Strojniški vestnik - Journal of Mechanical Engineering 52(2006)4, 225-236 UDK - UDC 681.892:531.23 Izvirni znanstveni članek - Original scientific paper (1.01) Dušenje torzijskih vibracij in raziskava učinkovitosti A Damper of Torsional Vibrations and an Investigation of Its Efficiency Bronislovas Spruogis - Vytautas Turla (Vilnius Gediminas Technical University, Vilnius) V prispevku so prikazani izvirni dušilniki torzijskih vibracij, ki učinkovito delujejo v širokem območju motilnih frekvenc. Raziskali smo tudi nekaj možnih oblikovalskih različic. Osnovni sestavni del dusilnika je rotacijski upogibni obroč. V prispevku raziskujemo gibajoči se sistem na osnovi sistema nelinearnih diferencialnih enačb. Z razdelitvijo gibanja na enakomerno vrtenje in nihanje, razvojem koeficientov enačbe v Taylorjevo potenčno vrsto in izključitvijo ustaljenih delov, dobimo sistem enačb za majhna nihanja. Sistem vsebuje vztrajnostne, upogibne in žiroskopske člene. Izpeljali smo gibalne enačbe in formulirali stabilnostne pogoje za dinamično ravnotežje sistema. Ustrezno pozornost smo namenili tudi drugim možnim nestabilnim oblikam in področjem upogibnega obroča. Analizirali smo lastne frekvence sistema. Oceno učinkovitosti dusilnika glede na različne parametre smo pridobili iz izraza za ekvivalentni vztrajnostni moment in njegovih mejnih vrednosti. © 2006 Strojniški vestnik. Vse pravice pridržane. (Ključne besede: absorberji vibracij, obroči krožni, vibracije, stabilnost, ekvivalent vztrajnostnih momentov) This paper reviews an original torsional vibration damper retaining its efficiency over a wide disturbing frequency band. Some potential design alternatives are considered. The basic structural element of the damper is a rotary flexible ring. The paper investigates the motion system on the basis of a set of nonlinear differential equations. By separating the motion into uniform rotary and oscillatory, expanding the equation coefficients into a Taylor s power series and excluding the static members, a system of equations for insignificant oscillations is derived. The system contains inertia, flexible and gyroscopic terms. The equations of motion are derived and the stability conditions for the system’s dynamic balance are formulated. Proper consideration is given to other possible loss-ofstability forms and regions of the flexible ring. An analysis of the system s natural frequencies is made. The efficiency estimation of the damper versus various parameters is effected as a result of the expression of the equivalent inertia moment and its limit values. © 2006 Journal of Mechanical Engineering. All rights reserved. (Keywords: torsional vibration dampers, rotary flexible rings, vibrations, stability, equivalent inertia moments) 0 INTRODUCTION New devices, mechanisms, assemblies and machines should be very efficient. High efficiency can be ensured by power- and speed-related properties. An increase in transmittable powers and speeds of motion is accompanied by an intensification of vibrations in the systems, and such vibrations frequently exceed their dynamic loads. The level of vibrations becomes one of the key criteria of quality and reliability of machines. Because of this, a limitation of the dynamic overloads of machine assemblies is an urgent problem, directly related to an increase of efficiency, reliability, accuracy and longevity of machines, mechanisms, assemblies and devices. The authors have worked on damping the torsional vibrations of complicated rotating rotor systems for several decades. They explored various methods and measures, for example, the first of them [1] also carried out theoretical and experimental investigations on the development of effective dampers of torsional vibrations. There is a variety of designs of dampers of torsional vibrations that can be naturally inserted into