Nonlinear Dynhttps://doi.org/10.1007/s11071-020-05502-z

ORIGINAL PAPER

Design and validation of a nonlinear vibration absorberto attenuate torsional oscillations of propulsion systems

A. Haris · P. Alevras · M. Mohammadpour ·S. Theodossiades · M. O’ Mahony

Received: 12 June 2019 / Accepted: 21 January 2020© The Author(s) 2020

Abstract Recent developments in propulsion sys-tems to improve energy efficiency and reduce haz-ardous emissions often lead to severe torsional oscil-lations and aggravated noise. Vibration absorbers aretypically employed to palliate the untoward effects ofpowertrain oscillations, with nonetheless an adverseimpact on cost and constrained efficacy over a lim-ited frequency range. Recently, the authors proposedthe use of nonlinear vibration absorbers to achievemore broadband drivetrain vibration attenuation withlow complexity and cost. These lightweight attach-ments follow the concept of targeted energy transfer,whereby vibration energy is taken off from a primarysystem without tuning requirements. In this paper, thedesign and experimental investigation of a prototypeabsorber is presented. The absorber is installed on adrivetrain experimental rig driven by an electric motorthrough a universal joint connection placed at an angle,thus inducing the second-order torsional oscillations.Vibration time histories with and without the absorber

A. Haris · M. Mohammadpour · S. Theodossiades (B)Wolfson School of Mechanical, Electrical andManufacturing Engineering, Loughborough University,Loughborough LE11 3TU, UKe-mail: S.Theodossiades@lboro.ac.uk

P. AlevrasDepartment of Mechanical Engineering, School ofEngineering, University of Birmingham,Birmingham B15 2TT, UK

M. O’ MahonyFord Motor Company Ltd, Dunton Technical Centre, Laindon,Basildon, Essex, UK

acting are recorded and compared. Frequency–energyplots are superimposed to the system nonlinear nor-mal modes to verify the previously developed designmethodology, whereas the achieved vibration reduc-tion is quantified by comparing the acceleration ampli-tudes of the primary system and monitoring the dis-tribution of energy damped in the primary system andthe absorber. The absorber prototype was found to leadto significant vibration reduction away from resonanceand near resonance with the additional feature of acti-vation over a relatively broad frequency range.

Keywords Nonlinear vibration absorber · Targetedenergy transfer · Torsional oscillations · Propulsionsystem

1 Introduction

Global legislations require the development of propul-sion systems that produce lower levels of hazardousemissions, with particular emphasis on diesel engines.Lower emissions and improved energy economy areachieved through recent technological advancements,such as downsized engines [1]. However, there is anincreased propensity for severe torsional oscillationsdue to reduced system weight. To mitigate these, man-ufacturers are employing various palliative measures.These include tuned vibration absorbers, such as clutchpre-dampers, the dual mass flywheel (DMF) [2] andDMF with centrifugal pendulum vibration absorbers(CPVA) [3,4]. These palliatives are tuned to specific

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frequency range, and consequently, they are not actingin a broadband manner over the powertrain response.Thus, there is a requirement for vibration absorbersthat would respond effectively over a broad frequencyrange without precise prior tuning. Nonlinear energysinks (NESs) can play this role. The NES is a pas-sive absorber, which follows the concept of targetedenergy transfer (TET). The energy of mechanical oscil-lations is shifted in an almost irreversible manner fromthe donor (linear primary system) to a recipient (NES),where it may be absorbed, redistributed or dissipated[5].

Numerous publications in the available literature arededicated to the understanding of TET and its poten-tial applications in systems undergoing translationalmotion. Vakakis et al. [6] andGendelman et al. [7] havestudied TET in engineering systems that were excitedby impulses. 1:1 stable sub-harmonic orbits (i.e. p:1periodic orbits with period equal to p-times the periodof the forcing term) were noted to act effectively ontransferring energy from the primary system donor tothe nonlinear vibration absorber. Vakakis et al. [8] stud-ied the nonlinear normal modes (NNMs) activated inorder to describe how the nonlinear absorbers interactwith energy donors (linear primary systems). Kerschenet al. [9] studied linear systems attached to groundedand ungrounded NES to understand their nonlineardynamics and the absorbers’ effectiveness. Luo et al.[10] designed a passive nonlinear vibration absorberto mitigate vibrations of a large-scale nine-floor struc-ture. The design of NES nonlinear springs was realisedthrough the use of pyramid-shaped elastomers, provid-ing effective control on achieving the desired nonlinearforce profile.

