targeted energy transfer and modal energy redistribution ... › content › pdf › 10.1007 ›...

Nonlinear Dyn (2017) 87:169–190DOI 10.1007/s11071-016-3034-4

ORIGINAL PAPER

Targeted energy transfer and modal energy redistributionin automotive drivetrains

E. Motato · A. Haris · S. Theodossiades ·M. Mohammadpour · H. Rahnejat ·P. Kelly · A. F. Vakakis · D. M. McFarland ·L. A. Bergman

Received: 26 April 2016 / Accepted: 17 August 2016 / Published online: 2 September 2016© The Author(s) 2016. This article is published with open access at Springerlink.com

Abstract The new generations of compact high out-put power-to-weight ratio internal combustion enginesgenerate broadband torsional oscillations, transmittedto lightly damped drivetrain systems. A novel approachto mitigate these untoward vibrations can be the useof nonlinear absorbers. These act as Nonlinear EnergySinks (NESs). The NES is coupled to the primary(drivetrain) structure, inducing passive irreversible tar-geted energy transfer (TET) from the drivetrain systemto the NES. During this process, the vibration energy isdirected from the lower-frequency modes of the struc-ture to the higher ones. Thereafter, vibrations can beeither dissipated through structural damping or con-sumed by the NES. This paper uses a lumped parame-ter model of an automotive driveline to simulate theeffect of TET and the assumed modal energy redistri-

E. Motato · A. Haris · S. Theodossiades (B) ·M. Mohammadpour · H. RahnejatWolfson School of Mechanical, Electrical andManufacturing Engineering, Loughborough University,Loughborough LE11 3TU, UKe-mail: [email protected]

P. KellyFord Werke GmbH, Cologne, Germany

A. F. VakakisDepartment of Mechanical Science and Engineering,University of Illinois at Urbana-Champaign, Urbana,IL 61801, USA

D. M. McFarland · L. A. BergmanDepartment of Aerospace Engineering, University ofIllinois at Urbana-Champaign, Urbana, IL 61801, USA

bution. Significant redistribution of vibratory energyis observed through TET. Furthermore, the integratedoptimization process highlights the most effective con-figuration and parametric evaluation for use of NES.

Keywords Targeted energy transfer · Nonlinearenergy sink · Automotive drivetrain · Modal energyredistribution

1 Introduction

Recent developments have resulted in downsized turbo-charged engines replacing the large capacity natu-rally aspirated variety [1]. Higher output power-to-weight ratio is the trend inmodern powertrain engineer-ing [2–4]. Furthermore, turbo-charged, direct injectionengines achievebetter fuel efficiency and reduced emis-sions (e.g. the new generation 1.0 l turbocharged FordEcoBoost three-cylinder engine produces the samepower as that of an older naturally aspirated 1.6 l four-cylinder engine). However, there is an inherent ten-dency for increased torsional oscillations, thus exacer-bated Noise, Vibration and Harshness (NVH) becauseof larger variations in combustion torque. There are aplethora of engine-induced NVH phenomena such asbody boom [5], clutch whoop [6], transmission rattle[7–9], axle whine [10,11] and driveline clonk [12,13]which are resolved during the Development process.Much of these phenomena are induced by engine ordervibration [2] or impulsive action leading to elasto-acoustic modal response [14]. There have been various

123

http://crossmark.crossref.org/dialog/?doi=10.1007/s11071-016-3034-4&domain=pdf

170 E. Motato et al.

palliative measures, such as clutch torsional dampers[15], Dual Mass Flywheel (DMF) [16] and DMFs withcentrifugal pendulum vibration absorbers in order toreduce automotive drivetrain vibrations [17]. Most ofthese remedial actions are quite expensive or add tothe powertrain inertia, which is contrary to the pre-vailing light weight and compact philosophy. Further-more, most palliativemeasures are tuned to operate in anarrow frequency band, usually targeting the dominantengine order frequency.

Nonlinear vibration absorbers (NESs) are lightweight devices, possessing essentially nonlinear stiff-ness and low damping [18–21]. When connected to alinear (primary) system, the NES nonlinearity coupleswith the vibration modes of the primary system, allow-ing irreversible energy transfer between them [22].Thus, the excess energy of the primary system is irre-versibly scattered from low-frequency modes to high-frequency ones, which is then dissipated through struc-tural damping. Some of the excess energy is retainedwithin the NES itself [23]. This action leads to a reduc-tion in the vibration amplitudes of the primary system ina broadband manner, which is due to the strong nonlin-earity of the NES [24–26]. The energy dissipation rateof the primary structure can be controlled by adjustingthe damping content of the NES [27].

Numerical and experimental investigations of sys-tems exhibiting translational motions have alreadyshown that an NES can induce energy redistributionbetween assumed modes, resulting in targeted energytransfers from low-to-high frequencies. In the avail-able literature [28–34], the mechanism of this targetedenergy transfer (TET) or “energy pumping” iswell doc-umented for structures coupled to strongly nonlinearmechanical oscillators. The effects of NES on the dis-sipation and redistribution of energy were studied byQuinn et al. [22] in a two-DoF linear structure, sub-jected to impulsive excitation. It has been shown thatthe energy redistribution is directly related to changesin the overall damping capabilities of the system. Sapsiset al. [35] defined two appropriate measures to quan-tify these effects due to variation in damping and stiff-ness during the operation of the primary linear systemcoupled to a NES. Three different types of NES wereintroduced, and their effects on the instantaneous andaverage damping were analysed. Wierschem et al. [36]complemented the work reported in [35] by providingexperimental evidence for enhanceddampingproducedby the NES in a two-DoF structure. It was demon-

strated that the proper selection of the absorber’s stiff-ness and inertia allowed it to efficiently attenuate vibra-tion energy over a broad frequency band. In a studywith increased complexity Al-Shudeifat et al. [27] usedtwo vibro-impact NESs to induce energy redistribu-tion between the modes of a nine-story building struc-ture. Numerical evidence showed that in regions whereNESs are active, the high-frequencymodes of the struc-ture are net recipients of energy, implying that thevibration energy is efficiently dissipated at thesemodesthrough structural damping.