the structure of a relevant unit. Seeking a natural arrangement is one of the causes of the above- 225 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)4, 225-236 mentioned variety of designs; however, successful designs of vibration dampers are rare. An advantage of the frictional dampers of torsional vibrations is their capability to preserve their efficiency in a certain frequency range. However, seeking for essential efficiency in such a case leads to a non-proportional increase of sizes, weight, etc., the more so that frictional damping of the vibrations is bound with the elimination of heating energy, wear and the use of special materials. Many works – both theoretical and experimental – have been devoted to investigations of vibration processes in mechanical rotor systems ([2] to [7]). The well-known pendulous vibration damper has an excellent feature of self-tuning for one harmonic of the torsional vibrations on any change of the speed of rotation of the system. However, it almost does not affect the adjacent harmonics and other torsional vibrations (for example, ones of random character). In addition, a pendulous vibration damper is completely discussed in transitional modes of motion, in particular during the starting period. The authors set the task to develop such a dynamic damper of torsional vibrations that would be tunable for a wide range of disturbing harmonics (frequencies), remaining a natural element of the rotating system. It became clear that the set task may be solved to a certain extent by the use of a vibration damper, based on the rotating elastic ring situated on two pendulous rings in the shape of elastic frames. The torsional vibrations of the system generate transversal bending vibrations of the elastic ring. The damping system includes an elastic ring, special masses that can be additionally fixed on it and elastic frames. 1 THE SCHEMES OF CONSTRUCTIONS OF VIBRATION DAMPERS The design scheme of the simplest damper of torsional vibrations in the shape of a rotating elastic ring [8] is presented in Fig. 1,a. The key element of such a vibration damper is the ring 2 connected to the principal system 1 with two Fig. 1. Schemes of dampers of torsional vibrations based on an elastic ring 226 Spruogis B. - Turla V. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)4, 225-236 opposite frames 3. The ring can be equipped with supplemental masses 5 (Fig. 1,b,c), perpendicular to masses 4. 1.1 The principle of operation of a vibration damper on the base of an elastic ring In the case of the absence of rotation or ideally uniform rotation, the axial line of the ring is an ideal circle in the limits of stability. Torsional vibrations of the principal system cause bending of the elastic frames 3 and periodic compression of the ring 2 in the transversal direction. The ring 2, because of its elasticity and centrifugal mass, can efficiently damp the torsional vibrations of the shaft 1 upon certain parameters across a wide range of harmonics. The technical realization of the vibration damper according to this scheme is presented in Fig. 1, a-1, a-2. In Fig. 1, a-2, the elastic frames are connected to the ring with swivel clamps. If two supplemental masses of a particular size are fixed to the ring, symmetrically with the axis of rotation in the plane perpendicular to the plane of the elastic frames (see Fig., 1,b), the ring is extended into an ellipse-shaped body on the rotation. In many cases, such an extended ring exhibits improved vibration-damping properties. The bent centrifugal pendulums 3 can be stabilized by a swivel parallelogram (Fig. 1, b-1) or replaced with symmetric elastic frames 3, tilted by a certain angle with respect to the radius (Fig. 1, b-2). In many cases the efficiency of the vibration damper can be increased by fixing the elastic ring on two symmetrically tilted pendulums (see Fig. 1, c). The tilted pendulums can be elastic frames (Fig. 1, c- 1) or elastically fixed tilted pendulums (Fig. 1, c-2). The vibration damper presented in Fig. 1, c-2 includes one more elastic rings of small diameter 3 in the middle, clamped at two opposite points. The ring is an elastic swivel. 