Limited number of studies exists on applicationsthat include rotational NES, which are more suitablefor attenuating torsional oscillations. In this regard,the application of a torsional NES to stabilise a drill-string system was examined in [11]. Gendelman et al.[12] studied the dynamics of a rotational NESmountedwithin a linear oscillator. Hubbard et al. [13,14] stud-ied experimentally the suppression of aeroelastic insta-bilities on a flexible wing using a rotational nonlinearvibration absorber. The NES was housed on the tipof the wing with piano steel wires as a mechanismfor generating the cubic force nonlinearity. Recently,Haris et al. [15–17] showed numerically that nonlin-ear vibration absorbers with significantly low inertia

can be effective in attenuating torsional vibrations ofdrivelines over a broader frequency range.

This paper presents the design and experimentalanalysis of an NES for suppressing torsional oscilla-tions in a propulsion system. To the best knowledge ofthe authors, this is the first numerical and experimen-tal demonstration of oscillation reduction for propul-sion systems using NES hardware. In the next section,numerical analysis of the subsystem model with NESis presented, highlighting vibration attenuation over abroad frequency range. The design and experimentaltesting of the NES is then presented, correlating wellwith the model results for operation away and near res-onance.

2 Subsystem model equipped with NES

In recent publications, the authors demonstratednumer-ically that a single NES with cubic stiffness nonlinear-ity can attenuate the torsional oscillations of a drive-line [15,16]. In comparison with current state-of-the-art palliatives, the NES has the potential of being sig-nificantly lightweight, nevertheless capable of achiev-ing more broadband vibration attenuation. Other typesof stiffness nonlinearity were analysed, whereby itwas conjectured that the lack of impacts in the drive-line leads to ineffectiveness of vibro-impact NES. Theauthors have also demonstrated that the NES func-tions either by absorbing energy from the primary sys-tem while dissipating a fraction of energy locally orthrough redistribution of energy to the higher modeswhere it is dissipated easier due to their higher struc-tural damping content [17]. These criteria were usedin [15] to study the effect of different types of non-linearity on reducing the amplitude of torsional vibra-tions, including cubic, quintic and vibro-impact. Alarge number of numerical simulations led to the con-clusion that an NES with cubic stiffness can efficientlyreduce the amplitude of torsional vibrations of automo-tive drivetrains within a higher frequency range. NESabsorberswith vibro-impacts and fifth-order nonlinear-ity did not demonstrate better performance. For prac-tical reasons related to the realisation of the nonlin-earity, the cubic NES has been prioritised for experi-mental validation in this paper. Owing to the effectivenumerical performance of the NES, a prototype wasdesigned and built for testing in a small-scale propul-sion system rig, similarly to the model layout depicted

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(a) (b)

Fig. 1 a Subsystem rig schematic, b Inertia 2 acceleration amplitudes of the 1.5 Engine Order contribution with NES parametersJNES = 10.7% of Inertia 2, kn = 2.2 × 103 Nm/rad3 and cn = 0.001 Nms/rad

Fig. 2 Angular velocitytime history of Inertia 2 withActive and Locked NES(JNES = 10.7% of Inertia 2,kN = 2.2 × 103 Nm/rad3,cN = 0.001 Nms/rad)

in Fig. 1a. The two main inertias of the propulsionsystem are excited by the cyclic irregularities origi-nating from an engine or motor. Brief description ofthe numerical model (originally presented in refer-ence [16]) comprising the equations of motion, keyparameter values and methods employed to generatethe results presented in Figs. 1, 2 and 3 is provided in“Appendix A”.

Considering that low vibration absorber inertia isdesirable in propulsion systems, the following set ofNES characteristics has been selected: JNES = 10.7%

of Inertia 2, kN = 2.2 × 103 Nm/rad3 and cN = 0.001Nms/rad. A vehicle manoeuvre in first gear at 100%throttle is used to analyse the performance of the NESthrough the entire engine speed range. The 1.5 EngineOrder (EO) acceleration amplitudes observedon Inertia2 with (Active) and without (Locked) the NES actingare depicted in Fig. 1b, clearly indicating significantvibration attenuation in the frequency range 55–70 Hz.The time history of Inertia 2 angular velocity is shownin Fig. 2 for the aforementioned vehicle manoeuvre.The input speed range where the NES suppresses the

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Fig. 3 NES torque–deflection curve

1.5 EO torsional oscillations is ranging approximatelybetween 2200 and 3000 rpm (55–70 Hz). In the inset ofFig. 2 (starting around 1.8 s), the angular velocity fluc-tuations for the system with Active NES are noticeablyreduced when compared to the system with LockedNES. The torque–deflection curve (generated by thenumerical model) resulting in this effective NES per-formance is shown in Fig. 3. During peak performance,the NES would be producing approximately 15 Nmtorque with a corresponding deflection of about 11◦.

3 NES identification (quasi-statically)

Based on the numerical analysis and the identified non-linear stiffness and inertia properties, a concept NESprototype was manufactured. The NES comprises twodiscs representing the inertia of the NES with two con-ical springs (Fig. 4) housed 180◦ apart for targetingboth the positive and negative regions of the torque–deflection characteristic curve (Fig. 3).