This paper presents numerical evidence of energyredistribution between the torsional structural modesof an automotive drivetrain. The normalized effectivedamping factor [35] is utilized to identify the struc-tural modes which absorb or release energy. This is aninitial study to determine the factors which maximizethe transferred energy from the primary (drivetrain)structure to the NES, whilst dissipating the energythrough structural damping. The main aim is to estab-lish whether the TET mechanism can be conceptuallyused in automotive drivetrains. This approach has nothitherto been reported in the literature in automotivepowertrain applications.

2 Drivetrain model

A front wheel drive (FWD) vehicle with a 3-cylinderengine is studied. The drivetrain is equipped witha clutch torsional damper and a solid mass fly-wheel. The powertrain comprises the engine, flywheel,clutch, gearbox, differential and transaxle half-shafts(Fig. 1). The engine-flywheel assembly (inertia J1)is not included in the model in order to reduce thetotal DoF. The drivetrain is powered by the trans-ferred engine torque variation as an input to the systemmodel.

The main torsional DOF are those of the clutchfriction disc θ2, the gearbox input/output shaft θ3/θ4,the differential input/output shaft θ5/θ6, the left axlehalf-shaft θ7, the right axle half-shaft θ8, the lefttyre θl and the right tyre θr . The constant ratio n1of the engaged gear pair couples the angular posi-tion of the gearbox input shaft (θ3) to that of thegearbox output shaft (θ4) as a constraint. Similarly,the constant ratio n2 couples the angular position ofthe differential input shaft (θ5) to its output shaft(θ6). The model parameters are the inertia of theclutch assembly (J2), the gearbox input shaft (J3),

123

Targeted energy transfer and modal energy redistribution 171

Fig. 1 The drivetrain model

the gearbox output shaft (J4), the differential inputshaft (J5), the differential output shaft (J6), the right(J7) and the left (J8) axle half-shafts and the vehi-cle inertia (J9). The torsional stiffness coefficientsk1, k2, k3, k4, k5, k6, k7 and k8 represent the con-nection restraints between the aforementioned drive-line components (Fig. 1).

The input torque at the flywheel end is given byk1θ1+ c1θ̇1, where θ1 is the flywheel rotation, k1 is thecoupling stiffness of flywheel to the clutch disc and c1is its corresponding structural damping coefficient. Therolling resistance torques from the left and right tyresare τl and τr , respectively, and are defined as [37]:

τr = τl = 12

(FD · rw + TR) (1)

where FD is the total aerodynamic drag force, rw is theladen rolling radius of the tyre and TR is the resistingtorque generated at the contact patch. The aerodynamicdrag resistance force is defined as:

FD = 12ρ · V 2 · CD · Af (2)

where ρ is the density of air, V is the vehicle forwardspeed, CD is the dimensionless coefficient of drag and

Af is the frontal area of the vehicle. The rolling resis-tance torque is:

TR = 4{(P)α (N )β

(A + B · V + C · V 2

)}rw (3)

where P = 250kPa is the tyre pressure, N = 3374.64(N ) is the normal load (a quarter of vehicle’s weight),α = −0.003 is the tyre pressure coefficient, β = 0.97is the normal force coefficient and the constants A =84e−4m2, B = 6.2−4m-s andC = 1.6e−4 s2. Typi-cal values for the vehicle component inertias, transmis-sion ratios and the stiffness parameters were providedby the industrial partners. The equations of motion takethe matrix form:

J

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

θ̈2θ̈4θ̈6θ̈7θ̈8θ̈lθ̈r

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦+ C

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

θ̇2θ̇4θ̇6θ̇7θ̇8θ̇lθ̇r

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦+K

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

θ2θ4θ6θ7θ8θlθr

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

k1θ1 + c1θ̇10000τlτr

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

(4)

where the inertial matrix is:

123


J =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

J2 0 0 0 0 0 00

(J3η21 + J4

)0 0 0 0 0

0 0(J5η22 + J6

)0 0 0 0

0 0 0 J7 0 0 00 0 0 0 J8 0 00 0 0 0 0 Jl 00 0 0 0 0 0 Jr

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

(5)

And the stiffness matrix is:

K =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

k1 + k2 −k2η1 0 0 0 0 0−k2η1

(k2η21 + k3

) −k3η2 0 0 0 00 −k3η2

(k3η22 + k4 + k5

) −k4 −k5 0 00 0 −k4 (k4 + k6) 0 −k6 00 0 −k5 0 (k5 + k7) 0 −k70 0 0 −k6 0 k6 00 0 0 0 −k7 0 k7

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

(6)

The structural damping matrix C is calculated usingthe Caughey method [38] as:

C = J× V× Z× VT × J (7)

whereV is the mass normalized modal matrix obtainedthrough the generalized eigenvalue problem, J is theinertia matrix and Z is a diagonal matrix containingthe modal damping ratios in the form:

Z =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

2ζ1ω1 0 0 0 0 0 00 2ζ2ω2 0 0 0 0 00 0 2ζ3ω3 0 0 0 00 0 0 2ζ4ω4 0 0 00 0 0 0 2ζ5ω5 0 00 0 0 0 0 2ζ6ω6 00 0 0 0 0 0 2ζ7ω7

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

(8)

with ζi as the i th damping ratio, corresponding to thenatural frequency.

Matlab/Simulink is used to integrate the equations ofmotion (4). Flywheel angular displacement data weremeasured from a vehicle equipped with a similar drive-train. For the operating conditions presented in here,the vehicle is driven on a race track with a sweepspeed range with a steady 25% throttle. Experimentalmeasurements of the vehicle acceleration (longitudi-nal and lateral), throttle position, engine mean torqueand speed were acquired through a Controller AreaNetwork (CAN) protocol and stored. Two additionalsensors were also used to directly measure the angularvelocities of the flywheel and the gearbox input shaft at

variable sampling rate. The corresponding angular dis-placements and accelerations were obtained throughsignal integration and differentiation.

