Modal Frequency Response of a Four-Pad Tilting Pad Bearing With Spherical Pivots, Finite Pivot Stiffness, and Different Pad Preloads

2013 ◽  
Vol 135 (4) ◽  
Author(s):  
Timothy Dimond ◽  
Amir A. Younan ◽  
Paul Allaire ◽  
John Nicholas
Keyword(s):  
Frequency Response ◽  
High Speed ◽  
Degrees Of Freedom ◽  
Natural Frequencies ◽  
Theoretical Calculations ◽  
Leading Edge ◽  
Damping Coefficients ◽  
Tilting Pad ◽  
The Subject ◽  
Stiffness And Damping

Tilting pad journal bearings (TPJBs) provide radial support for rotors in high-speed machinery. Since the tilting pads cannot support a moment about the pivot, self-excited cross-coupled forces due to fluid-structure interactions are greatly reduced or eliminated. However, the rotation of the tilting pads about the pivots introduces additional degrees of freedom into the system. When the flexibility of the pivot results in pivot stiffness that is comparable to the equivalent stiffness of the oil film, then pad translations as well as pad rotations have to be considered in the overall bearing frequency response. There is significant disagreement in the literature over the nature of the frequency response of TPJBs due to nonsynchronous rotor perturbations. In this paper, a bearing model that explicitly considers pad translations and pad rotations is presented. This model is transformed to modal coordinates using state-space analysis to determine the natural frequencies and damping ratios for a four-pad tilting pad bearing. Experimental static and dynamic results were previously reported in the literature for the subject bearing. The bearing characteristics as tested are compared to a thermoelastohydrodynamic (TEHD) model. The subject bearing was reported as having an elliptical bearing bore and varying pad clearances for loaded and unloaded pads during the test. The TEHD analysis assumes a circular bearing bore, so the average bearing clearance was considered. Because of the ellipticity of the bearing bore, each pad has its own effective preload, which was considered in the analysis. The unloaded top pads have a leading edge taper. The loaded bottom pads have finned backs and secondary cooling oil flow. The bearing pad cooling features are considered by modeling equivalent convective coefficients for each pad back. The calculated bearing full stiffness and damping coefficients are also reduced nonsynchronously to the eight stiffness and damping coefficients typically used in rotordynamic analyses and are expressed as bearing complex impedances referenced to shaft motion. Results of the modal analysis are compared to a two-degree-of-freedom second-order model obtained via a frequency-domain system identification procedure. Theoretical calculations are compared to previously published experimental results for a four-pad tilting pad bearing. Comparisons to the previously published static and dynamic bearing characteristics are considered for model validation. Differences in natural frequencies and damping ratios resulting from the various models are compared, and the implications for rotordynamic analyses are considered.

Modal Frequency Response of a Four-Pad Tilting Pad Bearing With Spherical Pivots, Finite Pivot Stiffness and Different Pad Preloads

2010 ◽  
Author(s):  
Timothy W. Dimond ◽  
Amir A. Younan ◽  
Paul E. Allaire ◽  
John C. Nicholas
Keyword(s):  
Frequency Response ◽  
High Speed ◽  
Degrees Of Freedom ◽  
Natural Frequencies ◽  
Theoretical Calculations ◽  
Leading Edge ◽  
Damping Coefficients ◽  
Tilting Pad ◽  
The Subject ◽  
Stiffness And Damping

Tilting pad journal bearings (TPJBs) provide radial support for rotors in high-speed machinery. Since the tilting pads cannot support a moment about the pivot, self-excited cross-coupled forces due to fluid-structure interactions are greatly reduced or eliminated. However, the rotation of the tilting pads about the pivots introduces additional degrees of freedom into the system. When the flexibility of the pivot results in pivot stiffness that is comparable to the equivalent stiffness of the oil film, then pad translations as well as pad rotations have to be considered in the overall bearing frequency response. There is significant disagreement in the literature over the nature of the frequency response of TPJBs due to non-synchronous rotor perturbations. In this paper, a bearing model that explicitly considers pad translations and pad rotations is presented. This model is transformed to modal coordinates using state-space analysis to determine the natural frequencies and damping ratios for a four-pad tilting pad bearing. Experimental static and dynamic results were previously reported in the literature for the subject bearing. The bearing characteristics as tested are considered using a thermoelastohydrodynamic (TEHD) model. The subject bearing was reported as having an elliptical bearing bore and varying pad clearances for loaded and unloaded pads during the test. The TEHD analysis assumes a circular bearing bore, so the average bearing clearance was considered. Because of the ellipticity of the bearing bore, each pad has its own effective preload, which was considered in the analysis. The unloaded top pads have a leading edge taper. The loaded bottom pads have finned backs and secondary cooling oil flow. The bearing pad cooling features are considered by modeling equivalent convective coefficients for each pad back. The calculated bearing full stiffness and damping coefficients are also reduced non-synchronously to the eight stiffness and damping coefficients typically used in rotordynamic analyses and are expressed as bearing complex impedances referenced to shaft motion. Results of the modal analysis are compared to a two degree-of-freedom second-order model obtained via a frequency-domain system identification procedure. Theoretical calculations are compared to previously published experimental results for a four-pad tilting pad bearing. Comparisons to the previously published static and dynamic bearing characteristics are considered for model validation. Differences in natural frequencies and damping ratios resulting from the various models are compared, and the implications for rotordynamic analyses are considered.