In order to ascertain that the designed NES con-forms to the required parameters, quasi-static torque–deflection tests were performed. The NES was loadedand increasing torque was applied. The recordedtorque–deflection curve is shown in Fig. 5a. It can beobserved that the experimental curve follows the theo-retical design characteristics closely. Nevertheless, theexperimental torque–deflection curve shows a linearcomponent at small deflections, which is expected toinfluence the NNMs of the system. Moreover, it is seenthat a quintic function is more accurate than the cubicone. Therefore, the NES torque function selected forthe computations to follow reads TNES = kn θ +knl θ5.The experimental curve has a small amount of hys-teretic damping, which can be computed by evaluatingthe area enclosed by the hysteretic loop. The energyloss in the NES (Fig. 5a) during quasi-static testing

Conical springs Spring

compression arm

NES Disks

(a) Rivets

BearingShaft 1 mounting hub

(b)

Fig. 4 Schematic of the NES prototype a front view and b side view

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Fig. 5 a Comparison of the NES torque–deflection curves (model vs. experiment), b testing machine hysteretic curve

(a) (b)UJ NES hub END NES

Jaw coupling UJ

Fig. 6 a CAD drawing of the experimental rig and b rig photograph showing the NES, jaw coupling and universal joint

was 0.3934 Nm. The energy loss of the testingmachineitself was evaluated to be 0.3933 Nm (Fig. 5b) for thesame test. This signifies that the NES has indeed lowdamping content.

Following the successful identification of the NEStorque–deflection characteristics, it is essential to eval-uate the designed prototype inertia. The NES devicewas loaded on a pendulum machine, where the pendu-lum plate was initially displaced and then released toperform 100 oscillations for which the total time wasnoted down. The process was repeated a few times, andan average inertia value was found to be within 2% ofthe designed value. This difference is not expected tosignificantly affect the NES performance.

4 Experimental set-up

The identification tests validated the designed NESstiffness and inertia to match the performance exhib-ited in the numerical simulations presented in Sect. 2.An experimental rig was built capable of producingspeed harmonics of a rotating assembly driven by anelectric motor. The set-up, shown in Fig. 6, includes anelectric motor with a universal joint (UJ) connected tothe motor’s output shaft. This type of connection intro-duces the second-order harmonics of the input speed,and it is therefore consistentwith engine ormotor-ordervibrations typically observed in propulsion systems.The UJ output shaft is connected to a jaw coupling,which is used to induce an independent degree of free-dom at the end of the set-up. This is required to assess

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the NES effectiveness in reducing the vibrations of acomponent that is not directly driven by the motor. Theend connection of the jaw coupling is attached to asolid steel shaft, which will be hereafter referred toas the coupling shaft. The latter accommodates a hubto mount the NES onto the rotating component. Thecoupling shaft is connected to a weight disc represent-ing the END inertia (see Fig. 6a) via a bellow cou-pling with high torsional stiffness. The whole rotatingassembly was supported by self-aligning pillow blockbearings.

The purpose of the experiments is to assess thetorsional vibration reduction experienced by the cou-pling shaft due to the NES action. This is accom-plished by running speed (frequency) ramps between0-2000 rpm motor speed under two scenarios (namely,Active NES and Locked NES). In the first case of theActive NES scenario, the END inertia disc is removedand the system is excited at a range away from anyresonant frequency. In the second of the same sce-nario, the END inertia is attached and resonance canbe observed in the primary system. In both cases, theNES is attached to the coupling shaft and measure-ments are recorded using non-contact, rotational laserDoppler vibrometers at three of the following locationsdepending on the scenario: (i) the UJ output; (ii) thecoupling shaft; (iii) the NES inertia disc and iv) theEND inertia disc. In the LockedNES scenario, theNESis replaced by a mock solid disc (with a centred mount-ing hole) that matches the NES inertia. In this way,any change in the coupling shaft oscillations is due tothe NES action, rather than to any additional inertiaon the system. During the Locked NES scenario, thevibrations of the UJ output, coupling shaft and ENDinertia disc (in the second resonant case) are recordedand then used as baseline data to assess the NESperformance.

