International Journal of Mechanical Sciences 44 (2002) 475–488
Piecewise approximate analytical solutions fora Je!cott rotor with a snubber ring
Evgueni V. Karpenkoa, Marian Wiercigrocha ; ∗, Ekaterina E. Pavlovskaiaa,Mathew P. Cartmellb
aDepartment of Engineering, University of Aberdeen, Aberdeen AB24 3UE, UKbDepartment of Mechanical Engineering, University of Glasgow, Glasgow G12 8QQ, UK
Received 19 March 2001; received in revised form 10 November 2001
Abstract
In the paper two approximate analytical methods for calculating nonlinear dynamic responses of an idealisedmodel of a rotor system are devised in order to obtain robust analytical solutions, and consequently speedup the computations maintaining high computational accuracy. The physical model, which is similar to aJe!cott rotor, assumes a situation where gyroscopic forces can be neglected and concentrates on the dynamicresponses caused by interactions between a whirling rotor and a massless snubber ring, which has much highersti!ness than the rotor. The system is modelled by two second-order di!erential equations, which are linearfor non-contact and strongly nonlinear for contact scenarios. The 6rst and the simpler method has been namedone point approximation (1PA) and uses only one point in the 6rst-order Taylor expansion of the nonlinearterm. It is suitable for soft impacts and gives a reasonable prediction of responses ranging from period one toperiod four motion. The second and more accurate method of multiple point approximation (MPA) expandsthe nonlinear term many times when the rotor and the snubber ring are in contact and it can even be used forcalculating chaotic responses. The methods are evaluated by a comparison with direct numerical integrationshowing an excellent level of accuracy. ? 2002 Elsevier Science Ltd. All rights reserved.
Keywords: Rotordynamics; Je!cott rotor; Periodic motion; Discontinuously nonlinear system; Clearance
1. Introduction
Mechanical systems where moving components intermittently contact each other are very commonin mechanical engineering practice, and in most cases can generate dangerous vibration. For example,rub–impact interactions in rotating machinery may cause serious malfunctions in their operation.Problems of this nature are present in a wide variety of commercial and defence machinery such
∗ Corresponding author.E-mail address: [email protected] (M. Wiercigroch).
0020-7403/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved.PII: S 0020-7403(01)00108-4
476 E.V. Karpenko et al. / International Journal of Mechanical Sciences 44 (2002) 475–488
Nomenclature
c viscous damping coeAcient of the rotork1 rotor sti!nessk2 snubber ring sti!nessK sti!ness ratio, k2=k1m� out-of-balanceM mass of the rotorR radial displacement of the rotor relative to the equilibrium position of the snubber ringt timevx0 the velocity of the rotor at the point of contact in x-directionvy0 the velocity of the rotor at the point of contact in y-directionx displacement of the rotor in the horizontal directionx0 horizontal co-ordinate of the point of contactx displacement ratio of the rotor in the x-direction, x=�y displacement of the rotor in the vertical directiony0 vertical co-ordinate of the point of contacty displacement ratio of the rotor in the y-direction, y=�z radial displacement ratio, R=�
Greek symbols
�x eccentricity of rotor in the x-direction�y eccentricity of rotor in the y-direction�x dimensionless eccentricity in the x-direction, �x=��y dimensionless eccentricity in the y-direction, �y=�� frequency ratio, �=!1
�m mass ratio, m=M� radial clearance between the rotor and the snubber ring� damping ratio of the rotor, c=2
√k1M
!1 natural frequency of the rotor,√
k1=M� shaft rotational velocity’0 initial phase shift the angle between the radial displacement of the rotor and the horizontal direction� dimensionless radius, �=�� dimensionless time, !1t�0 the moment in time when there is contact between the rotor and the snubber ring
as gas turbines, pumps, centrifuges, compressors, generators, etc. The main cause of these unwantede!ects is nonlinear dynamic interactions between rotor and stator.