Themodal damping ratios inmatrixEq. (8) are tunedso that the numerical simulations match the measure-ments both qualitatively and quantitatively (in bothspectral and temporal domains). Thus, a variety ofdamping ratio combinations is obtained with the objec-

tive of matching the responses of the gearbox inputshaft. Table 1 lists the selected numerical dampingratios, the approximate natural frequencies and the cor-responding mode shapes:

The model validation in the frequency domain iscarried out using the continuous wavelet transform(CWT). This is because of the transient nature of theexamined powertrain manoeuvres. The CWT analysisshows the fundamental engine order (EO) response tobe at 1.5 multiple of crankshaft rotational frequency,which would be expected for a 3-cylinder 4-strokeengine (combustion occurs thrice over two crankshaftrevolutions) [2]. The experimental and numerical pre-dicted CWT spectra of the gearbox input shaft velocityfor the drivetrain accelerating in first gear with 25%open throttle are shown in Fig. 2. The predictions con-form well to the experimental results. The analysis

Table 1 Damping ratios, approximate natural frequencies andthe corresponding mode shapes

Damping ratio Natural frequency(Hz)

Main contributingDoF

ζ1 = 0.0065 ω1 = 1 θ2, θlζ2 = 0.0065 ω2 = 2 θ8, θrζ3 = 0.055 ω3 = 17 θ2, θ7ζ4 = 0.065 ω4 = 18 θ8, θ2ζ5 = 0.75 ω5 = 60 θ2, θ4ζ6 = 0.65 ω6 = 700 θ2, θ4ζ7 = 0.75 ω8 = 1100 θ2, θ6

123


Fig. 2 CWT forexperimental and simulatedgearbox input velocity infirst gear with 25% openthrottle

Fig. 3 Comparison of the gearbox input shaft velocity for numerical and experimental time histories (corresponding to 1.5 EOHarmoniccontributions)

is extended to different manoeuvres in higher gears,showing the same degree of conformance. Figure 3shows the prediction-measurement comparison in thetime domain, where the insets to the figure are for dif-ferent instants in the manoeuvre, exhibiting the pre-

dictions (in discontinuous line) and the experimentalmeasurements (continuous line) for the gearbox inputshaft velocity with the drivetrain operating in the firstgear. The scale of the corresponding engine harmonicsis provided below the time axis.

123


Fig. 4 The drivetrain model with two nonlinear absorbers in parallel with the clutch disc

3 Drivetrain model with two coupled nonlinearvibration absorbers

The drivetrainmodel represented by Eq. (4) ismodifiedto accommodate two cubic NESs connected in parallelto the clutch inertia (Fig. 4).

The differential equations of motion become:

JN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

θ̈2θ̈4θ̈6θ̈7θ̈8θ̈lθ̈rθ̈n1θ̈n2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

+ CN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

θ̇2θ̇4θ̇6θ̇7θ̇8θ̇lθ̇rθ̇n1θ̇n2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

+KN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

θ2θ4θ6θ7θ8θlθrθn1θn2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

+

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

kn2 (θ2 − θn2)3 + kn1 (θ2 − θn1)3000000

kn1 (θn1 − θ2)3kn2 (θn2 − θ2)3

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

k1θ1 + c1θ̇10000τlτr00

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(9)

The new inertia and stiffness matrices are:

JN =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

J2 0 0 0 0 0 0 0 00

(J3η21 + J4

)0 0 0 0 0 0 0

0 0(J5η22 + J6

)0 0 0 0 0 0

0 0 0 J7 0 0 0 0 00 0 0 0 J8 0 0 0 00 0 0 0 0 Jl 0 0 00 0 0 0 0 0 Jr 0 00 0 0 0 0 0 0 Jn1 00 0 0 0 0 0 0 0 Jn2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(10)

123


KN =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

(k1 + k2) −k2η1 0 0 0 0 0 0 0−k2η1

(k2η21 + k3

) −k3η2 0 0 0 0 0 00 −k3η2

(k3η22 + k4 + k5

) −k4 −k5 0 0 0 00 0 −k4 (k4 + k6) 0 −k6 0 0 00 0 −k5 0 (k5 + k7) 0 −k7 0 00 0 0 −k6 0 k6 0 0 00 0 0 0 −k7 0 k7 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(11)

The drivetrain model in Eq. (9) is hereinafter referredto as the Active NES model. This designation shouldnot be confused with the purely passive nature ofthe NES; rather, the characterization as “active” ismeant to denote an operational NES, inducing stronglynonlinear effects under transient dynamic conditions.This point is emphasized due to the nonlinear (nonlin-earizable) nature of the stiffness of the NES. There-fore, it can interact with arbitrary modes of the drive-train, passively absorbing and redistributing vibrationenergy over a broad range of frequencies. It followsthat although local, the NES can induce global effectsin system dynamics to which it is attached (in this casethe drivetrain system) [18].

The parameters θn1, kn1, Jn1 and cn1 are the angu-lar position, stiffness, inertia and damping coefficientsassociated with the first NES. In a similar manner, theparameters θn2, kn2, Jn2 and cn2 correspond to the sec-ond NES. The (9× 9) linear damping matrix of theActive NES model is given by:

CN =[Cx cTaca cd

](12)

where the matrices ca and cd are defined as:

ca =[−cn1 0 0 0 0 0 0−cn2 0 0 0 0 0 0

](13)

cd =[cn1 00 cn2

](14)

The matrix Cx is based on the operation of a drive-train with locked (non-active) NESs, since the perfor-mance of the absorbers are evaluated by comparing theresponse of the Active NES model to that of a modelwith the two NES inertias simply added to the clutchin the inertia matrix J (locked model). The equationsof motion describing the locked NES model take thefollowing form:

Ĵ

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣


⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦+ Cx

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣


⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦+K

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣


⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

k1θ1 + c1θ10000τlτr

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

(15)

with the inertia matrix given as:

Ĵ =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

J2 + Jn1 + Jn2 0 0 0 0 0 00

(J3η21 + J4

)0 0 0 0 0

0 0(J5η22 + J6

)0 0 0 0

0 0 0 J7 0 0 00 0 0 0 J8 0 00 0 0 0 0 Jl 00 0 0 0 0 0 Jr

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

(16)

and the damping matrix defined as:

Cx = Ĵ× V̂× Z× V̂T × Ĵ (17)

where V̂ is the mass normalized modal matrix of thelocked system, obtained through the generalized eigen-value problem for Ĵ and K.