4.1 Universal joint harmonics

The kinematics of the universal joint running at anangle β is governed by the relationship : tan θin =cosβ tan θout [18], with respect to the angles of rota-tion of the input shaft, θin, and the output shaft, θout.Differentiating this equation leads to a formula for theangular velocities:

ωin = ωout cosβ

1 − sin2 β sin2 θout. (1)

A Taylor expansion of Eq. (1) leads to:

ωin = ωout cosβ

∞∑

n=0

sin2n β sin2n θout. (2)

Then, recall the following trigonometric identity [19]:

sin2n x = 1

22n

(2nn

)+ (−1)n

22n−1

n−1∑

k=0

(−1)k

(2nk

)cos 2 (n − k) x, (3)

where

(ab

)is the binomial coefficient. Constraining

the expansion to terms of the second harmonic, i.e.n − k = 1, we get:

ωin = ωout + ωout cosβ

∞∑

n=1

sin2n β(−1)n

22n−1 (−1)n−1

(2n

n − 1

)cos 2θout, (4)

which is a converging series, yielding:

ωin = ωout

[1 − 2 sin2 β

(1 + cosβ)2cos 2θout

]

= ωout [1 − A cos 2θout] , (5)

where A is a constant depending on angle β.

4.2 Numerical model of the experimental layout

A model has been built in MATLAB® environmentto simulate the torsional response of the system (withand without the NES), as shown in Fig. 7. The UJ inputshaft is connected to amotorwith large inertia, which isconsidered as ideal excitation for modelling purposes.Three inertias constitute the independent degrees offreedom: J1, J2 and Jn . The equations of motions arethen expressed as follows:

J1θ̈1 + c1(θ̇1 − θ̇out

) + k1 (θ1 − θout) − c2(θ̇2 − θ̇1

)

− k2 (θ2 − θ1) − cn(θ̇n − θ̇1

) − kn (θn − θ1)

− knl (θn − θ1)5 = 0,

Jn θ̈n+cn(θ̇n−θ̇1

)+kn (θn − θ1)+knl (θn−θ1)5=0,

J2θ̈2 + c2(θ̇2 − θ̇1

) + k2 (θ2 − θ1) = 0, (6)

where J1 is the coupling shaft inertia; J2 is the End iner-tia; Jn is the NES inertia; θi (i = 1, 2, n) are angulardisplacements; k1 and k2 are linear stiffness coefficients

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k1, c1

JUJ, out

JUJ, in kn, knl, cn

in

out 1

J1

Jnn

k2, c2

2

J2

Fig. 7 Sketch of the experimental rig model

Table 1 Model parameters in the two experimental set-ups

Parameter Away from resonance Near resonance

J1 (kgm−2) 2.08 × 10−4 2.08 × 10−4

J2 (kgm−2) – 2.931 × 10−3

Jn (kgm−2) 4.0085 × 10−4 4.0085 × 10−4

c1 (Nm s rad−1) 0.15 0.15

c2 (Nm s rad−1) – 0.2

cn (Nm s rad−1) 0.015 0.015

k1 (Nm rad−1) 540 270

k2 (Nm rad−1) – 8500

kn (Nm rad−1) 10.89 10.89

knl (Nm rad5) 38870 38870

β (◦) 20 20

of the corresponding shaft and coupling combinations;kn and knl are the NES linear and nonlinear stiffnesscoefficients, respectively; ci (i = 1, 2, n) are dampingcoefficients. Table 1 shows the values used for the sys-tem parameters. The driving angle θout is acting as baseexcitation of the torsional system and is taken from Eq.(5). Equation (6) describes the dynamicswhen theENDinertia is attached (case near resonance). In the non-resonance case, the third equation in Eq. (6) is mute.Thus, the stiffness and damping forces (proportional toc2 and k2) entering the first equation are also omitted.

5 Results and discussion

This section presents the experimental results obtainedwith the rig shown in Fig. 6 away and near resonance.

The latter is practically induced by the presence of theEND inertia disc. In what follows, time histories of theNES and the corresponding primary system are pre-sented, the acceleration amplitudes are processed andcompared between the Locked and Active NES cases,whereas the system energy is calculated and plottedalong with the system’s NNMs.

5.1 Away from resonance

Figure 8 exhibits the rotational velocity time historiesduring a forward speed sweep between 0 and 2000 rpm,with an additional backward sweep in Fig. 8e showingthe NES nonlinearity. Equation (5) signifies that thetorsional excitation depends on motor’s speed not onlyregarding the frequency content but for the amplitudeas well. The system is therefore subjected to an excita-tion of increasing amplitude as the sweep progresses.The results correspond to a UJ angle of β = 20o. Com-paring the Active NES results with the time history ofthe Locked NES case, a vibration reduction of the pri-mary system is observed, located around 20–35 s in thetime domain. The magnitude of the reduction is moreclearly shown in the zoom-in extracts of Fig. 8c, d. Itcan also be noted that multiple co-existing solutions forthe NES are realised in this region (see Fig. 8e, f).