1.2 Investigation of the operation of the vibration damper Let us start the investigation of the operation of the vibration dampers (see Fig. 1) with a calculation of the potential energy of the deformed ring and the kinetic energy of the system. The potential energy of a half-ring as an elastic body, deformed by the impact of concentrated forces (Fig. 2), can be found from the following expression according to [8]: P 1/ 2 = a B P 2 + b B P Q + a B Q 2 (1), where aB and bB are coefficients, and P and Q are fictitious forces. Let us find the coefficients aB and bB. For this purpose, the shifts of the ring in the direction of the forces P and Q should be found. In accordance with [8]: 1/2 = 2aP + bQ = 2aP = -\-------------- dP BBB U 8 J 11 = bP + 2aQ = bP = -\-------------- 8Q BBB U 4) 1Q=0 2Q=0 Thus: (2). 1(n 2\R3 1f 2 1R3 a B = 4{4 -*)EI , b B = - 2b-~2)EI (3). Fig. 2. For determining the potential energy of a deformed elastic ring Dušenje torzijskih vibracij - A Damper of Torsional Vibrations 227 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)4, 225-236 Let us consider that the shift of point A consists of two components, i.e.: u = u D - k u (4), where uD is the shift of point A caused by the force P , u 0 is the shift of the point B caused by the force Q: bB 2(4-p ) correspondingly: 2a p2 u2 = u0 - kuD (5), (6). u2: On the basis of Equations (4) and (6), we find u 2 = u 0 1 - k 2 - k u 1 () (7). Let us find the potential energy of the deformed ring as a function of the shifts u1 and u2. From the above expressions (2) and (3), the forces P and Q are expressed as follows: u +ku 2aB u0 2a (8) (9). After the insertion of the expressions of the forces from (8) and (9) into the expression (1), we find the full potential energy of the elastic ring from the forces P and Q as follows: 1 2 4aB2 -bB2 2 1 2 1 2 P=2P1/2 = u1 + 3 u0 = Cn1u1 + Cn2u0 2aB 8aB 2 2 (10), where: 1 4aB2 -bB2 C n 1 = an d C n 2 = 3 (11). aB 4aB The total potential energy of the system (PS) consists of the potential energy of the elastic ring (Pk), the potential energy of other deformable elements of the vibration damper (Pi) and the potential energy of the torsion of the shaft (Pv), i.e.: P= P + P + P (12), where: Pk=f j1-j2,u0 , Pi=f j1-j2 = C j1-j22 ( ) ()1() 2 Pv=f j1 (13). () In order to calculate the kinetic energy, let us consider the system to be a system with three degrees of freedom. Fig. 3 presents the estimated scheme of the vibration damper, where j1 and j2 are independent of the angular coordinates, R is the initial radius of the elastic ring, m is the reduced mass of the part of the ring with the relevant concentrated mass, m0 is the reduced mass of the part of the ring with the supplemental mass, r is the distance of the mass m from the axis of rotation, r0 is the distance of the mass m0 from the axis of rotation, r and r0 in the general case are a function of the rotational deformation of the vibration damper. If the radial shift of the mass m(u1) is a function of (j1 - j2), the radial shift of the mass m0 also includes the independent component u0 that is the third generalized coordinate. The kinetic energy of the system, taking into account the existence of two couples of masses, is Fig. 3. The estimation scheme of the vibration damper 228 Spruogis B. - Turla V. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)4, 225-236 found from the following expression: T = mv +m0v1 +-I1j 1 (14), where: 2 2-2 v =rj2 + dr d( j1-j2 (j 1-j2 2 2j2 r2 I1 is moment of inertia of the rotated mass; the super point means differentiation with respect to time. In order to simplify the estimated dependences, we consider that the radius of the string clamping (r) equals a half of the radius of the elastic ring (R). Then, the following is concluded from Fig. 3: r = Rcos(j1 -j2) r 0 = R + 2 k r[1 - c o s ( j 1 - j 2 )] + u 0 ( 1 - k 2 ) (15) r 0 = 2kr sin (j1 - j2) (j 1 - j 2) + ui0 (1 - k2) Finally, we find the following expression for the kinetic energy: T = {-I+m 2 1 -2 m dr _d(j1-j2)_ I2 dr dr0 d(j1-j2) dr d(j1-j2) dr0 d( j1-j2) + r + m d(j1-j2) dr0 d( j1-j2) j 1j2 + (16). + r02 \j2 After relevant transformations, we find the kinetic energy as a homogenous quadratic form of the generalized speeds [8]: 1 2 1 2 (17), where the values of the relevant coefficients are as follows: A11 =I1+ 8mr2 sin2 (j1 -j2) + 8m0k2r2 sin2 (j1-j2) A12 =-\ 8mr2sin2(j1 -j2) + 8m0k2r2sin2(j1 -j2) A22 =8mr2 + 2m { R + 2kr1-cos ( j1 -j2 )]+u0 ( 1-k2 )} + +8m0k2r2 sin2 (j1-j2) (18). A33 = 2m0(1-k2)2 A13 = 4m0kr sin (j1 - j2) (1 - k2 ) A23 = -4m0kr sin (j1 - j2) (1 - k2 ) For the formation of differential equations of motion we will use Lagrange equations of the second order [9]: d dT dT aP (19), + M (i = 1,2) dt 5j i 5ji 5ji ddT -dT =-5P where M is the generalized moment of potential forces. i After the differentiation, we find the equations of motion of the system, neglecting the frictional forces: 1j 1 + A12j2 + A13u0+-A11 ( j 1-j2) - 2 5(j1-j2 dw 5 (j1-j2 j&2 5A 5A23 5(j1-j2) 5(j1-j2) u 9P 9Pv +--------- +— = M1 5(j1-j2) 5j1 2 5(j1-j2) + 9w o(j1-j2) 5u 5A 2 5A j&1u&0 5P 5(j1-j2) 5(j1-j2)J10 5(j1-j2 5A A13j 1+A23j 2 + A33u0 + 5 (j1-j 13 2 j&1 5A 5A23 . . + j 1j 2 5(j1-j2) 5(j1-j2) 5A3 1 5A . 2 5Pk 2 +----22 j 2 +— = 0 5(j1 -j2) 2 du0 du0 here w1 = ~A11 + A12 +-A22. After the disintegration of the coefficients of the equations (20) into a Taylor‘s series, we find according to [9]: u0 = u0c + v (u0c is the statistic component, v is a small varying value), where each equation is divided into two parts, corresponding to stationary and vibrating motion. 1.3 Investigation of the stability of the dynamic balance Taking into account a certain identity of two first equations of the system (20), its is described by Dušenje torzijskih vibracij - A Damper of Torsional Vibrations 229 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)4, 225-236 two equations only: P'' -w''w2=0 1 2 Px 2 x xw 2 a22 = 0 (21). Let us check the stability of the positiveness of the matrix determinant: P-ww2 Px - w'xw2 Px thus: ( P"-ww2 )[Pxx-- where: aw (P x > 0 (22), w'xw2)2)0 1 - - 1 - a11 + a12 + a22 1 1 2 a11+a12+-a2 d(j1-j2) d( j1-j2)2 d xx = d ' = d The system will be stable if D > 0 and P - w w 2<0 (23). With an experimental investigation of the various schemes of dynamic vibration dampers on the basis of an elastic ring, some other forms of loss of stability were obtained: 1) because of an excessive increase in the ring‘s radius for its extension, 2) because of a symmetrical deflection of the ring from the axis of rotation, 3) because of non-symmetrical sideways deflections, etc. On the basis of the experimental data, an analytic investigation on the stability of dynamic balances of the elastic ring was carried out at the preset static deformation, and some peculiarities were cleared up. The criterion of the stability of the dynamic balance shall be considered as the existence of the maximum of kinetic potential in the preset position (point). In such a case, if the kinetic energy itself is equal to the maximum, the stability shall be considered natural and the position of the balance will not depend on the mode of the speed. If for the point under investigation the kinetic energy is equal to the minimum or at least does not depend on the disturbance under discussion, the forced stability will only