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Article history:Received 18 April 2013Received in revised form26 September 2013Accepted 30 September 2013Handling Editor: Ivana KovacicAvailable online 9 November 2013

a b s t r a c t

and reciprocatingstion engines [4].linear theory forer was proposed

In Fig. 1 the schematic diagram of a bifilar CPVA rotating with angular velocity and consisting of three pendula

Contents lists available at ScienceDirect

journal homepage: www.elsevier.com/locate/jsvi

Journal of Sound and Vibration

Journal of Sound and Vibration 333 (2014) 7117290022-460X/$ - see front matter & 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.jsv.2013.09.042n Corresponding author. Tel.: 49 8928915201.E-mail addresses: johannesmayet@tum.de (J. Mayet), ulbrich@tum.de (H. Ulbrich).URLS: http://www.amm.mw.tum.de (J. Mayet), http://www.amm.mw.tum.de (H. Ulbrich).is shown. Due to bifilar suspensions, the absolute angular velocity of the pendula is equal to the angular velocity of the rotorwhich allows us to describe the pendula as point masses moving along a circular path. Since oscillations of the pendula1. Introduction

Centrifugal pendulum vibration absorbers (CPVAs) are passive vibration reduction devices for rotatingmachines. Typical cases of application include helicopter rotors [1,2], radial aircraft engines [3] and combuThe history of CPVAs began several decades ago with a variety of patented designs [5,6]. The underlyingdimensioning, e.g. presented in [3,711], was used until circa 1980, when a constant frequency bifilar absorb[12]. Since then the bifilar absorber type is believed to be most effective.shows how to analyze this class of centrifugal vibration absorbers using a Hamiltonianformulation. Successive canonical transformations lead to nonlinear equations in action-Since the 1930s, centrifugal pendulum vibration absorbers have been used in rotating andreciprocating machinery for the attenuation of torsional vibrations. A large variety ofabsorber types were suggested and the design was done by linearization theory until theintroduction of the tautochronic bifilar pendulum absorbers. Since then, the performanceand dynamic stability of this specific absorber type have been considered in analytical andnumerical investigations. Different perturbations, e.g. nonlinear mistuning, were consid-ered in order to optimize the system performance, but the characteristic bifilar designremained unchanged. In this paper, a general approach for the design of tautochronicpendulum vibration absorbers is proposed. As a result, it is possible to deal with a largevariety of non-bifilar centrifugal vibration absorber designs which provide application-related optimal performance and resolve some of the existing design limitations.

Established analytic predictions that show a satisfactory agreement with numericalas well as experimental investigations for bifilar absorbers are not applicable for thecomparison of different tautochronic absorbers. Therefore, the second part of this work

angle variables, which are then approximated to first order and analyzed by usingthe method of averaging. These results provide a basis for the design and analysis oftautochronic bifilar and non-bifilar vibration absorbers.

& 2013 Elsevier Ltd. All rights reserved.Tautochronic centrifugal pendulum vibration absorbersGeneral design and analysis

J. Mayet n, H. UlbrichTechnische Universitt Mnchen, Institute of Applied Mechanics, Boltzmannstr. 15, D - 85748 Garching, Germany

J. Mayet, H. Ulbrich / Journal of Sound and Vibration 333 (2014) 711729712Nomenclature

c0, cl, blAR constantsdl, d scaled damping coefficientsD scaled viscous damping matrixE total energy of the mechanical systemf lsl, glsl design functions of the lth absorberF vector of non-conservative forcesh vector of the transformed

non-conservative forces~H scaled Hamiltonian~I rotor momentum deviation

TR kinetic energy of the rotorT tot total kinetic energy of the mechanical systemw0 constant scaled torsional torque (time-based)w^ scaled torsional torque (time-based)w^j jth scaled torsional harmonic torque

amplitude (time-based)

Rx, Ry, Rz Cartesian coordinates w.r.t. coordinatesystem R

is relative rotation of the ith pendulum scaled inertia of the rotorkl small deviation from a l multiple of the

excitation orderabout their vertex induce a periodic torque acting on the rotor, it is possible to counteract external torque excitations [13,14].If the external torque M is completely counteracted, the angular velocity of the rotor remains constant, corresponding witha vibration annihilation. Taking into account that the natural frequency of a CPVA (linearized about the vertex) is ageometrical factor times , a properly tuned absorber is effective for external torques that are harmonics of the nominalrotation frequency [13,15]. Nonlinear kinematic effects, however, are the sources of amplitude depending absorberfrequencies and amplification of higher harmonics [13]. The nonlinear mistuning of the circular paths (hardening) isavoided by more effective absorbers that are guided by roller suspensions (see Fig. 3) on cycloidal [12] or epicycloidalpaths [16,17]. In particular, one specific epicycloidal path, called tautochronic (gr. tauto same and chronos time) path, thatmaintains the absorber frequency regardless of the absorber amplitude [18] for constant rotor rotations seems to be mostdesirable [13,19]. Due to nonlinear effects, even tautochronic absorbers cannot completely counteract a single harmonic inthe external torque. In addition, the unison motion of (nearly) identical absorbers may become unstable resulting inresponses that are detrimental to the system [2022]. Recent research work addresses linear and nonlinear mistuning aswell as relative imperfections among the pendula and provides design guidelines [23,24].