The performance of the two parallel NESs isassessed with respect to the reduction in accelerationamplitude of the 1.5 EO harmonic at the location ofthe gearbox input shaft. This frequency harmonic car-ries most of the vibration energy of the drivetrain, and,therefore it is an acceptable benchmark for assessingthe induced oscillations in the drivetrain [39]. By com-paring the amplitude of gearbox input shaft accelera-tion for the 1.5 EO harmonics when (a) the drivetrainis operating with the two parallel NESs active against(b) the drivetrain operating with the inertias of the twoparallel NES simply added to the clutch inertia (locked

123


Fig. 5 Area of effective acceleration reduction (AEAR)

NESs), the regions of effective operation for each NEScan be identified (Fig. 5). Subtracting the area underthe curve defined in (a) from the area under the curvedefined in (b), the area of effective acceleration reduc-tion (AEAR) is obtained (Fig. 5).

The higher the value of AEAR, the better the perfor-mance of the NES is in reducing the system torsionalvibrations. For both cases (i.e. theLocked and theActiveNES), the acceleration amplitude corresponding to the1.5 EO harmonic is calculated using:

Ai =√Pi × fi (18)

where fi is the frequency and Ai is the accelerationamplitude of the 1.5 EO harmonic of the gearbox inputshaft and Pi is its corresponding power spectral den-sity. The power spectral density is computed using theMatlab command pwelch, which uses Welch’s method[40].

A large number of simulations are carried out inMatlab/Simulink for a range of NES stiffness, damp-ing and inertia combinations with the objective of max-imizing AEAR. In total, 140,000 parameter combi-

nations are examined for a typical vehicle manoeu-vre which lasts 8 s, corresponding to the accelerativemotion of the vehicle in first gear with 25% open throt-tle. During this manoeuvre, the speed of the engineincreases from (approximately) 1000 to 6000 rpm. Thedata obtained from these simulations are introduced inthe optimization software CAMEO� (of AVL), whichuses a neural network/genetic algorithm [41] subjectto pre-defined parameter constraints in order to deter-mine the NES parameter combination which wouldprovide the highest AEAR. Initially, CAMEO gener-ates a multi-layer perceptron neural network whichdescribes the input/output map of the data provided.For this particular case, the NES parameters are usedas inputs and the corresponding AEAR is assumed asthe output (objective function). The valid neural net-work model obtained is used by CAMEO in conjunc-tion with external parameter constraints (for stiffness,damping and inertia) to generate a hyper-dimensionalsurface. Finally, a genetic algorithm locates the point onthe surface that maximizes AEAR. The total inertia forthe two NESs added is constrained to be smaller than15% of the gearbox input shaft inertia, thus represent-

123


Table 2 Nonlinear absorber parameters

NES 1 NES 2

kn1 = 3.9e4N m/rad3 kn2 = 9.5e5N m/rad3Jn1 = 6% of the input shaftinertia

Jn2 = 9% of the input shaftinertia

cn1 = 0.001N m s/rad cn2 = 0.001N m s/rad

ing a modest and acceptable addition to the drivetrainoverall inertia and mass. The optimized NES parame-ters, which lead to the maximum reduction in the 1.5EO acceleration (at the input shaft), are presented inTable 2.

The frequency rangewhere a significant reduction inthe acceleration amplitude of the 1.5EO input shaft har-monic occurs can be divided into two regions. The firstregion is covered by the NES with the lower stiffness(NES1) between 60–110Hz, whilst the second regionis covered by the NES with the higher stiffness (NES2)between 80 and 140Hz. These regions are identifiedby simulating the drivetrain model equipped first onlywithNES1 (Fig. 6a) and then onlywithNES2 (Fig. 6b).

The corresponding spectrum of the 1.5 EO acceler-ation amplitude of the input shaft for both the NESsacting simultaneously is shown in Fig. 7. It can be seen

that the use of two NESs, coupled to the drivetrain,increases the frequency range effectiveness in a syner-gistic manner.

The time history of the gearbox input shaft angu-lar velocity for the optimized system with two NESs(data presented in Table 2) is shown in Fig. 8. Theamplitude of oscillations is reducedwithin a significantpart of the manoeuvre, starting approximately at 3 s.This specific timeframe corresponds to the frequencyrange, where the influence of the 1.5 EO is diminished(approximately at 60–140Hz). As the inset to the fig-ure shows, around 4s into the manoeuvre the angularvelocity fluctuations of the gearbox input shaft (dueto the 1.5 EO harmonic) for the active system is sub-stantially reduced, when compared to the locked sys-tem. This reduction corresponds to the minimum valueof the gearbox input shaft acceleration amplitude (upto 200 rad/s2), which occurs around 100Hz (the 1.5EO harmonic, Fig. 7). At around 6s, the amplitude ofoscillations of the active system increases again, simi-lar to that of the locked system. This behaviour occursaround 110Hz (Fig. 7), where the gearbox input shaftacceleration increases to 420 rad/s2. The other timedomain insets (at 7 and 8s) show that the reductionin oscillations is effective for the rest of the manoeu-vre.

Fig. 6 Spectra of the 1.5 EO harmonic acceleration amplitude of the transmission input shaft for a NES1 and b NES2 in 1st gear with25% open throttle

123


Fig. 7 Spectrum of the 1.5EO harmonic accelerationamplitude of thetransmission input shaft forboth NESs acting in firstgear with 25% open throttle

Fig. 8 Transmission input shaft velocity time history for the drivetrain model with active and locked NES operating in first gear with25% open throttle

123


Fig. 9 Spectrum of theacceleration amplitude of1.5 EO harmonics for thetransmission input shaft forboth active NESs in thirdgear with full throttle

The effectiveness of the absorbers in suppressingvibrations in different manoeuvres is also examined.Figure 9 shows the spectrum of the acceleration ampli-tude of the transmission for 1.5 EO harmonic in amanoeuvre, lasting 34s in third gear with full throt-tle. It can be seen that in this case the two NESs alsoreduce the amplitude of oscillations over a broad rangeof frequencies (70–160Hz).