The rotor–stator dynamic interactions have attracted a large interest from researchers sincethe famous paper by Je!cott [1], proposing a simple but robust model of the rotor bearing
E.V. Karpenko et al. / International Journal of Mechanical Sciences 44 (2002) 475–488 477
assembly, where it is assumed that the rotor is free from gyroscopic forces. This work has beenfollowed by a number of excellent papers on the rotor–casing (rotor–stator) rub interactions,where both friction and impact e!ects were considered, see for example Refs. [2–10]. It hasbeen pointed out by many investigators (e.g. Ref. [11]), that the dynamic responses can bechaotic, i.e. extremely sensitive for small changes in the initial conditions and the system para-meters. To gain a deeper insight into the main mechanism of these complex dynamic responses,a separation of rubbing from impacts and then studying these nonlinear e!ects alone seem tobe a reasonable solution. The 6rst steps in this direction were made by Gonsalves et al. [12],where the chaotic vibrations for a Je!cott rotor purely due to impacts were determinednumerically and experimentally. Further numerical investigations on the same model byKarpenko et al. [13] have con6rmed these 6ndings and have showed an existence of multiple at-tractors and fractal basins of attractions. Computation of dynamic responses for this system by adirect numerical integration proved to be very laborious. This is due to the necessity of determiningtimes of contact with high accuracy, which is normally done by decreasing the integration timestep.
Classical texts on nonlinear oscillations, such as Minorsky [14] and Andronov et al. [15], containanalyses of simple systems (mostly one-degree-of-freedom) with continuous nonlinearities, althoughmostly it is the free vibrations that are fully analysed. More recent studies such as those by Thompsonand Gha!ari [16] and Shaw and Holmes [17] on forced piecewise oscillators have provided solidfoundations for a better understanding of the so-called dynamic systems with motion dependentdiscontinuities, and this has stimulated further theoretical and experimental research con6rming theusefulness of piecewise linear models (e.g. Refs. [18–23]).
It has been discussed above that the rotor–stator dynamic interactions can be studied using asimple Je!cott rotor model. As the intermittent impacts between the rotor and the snubber ring arethe main contributors to the system dynamics, an eAcient method to calculate dynamic responses isrequired. Hence the main objective of this paper is to develop robust (i.e. precise and much fasterthan a direct numerical integration) approximate analytical solutions, which can be also useful forthe stability analysis.
2. Description of the system
A two-degree-of-freedom model of a Je!cott rotor system with a snubber ring, as shown inFig. 1, is considered in this study. A rotor of mass M , subjected to out-of-balance �m, rotatesinside a massless elastic snubber ring. During operation the rotor makes intermittent contacts withthe snubber ring, resulting in complex dynamic behaviour.
To derive the equations of motion the co-ordinate system shown in Fig. 2 has been chosen. Thecentres of the snubber ring, Os0, and the rotor, Or0 in the static equilibria are separated by a vec-tor � = (�x; �y). Hence, if the rotor is placed concentrically within the snubber ring, �x = �y = 0.Once the rotor hits the snubber ring, only the normal force FN is generated as it is assumed thatthere is no friction between the rotor and the snubber ring. This force depends on the displace-ment of the snubber ring relative to its static equilibrium position, which is determined from theprinciple of minimum energy stored in the springs supporting the ring. The horizontal and vertical
478 E.V. Karpenko et al. / International Journal of Mechanical Sciences 44 (2002) 475–488
m
M
k1
k2
c
k1
c
k2
γRotor
SnubberRing
y
x
ρ
Fig. 1. Physical model of a rotor system with a bearing clearance.
