration, vol. 323, no. 35, (2009), pp. 677–696.


experimental identification of high-frequency gear mesh...

Documents