be possible at this point, i.e., we will consider that rigid forced stabilization is possible due to elastic elements. In other cases, we will ensure forced stability is possible on a certain shift of the point of dynamic balance. The size of such a shift depends, among other factors, on the mode of the speed. However, such a shift usually is bound with the appearance of a certain instability that is not allowed in a vibration damping system. Let us discuss various cases: 1. Stability of an ideally symmetric concentric ring in the case of its uniform rotation Let us suppose that the concentricity of the ring is ensured in any case and no static bending exists. The kinetic energy of the ring is: T = p R g n w 2\R 2p (24), where R is the initial radius of bending of the elastic ring, gn is the mass of the unit of length of the ring, w is the average rotational frequency of the system, Dl is the absolute elongation of the ring. The potential energy of the ring is: P= -—( Dl ) 2 ( 2 5 ), where F is the area of the cross-section of the elastic ring, E is the module of longitudinal elasticity. Let us consider that a stable extension of the ring corresponds to the maximum kinetic potential, i.e., the following condition should be satisfied: a(T-P) a(Dl) a2(T-P) a ( Dl ) 2 (0 (26). In such a case, the extension of the ring will be as follows: D l 2p R3gnw2 EF-gnR2w2 and the limit value of the angular speed will be: EF (27) w 2{ (28). The condition (28) identifies the limit of the zone of a stable extension of the ring. 2. Symmetrical longitudinal extension of the ring As our investigations showed, the maximum efficiency of the vibration damping is achieved when z > 1 (in Fig. 3, z = m/m) and the ring operates as extended. If we consider that the ring is deformed 230 Spruogis B. - Turla V. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)4, 225-236 according to the scheme provided in Fig. 3, the element and a more precise investigation provides that the angle a = 0, if z < 1, and a > 0, if z > 1, i.e., the deformation, identified with the torsion angle a, when z > 1, has a stable fixed value, dependent on the rigidity of the elastic ring and other elements. It is notable that the bending rigidity of the ring itself does not affect the limits of the zones of stability (z = 1), it only defines (together with other parameters) the value of the static deformation. 3. Symmetrical sideways deflection of the ring The element investigation of stability of the positions of dynamic balance upon the condition of the extremity of the kinetic potential allowed us to make the following conclusions: a) zero deformation of the ring is possible only on z < 1/3 (if the ring is fixed on swivels of pendulum frames); b) if z = 1/3, a certain statistical deformation of the ring is set, depending on the mode of the speeds and the rigidity of the rings; c) if z > 1/3, the ring does not achieve a natural balance, so a symmetric deflection may transform into a non-symmetrical one. Some other possible disturbances of the ring were discussed, and the conditions of balance were explored as well. 2 INVESTIGATION OF THE EFFICIENCY OF OPERATION OF THE VIBRATION DAMPER Let us divide the motion along the cyclic coordinates % , 1=a1+wt + ji1 - a'22co j32+ \Pk -a'22co v = 0 The linearized equations of motion (31) include inertial, gyroscopic and quasi-elastic members. Thus, the dynamic link of the vibration damper is a rather complicated link between rotating objects, and in the general case it cannot be reduced to the usual (linear or non-linear) elasticity. 