Fig. 2. Geometrical representation of a rotor system with a bearing clearance.
components of the snubber ring displacement can be calculated as
xs = k2(R− �) cos ;
ys = k2(R− �) sin ;
where R=√
(x − �x)2 + (y − �y)2 is the displacement of the rotor relative to the equilibrium positionof the snubber ring, � is the gap, (R− �) is the radial displacement of the snubber ring relative toits equilibrium position, and cos =(x− �x)=R, sin =(y− �y)=R. If R¡�, the rotor does not touchthe ring, otherwise they are in contact. The equations of motion of the modelled system are linear
E.V. Karpenko et al. / International Journal of Mechanical Sciences 44 (2002) 475–488 479
for non-contact, and nonlinear for the contact situation, and take the following form:
M Mx + cx + k1x +
{k2(R− �) cos ; R¿ �
0; R¡�
}= m�O2 cos(’0 + Ot); (1)
M My + cy + k1y +
{k2(R− �) sin ; R¿ �
0; R¡�
}= m�O2 sin(’0 + Ot); (2)
where k1 and c denote the sti!ness and the viscous damping of the rotor, k2 is the snubber ringsti!ness, and ’0 is the initial phase shift.
Eqs. (1) and (2) were derived after having made the following series of assumptions. Firstly,the dry friction between the ring and rotor has been neglected and it has been assumed that thesnubber ring itself is massless. Secondly, the equations of motion do not take into account anydamping e!ect from the snubber ring. In order to develop approximate solutions to Eqs. (1) and(2), a typical non-dimensionalisation procedure was carried out, so the equations of motion are asfollows:
x′′ + 2�x′ + x +
{K(R− �) cos ; R¿ �
0; R¡�
}= �m��2 cos(’0 + ��); (3)
y′′ + 2�y′ + y +
{K(R− �) sin ; R¿ �
0; R¡�
}= �m��2 sin(’0 + ��); (4)
where
!1 =
√k1M
; �= !1t; ′ = d=d�; �=c
2!1M; �=
O!1
; �m =mM
; K =k2k1
:
The displacements are also non-dimensionalised by using the radial clearance, �, as a referencedisplacement. The dimensionless variables and parameters, which result, are as follows:
x =x�; y =
y�; �=
��; �x =
�x�; �y =
�y�:
In terms of the dimensionless quantities quoted above the radial displacement of the rotor, R, canbe written as
R= �√
(x − �x)2 + (y − �y)2 = �z:
Thus, using dimensionless variables, the system of equations (1) and (2) can be written in thefollowing 6nal form. For a freely rotating rotor (z ¡ 1) one can get
x′′ + 2�x′ + x = �m��2 cos(’0 + ��);
y′′ + 2�y′ + y = �m��2 sin(’0 + ��):(5)
480 E.V. Karpenko et al. / International Journal of Mechanical Sciences 44 (2002) 475–488
For the rotor in contact with the snubber ring (z¿ 1), the set of equations is modi6ed to:
x′′ + 2�x′ + x + K(x − �x)(1− 1=z) = �m��2 cos(’0 + ��);
y′′ + 2�y′ + y + K(y − �y)(1− 1=z) = �m��2 sin(’0 + ��):(6)
3. Approximate analytical solutions
The solution to Eqs. (5) is found for x(�0) = x0 and x′(�0) = vx0 , and for �¡ 1, it is
x(�; �0; x0; vx0) = exp(−��)(C1 cos("�) + C2 sin("�)) + C1 cos(’0 + ��) + C2 sin(’0 + ��);(7)
where C1 = �m�2�(1− �2)=(4�2�2 + (1− �2)2), C2 = 2��m�3�=(4�2�2 + (1− �2)2),
C2 = exp(��0)((vx0 + �(C1d2 − C2d1))g1 + R1(x0 − C1d1 − C2d2))=(R1g2 − R2g1);
C1 = (exp(��0)(x0 − C1d1 − C2d2)− C2g2)=g1
and where "=√1− �2, R1=�g1+"g2, R2=�g2−"g1, g1=cos("�0), g2=sin("�0), d1=cos(’0+��0)
and d2 = sin(’0 + ��0).Similarly, the solution of Eqs. (5) for y(�), based at y(�0) = y0, y′(�0) = vy0 is
y(�; �0; y0; vy0) = exp(−��)(C3 cos("�) + C4 sin("�)) + C4 cos(’0 + ��) + C5 sin(’0 + ��);(8)
where C4 =−2��m�3�=(4�2�2 + (1− �2)2), C5 = �m�2�(1− �2)=(4�2�2 + (1− �2)2),
C4 = exp(��0)((vy0 + �(C4d2 − C5d1))g1 + R1(y0 − C4d1 − C5d2))=(R1g2 − R2g1);
C3 = (exp(��0)(y0 − C4d1 − C5d2)− C4g2)=g1:
If z¿ 1 (for which the rotor is in contact with the snubber ring) the equations describing the dy-namics of the rotor become nonlinear. The nonlinear functions fx(x; y) = (x − �x)(1 − 1=z) andfy(x; y) = (y − �y)(1 − 1=z) describing the restoring forces in the snubber ring in the x andy directions, respectively, were expanded at the point of contact �0 between the rotor and the snubberring. For a 6rst approximation only the 6rst-order terms were retained. A Taylor expansion of thesefunctions in the vicinity of the point (x0; y0) is given next.