The time history of the transmission input shaft’sangular velocity for the optimized system operatingin third gear with full throttle is shown in Fig. 10.The amplitude of oscillations is reduced within a sig-nificant part of the manoeuvre, commencing approxi-mately at 15 s. This timeframe corresponds to the fre-quency range, where the influence of 1.5 EO is dimin-ished (70–160Hz, as shown in Fig. 9). In the corre-sponding inset to the figure at 20 s the angular velocityfluctuations of the gearbox input shaft for the activesystem are reduced substantially, when compared withthose of the locked system. This reduction continues tothe minimum value of the gearbox input shaft acceler-ation at 1.5 EO harmonic (approximately 350 rad/s2),which occurs at around 130Hz (Fig. 9). The other timedomain insets (at 20 and 30s) show that the reduction inoscillations induced by the NES is effective throughoutthe rest of the manoeuvre.

4 Modal energy redistribution, effective dampingand energy dissipation

It has already been shown that the essential non-linear stiffness of the two NESs induces significantchanges in the dynamics of the primary structure (drive-train), reducing the torsional vibrations over a broadband of frequencies. This may occur in two possi-ble ways. Firstly, the NES operates in a 1:1 reso-nance with the primary system. This resonance canoccur in isolation, involving a single frequency or incascade resonant captures, where the response jumpsbetween different frequencies. This phenomenon gen-erates one-directional energy transfer from the highlyenergetic structural modes to the NES, where theenergy is trapped and dissipated. Secondly, the NESinduces energy transfer from the primary structure’slow-frequency modes to the high-frequency regions.Since these frequencies are characterized by higherdamping, the excess vibration energy is dissipated bythe internal damping of the structure. Employing themethodology described in [27] on the examined drive-train, numerical evidence for the second mechanism isinitially presented. The model represented by Eq. (6)is re-arranged as:

123


Fig. 10 Transmission input shaft velocity time history for the drivetrain model with active and locked NES operating in third gear withfull throttle

J

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣


⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦+ C

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣


⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦+K

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣


⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

+

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

kn2 (θ2 − θn2)3 + kn1 (θ2 − θn1)3000000

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣


⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

(19a)

[Jn1 00 Jn2

] [θ̈n1θ̈n2

]+

[cn1

(θ̇n1 − θ̇2

)cn2

(θ̇n2 − θ̇2

)]

+[kn1 (θn1 − θ2)3kn2 (θn2 − θ2)3

]

=[00

](19b)

Equation (19a) describes the dynamics of the drive-train, whereas Eq. (19b) represents the dynamics ofthe two NESs. These equations are coupled throughthe force produced by the two nonlinear springs kn1and kn2 and the two linear dampers cn1 and cn2. Equa-tion (19a) is then changed intomodal coordinates usingthe following transformation:

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣


⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

= [J]−1/2 [V] q (20)

123


where q is the 7x1 vector of modal coordinates and Vis the 7x7 mass normalized modal matrix. SubstitutingEq. (19b) into (19a) and pre-multiplying both sides ofthe resultant equation by [V]T [J]−1/2 yields:

q̈+[Ĉ

]q̇+

[K̂

]q = [V]T [J]−1/2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣


⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

−

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

kn2 (θ2 − θn2)3 + kn1 (θ2 − θn1)3000000

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎭

(21)

where Ĉ = VT J−1/2CJ−1/2V and K̂ = VT J−1/2KJ−1/2V are 7x7 diagonal matrices. Thus, the instan-taneous total mechanical energy for the qi mode; Ei iscomputed as:

⎡⎢⎣E1...

E7

⎤⎥⎦ = 1

2q̇2 + 1

2qT

[K̂

]q (22)

Energy redistribution between the drivetrain modescan be observed, when the modal energy content iscompared for two systems; the active and the lockedNES models (both represented in modal coordinatesfor this purpose). Both systems are excited using thevehicle manoeuvre already presented in first gear with25% open throttle. The optimized set of NES parame-ters (Table 2) is employed for this exercise. To facil-itate the analysis, the rigid body mode for each DoFis suppressed by subtracting the rotation due to thecorresponding mean velocity. The latter is computedusing the flywheel mean velocity and the correspond-ing reduction ratio (for the transmission and the differ-ential), where necessary. The first four lower modalcoordinates q1, q2, q3 and q4 (where the rigid bodymode has been supressed) are used to calculate thecorresponding instantaneous mechanical energy of thedrivetrain (Fig. 11) for the locked system (solid blackline) and for the active system (dotted grey line). Forthese modes and within the frequency region, where

the two NES are effective (2–8s of the manoeuvre), itcan be seen that the mechanical energy of the lockedsystem is slightly higher than that of the active sys-tem. Snapshots during 2s time intervals reveal thissmall difference. Conversely, Fig. 12 shows that themechanical energy of the three higher-frequencymodalcoordinates (with suppressed rigid body modes) foractive NES (dotted grey line) is slightly higher forq5 and significantly higher for q6 and q7, when com-pared with that of the locked system (solid black line).This behaviour clearly demonstrates that the nonlin-ear vibration absorbers induce energy redistributionbetween the drivetrain modes (i.e. transferring energyfrom the lower to higher-frequency modes).

