Stability Analysis of a Gear Pair System Supported by Squeeze-Film

This paper examines the stability of the steady-state periodic response of a gear pair system supported by squeeze-film dampers. The steady-state response of the system is obtained by using the hybrid technique of Harmonic Balance Method and Time Collocation. The Fioquet-Liapunov theory is used to perform the stability analysis of the first variation equations with periodic coefficients, which is generated by the perturbation technique. The stability charts on gear mesh stiffness, spin ratio, disk unbalance, gravity, and squeeze-film damper are used to perform parameter studies. The numerical results show that the unstable region always occurs when the spin ratio is near the second coupled mode of the gear pair system. Furthermore, the mesh stiffness has a significant influence on the coupled critical speeds. Therefore, it plays an important role in determining the spin ratio stability range.


INTRODUCTION
Gear rotor-bearing systems are one of the most common mechanisms for modern power transmis- sion.On account of the ever increasing demand for power and high speed transportation, research in the field of geared rotor dynamics has become very important.The dynamic analysis of a geared rotor- bearing system is an essential step in the development of high performance machinery.Rao et al. (1995) included the effect of pres- sure angle, developed a finite element model for determining the coupled bending-torsion natural frequencies and mode shape of geared rotor.Squeeze-film dampers are now commonly used to suppress the lateral vibrations due to rotor unbal- ance.Mohan and Hahn (1974)considered the design of squeeze-film damper supports for rigid rotors.Cookson and Kossa (1979) considered the effectiveness of squeeze-film dampers for rigid rotors without a centralizing spring.Hwang and  Shiau (1991) developed a Generalized Polynomial   Expansion method to study the nonlinear effects of squeeze-film forces on flexible rotors.They applied the Harmonic Balance Method (HBM) and the Collocation Method to determine the system Corresponding author.Tel." 886-5-2720411-6105.Fax: 886-5-2720589.E-mail: imetls@ccunix.ccu.edu.tw.response.Shiau and Hwang (1993) studied the stability of a nonlinear rotor supported by squeeze-film dampers by using Floquet-Liapunov theory.Chen et al. (1993) considered the applica- tion of squeeze-film dampers to control lateral vibrations of a gear pair system.They reported one result using a Poincar6 section from Direct Integration Method which indicated chaos in the system.Shiau et al. (1999) considered coupled bending-torsion vibrations of a gear pair due to unbalance and gravity.Considering the nonlinear squeeze-film damper forces, the route to chaos was established by using bifurcation maps.In geared rotor systems, it was also shown that torsional excitation can induce lateral vibrations and squeeze-film dampers can suppress large ampli- tudes of whirl due to the torsional excitation.
In this paper, a gear pair system supported by squeeze-film dampers with six degrees of freedom is considered.The hybrid technique, which com- bines the merits of the HBM and Time Collocation (TC), is employed to determine the steady-state periodic response due to unbalance and gravity.With the perturbation technique, the first variation equations with periodic coefficients are generated.
The Floquet-Liapunov theory is used to examine the system stability.The stability charts based on the parameters of mesh stiffness, spin ratio, disk unbalance, gravity effect, and squeeze-film damper force are plotted.