fx| x=x0y=y0
= (x − �x)(1− 1=z)| x=x0y=y0
= (x0 − �x)(1− 1=√%)
+ (1− 1=√%+ (x0 − �x)2=%3=2)(x − x0)
+ (x0 − �x)(y0 − �y)(y − y0)=%3=2; (9)
fy| x=x0y=y0
= (y − �y)(1− 1=z)| x=x0y=y0
= (y0 − �y)(1− 1=√%)
+ (1− 1=√%+ (y0 − �y)2=%3=2)(y − y0)
+ (y0 − �y)(x0 − �x)(x − x0)=%3=2; (10)
E.V. Karpenko et al. / International Journal of Mechanical Sciences 44 (2002) 475–488 481
where % = (x0 − �x)2 + (y0 − �y)2. After substitution of the approximate equalities of Eqs. (9) and(10) into Eq. (6) one obtains
x′′ + 2�x′ + A1x + By + D1 = �m��2 cos(’0 + ��);
y′′ + 2�y′ + A2y + Bx + D2 = �m��2 sin(’0 + ��);(11)
where
A1 = (%3=2(K + 1) + ((x0 − �x)2 − %)K)=%3=2; A2 = (%3=2(K + 1) + ((y0 − �y)2 − %)K)=%3=2;
B= K(x0 − �x)(y0 − �y)=%3=2;
D1 = K(−�x%(√%− 1)− (x0 − �x)(x0(x0 − �x) + y0(y0 − �y)))=%3=2;
D2 = K(−�y%(√%− 1)− (y0 − �y)(x0(x0 − �x) + y0(y0 − �y)))=%3=2:
In the context of this paper it is impractical to write full analytical solutions for Eq. (11) due to theawkward form of the equations which result. However, for the purpose of clarity, the main stepsare discussed here. The di!erential equations (11) can be transformed to one fourth-order di!erentialequation:
x′′′′ + 4�x′′′ + (A1 + A2 + 4�2)x′′ + 2�(A1 + A2)x′ + (A1A2 − B2)x = BD2
+ �m�2�(A2 − �2) cos(’0 + ��)− �m�2�(B+ 2��) sin(’0 + ��)− A2D1: (12)
The solution to Eq. (12) is obtained in the following form:
x(�) = xg(�) + xp(�) = exp(−��)(C1 cos(�1�) + C2 sin(�1�) + C3 cos(�2�) + C4 sin(�2�))
+ C1 cos(’0 + ��) + C2 sin(’0 + ��) + C3; (13)
where
�1 = 12
√2(A1 + A2)− 2
√(A1 − A2)2 + 4B2 − 4�2;
�2 = 12
√2(A1 + A2) + 2
√(A1 − A2)2 + 4B2 − 4�2;
C1 =%6 − C2(%2�− 4��3)
�4 − %1�2 + %3; C2 =
−%5(�4 − %1�2 + %3)− %6(4��3 − %2�)(4��3 − %2�)2 + (�4 − %1�2 + %3)2
; C3 =%4%3
and where
%1 = A1 + A2 + 4�2; %2 = 2�(A1 + A2); %3 = A1A2 − B2;
%4 = BD2 − A2D1; %5 = �m�2�(B+ 2��); %6 = �m�2�(A2 − �2):
The y displacement is directly expressed from the second equation of Eq. (11) and is de6ned as
y(�) =1B(�m��2 cos(’0 + ��)− x′′ − 2�x′ − A1x − D1): (14)
482 E.V. Karpenko et al. / International Journal of Mechanical Sciences 44 (2002) 475–488
The constants within the general solution, C1, C2, C3 and C4, are found using the following set ofinitial conditions:
x(�0) = x0 x′(�0) = vx0 y(�0) = y0 y′(�0) = vy0: (15)
A diAculty which arises when attempting to join solutions (7)–(8) and (13)–(14) together,in order to obtain the global solution, is that the crossing times �i0 (i.e. when x(�i0) = xi0 andy(�i0) = yi0) are not known explicitly. These times are given by the roots of the contact equation:√