The achieved reduction is more apparent when theacceleration amplitudes are compared. In Fig. 9, theacceleration amplitudes are plotted against the instanta-neous excitation frequency, as this was computed fromthe spectrograms of the time histories. The observedreduction is consistent with the remarks made for thetime histories. As the NES vibration grows, energy is

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Fig. 8 Rotational velocity time histories without the End inertia: a Locked NES, b Active NES, c zoom-in plot from a, d zoom-in plotfrom b, e NES in forward and backward sweeps, f zoom-in plot from e

extracted from the primary system which results inthe corresponding vibration amplitude reduction. Assoon as the NES motion “jumps down” (detachingfrom the high amplitude response), the primary system

oscillations in the Active NES case are almost identi-cal to these of the Locked NES cases. This indicatesthat the achieved reduction is due to the NES action,and further investigation using the frequency–energy

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Fig. 9 End shaft acceleration amplitude for the Active NES caseversus the Locked NES case

Fig. 10 Frequency–energy plot of the Active NES case com-puted using Eq. (7), superimposed to the NNMs branch

plots becomes pertinent. The system’s vibratory energy(excluding the rigid body motion) reads:

EV = KV +UV = 1

2J1

(θ̇1 − θ̇RB

)2

+1

2Jn

(θ̇n − θ̇RB

)2 + 1

2k1 (θ1 − θout)

2

+1

2kn (θn − θ1)

2 + 1

6knl (θn − θ1)

6 , (7)

where θ̇RB is the rigid body motion velocity of the sys-tem. Figure 10 shows the frequency–energy plot of thesystem with the branch of NNMs superimposed. Thelatter describes the locus of the conservative solutionsof the system in the frequency–energy space. The dis-tribution of energy in the NNMs has been shown to

influence TET in impulsive and forced systems withlight damping and is therefore essential to design anNES. Figure 11 shows examples of these NNMs forselected cases. Note that this is not the only branchthat exists for this system. A second branch exists ina higher frequency range due to the relatively highstiffness of the primary system; however, this sec-tion explores TET away from resonance, and there-fore, the higher frequency NNM branch is only rele-vant for Sect. 5b, where TET in a resonant region isdiscussed. NNMs were computed using the NNMcon-t®computational package [20]. It is noticed that theNES amplitude grows (see Fig. 8e) when the excita-tion frequency drives the system energy close to over-lapping with the NNMs. The region where the NNMcrosses through the system energy is the range of theobserved vibration reduction. This confirms previousobservations in the development of a graphical methodto design an NES vibration absorber [15,16], where anoverlap of the system energy with the NNMs has beenidentified as key to the design of effective absorbers.Moreover, Fig. 10 also suggests that vibration reduc-tion is most effective when the forced response of thesystem is closely related to the conservative system’sNNMs. When the NES detaches from the high ampli-tude response and thus, its vibration mitigation capa-bility diminishes, the system energy also recedes fromthe NNMs in the frequency–energy plot, indicating adirect correlation of the system’s NNMs with the NEScapability to absorb energy.

The above described correlation is more evidentat the system’s time histories and the correspondingNNMs, shown in Fig. 11. Time histories from threetime snapshots corresponding to qualitatively differentresponses are shown in the left-hand side column, withthe NNMs acting at that time in the right-hand sidecolumn. The top set of figures is taken before the NESreaches the high amplitude response at about 15.5 s.The system energy is not broadly affected by the NNMdue to the frequency mismatch: energy is supplied ata frequency of about 21 Hz, whereas the frequency ofthe NNM is monochromatic and close to 35 Hz. As thesweep progresses, the NES gains energy and reacheshigher oscillating amplitudes. Themiddle set of figurescorresponds to the region of effective vibration reduc-tion at about 34 s. The NES NNM amplitude is morethan an order of magnitude larger than primary sys-tem’s NNM and out of phase. Due to the overlap of thesystem energy with the NNM locus, this is manifested

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Fig. 11 a–c–e NES and End shaft velocity time histories, b–d–f corresponding dominant NNMs

as strong vibration reduction in the forced response.When the system detaches from the NNM branch (bot-tom set of figures), the influence of the conservativedynamics diminishes, leading to the primary systemvibrating at amplitudes similar to the locked case.

Another important metric worth considering is theenergy that the NES damping consumes, given bythe integral over time of the NES damping power,cn

(θ̇n−θ̇1

)2. Figure 12 shows the ratio of the total

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Design and validation of a nonlinear vibration absorber

Fig. 12 Percentage of total energy damped by theNES over totalinput energy

energy damped by the NES over the total input (oscil-latory) energy, as described by Eq. (8), where Ein(t) isthe instantaneous input energy.

Er =∫ tend0 cn

(θ̇n−θ̇1

)2dt

∫ tend0 Ein(t)dt

. (8)

A steep increase in the plotted ratio occurs between 20and 35 s, which is the region identified from previousgraphs as the vibration reduction area. This plot shows aremarkable ability of theNES to absorb energy from theprimary systemwhen it is operatingwithin its effective,overlapping frequency range. Up until 35 s, the NEShas damped more than 70% of the total energy inputbetween 0 and 35 s. After 35 s, the primary systemhas much higher instantaneous damped energy, whichleads to the ratio slightly declining.