The modal energy redistribution is further extendedwhen calculating the normalized effective dampingratio λeff,i/λi [35], where λi is the modal dampingof the system with the locked NES. The time-varyingeffective damping measure [22] becomes:

λeff,i =ddt

(〈q̇2i

〉)〈q̇2i

〉 (23)

The operator〈q̇2i

〉is the envelope of q̇2i , calculated as

a spline fit to the local maxima of q̇2i . According to[27] if for a particular mode the ratio λeff,i/λi > 1,then energy is transferred from this mode to others,whereas if the ratio λeff,i/λi < 1, then energy isimported from the other modes. Figure 13 shows thenormalized effective damping for the seven drivetrainmodal coordinates. The first four modes satisfy thecondition λeff,i/λi > 1, and it can be concluded thatenergy is transferred from these low-frequency modesto the higher frequencies. The last three modes satisfythe condition λeff,i/λi < 1, indicating that energy isabsorbed by them. These results corroborate the find-ing that nonlinear absorbers redistribute the energybetween the drivetrain modes.

The previous analysis was repeated for additionalmanoeuvres engaged in higher gears, and similarresults were obtained. The plots in Fig. 14 exhibit theinstantaneousmechanical energy of the first fourmodalcoordinates (rigid bodymodes have been supressed) forlocked and activeNES cases for 35 s in third gear at fullthrottle. For the first fourmodes, themechanical energyof the locked system is generally higher than that of theactive system. The insets in the figure during the 9 stime interval clearly show this small difference. Con-

123


Fig. 11 Mechanical energy of the first four modal coordinates for systems with active (dashed line) and locked (solid line) NES in firstgear with 25% open throttle manoeuvre (the rigid body modes have been supressed)

versely, Fig. 15 shows themechanical energy of the twohigher frequency modal coordinates q6 and q7 (rigidbody mode supressed) of the system with active NESsagain substantially higher, when compared with thoseof the locked system. This behaviour for the third gearat 100%open throttlemanoeuvre is similar to the previ-ously described manoeuvre in first gear and with 25%open throttle. Furthermore, the NESs-induced energy

redistribution between the drivetrainmodes follows thesame trend as previously, which is an indication ofrobust NESs’ operation.

Figure 16 shows the normalized effective dampingof four drivetrain modal coordinates for a manoeu-vre in third gear engaged with full throttle. Since forthe first two modes (i.e. q1 and q2): λeff,i/λi > 1,then it can be concluded that these modes spread

123


Fig. 12 Mechanical energy of modal coordinates q5, q6 and q7 for systems with active (dashed line) and locked (solid line) NES infirst gear with 25% open throttle manoeuvre (the rigid body modes have been supressed)

their energy elsewhere throughout the manoeuvre. Forthe last two modes (i.e. q6 and q7): λeff,i/λi < 1.Therefore, these modes mainly absorb energy through-out the manoeuvre. Thus, it can be concluded thatfor this manoeuvre, as in the previous cases, there isenergy flow from the lower-frequency modes to thehigher ones. These results further verify the effective-ness and consistent action of the nonlinear absorbersin redistribution of energy between the drivetrainmodes.

In addition to the modal energy distribution, eachNES also induces a single-directional energy trans-fer from the highly energetic structural modes to the

NES, where the energy is then dissipated [42,43]. Evi-dence of this mechanism when in third gear at fullthrottle can be obtained by computing the percentageenergy entering into the powertrain which is dissipatedby the NES damper(s) [18]. In Eq. (24), the numera-tor represents the instantaneous dissipated energy bythe i th damper and the denominator represents theamount of instantaneous entrant energy into the drive-train.

Ediss,NES(i)% = 100(

cni(θ̇n1 − θ̇2

)2k1 (θ1 − θ2)2 + J2θ̇22

)

(24)

123


Fig. 13 Normalized effective damping of the drivetrain modal coordinates (first gear with 25% open throttle)

Figure 17a shows the percentage entrant energy intothe powertrain system which is dissipated by the NES1damper. In Fig. 17b, NES1 exhibits strong oscillationsduring the first 25 s of the manoeuvre, but the max-imum percentage instantaneous dissipated energy bythe absorber in this region is only 0.25% of the total

entrant energy (see the inset to Fig. 17a captured around2.8 s). Similarly, Fig. 18a shows the percentage entrantenergy into the powertrain system, dissipated by theNES2 damper. In this case, NES2 exhibits strong oscil-lations during the entire manoeuvre, but the dissipatedenergy is even smaller than in the previous case (only

123


Fig. 14 Mechanical energy of the first four modal coordinates for systems with active (dashed line) and locked (solid line) NES inthird gear with full throttle (the rigid body modes have been supressed)

0.08% of the total entrant energy, as shown in the insetto Fig. 18a, corresponding to 2.46 s). The above resultsimply that during the specified manoeuvre most of theexcess vibrating energy is dissipated by the structuraldamping of the drivetrain throughmodal redistribution.

5 Conclusions and future work

The new generations of internal combustion enginesinduce broadband, high amplitude torsional oscilla-tions, transmitted into the driveline. A novel approach

123


Fig. 15 Mechanical energy of the modal coordinates q6 and q7 for systems with active (dashed line) and locked (solid line) NES inthird gear with full throttle (the rigid body modes have been supressed)

Fig. 16 Normalizedeffective damping of thedrivetrain modal coordinatesq1, q2, q6 and q7 (third gearat full open throttle)

123


Fig. 17 a Percentage entrant energy to the drivetrain, dissipated by the NES1 damper. b Relative displacement between the NES1 andthe clutch (3rd gear full throttle)

employs the use of nonlinear vibration absorbers cou-pled to the driveline primary structure. This is inorder to reduce the velocity fluctuations by redi-recting the excess vibration energy from the lower-frequency drivetrain modes to the higher ones. Thepaper presents numerical evidence for this energy redis-tribution process. A validated model of an automo-tive drivetrain is used to demonstrate the effect ofa pair of nonlinear absorbers on the modal energyredistribution of the primary (linear) drivetrain system.The study of the energy content in each modal coor-dinate shows that in regions where the two parallelNESs reduce the level of vibrations, energy is trans-ferred from lower-frequency to the higher-frequencymodes. Since the latter are typically modes with higher

damping, the reduction in vibration amplitudes can bedirectly associated with greater levels of energy dissi-pation at the higher modes of the structure. This con-clusion is further illustrated through estimation of thenormalized effective damping coefficients of the dif-ferent modal coordinates. In particular, the first fourlowest drivetrain modes have coefficients exceedingunity, indicating that energy is pumped out of them.Conversely, the higher-frequency modes have normal-ized effective damping values lower than unity, con-firming that energy is absorbed by them. The futuredirection of this research includes experimental proofof concept, as well as detailed examination of theinfluence of gear ratio and input load in activation ofTET.