(x(�; �i0; vxi0)− �x)2 + (y(�; �i0; vyi0)− �y)2 = 1; (17)
where expressions (7) and (8) are used to calculate functions x and y. An eAcient determination oftimes of contact in terms of speed and accuracy is a complex task, therefore a full account will begiven in a separate publication. In short, it is based on an estimation of times of contact dependingon the expected dynamic response or the shape of the spatial orbit. Once it is estimated, i.e. thetrajectory has crossed the ellipse described by Eq. (17), the time of contact is precisely calculatedusing a bisection method.
4. One point and multiple points approximation methods
E!ectiveness of the devised approximate methods can only be assessed for a complex behaviourof the system. The range of parameters for which the complex behaviour occurs was determined byconstructing bifurcation diagrams. A bifurcation diagram shown in Fig. 3 was plottedin the form of the x displacement of the rotor, as a function of the frequency ratio �, together with
Fig. 3. Bifurcation diagram for �= 0:125, K = 30, �x = 1, �y = 0, �m = 0:001, �= 70.
E.V. Karpenko et al. / International Journal of Mechanical Sciences 44 (2002) 475–488 483
0.00 0.06 0.12 0.18 0.24
-0.10
-0.05
0.00
0.05
0.10
(c) (d)
(b)(a)^
^x displacement
DNI 1PA 2PA
0.0 0.1 0.2 0.3-0.10
-0.05
0.00
0.05
0.10
^ y di
spla
cem
ent
^x displacement
DNI 1PA 2PA
0.0 0.1 0.2 0.3 0.4
-0.10
-0.05
0.00
0.05
0.10
^ y di
spla
cem
ent
^x displacement
DNI 1PA 2PA
-0.1 0.0 0.1 0.2 0.3 0.4 0.5
-0.10
-0.05
0.00
0.05
0.10
^ y di
spla
cem
ent
^x displacement
DNI 1PA 2PA
y d
ispl
acem
ent
Fig. 4. Spatial orbits of periodic motion obtained using 1PA, 2PA and DNI methods for: (a) � = 2:2; (b) � = 2:6;(c) �=2:44; and (d) �=2:72 keeping all other parameters 6xed at K =30, �=0:125, �x =1, �y =0, �m =0:001, �=70.
the following set of parameters: K = 30, � = 0:125, �x = 1, �y = 0, �m = 0:001, � = 70. Becauseperiod one motion was predominant below �= 2, the frequency ratio range was chosen from �= 2to 3.25. First, period one motion jumps into period three as the forcing frequency is increased. Thisperiod three motion then coexists with the period one motion over a narrow frequency range beforereturning to period one motion. After that a period of doubling bifurcation occurs and the periodone motion transforms to period two. In the vicinity of �= 2:8 another period-doubling bifurcationis observed and period four motion is generated. At � = 2:92 this form of motion disappears by ameans of Qip bifurcation into period two motion.