A comparison is also carried out between the exper-imentally obtained data versus model predictions toverify the predictive capabilities of the model and thevalidity of the design method. Figure 13 depicts thiscomparison, starting with NES and primary systemtime histories in Fig. 13a, b. The region of strong NESperformance is well captured by the model for boththe NES and primary system, whereas a minor devia-tion in the NES response is observed at higher runningspeeds. Recalling that NES damping is dominated bymetallic interactions in the conical springs, it is antic-ipated that a viscous damping model would not accu-rately describe all forcing conditions. However, this isnot concerning since the NES vibrates at low ampli-tudes away from the region of interest. Moreover, the

acceleration amplitudes predicted by the model matchwell the experimental data (Fig. 13c), whereas the rela-tion of the predicted amplitudes of the Active NES caseto the LockedNES case is similar towhat is experimen-tally obtained (Fig. 13d).

This section has revealed that the NES is capable ofextracting energy from the primary system, even awayfrom the main resonance. The methods employed havebeen verified, and the model’s predictive power hasbeen reaffirmed. The next section will consider the sys-tem response near resonance.

5.2 Near resonance

This section examines the NES capability to mitigatethe primary system resonance. In order to induce reso-nant conditions (within the frequency limitations of theexperimental apparatus), twomodifications aremade inthe experimental set-up. First an inertia disc is attachedat the end of the coupling shaft using a bellow couplingwith torsional stiffness coefficient of 20,000 Nm/rad.Combining this with the additional shaft’s torsionalstiffness yields a value of equivalent stiffness coeffi-cient k2 = 8, 500 Nm/rad. A second identical jaw cou-pling is added in series to the existing one, reducingthe torsional stiffness coefficient to k1 = 270Nm/rad,whereas the END inertia disc further reduces themodalfrequencies of the primary system. The rig layout cor-responds to the set of Eq. (7)

Figure 14 shows the recorded time histories during aforward motor speed sweep from 0 to 2000 rpm. Eventhough the excitation amplitude is increasing during thesweep, there is a clear resonant region in the vibrationsof the END inertia at around 1400 rpm, as shown in theLocked NES case time history of Fig. 14a. The aim isto evaluate the NES effectiveness on mitigating the pri-mary system resonance oscillations by performing anidentical sweep with the NES attached to the mountinghub (instead of the mock disc). Figure 14b, c includesthe time histories of the END inertia and NES, respec-tively. It is readily observed that the NES vibrates withrelatively large amplitudes for awide range of input fre-quencies. Moreover, it appears that the NES engageswith—qualitatively—two different response regimes.Up until 1400 rpm, the response builds up in the sameway as in the non-resonance case. After that, insteadof jumping down, the NES continues to vibrate withalmost constant amplitude for about 200 rpm, before

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Fig. 13 Experiment versus numerical model response comparisons a, b NES and End shaft velocity time histories, c End shaftacceleration amplitude, d ratio of End shaft acceleration amplitude for Active over Locked NES

starting a beating-like motion. Eventually, just before1800 rpm, its response reduces to low amplitude vibra-tions. Cross-checking with Fig 14b, the effect of theNES on the END inertia time history can be concurred.Interestingly, the initialNESactivation leads to increas-ing amplitude of the END inertia vibrations. In orderthough to evaluate the magnitude of this increase, acloser look is taken on the acceleration amplitudes inFig. 15a. In fact, the vibration amplitudes of the ENDinertia are almost identical to the locked case, leadingto the conclusion that the NES is not affecting the pri-mary system to this point (∼ 1350 rpm). As soon asthe NES reaches its peak amplitude (and for the secondregion), a simultaneous significant reduction in the Endinertia vibrations is observed, which lasts until the thirdregion of the beating and can be distinguished even in

the time history of the response. This effect is more evi-dent if the primary system’s kinetic energy is consid-ered. In Fig. 15b, the energy transferred away from theprimary system ismore prominent, reaching a peakmit-igation of 79.76%, calculated based on the maximuminstantaneous difference between the active and lockedcurves:

(KP,Locked − KP,Active

)/KP,Locked|t=t0 , where

KP is the primary kinetic energy and t0 is the timeinstance when the maximum difference occurs.