123


Fig. 18 a Percentage entrant energy to the drivetrain, dissipated by the NES2 damper. b Relative displacement between the NES2 andthe clutch (3rd gear at full throttle)

Acknowledgements The authors wish to express their grat-itude to the UK Engineering and Physical Sciences Council(EPSRC) for the financial support extended to the project enti-tled “Targeted energy transfer in powertrains to reduce vibration-induced energy losses” Grant (EP/L019426/1), under which thisresearch was carried out. Thanks are also due to Ford MotorCompany and Raicam Clutch for their technical support, as wellas to AVL List for providing access to the multi-objective opti-mization software: CAMEO.

Open Access This article is distributed under the terms ofthe Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permitsunrestricted use, distribution, and reproduction in any medium,provided you give appropriate credit to the original author(s) andthe source, provide a link to the Creative Commons license, andindicate if changes were made.

Funding Research data for this paper are available on requestfrom Stephanos Theodossiades.

References

1. Turner, J., Popplewell, A., Patel, R., Johnson, T., et al.: Ultraboost for economy: extending the limits of extreme enginedownsizing. SAE Int. J. Engines 7(1), 387–417 (2014)

2. Rahnejat,H.:Multi-bodyDynamics:Vehicles,Machines andMechanisms. Professional Engineering Publishing, Bury StEdmunds (1998)

3. Gheorghiu, V.: Ultra-downsizing of internal combustionengines. SAE Technical Paper, Pap. No. 2015-01-1252(2015)

4. Borrmann, D., Pingen, B., Mueller, B., Kelly, P., Wirth, M.:The powertrainwith a small downsized engine: design strate-gies and system components. In: 30th International ViennaMotor Symposium, Vienna, Austria (2009)

5. Wellmann, T., Govindswamy, K., Eisele, G.: Driveline boominterior noise prediction based on multi body simulation.SAE Technical Paper, Pap. No. 2011-01-1556 (2011)

6. Kelly, P., Rahnejat, H., Biermann, J.W.: Multi-body dynam-ics investigation of clutch pedal in-cycle vibration (whoop).

123

http://creativecommons.org/licenses/by/4.0/


In: IMechE Conference Transactions C553/013/98, vol. 13,pp. 178–178 (1998)

7. Tangasawi, O., Theodossiades, S., Rahnejat, H.: Lightlyloaded lubricated impacts: idle gear rattle. J. Sound Vib.308(3), 418–430 (2007)

8. Wang, M.Y., Manoj, R., Zhao, W.: Gear rattle modelling andanalysis for automotivemanual transmissions. Proc. IMechEPart D J. Automob. Eng. 215(2), 241–258 (2001)

9. De la Cruz, M., Theodossiades, S., Rahnejat, H.: An inves-tigation of manual transmission drive rattle. Proc. IMechEPart K J. Multi Body Dyn. 224(2), 167–181 (2010)

10. Kim, S.J., Lee, S.K.: Experimental identification of a gearwhine noise in the axle system of a passenger van. Int. J.Automot. Technol. 8(1), 75–82 (2007)

11. Koronias, G., Theodossiades, S., Rahnejat, H., Saunders, T.:Axle whine phenomenon in light trucks: a combined numer-ical and experimental investigation. Proc. IMechE Part D J.Automob. Eng. 225(7), 885–894 (2011)

12. Menday, M.T., Rahnejat, H., Ebrahimi, M.: Clonk: an ono-matopoeic response in torsional impact of automotive driv-elines. Proc. IMechE Part D J. Automob. Eng. 213(4), 349–357 (1999)

13. Farshidianfar,A., Ebrahimi,M.,Rahnejat,H.,Menday,M.T.:High frequency torsional vibration of vehicular drivelinesystems in clonk. Int. J. Heavy Veh. Syst. 9(2), 127–149(2002)

14. Theodossiades, S., Gnanakumarr,M., Rahnejat, H.,Menday,M.: Mode identification in impact-induced high-frequencyvehicular driveline vibrations using an elasto-multi-bodydynamics approach. Proc. IMechEPartK J.Multi BodyDyn.218(2), 81–94 (2004)

15. Drexl, H.: Torsional dampers and alternative systems toreduce driveline vibrations. SAE Technical Paper, Pap. No.870393 (1987)

16. Theodossiades, S., Gnanakumarr, M., Rahnejat, H., Kelly,P.: Effect of a dual-mass flywheel on the impact-inducednoise in vehicular powertrain systems. Proc. IMechE Part DJ. Automob. Eng. 220(6), 747–761 (2006)

17. Haddow, A.G., Shaw, S.W.: Centrifugal pendulum vibra-tion absorbers: an experimental and theoretical investigation.Nonlinear Dyn. 34(3), 293–307 (2003)

18. Vakakis, A., Gendelman, O.V., Bergman, L.A., McFarland,D.M., Kerschen, G., Lee, Y.S.: Nonlinear Targeted EnergyTransfer inMechanical andStructural Systems. SpringerSci-ence & Business Media, Berlin (2008)

19. Gendelman, O.V., Sapsis, T., Vakakis, A.F., Bergman, L.A.:Enhanced passive targeted energy transfer in strongly non-linear mechanical oscillators. J. Sound Vib. 330(1), 1–8(2011)

20. Lee, Y.S., Vakakis, A.F., Bergman, L.A., McFarland, D.M.,Kerschen, G., Nucera, F., Panagopoulos, P.N.: Passive non-linear targeted energy transfer and its applications to vibra-tion absorption: a review. Proc. IMechE Part K J.Multi BodyDyn. 222(2), 77–134 (2008)