To examine the dynamic responses of the two-dimensional rotor system two approximate methodshave been devised. The 6rst of these two methods is henceforth de6ned as the one point approx-imation method (1PA method). As mentioned above the local solution is known explicitly for the
484 E.V. Karpenko et al. / International Journal of Mechanical Sciences 44 (2002) 475–488
0.0 0.2 0.4 0.6
-0.24
-0.12
0.00
0.12
0.24
^ y di
spla
cem
ent
^x displacement
DNI MPA 5PA
0.0 0.2 0.4 0.6
-0.24
-0.12
0.00
0.12
0.24
(a) (b)^ y
dis
plac
emen
t
^x displacement
1PA 2PA
Fig. 5. (a) False spatial orbits obtained by 1PA and 2PA; and (b) real ones calculated by 5PA, MPA and DNI methodsfor �= 0:125, K = 28, �x = 1, �y = 0, �m = 0:0017, �= 70, �= 2:6.
rotor prior contact. At the initial moment of contact the nonlinear functions (9) and (10), from Eqs.(6), were expanded in a Taylor series and only the 6rst order terms were retained. Thus, at thepoint of contact, the two solutions are joined together, and when the rotor moves in contact withthe snubber ring Eqs. (6) are solved using the initial conditions, which are the 6nal values of thesystem displacement and velocities for the previous non-contact stage. The second method, namedas the multiple point approximation (MPA), requires that when the rotor moves into contact withthe snubber ring the nonlinear function is expanded many times during that contact (ultimately itcan be expanded at each step of the calculation).
In Figs. 4 and 5, periodic orbits computed by 1PA, two points approximation (2PA), MPA meth-ods, and a direct numerical integration (DNI) are plotted for di!erent cross-sections of Fig. 3. InFig. 4, period one motion of the system was calculated for the value of frequency ratio �=2:2 (Fig.4a), period two for �=2:6 (Fig. 4b), period three for �=2:44 (Fig. 4c), and period four for �=2:72(Fig. 4d), and K = 30, �= 0:125, �x = 1, �y = 0, �m = 0:001, �= 70. Fig. 5 is obtained for K = 28,�= 0:125, �x = 1, �y = 0, �m = 0:0017, �= 70 and �= 2:6.As can be seen from Fig. 4a–c larger impacts produce more complex periodic orbits. Also it
is clear that both the 1PA and the 2PA methods are predicting the shape of the orbits correctly.However, an error between the exact solution and the 1PA method is larger than for the 2PA method.With the increasing complexity of the responses (see orbits in Fig. 4c), there is a signi6cant erroreven for the 2PA method. For more complex responses this may lead to a completely incorrectshape of the computed orbit (Fig. 5a) when using 1PA and 2PA methods. Fig. 5b unveils the realcomplexity of the orbit, which is a tangled period 14 motion, and confronts the 5PA and the MPAresults with the exact one. The orbit predicted by the MPA is virtually the same as the exact orbit,whilst the 5PA method gives a fairly good result (the shape and number of loops are correct).
As has been demonstrated, expanding the nonlinearity at one point (i.e. the initial moment ofcontact) can produce correct qualitative responses (see Fig. 4a–c), but the spatial trajectories di!erfrom the exact solution. Such di!erences can be explained by referring to the approximation methods,
E.V. Karpenko et al. / International Journal of Mechanical Sciences 44 (2002) 475–488 485
Fig. 6. 3D surfaces and cross-sections at the contact point of the nonlinear functions representing the restored forces:(a) fx function and the tangent plane; (b) a cross-section of fx with the tangential at y0 = 2:855× 10−2; (c) fy functionand the tangent plane; and (d) a cross-section of fy with the tangential at x0 =−2:925× 10−3.