I, vectors of the action-angle variablesIa, a vectors of the averaged action-angle variableskE order of excitationkE;l order of the lth absorber oscillationkI inertia based weighting of the scaled absorber

radius ~K scaled transformed Hamiltonian~L scaled Lagrangianmi mass of the ith pendulumM mass matrixM torsional torqueM0 torsional torque amplituden number of centrifugal absorbersna number of auxiliary bodiesnp number of pendulap vector of the (old) canonical momentaP vector of the (new) canonical momentaq vector of the (old) canonical/generalized

coordinatesQ vector of the (new) canonical coordinatesri maximal radius of the ith pendulum's center

of massriis radius of the ith pendulum's center of massscusp maximum absorber displacementsmax maximal absorber amplitudestASa absorber degree of freedomSa set of the absorber degree of freedomS set of the scaled absorber radiit timeT common averaging periodT constant oscillation periodTA;j kinetic energy of the jth auxiliary bodyTP;i kinetic energy of the ith pendulum

l small damping coefficient of the lth absorberjs relative rotation of the jth auxiliary body scaled torsional torque (angle-based)m;i scaled moment of inertia of the ith pendulumr;i scaled moment of inertia of the ith pendulumE order of resonancelAQ

irreducible fractionsj scaled moment of inertia of the jth auxiliary

body mean angular velocity of the rotorj phase of the jth scaled torsional torque

harmonic (time-based)t rotor degree of freedomis scaled radius of the ith pendulum's center of

masst scaled time~ rotor angle deviationRotor moment of inertia of the rotorA;j moment of inertia of the jth auxiliary bodyS;i moment of inertia of the ith pendulumScale moment of inertia for scalingi relative angle of the ith pendulum degree of irregularityCOM center of massCPVA Centrifugal Pendulum Vibration AbsorberEOM equations of motionideCPVA ideal (f s const:) CPVALCD lowest common denominatorrotCPVA rotating CPVAstaCPVA (non-rotating) standard bifilar CPVASCPA synchronous CPVA differentiation w.r.t. _ differentiation w.r.t.

equations of motion are introduced. In the last section, the steady-state responses of three different tautochronic absorbers areillustrated and the paper is closed with a conclusion and a outline of future work.

J. Mayet, H. Ulbrich / Journal of Sound and Vibration 333 (2014) 711729 7132. Mathematical formulation

A system consisting of one absorber and an idealized rotor is considered for design purposes.1 The absorber consists of at leastone pendulum that moves in plane and is mounted on a rotor with moment of inertia Rotor having only a rotational degree offreedom t. Consequently, the kinematically connected bodies which represent the additional degree of freedom constituteabsorber units. In order to give a comprehensive notation and description of kinematics, the synchronous centrifugal pendulumabsorber (SCPA), presented in Ref. [25] and schematically shown in Fig. 2, is used for the mathematical formulation of themechanical system.

The SCPA is a single absorber system which has one degree of freedom described by the generalized coordinate s(t)The intent of this paper is to give design solutions to avoid dynamic misbehavior and decreased performance but not to solveone of the dynamical issues mentioned above in detail. In contrast to the tautochronic design, derived by Denman in Ref. [19], thecentrifugal pendulum absorbers will not be considered as simple point masses. As a consequence, the absorber dynamics are notcaptured by observing the center of mass moving on arbitrary curves and therefore the already derived geometrical relations inRef. [19] are not applicable.

This paper is organized as follows. In Section 2 the mathematical formulation of the mechanical system, the used notationand scaling of system parameters are presented. In the following, the tautochronic design for a single absorber is carriedout and two practice-oriented design examples are given. In order to obtain averaged equations of motion for steady-stateresponses, in the second part of the paper the equations of motion, canonical transformations and Hamilton's modified

