It is stated that the work in the thesis entitled Analysis of Stability and Unbalance Response of flexible Rotor Supported on Hydrodynamic Porous Journal Bearing, by Swarup Kumar Laha, a student of the Department of Mechanical Engineering, Indian Institute of Technology Guwahati, India, for the grant of the degree of Doctor of Philosophy was completed under my supervision and that this work has not been submitted elsewhere for a degree. SK Kakoty) Professor Department of Mechanical Engineering Indian Institute of Technology Guwahati April 2010. I also thank the members of my doctoral committee: Professor R Tiwari, Professor SK Dwivedy and Professor S Talukdar whose helpful suggestions enhanced both the originality and quality of the thesis.
List of Tables
Notation
MR Non-dimensional element rotational mass matrix of the rotor Md Non-dimensional mass matrix of the disk. M Non-dimensional composite mass matrix of the rotor disk n Number of discrete points in Poincaré map.
Abstract
Introduction
- State of the Art
- Literature Review
- Instability in Rotor-Bearing System
- Effect of Unbalance on Rotor-Bearing System
- Non-linear Dynamic Analysis of Rotor-Bearing System
- Porous Oil Bearing
- Application of Finite Element Method in Rotordynamics
- Scope of the Present Work
- Organization of the Thesis
Ozguven and Ozkan [42] applied finite element method to study the unbalance response of a rotor deck system. An example MATLAB program for time-transient analysis of the rotor bearing system is given in Appendix 3.
Mathematical Formulation
- Introduction
- Assumptions
- Finite Element Formulation of the Rotor
- Non-dimensionalization of the System Equation of Motion The following non-dimensional scheme is introduced,
- Hydrodynamic Porous Oil Journal Bearing
- Governing Equations
- Boundary Conditions
- Time Transient Analysis
- Non-dimensional Parameters
- Summary
In the non-dimensionalization method (Appendix 1) of the rotor daer system, the following non-dimensional parameters are obtained:. In the present chapter, the non-dimensional equation of motion of the rotor based on Timoshenko beam theory is presented.
Stability Analysis of Flexible Rotor Supported on Finite Hydrodynamic Porous Journal Bearing using Non-linear
- Introduction
- Mathematical Model
- Stability Analysis
- Results and Discussions
- Effect of Eccentricity Ratio of Bearing
- Effect of bearing feeding parameter
- Effect of Sommerfeld number
- Effect of Slenderness Ratio
- Effect of Stiffness Parameter
- Effect of Clearance Ratio of the Bearing
- Effect of Bearing Aspect Ratio
- Effect of Ratio of Disk Mass to Shaft Mass
- Effect of Location of Bearings
- A Case Study
- Summary
It can be observed that the nature of the trajectory (i.e. stable, critically stable or unstable) remains the same throughout the length of the rotor. It may be mentioned that enlarged views of the trajectories are presented in Fig. Thus, it can be concluded that the greater the clearance ratio, the worse the stability of the rotor is.
Bearing aspect ratio L DB/ is another important non-dimensional parameter which affects rotor stability. Therefore, it can be concluded that the stability of the rotor bearing system is improved with the increase of the bearing size ratio. It has been observed from Table 3.1 that as the bearings are located further in from the outer ends of the shaft, stability improves.
Stability is lowest when the bearings are placed at the extreme ends of the shaft (node numbers 1 and 15). The methodology applied results in plotting the trajectory of the journal center at different locations of the rotor. This issue can be addressed later for better prediction of flexible rotor bearing system as it is not within the scope of the current work.
Introduction
Equation of Motion
The elementary equation of motion of a rotor element in non-dimensional form is given by [116]. The non-dimensional equation of motion of the disk is,. where { }Fd unbis the vector of unbalance forces due to the eccentricity of the disk. The system equation of motion of the rotor is obtained by putting together the element equations and would be as follows:
In the system equation of motion (Eq. 4.3), the right-hand side is the force vector consisting of the hydrodynamic forces at the ends of the bearings and the unbalance forces due to the disc eccentricity.
Solution Procedure
Waterfall charts are spectrum plots taken over a range of speeds of a machine, such as during startup or shutdown. Waterfall diagrams refer to the rotor spin speed, amplitude of vibration and swirl frequency in a 3-D plot. The state variables at the end of one integration cycle are taken as the initial condition for the next mass parameter.
The Sommerfeld number varies with the mass parameter according to the following relationship: if S1 and S2 are two Sommerfeld numbers and m1 and m2 are the corresponding mass parameters, then 122 is 1.
Results and Discussion .1 Validation
- Journal Centre Trajectory and Responses
- Waterfall Diagrams
- Effect of Bearing Feeding Parameter
- Effect of Slenderness Ratio
- Effect of Stiffness parameter
- Effect of Ratio of Disk Mass to Shaft Mass
- Effect of Eccentricity of Disk Mass
- Bifurcation Diagrams
- Effect of Slenderness Ratio
- Effect of Stiffness Parameter
- Effect of Bearing Feeding Parameter
- Effect of Ratio of Disk Mass to Shaft mass
- Effect of Bearing Aspect Ratio
The table shows that at high values of β, oil swirl occurs at lower mass parameters. Thus, it can be concluded that with the increase in slenderness ratio, oil swirl occurs at a higher mass parameter with respect to a higher spin speed. Therefore, it can be concluded that with the increase of slenderness ratio, oil swirl occurs at higher mass parameters.
It can be observed that, for all values of β, as the slenderness ratio increases, oil swirl occurs at higher mass parameter. These results clearly indicate that with an increase in the stiffness parameter, oil swirls occur at higher mass parameters. Table 4.4 shows that for all values of the bearing feed parameters, oil swirl occurs at higher mass parameters as the stiffness parameter increases.