2.1 Solution of the system of equations The natural frequencies of the system with the vibration damper can be found on the basis of the characteristic determinant: a11A2 -w''w2 +P'' a12 /t2 - 2w'wA + w''w2 -P'' a1A2 -a'x w2 2 aA2 +2w'wA + w''w2 -P'' a~A2 - w''w2 +P'' aA2 +ax22wA+a'x oj2 12 22 23 2 22 ak -a'xG) i7A - ax22wA+a'x ft) aA +Pxx-----a'^o 13 2 22 23 2 22 33 2 22w (32), where X = iwpc, pc is one of the natural frequencies. This determinant (32) provides three natural frequencies. For a determination of the natural frequencies of a vibration damping system only, the below reduced equation (33) can be used; this equation is the condition of frequency tuning for the vibration damper as well: (a a22\ Pxx-----a22xxw2 -a33 ( w''w 2 - P'' ) + ( a22 'x-a22 'x a23 ) - (w''w2-P'') Pxx-----axx22w2 \ + -a22'x2w X2- (33). For an assessment of the impact of the vibration damper on the vibration of the system, we find an expression of the equivalent moment of inertia Ie, the value of fictitious mass, rigidly connected with the principal system. The vibration damping effect of the mass with respect to the principal system is equivalent to the relevant effect of the supplemental vibration damping unit. Dušenje torzijskih vibracij - A Damper of Torsional Vibrations 231 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)4, 225-236 The periodic component of the reactive torsional moment is: M =-(a11-I1)b 1 -a12b2 -a13v + 2w'wb2 +w(a13 '+a23 ')v- -(w''w2 -P'') ( b1 -b2 ) +-a22 'x w2 v and the equivalent moment of inertia is: (34) (35). i=1,2 After insertion of the values: bi = Ai sin pt + Bi cos pt v = A3 sin pt + B3 cos pt as well as their fluxions b i, b?, v, v, and ratios A2/A1, B2/B1 and so on into (35), we find the following expression for the equivalent moment of inertia: [a11 -I1 ) (a22a23 -a232)-2a12a232 - a22a232 - a122a33 jp4 + (a22a33 -a232 )p4 - a22\ Pxx -a22xxw2 -a33 (w''w2 -P')-a22 'x a23w2 + + { ( 2w-I1 )[a33(w''aw2-P'')-a13a22'xw2]-[( a11-I1 ) a22-a122]} +a22x2w2 ]p2-(Pxx-2a22xxw2J(w''w2-P'')+4 a 22 ' x 2 w 4 (36) + Pxx-a22xxw2 \ + 4w'w2 (a13a22x -w'a23)-(a11-I1)a22x2w2p2 + 2w'w2 2w' Pxx-a22xxw2 \ + a22'x a22xw2 \-(w''w2 -P '')a222 w2 - \-(2w-I1){Pxx--a22xxw2(w''w2-P'')-(2w-I1)-a22'x2iw4 The limit values of the equivalent moment of inertia for low- frequency disturbances will be found from the following expression: 2w'w2\ 2w'\ Pxx-a22xxw2 + a22 'xa22xxw2 \- ( w''w2-P'' ) a22xx2w2 limIe= Pxx - a^sco2 ( w''w 2 - P'' ) - a 'x2 w4 2 22 j 4 22 (37) - ( 2w-I1 )(( Pxx--a22xxw2 ) ( w''w2 )( w''w2-P'' ) - ( 2w-I1 )-a22'xaw4 ) and for high-frequency disturbances: limIe V11 I J\fl22 a 33 a 23 / 2a 12 a 23 ^22 a 23 a 12 a 33 22 33 23 (38). In our case, the coefficients will be expressed as follows: a11=I1 + 2mR2 sin2 a (1 + zk2 ) a12 = -2mR2 sin2 a (1 + zk2 ) a22 = 2mR2(1 + z) + 4mR2zk(1-cosa)(1 + k) + + 2mu0z(1-k2){u0(1-k2) + 2R 1 + k ( 1-cosa )] a13 = 2mzkR(1 -k2)sina (39), a23 = -2mzkR (1-k2)sina a33 = 2mz ( 1-k2 ) where R = 2r, m = mz, a1 - a = a, and z is the ratio of the centrifugal masses m0 and m. Other coefficients will be expressed as follows: a11 ' = 2mR2 sin 2a (1 + zk2 ) a22' = 4mzkR(1 + k)sina\_R + u0(1-k)\ a12' = -2mR2 sin 2a (1 + zk2 ) w'' = 2mzkR ( 1 + k ) cos a\_R + u0 ( 1-k )\- 2mR2 cos 2a(1 + zk2) a11 '' = 4mR2 cos 2a (1 + zk2) a22'' = 4mzkR (1 + k) cos a [R + u0 (1 - k)] a12'' = -4mR2 cos 2a (1 + zk2) a22 'x = 4mzkR(1 -k2) sin a a22xx = 4mz ( 1-k2 ) a22 = 4mz ( 1-k2 ) {kR(1-cosa) + lR + u0 ( 1-k2 P'' = Cn1R2(cosa-cos2a) 8p2(p3 -20p +32) EI (40). Pxx = C C n1 16p EI p2-8R3 Fig.4. Dependence of the critical frequency of rotation (w) of the damper on the thickness of the elastic ring (tk), when b = 20 mm, z = 2,0, m = 0.03 kg, where the curve 1 corresponds to R = 80 mm and the curve 2 to R = 100 mm (the zones of stability are shaded) 232 Spruogis B. - Turla V. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)4, 225-236 From the equations of dynamic balance we find the average angle of torsion a-a2 of the vibration damper and uc is the static component of the independent variable u0: Cn1Cn2 - 2Cn2mzk ( 1 + k )co2- 2Cn1mz ( 1 - k2 )2 co2 Cn1Cn2-2mco2\Cn2 ( zk2+1 ) +z ( 1-k2 )2( Cn1-2mco2 ) (41 2mzRco2 [_Cn1 - 2mco2 (1 + k)] (1 - k)2 a-a= arccos u0 Cn1Cn2-2mco2\Cn2 ( zk2 +1 ) + z ( 1-k2 )2 ( Cn1-2ma)2 )\ (42). The complete expression of the equivalent moment of inertia (36) of the vibration damper under discussion is not presented here because of its length. The equivalent moment of inertia in the high-frequency zone of disturbances is expressed as follows: p->» C2 limIe = u0 ( 1-k2 )2 \2Ru0z(1-k02)\_k ( 1-cosa ) + 1 C1=2mR2sin2a\ \cos2 a + z(l-k2 sin2 a ) + \+R 2\ v I \_+2zk(1-cosa)(1-k) R2(1 + z) + 2R2zk(1-cosa)(1 + k) + ]+zu0 (1-k2)\u0 (1-k2)+2R ( 1 + k )( 1-cosa )~\-zk2R2sin2 a (43). C As specific quantitative calculations showed, it is sufficient to describe the deformed ring in many constructions of vibration dampers with only two generalized coordinates, i.e., it can be considered that a transversal compression of the ring is proportional (in some cases equal) to its longitudinal extension. In such a case, the expression for kinetic energy is reduced to the well-known quadratic trinomial [11], and its coefficients are as follows: A11=I1 +8mr2sin2(^1 -(p2) + 8m0k2r2sin2((p1 -1. Based on the methods described in [12], we find linearized equations for the small torsional vibrations that may be used for an assessment of the stability of dynamic balance and the efficiency of the vibration Fig.5. Dependence of the natural frequencies of vibration damper (pc) on its structural parameters and the number of revolutions, when the radius of the elastic ring R = 80 mm, m = 0.03 kg, n = 1500 rpm, z = 1.25, tk = 1.1 mm, b = 20 mm (the solid lines are obtained by using the equations (33) and the dotted ones by using the simplified calculation (48)), where the curves 1, 1, 1 - pc = f(n), 2, 2’, 2’ - pc = f(z), 3, 3’, 3’ - pc = f(tk), the index ' corresponds to the first frequency and the index “ - to the second frequency (in the formula (33)) Dušenje torzijskih vibracij - A Damper of Torsional Vibrations 233 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)4, 225-236 damper. To a certain extent, such cases are described in [7] to [9] as well. In this case, the stable positions of the dynamic balance of the ring are described by one of the following conditions: a) if z < 1, a1 - a2 = 0, f 1 C-4zmw b) i z >, a1 - a2 = arccos n1 (46). Cn1-2(1 + z)m2 For the case b), the minimum cross-section of the ring (its axial moment of inertia) will be found from the following inequality: p 2 - 8 ( 3 z + 1 ) m w 2 I~16p E 3 RESULTS (47). The possible combinations of the parameters are illustrated in Fig.4. The frequency of resonance tuning of the vibration damper is: ^w (cosa - cos2a - zk ( 1 + k ) cosa-(1 + zk2 )cos2a ) p =w\ \2mw 1 + z + 2zk (1 - cos a) (1 + k) (48). Fig. 5 illustrates some curves of natural frequencies. The expression for the equivalent moment of inertia will be as follows: ( A-Bsin 2 a )\ Bsin 2 a — + C cosa - B cos2a - DE -4sin 2 a (C-B cos a) A\^\ +Ccosa-Bcos2a-DE where: (49), A = 1 + z + 2zk(1 + k)(1 -cos a) B = 1 + zk2 C = zk(1 + k) 2mw E= cos a - cos 2a The dependence of equivalent moment of inertia structural and performance parameters is presented in Fig.6. The limit value Ie in the low-frequency zone: 8m sin2 a (C-B cos a) w2 limIe =2mR2 A-Bsin a 2m (C cos a -B cos 2a) w2 -Cn1E (50), in the high frequency zone: lim I =2BmR2 sin2 a Bsin a (51). Fig. 6. Equivalent moment of inertia of the damper, where 1 - I = f(m), 2 -1 = f(R), 3 -1 = f(z), 4 -1 = f(t), 5 - Ie = f(w), and 6 - Ie = f(EI) e e e 234 Spruogis B. - Turla V. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)4, 225-236 Fig.7. Comparison of the efficiency of the damper for various estimated models (the dotted lines correspond to the approximate calculation)p, if R = 80 mpm, m = 0.0p3 kg, z = 2.0, tk = 1.2 mm, b = 20 mm, where 1 - = 0,1, 2- =1,0, 3- =10. w ww The comparison of the values of the of centrifugal pendulums with symmetric elastic equivalent moment of inertia, found from the equation (36) and the simplified calculation (49), is provided in Fig. 7. 