Fig. 1. Bifilar schematic CPVA.considering (bilateral) kinematic constraints. The degrees of freedom of the kinematic pairs, e.g. 1 1s of the revolute joint,are therefore parameterized by the absorber degree of freedom s(t). The absorber system primarily consists of the na1 auxiliarybody, the rollers (massless) and the np4 pendula. Each pendulum (mass mi, moment of inertia S;i w.r.t. its center of mass(COM) and radial distance rii from the rotation center of the rotor) is able to perform a rotational movement is relative to therotor fixed coordinate system Rx; Ry; Rz. These pendulummovements are directly coupled via rollers with the relative rotationalmovement 1s of the auxiliary body (moment of inertia A;1 about the axis through the center of mass). The center of mass ofthe auxiliary body is located on the rotation center and therefore (centrifugal) accelerations are zero contrary to accelerations ofthe pendula. In the general case bodies are defined as auxiliary bodies if their COM is in a state of constant, rectilinear motionw.r.t. an inertial frame of reference or fixed to the rotor for all times. All other bodies are then automatically defined as pendula. Thecontour design of the roller cutouts realizes an arbitrary nonlinear kinematic relation between is and 1s consideringconstructive restrictions and rolling rollers assuming bilateral contacts. The time t is scaled by the mean angular velocity of therotor and as indicated the rollers' dynamic will be neglected for simplicity. The scaled time will be denoted by t and forderivatives with respect to the abbreviation _ d=d is used. The total kinetic energy T tot of the system is given by2

T tot TR np

i 1TP;i

na

j 1TA;j; (1)

1 The analysis of the dynamic behavior will be carried out with an arbitrary number n of centrifugal absorbers.2 Note that in general a finite number naa1 of auxiliary bodies are possible.

J. Mayet, H. Ulbrich / Journal of Sound and Vibration 333 (2014) 711729714where TR 12Rotor2 _ 2 is the kinetic energy of the rotor, TP;i is the kinetic energy of the ith pendulum and TA;j in Eq. (2) is thekinetic energy of the jth auxiliary body:

TA;j 12A;j

js

_s _ 2

2 (2)

The center of mass of each pendulum is given in polar coordinates and the kinetic energy of the ith pendulum including therotation of the pendulum can be expressed as

TP;i 12mir2i

is

2_s22i

is

_s _ 2 !

212S;i

is

_s _ 2

2; (3)

where is with i0 1 is the scaled distance along the Pxi-axis. The angle i is the angle between the rotor fixed coordinatesystem Rx; Ry; Rz and the coordinate system Pxi; Pyi; Pzi pointing with the Pxiaxis to the COM of the ith pendulum. Since arotation of the ith pendulum about its revolute joint implicates a motion of its COM, kinematic constraints are necessary. In thegeneral case all kinematic restrictions, e.g. rolling rollers are formally represented as kinematic constraints.

Due to the special field of application, e.g. a four-stroke internal combustion engine where the engine cycle repeats everytwo revolutions, the disturbing torque is given in terms of the rotor angle . Neglecting gravitational effects, the potential

Fig. 2. Schematic diagram of the SCPA [25]: (a) CAD design; (b) schematic diagram of the SCPA.energy V becomes

V Z 0

M d M0 sin kE; (4)

where the disturbing torque3 M kEM0 cos kE acting on the rotor is assumed to be a single harmonic of orderkEAQ

and forcing amplitude kEM0AR. The scaled Lagrangian ~L T totV=Scale2 can be expressed as

~L 121 _ 21

2np

i 1m;i

is

2_s22i

is

_s _ 2 !

r;iis

_s _ 2 !

12na

j 1j

js

_s _ 2

sin kE (5)

by using the replacements given in Eqs. (6):

Scale maxm1r21;m2r22;;mnpr2np ; M0=Scale2; j A;j=Scalem;i mir2i =Scale; r;i S;i=Scale; Scale=Rotor (6)

The scaled Lagrangian in Eq. (5) is referred to as general scaled Lagrangian since it is suitable for every planar centrifugalabsorber. Additional elements, e.g. rollers, are classified as pendula or auxiliary bodies depending on the acceleration of theCOM w.r.t. an inertial frame of reference.

The scaled radii i of a periodically acting absorber system is restricted to satisfy the properties

i : Sa-SD 0;1; i0 1; (7)

3 The disturbing torque M can be replaced by a series of harmonics without changing the design procedure.

with

Sa fsARjjsjrscuspAR ; 8AR g; (8)

J. Mayet, H. Ulbrich / Journal of Sound and Vibration 333 (2014) 711729 715where scusp is the geometrical boundary of the absorber amplitude. The geometrical boundary scusp is dependent on thespecific design of the absorber and property (7) requires the scaled radii to be bounded. Furthermore, the scaled radii arechosen to be equal to one for zero absorber displacement s0 to unify and simplify the design procedure.

3. Tautochronic design

In this section the conditions for the tautochronic design are derived. Assuming that the system is conservative and that theabsorber exactly counteracts the external torque, the angular velocity of the rotor remains constant and the differential equationfor the rotor...