These waterfall diagrams clearly indicate that oil vortices occur at lower mass parameters as the ratio of disk mass to axle mass increases. Thus, it can be concluded that as md increases, oil swirl occurs at lower mass parameter. Table 4.5 shows that as the md increases, oil swirl occurs at a lower mass parameter.
It can also be seen that as β increases, bifurcation occurs at lower mass parameter. It can therefore be inferred that with increase in md, bifurcation occurs at lower mass parameter.
Summary
The effect of varying bearing aspect ratio on system bifurcation is now tabulated in Table 4. The following conclusions can be drawn from the bifurcation analysis of a flexible rotor supported by a fluid film bearing. By increasing the bearing feed parameter, bifurcation of the system occurs at a lower mass parameter.
As the slenderness ratio of the rotor increases, the rotor bearing system undergoes Hopf bifurcation at higher mass parameter. With increase in the ratio of disc mass to shaft mass, the rotor bearing system undergoes bifurcation at lower mass parameter. Effect of Tangential Velocity Slip on Stability and Dynamic Response of a Flexible Rotor Supported on Porous Bearings.
Effect of Tangential Velocity Slip on Stability and Dynamic Response of a Flexible Rotor Supported on Porous Bearing
- Introduction
- Effect of Velocity Slip on the Stability of a Flexible Rotor
- Effect of Slip on Nonlinear Dynamic Response of a Flexible Rotor
- Summary
The effect of tangential velocity slip on rotor stability can be studied in Fig. The nonlinear dynamic response of the rotor bearing system in the unbalance condition is very important. The effect of speed slip on the dynamic response of the rotor bearing system is then analyzed.
The effect of slip coefficients on the dynamics of rotor bearings can be studied by changing its value while keeping other parameters constant as shown in fig. It can thus be deduced that slip has little influence on the dynamic behavior of the rotor bearing system. It can be observed that for all values of slip parameter, α, oil vortex starts at m= 1.4.
Thus it is clear that speed slip does not have any significant effect on the bifurcation characteristics of the rotor bearing system. From the bifurcation diagrams it can be seen that for all values of the slip parameter, α, the rotor bearing system undergoes the Hopf bifurcation at m=1.4. Speed slip has no significant effect on the nonlinear dynamic responses of the rotor bearing system.
Concluding Remarks
- Introduction
- Important Results
- Scope for Future Works
- Summary
A non-dimensional parametric study has been carried out and therefore the effect of the non-dimensional parameters can be studied. Furthermore, the non-dimensional results have been used to study a wide range of combinations of rotor-bearing systems with different lengths, diameters, loads, material, eccentricity, speed, bearing porosity etc. It can be observed that these parameters have profound effect on the stability and dynamic response of the rotor bearing system.
The method has the advantage that it traces the trajectories at any point on the rotor, and thereby the behavior of the rotor can be studied in detail. Another important issue worth mentioning is that the method also allows to study not only the effect of hydrodynamic bearings, but also the effect of the rotor imbalance. The dynamic behavior of the rotor bearing system during speed increase and speed decrease can also be studied.
Also, the rotor spin speed at which oil whirl occurs increases with the increase in rotor slenderness ratio. Therefore, to avoid high amplitude of vibrations, the permeability of the porous bush should be low. The effect of rotor acceleration or deceleration on the dynamic response, especially on the hysteresis phenomena of the rotor bearing system, can be studied in more detail.
C., "Effect of Fluid Inertia on Stability of Oil Oil Bearings", ASME Transaction: Journal of Tribology, 2000, Vol. J., "Instability threshold and stability limit of rotor-carrying systems", ASME Transaction: Journal of Engineering for Gas Turbine and Power, 1996, Vol. C., "On the Stability of a Rigid Rotor in Finite Porous Journal Bearings with Slip", ASME Transaction: Journal of Tribology, 1986, Vol.
K., “Calculations and Experiments Relating to the Unbalance Response of a Flexible Rotor,” ASME Transaction: Journal of Engineering for Industry, 1967, Series B, Vol. T., “Experimental verification of subcritical vortex branching of a rotor supported by a fluid film bearing”, ASME Transaction: Journal of Tribology, 1998, Vol.120, pp. M., “Bifurcation Analysis of a Flexible Rotor Supported by Two Fluids -Film Journal Bearings”, ASME transaction: Journal of Tribology, 2006, Vol.
M., "On the Hysteresis Phenomenon Associated with Instability of Rotor Bearing Systems", ASME Transaction: Journal of Tribology, 2006, Vol. M., "The Dynamics of Rotor-Bearing Systems using Finite Elements", ASME Transaction: Journal of Engineering for Industry, 1976, Vol. D., "Finite Element Simulation of Rotor Bearing Systems with Internal Damping" ASME Transaction: Journal of Engineering for Power, 1977, Vol.
Non-dimensionalization and Elemental Matrices
It has been observed from the above matrices that the following non-dimensional parameters are obtained, stiffness parameter, . ρEgL); clearance ratio, R C; slenderness ratio,. A2), the following non-dimensional parameters are obtained: mass parameter, C 2. Ω and the ratio between disk mass and axle mass,. The nondimensional parameters obtained from Eqn. 2RL and the ratio between disc mass and axle mass,.
The non-dimensional parameters obtained from Eq. 2RL and and the ratio of the mass of the disk to the mass of the shaft,. Ω where, FB is obtained by solving the modified dimensionless Reynolds equation for the pressure and then double integrating the dimensionless pressure over the space interval.
Relation between Sommerfeld Number and Mass Parameter
A Sample MATLAB Program for Time-transient Analysis of Rotor-Bearing System with Velocity Slip
List of Publications
Vitae