21. Sigalov, G., Gendelman, O.V., Al-Shudeifat, M.A.,Manevitch, L.I., Vakakis, A.F., Bergman, L.A.: Resonancecaptures and targeted energy transfers in an inertially-coupled rotational nonlinear energy sink. Nonlinear Dyn.69(4), 1693–1704 (2012)

22. Quinn, D.D., Hubbard, S., Wierschem, N., Al-Shudeifat,M.A., Ott, R.J., Luo, J., Bergman, L.A.: Equivalent modal

damping, stiffening and energy exchanges in multi-degree-of-freedom systems with strongly nonlinear attachments.Proc. IMechE Part K J. Multi Body Dyn. 226(2), 122–146(2012)

23. McFarland, D.M., Kerschen, G., Kowtko, J.J., Lee, Y.S.,Bergman, L.A., Vakakis, A.F.: Experimental investigation oftargeted energy transfers in strongly and nonlinearly coupledoscillators. J. Acoust. Soc. Am. 118, 791–799 (2005)

24. Paresh, M., Dardel, M., Ghasemi, H.M.: Performance com-parison of nonlinear energy sink and linear tuned massdamper in steady-state dynamics of a linear beam. NonlinearDyn. 81(4), 1981–2002 (2015)

25. Nucera, F., Vakakis, A.F., McFarland, D.M., Bergman, L.A.,Kerschen, G.: Targeted energy transfers in vibro-impactoscillators for seismic mitigation. Nonlinear Dyn. 50(3),651–677 (2007)

26. Petit, F., Loccufier, M., Aeyels, D.: The energy thresholds ofnonlinear vibration absorbers. Nonlinear Dyn. 74(3), 755–767 (2013)

27. Al-Shudefait, M., Vakakis, A.F., Bergman, L.A.: Shock mit-igation bymeans of low to high frequency nonlinear targetedenergy transfer in a large-scale structure. J. Comput. Non-linear Dyn. 11(2), 021006 (2015). Paper No: CND-15-1015

28. Gendelman, O., Manevitch, L.I., Vakakis, A.F., M’closkey,R.: Energy pumping in nonlinear mechanical oscillators:part I—dynamics of the underlying Hamiltonian systems.J. Appl. Mech. 68(1), 34–41 (2001)

29. Vakakis, A.F., Gendelman, O.: Energy pumping in nonlinearmechanical oscillators: part II—resonance capture. J. Appl.Mech. 68(1), 42–48 (2001)

30. Gendelman, O.V.: Transition of energy to a nonlinear local-ized mode in a highly asymmetric system of two oscillators.Nonlinear Dyn. 25(1), 237–253 (2001)

31. Al-Shudeifat, M.: Highly efficient nonlinear energy sink.Nonlinear Dyn. 76(4), 1905–1920 (2014)

32. Guo, C., Al-Shudeifat, M., Vakakis, A., Bergman, L.A.,McFarland,M.D., Yan, J.: Vibration reduction in unbalancedhollow rotor systems with nonlinear energy sinks. NonlinearDyn. 79(1), 527–538 (2015)

33. Zhang, Y.W., Zhang, Z., Chen, L., Yang, T., Fang, B., Zang,J.: Impulse-induced vibration suppression of an axiallymov-ing beam with parallel nonlinear energy sinks. NonlinearDyn. 82(1), 61–71 (2015)

34. Lin, D.C., Oguamanam, D.C.D.: Targeted energy transferefficiency in a low-dimensional mechanical system withan essentially nonlinear attachment. Nonlinear Dyn. 82(1),971–986 (2015)

35. Sapsis, T.P., Quinn, D.D., Vakakis, A.F., Bergman, L.A.:Effective stiffening and damping enhancement of structureswith strongly nonlinear local attachments. J. Vib. Acoust.134(1), 011016 (2012)

36. Wierschem, N.E., Quinn, D.D., Hubbard, S.A., Al-Shudeifat, M.A., McFarland, D.M., Luo, J., Bergman, L.A.:Passive damping enhancement of a two-degree-of-freedomsystem through a strongly nonlinear two-degree-of-freedomattachment. J. Sound Vib. 331(25), 5393–5407 (2012)

37. Gillespie, T.D.: Fundamentals of Vehicle Dynamics. SAE,Warrendale (1992)

38. Caughey, T.K., O’Kelly, M.E.J.: Classical normal modes indamped linear dynamic systems. J. Appl. Mech. 32(3), 583–588 (1965)

123


39. Haris, A., Motato, E., Theodossiades, S., Vakakis, A.F.,Bergman, L.A., McFarland, D.M.: Targeted energy transferin automotive powertrains. In: 13th International Conferenceon Dynamical Systems—Theory and Applications (DSTA),Lodz, Poland, 7–10 December, vol. 1, pp. 245–254 (2015)

40. Welch, P.D.: The use of fast Fourier transform for the esti-mation of power spectra: a method based on time averagingover short, modified periodograms. IEEETrans. Audio Elec-troacoust. AU–15, 70–73 (1967)

41. Fortuna, D.I.T., Koegeler, D.I.D.H.M., Vitale, D.I.G.: DoEand beyond—evolution of the model-based developmentapproach. ATZ Worldw. 117(2), 30–35 (2015)

42. Andersen, D., Starosvetsky, Y., Vakakis, A., Bergman, L.:Dynamic instabilities in coupled oscillators induced by geo-metrically nonlinear damping. Nonlinear Dyn. 67(1), 807–827 (2012)

43. Palacios, J.L., Balthazar, J.M., Horta,M.J.: On energy pump-ing, synchronization and beat phenomenon in a nonidealstructure coupled to an essentially nonlinear oscillator. Non-linear Dyn. 56(1), 1–11 (2009)

123

Targeted energy transfer and modal energy redistribution in automotive drivetrainsAbstract1 Introduction2 Drivetrain model3 Drivetrain model with two coupled nonlinear vibration absorbers4 Modal energy redistribution, effective damping and energy dissipation5 Conclusions and future workAcknowledgementsReferences

targeted energy transfer and modal energy redistribution ... › content › pdf › 10.1007 ›...

Documents