GOVERNING EQUATION
A simple gear pair, Rao et al. (1995), mounted on squeeze-film damper supports is considered.The nonlinear squeeze-film damper, unbalance, and gravity force are the excitations on the system.The model of the squeeze-film damper is based on short bearing theory with the assumption of a fully cavitated oil film.The equations of motion can be expressed as where M33 Ip1, M66 Ip2, Cll C22 C1, C44 C55 C2, Kll K22 K1, K44 K55 K2 (3) The matrices [M] and [C] denote the mass matrix and the damping matrix, respectively.The matrix [K] is stiffness matrix associated with the centering spring.The gear mesh matrix is denoted as [Sh ].The vector {Q} is a linear force vector composed of the excitation of unbalance and gravity.The vector {Fd} denotes the nonlinear squeeze-film damper forces.
The nonlinear squeeze-film damper forces in radial and tangential direction based on short bearing theory, Mohan and Hahn (1974), are given by F.
Substitution of Eq. ( 13) into Eq.( 11) with rearrange- ment yields first variational equations as STABILITY ANALYSIS [1] (14)   A hybrid method, Hwang and Shiau (1991), which combines the merits of both the HBM and TC is employed to obtain the steady-state response of Eq. (7).To investigate the stability of steady-state periodic solution {q}ss, a perturbation technique is used.Let the response {q} be expressed in the form as {g/} {g/}ss + {6g/} (10) where {6g/} represents a perturbation of the original steady-state response.The stability will depend on perturbation response Substituting Eq. (10) into Eq.( 7) and knowing that {q}ss satisfies Eq. ( 7), the equations of motion governing the perturbation response can be expressed as (5) The stability of the nonlinear steady-state response is governed by Eq. ( 14) which is a set of homogeneous ordinary differential equations.Since the coefficient matrices in the system equations are periodic functions of time, the stability of the sys- tem can be examined using Floquet-Liapunov theory, Shiau and Hwang (1993).The transition matrix of the periodic system is established and its eigenvalues are extracted.The stability criterion states that the system is stable only if the maximum absolute eigenvalue of the transition matrix is smaller than one.This can be expressed as A max < 1.0" Stable A max --1.0 Unstable ( 16) {gd({,}, {0'}, -)} {kd({O}, {<}, )} (11) NUMERICAL EXAMPLES With the Taylor series expansion in the first order approximation, the right-hand side of Eq. ( 11) can A six degree of freedom gear pair system supported by centralized squeeze-film dampers, as a sample configuration to investigate the effect of nonlinear dynamic behaviors.The non-dimen- sional parameters of the system are listed in Table I and (he natural frequency diagram is shown in Fig. 2. It shows that the critical speeds due to syn- chronous excitation occur at r equal to 0 (rigid Semi-major axis Semi-minor axis 0 0.5 1.5 2 2.5 3 FIGURE 3 (a) Maximum absolute eigenvalues A for U U2 0.1, Wa, 0.5, SD 0.1, (; 1. (b) Lateral response of driving gear for U U2 0.1, Wg 0.5, SD 0.1, (K 1. body torsional), 0.792 (coupled), 1.0 (three lateral), 2.524 (coupled).
The steady-state periodic response is obtained by using the hybrid method (HBM / TC).The stabil- ity of the periodic responses is obtained using Floquet theory.Figure 3(a) shows the maximum absolute value, IAImax, of the eigenvalues of transi- tion matrix for two different damping ratios, 0.01 and 0.005.The unstable zone shown in Fig. 3(a) occurs in the range of 2r=2.46-2.71for the case, Ch = =0.01,and that of r 2.43-2.74for the case, 'h ==0.005.For light damping, the unstable region is a little wider.The unstable zone for both cases is around the second coupled mode.
The first three harmonic components of both the non-dimensional semi-major and semi-minor axes, leo, 2c, and 3c, associated with a constant offset, 0c, versus spin ratio are plotted to illustrate the steady-state periodic responses of the driving gear for h 0.01 as shown in Fig. 3(b).The higher harmonic components of response are found very small compared with the first harmonic component of response.It is seen that the semi-major axis amplitude of the first harmonic component is larger than the constant offset only when the is near the unstable region which is around the second coupled mode.
The orbital response of the driving gear at f,-= 1.0, using the hybrid method, is shown in Fig. 4(a) and by using the Newmark method with two different initial conditions is also shown in Fig. 4(b) and (c).Two initial conditions (I, II) are considered.Initial condition 1 is zero initial condi- tion.The initial lateral responses and velocities of value 0.6 and the initial torsional responses and velocities of value 6 10 -6 are considered for ini- tial condition II.As shown in Fig. 3(a), the system periodic response at fr= 1.0 is stable, and the steady-state orbits shown in Fig. 4(a)-(c) approach the same limit cycle.When the system is operating at fr=2.6, the steady-state periodic response obtained by using the hybrid method is unstable as was shown in Fig. 3(a).The motion orbit of the driving gear at f= 2.6 is shown in Fig. 5(a)-(d).
The orbit shown in Fig. 5(a) is obtained by using the hybrid method and thus used in the stability analysis.Figure 5(b) and (c) show the transient motion of the driving gear calculated by the numerical integration method with different initial conditions, I and II, for the first 40 consecutive rotations.The orbits obtained with two different initial conditions approach the same steady-state response, which is also shown in Fig. 5(d).How- ever it is quite different from that by using the hybrid method as shown in Fig. 5(a).To classify the orbit of Fig. 5(d), the FFT semi-major axis spectrum corresponding to the time history and the associated Poincar6 map are also shown in Fig. 5(e) and (f), respectively.It is seen that the stable steady-state response of Fig. 5(d), obtained by using Newmark's method, is essentially quasi- periodic, Shiau et al. (1999).
Further analysis of the parameter influence on the stability is studied.It is difficult to determine the unstable zones from the above figures.There- fore, stability charts are made to show the instability zones for different parameter combina- tions.From these charts it is much easier to see the influence of a certain parameter on the stability.
The number of plots can also be reduced since two variables can be shown in the same chart.
The parameters of the gear pair system given in Table I and Wg 0.3 are used.The stability chart in Fig. 6 shows the influence of unbalance and spin ratio with the unbalance values ranging from 0.05 to 0.2, and fr from 0 to 3. Two major unstable zones are found, one is near the lateral mode and the other is around the second coupled mode.The unstable zone for the left one and the right one occurs when the unbalance values U1 and U2 are    larger than 0.137, and 0.087, respectively.It is also seen that the interior of the unstable zone around the second coupled mode has a narrow stable region.
In Fig. 7 the gravity effect has been studied.The solution becomes unstable in the interval from Ctr-2.46-2.71 at Wg-0.5 of Fig. 7.This is the same as that of Fig. 3(a).instability zone only occurs around the second coupled mode and the unstable range of fr gets wider if gravity W e increases.If the gravity is smaller than 0.215, the solution is stable regardless of the value of The stability chart for investigating the influence of squeeze-film damping is shown in Fig. 8.In this chart, only one unstable zone is found around the second coupled mode and it is stable when the value of SI) is larger than 0.155.For smaller values of the unstable range of ,. increases.
Figure 9 shows a stability chart for the variables of mesh stiffness ratio, c/<, and spin ratio, 2r.It is clear that the value of fZr near which an unstable zone occurs is quite different from the previous stability charts and is very sensitive to the value of :.An explanation may be that the range of [2r at which the unstable zone occurs is near the second coupled mode.It can be verified from the critical speeds for different value of c: as given in Table II.
From Table II, e.g.c:=2, the critical speeds are 0.80 (coupled), 1.0 (three lateral), and 3.51 (coupled) and from Fig. 9 it is shown that the unstable region for (/ 2 ranges from f 3.54- 3.58.This is very near the second coupled mode, fr= 3.51.

CONCLUSIONS
A stability analysis is performed by using the hybrid method to obtain the steady-state periodic response and then solving the eigenvalues of the Floquet transition matrix for the perturbed response.When the solutions are stable, the steady-state orbits obtained by using the Newmark method and the hybrid method are in good agreement.From the stability charts, it may be concluded that the unstable region always occurs when the value of the spin ratio is near the second coupled mode.Furthermore, the mesh stiffness has a significant influence on the coupled critical speeds.Therefore, it strongly influences the spin ratio at which the system is stable.