4 CONCLUSIONS The general results of the investigation are provided in the presented schemes of vibration dampers: - the schemes a, a-1, a-2 (Fig. 1) are distinguished by natural stability, if z < 1/3. In such a case, the frames 3 may be designed in the shape of strings that only resist extension, - if z > 1/3, a rigid forced stabilization can be achieved because of the elasticity of the frames 3, - the vibration dampers, showed in the schemes b and c, can be stabilized by the introduction of a relevant elastic element, resistant to the deflection of the frames 3, - the remaining schemes are distinguished by rigid stability and preserve their strict symmetry on the relevant rigidity of the elastic elements. The side stabilization of the extended elastic ring can be ensured by a connection of the deflected centrifugal pendulums 3 with the swivel parallelogram (see Fig. 1, b-1) or the replacement frames 3, situated at a certain angle with respect to the radius (see Fig. 1, b-2), - in many cases the efficiency of a vibration damper may be increased by fixing the elastic ring on two symmetrically deflected centrifugal pendulums (Fig. 1, c). In such a case, the vibration damper is not a stable system; it is inclined to a non-symmetric sideways “deflection”. The stabilization of the ring can be ensured if the deflected pendulums are elastic frames (see Fig. 1, c-1) or elastically fixed pendulums (see Fig. 1, c-2). - the analytical investigation of the efficiency of the vibration damper, applying special sets of programmes, allows us to state that a vibration damper can be sufficiently precisely presented as a vibrating system with two degrees of freedom for most practically important ranges, - in many cases the simplified calculation ensures sufficient accuracy, - with an increase in the radius of the elastic ring, the critical frequencies of rotation of the vibration damper decrease, - the value of the equivalent moment of inertia is mostly affected by the thickness of the elastic ring, - the efficiency of a vibration damper increases with an increase in its natural frequencies. Dušenje torzijskih vibracij - A Damper of Torsional Vibrations 235 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)4, 225-236 5 REFERENCES [I] Kavolelis, A.-P., Spruogis, B. (1973) Damper of torsional vibrations. The USSR Authorship Certificate No. 375423, The Bulletin of Inventions, 1973, No. 16 (in Russian). [2] G. Adiletta, A.R. Guido (2000) Dynamical behaviour of a torsional system with parametric and external excitations. Journal of Mechanical Engineering Science. Vol. 214, 2000, p.955-973. [3] Muszynska A. (1995) Vibrational diagnostics of rotating machinery malfunctions. International Journal of Rotating Mashinery, 1995, Vol.1, No.3-4, p.237-266. [4] J.Shaw (2001) Active vibration isolation by adaptive and control. Journal of Engineering Mechanics, 2001, No.7, p.19-31. [5] S.Xue, J.Tobita, S.Kurita, M.Izumi (1997) Mechanics and dynamics of intelligents passive vibration control system. Journal of Engineering Mechanics, April 1997, p.322-330. [6] R.N. 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Self-synchronization, Simulation). Mintis, Vilnius (in Russian). Authors’ Addresses: Prof.Dr. Bronislovas Spruogis Vilnius Gediminas Technical University Faculty of Transport Engineering Lithuania bs@tti.vtu.lt Doc.Dr. Vytautas Turla Vilnius Gediminas Technical University Mechanical Faculty Lithuania vytautas.turla@me.vtu.lt Prejeto: Sprejeto: Odprto za diskusijo: 1 leto 12.7.2005 16.11.2005 Received: Accepted: Open for discussion: 1 year 236 Spruogis B. - Turla V.