which have been used. Three-dimensional plots (Figs. 6a and 6c) of the functions (9) and (10) areconstructed together with the tangent plane as built in at the point of contact (x0; y0) to illustratetheir geometry. The actual point of contact x0 =−2:925× 10−3 and y0 =2:855× 10−2 is considered.As can be seen from Figs. 6a and 6c, the nonlinearities at the point of contact are very strong, andfor a better understanding of what happens at the point of contact in Figs. 6b and 6d two-dimensionalcross-sections of Figs. 6a and 6c are given. From Fig. 6b it can be clearly seen that the tangent ischanging signi6cantly at the vicinity of the contact point (x0; y0). Thus, the 1PA method will producea large error in approximating the tangentials to the slope in an operating range (say y∈ (−0:2; 0:3))by a single tangential evaluated once at the contact point. This error may accumulate, leadingeventually to completely erroneous orbits (e.g. Fig. 5a). As has been demonstrated for period 14motion, the MPA method can handle complex responses. If the expansion when the rotor is incontact with the snubber ring is undertaken densely enough, the MPA method can even accuratelypredict the chaotic responses, as shown in Fig. 7, where the spatial orbit (Fig. 7a) is virtually thesame as the exact one (Fig. 7b). Also the PoincarRe sections (Fig. 7c and d) show almost the sametopology of the strange attractor.
486 E.V. Karpenko et al. / International Journal of Mechanical Sciences 44 (2002) 475–488
0.0 0.2 0.4 0.6
-0.12
0.00
0.12
0.24
(d)(c)
(b)(a)
^ y di
spla
cem
ent
^x displacement
DNI
0.0 0.2 0.4 0.6
-0.12
0.00
0.12
0.24
^ y di
spla
cem
ent
^x displacement
MPA
-0.05 0.00 0.05 0.10-0.12
-0.08
-0.04
0.00
0.04
0.08
^ y di
spla
cem
ent
^x displacement
DNI
-0.05 0.00 0.05 0.10-0.12
-0.08
-0.04
0.00
0.04
0.08^ y
disp
lace
men
t
^x displacement
MPA
Fig. 7. Chaotic spatial orbits obtained using: (a) MPA and (b) DNI methods; PoincarRe sections of the spatial orbitsobtained using (c) MPA and (d) DNI methods for K = 30, �= 0:124, �x = 1, �y = 0, �m = 0:0017, �= 70.
5. Conclusions
Two approximate analytical methods (1PA and MPA) for calculating nonlinear dynamic responsesof an idealised model of a rotor system are devised and tested. The analytical solutions obtainedspeed up the computations signi6cantly while maintaining high computational accuracy. The modelconsidered, which is similar to the Je!cott rotor, neglects gyroscopic forces and concentrates onthe dynamic responses caused by interactions between a whirling rotor and a massless snubber ringwith a much higher sti!ness than the rotor. The dynamic system is modelled by two second orderdi!erential equations, which are linear for non-contact and strongly nonlinear for contact scenarios.The 6rst method has been named the one point approximation (1PA) and expands only once thenonlinear term. It is suitable for soft impacts and gives a reasonable prediction of responses ranging
E.V. Karpenko et al. / International Journal of Mechanical Sciences 44 (2002) 475–488 487
from period one to period four motion. The limitation of this method is shown for a complex orbit,where 1PA and 2PA method predictions were very inaccurate. The second and more accurate methodof multiple point approximation (MPA) expands the nonlinear term many times when the rotor andthe snubber ring are in contact. The robustness of the methods is evaluated by a comparison withdirect numerical integration showing excellent accuracy when used appropriately. The MPA methodcan be even used to determine chaotic motion.
Acknowledgements
The authors would like to thank the anonymous referees for constructive comments on the manu-script. The 6nancial support provided by EPSRC and Rolls-Royce plc is gratefully acknowledged.
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