The energy interactions between the NES and theprimary systemcan be better elucidated via juxtaposingthe system’s response energy with the NNM branches.It is worth mentioning that in this case, two NNMbranches are falling within the speed range of inter-est (Fig. 16a). The system energy in the case nearresonance is computed from the experimental data by

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Design and validation of a nonlinear vibration absorber

motor speed (rpm) motor speed (rpm)

motor speed (rpm)

EN

D in

ertia

spe

ed (r

pm)

EN

D in

ertia

spe

ed (r

pm)

NE

S s

peed

(rpm

)

(a) (b)

(c)

Fig. 14 Velocity time histories near resonance a, b End inertia for the Locked and Active NES cases, c NES

Fig. 15 a Primary system acceleration amplitude and b kinetic energy ratio for the Active and Locked NES cases

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0 200 400 600 800 1000 1200 1400 1600 1800 2000speed (rpm)

0

500

1000

1500

2000

2500

spee

d (r

pm)

0

0.2

0.4

0.6

0.8

1

Rat

io o

f dam

ped

ener

gies

NES

END inertia

NES damping (%)

END inertia (%)

(a) (b)

(c)

Fig. 16 a Frequency–energy plot of the system energy with NNMs, b damped energy ratios of the NES and End inertia with the NNMs,c damped energy ratios for the NES and End inertia, superimposed to their time histories

adding to Eq. (8) the End inertia kinetic and potentialenergies: 0.5 J2

(θ̇2 − θ̇RB

)2 + 0.5 k2 (θ2 − θ1)2. The

damped energy ratios for the NES and the END iner-tia are computed in an analogous way to the previoussection. Figure 16a exhibits the system energy super-imposed to the system NNMs. The first region of NESactivation is dominated by the lower frequency NNMbranch. The system is drawn to follow this branch untilabout 1400 rpm (or 46.67 Hz in excitation frequencyterms), leading to the response observed in the time his-tories of Fig. 14. Even though the vibration amplitudes

are not mitigated in this region, it is interesting to seehow damping is distributed in the system (recall that ina forced system, the energy input depends on systemdynamics, in contrast with impulsively excited systemwhere the energy input is given). Indeed, as Fig. 16bshows, there is a lively interaction between the NESand the primary system (note that the bottom horizon-tal axis of this plot refers to the NNMs, whereas the tophorizontal axis refers to the damping ratios).

At the beginning of the sweep test, the NES followsa nearly rigid bodymotion andmost of the input energyis damped by the END inertia damper, indicating that

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Design and validation of a nonlinear vibration absorber

Fig. 17 a Zoom-in plot of the NES time history modulated response, b steady-state NES response at 1555 rpm

this energy is maintained within the primary system.Moving to higher frequencies and as soon as the sys-tem energy reaches the first NNM branch (at about 25Hz in Fig. 16b), there is a steep change in the dampedenergy distribution. This is a rapid energy shift from theprimary damper to the NES, leading the NES dampingratio to surpass the END inertia ratio, reaching a peakshortly above 30 Hz. After that, and for as long as theresponse is dominated from the first NNM, there is areversal of the damped energy distribution to the previ-ous state of the END inertia damper prevalence. This isconsistent with previous observation of targeted energytransfer via the 1:1 resonance, due to the magnitude ofthe NNMs along the first branch. After the system isdetached from the first NNM, there is a second regionof strong energy transfer from the primary system tothe NES, evidenced by the rapid recurrence of the NESdamped energy ratio located in the region of the sec-ond NNM (corresponding to the second response tierbased on the time histories of Fig. 14). The correlationof the damped energy ratios with the time histories isshown in Fig. 16c, where it is even clearer that the NESabsorbs energy from the primary system while vibrat-ing in the second response region. In fact, this is theregion of the most effective vibration mitigation wherethe primary system has been seen to vibrate with as lowas 79.76% reduced amplitudes.

The relatively high damped energy ratio persists inthe region of beating response as well, even though theNES damping ratio attains significantly lower values.The attachment of an NES to a forced primary sys-tem is known to induce modulated response, either in a

strong modulation regime (SMR) or in a weak modula-tion regime (WMR) [21,22]. Herein, the main aim wasto establish the effectiveness of the NES to mitigatevibrations based on the previously developed designtechniques targeting the 1:1 resonance and the actionof the relevant superharmonics. It is observed that theNES engages with a modulated response, shown in thezoom-in plot of Fig. 17a. This is an important observa-tion that corroborates the activation of nonlinear energytransfer, utilised in this work for achieving mitigationof torsional oscillations. To further validate the mani-festation ofmodulated response, steady-state testswererecorded with themotor running at 1555 rpm. TheNESmodulated response shown in Fig. 17b confirms thenonlinear energy transfers. However, a detailed analy-sis of the slow invariantmanifold thatwould be requiredto predict the forcing conditions and the design param-eters leading to this modulation is out of the scope ofthis paper.

6 Conclusions

Acombinednumerical and experimental studywaspre-sented on the design and validation of an NES for tor-sional vibration mitigation purposes of a propulsionsystem. The set-up comprised an electric motor drivingaprimary (propulsion) systemof shafts and inertia discsthrough a universal joint, which induces the second-order torsional oscillations. The NES was designedfollowing previously developed techniques, for the pur-pose of experimentally validating themethodology fol-

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A. Haris et al.

lowed and the predicted vibration reduction of the pri-mary system. The NES prototype included two coni-cal springs as the elements providing essential stiffnessnonlinearity. The prototype stiffness and dampingwereidentified to feed inputs to a model, used for validationpurposes. Two different forcing scenarios were consid-ered, away from resonance and near resonance bymod-ifying the set-up of the primary system accordingly. Inboth cases, the NES was found to lead to significantvibration reduction with the additional feature of acti-vation over a relatively broad frequency range, particu-larly for mitigating resonant oscillations. This capacityof the NES to suppress vibrations over broad frequencyranges is advantageous, especially when comparedwith typical tunedmass dampers. Post-processing anal-ysis was used to derive frequency–energy plots thatsignified the importance of the NNM branches in theforced response of the nonlinear system. Differentregions were identified where the conservative dynam-ics drive the system to increasing energy transfer to theNES, evidenced not only by the reduction of the pri-mary system’s kinetic energy, but also via the increasein the energy damped by the NES. Noteworthily, rapidenergy exchanges between the NES damper and theprimary damper were observed that could lead to effi-cient vibration mitigation. The nonlinear energy inter-actions were confirmed through the manifestation ofmodulated response—characteristic of the essentiallynonlinear dynamics—both in speed sweeps (transient)and steady-state (constant speed) tests. These resultsestablish an encouraging background for further devel-opment and optimisation of the nonlinear vibrationabsorber for real-world applications.

Acknowledgements The authors wish to express their grati-tude to the Engineering and Physical Sciences Research Coun-cil (EPSRC) for the financial support extended to the “Tar-geted energy transfer in powertrains to reduce vibration-inducedenergy losses” [Grant No. EP/L019426/1] and to the EnterpriseProjects Group (EPG) No. 102, under which this research wascarried out. The authors declare that they have no conflict of inter-est. Research data for this paper are available on request fromStephanos Theodossiades.

Open Access This article is licensed under a Creative Com-mons Attribution 4.0 International License, which permits use,sharing, adaptation, distribution and reproduction in anymediumor format, as long as you give appropriate credit to the originalauthor(s) and the source, provide a link to the Creative Com-mons licence, and indicate if changes were made. The images orother third partymaterial in this article are included in the article’sCreative Commons licence, unless indicated otherwise in a credit

line to thematerial. If material is not included in the article’s Cre-ative Commons licence and your intended use is not permitted bystatutory regulation or exceeds the permitted use, you will needto obtain permission directly from the copyright holder. To viewa copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Appendix A

The drivetrain dynamic model of Fig. 1a equipped withan NES has been presented in detail in reference [16]and is described by the following matrix set of equa-tions of motion:⎡

⎣J1 0 00 J2 00 0 Jn

⎤

⎦

⎡

⎣θ̈1θ̈2θ̈n

⎤

⎦ + [CN]

⎡

⎣θ̇1θ̇2θ̇n

⎤

⎦

+⎡

⎣k2 + k1 −k2 0−k2 k2 00 0 0

⎤

⎦

⎡

⎣θ1θ2θn

⎤

⎦

=⎡

⎣k1θF + c1θ̇F − kN(θ1 − θn)

3

−TReskn(θ1 − θn)

3

⎤

⎦

TRes is the resisting torque on the transmission inputshaft. More details about the derivation of the dampingmatrix [CN] and TRes can be found in reference [16].

The typical range of the drivetrainmodel parametersand the NES data are as follows [16]:

Inertia (kgm2) Stiffness (Nm/rad)

J1 = 0.0005 − 0.005 k1 = 500 − 5000J2 = 0.001 − 0.006 k2 = 5000 − 30000

Jn = 10.7% of transmission input shaft inertia (J2)and Kn = 2, 200 Nm/rad3.

The NES performance is evaluated for the three-cylinder engine case study by comparing the acceler-ation amplitude of the transmission input shaft at 1.5Engine Order (EO) for systems with (Active) and with-out (Locked) NES. Thismetricwas selected as themainperformance indexbecause the 1.5EO is the fundamen-tal firing order of a three-cylinder engine.

The Matlab command Pwelch is used to obtain theacceleration amplitudes corresponding to the 1.5 EOharmonics by calculating the power spectral density(PSD) of the transmission input shaft acceleration sig-nal using Welch’s overlapped segment-averaging esti-mator. The resultant PSD output (units rad2/s4/Hz)

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Design and validation of a nonlinear vibration absorber

is processed as√PSD × 1.5Engine Order Frequency,

where the nominated frequency is the 1.5 EOharmonic.

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