Comparison of Tilting-Pad Journal Bearing Dynamic Full Coefficient and Reduced Order Models Using Modal Analysis

2009 ◽  
Author(s):  
Timothy W. Dimond ◽  
Amir A. Younan ◽  
Paul E. Allaire
Keyword(s):  
System Identification ◽  
Modal Analysis ◽  
State Space ◽  
Degrees Of Freedom ◽  
Natural Frequencies ◽  
Reduced Order ◽  
Damping Coefficients ◽  
Modal Damping ◽  
Tilting Pad ◽  
Stiffness And Damping

There is significant disagreement concerning the frequency response of tilting pad journal bearings (TPJBs) due to non-synchronous excitations. Two linear models for the frequency dependence of TPJBs have been proposed. The first model, the full-coefficient or KC model, considers Np tilting pads and rotor motions for Np + 2 degrees of freedom. Dynamic reduction of the KC model results in eight frequency-dependent stiffness and damping coefficients. The second model, based on results from bearing system identification experiments, yields twelve frequency-independent stiffness, damping, and mass (KCM) coefficients. Experimental data has been presented to support both models. There are major differences in the two approaches. The analysis in this paper takes a new approach of considering the pad dynamics explicitly in a state-space modal analysis. TPJB shaft and bearing pad stiffness and damping coefficients are calculated using a well known laminar, isothermal analysis and a pad assembly method. The TPJB rotor and pad full system eigenvalues and eigenvectors are then evaluated using state-space methods, with rotor and bearing pad inertias included explicitly in the model. The full bearing coefficient results are also non-synchronously reduced to the 8 stiffness and damping coefficients are and expressed as shaft complex impedances. The system identification method is then applied to these complex impedances, and the state space modal analysis is applied to the resulting KCM model. The damping ratios, natural frequencies, and mode shapes from the two bearing representations are compared. Two example TPJBs are examined in detail. The analysis indicated that four underdamped modes, two forward and two backward, dominate the rotor response over excitation frequencies from 0 to running speed. The full coefficient, non-synchronously reduced model predicts additional critically damped or overdamped modes due to the additional degrees of freedom as compared to the identified KCM model. The KCM model results in natural frequencies that are 63–65 percent higher than the full coefficient model. The difference in modal damping ratio estimates depend on the TPJB considered, with KCM being 7–17 percent higher than the full coefficient model. The full coefficient model also indicates that the bearing pads contribute significantly to the underdamped modes. The results indicate that the system identification method results in a reduced order model of TPBJ dynamic behavior. Additionally, the differences in the modal calculated system natural frequency and modal damping have potential implications for rotordynamic analyses of flexible rotors, such as critical speed and stability analyses.

Comparison of Tilting-Pad Journal Bearing Dynamic Full Coefficient and Reduced Order Models Using Modal Analysis (GT2009-60269)

2010 ◽  
Vol 132 (5) ◽  
Author(s):  
Timothy W. Dimond ◽  
Amir A. Younan ◽  
Paul Allaire
Keyword(s):  
System Identification ◽  
Modal Analysis ◽  
State Space ◽  
Degrees Of Freedom ◽  
Natural Frequencies ◽  
Reduced Order ◽  
Damping Coefficients ◽  
Modal Damping ◽  
Tilting Pad ◽  
Stiffness And Damping

There is a significant disagreement in the literature concerning the proper evaluation of the experimental identification and frequency response of tilting-pad journal bearings (TPJBs) due to shaft excitations. Two linear models for the frequency dependence of TPJBs have been proposed. The first model, the full coefficient or stiffness-damping (KC) model, considers Np tilting pads and two rotor radial motions for Np+2 degrees of freedom. The dynamic reduction of the KC model results in eight frequency-dependent stiffness and damping coefficients. The second model, based on bearing system identification experimental results, employs 12 frequency-independent stiffness, damping, and mass (KCM) coefficients; pad degrees of freedom are not considered explicitly. Experimental data have been presented to support both models. There are major differences in the two approaches. The present analysis takes a new approach of considering pad dynamics explicitly in a state-space modal analysis. TPJB shaft and bearing pad stiffness and damping coefficients are calculated using a well known laminar, isothermal analysis and a pad assembly method. The TPJB rotor and pad KC model eigenvalues and eigenvectors are then evaluated using state-space methods, with rotor and bearing pad inertias included explicitly in the model. The KC model results are also nonsynchronously reduced to the eight stiffness and damping coefficients and are expressed as shaft complex impedances. The system identification method is then applied to these complex impedances, and the state-space modal analysis is applied to the resulting KCM model. The damping ratios, natural frequencies, and mode shapes from the two bearing representations are compared. Two sample TPJB cases are examined in detail. The analysis indicated that four underdamped modes, two forward and two backward, dominate the rotor response over excitation frequencies from 0 to approximately running speed. The KC model predicts additional nearly critically damped modes primarily involving pad degrees of freedom, which do not exist in the identified KCM model. The KCM model results in natural frequencies that are 63–65% higher than the KC model. The difference in modal damping ratio estimates depends on the TPJB considered; the KCM estimate was 7–17% higher than the KC model. The results indicate that the KCM system identification method results in a reduced order model of TPBJ dynamic behavior, which may not capture physically justifiable results. Additionally, the differences in the calculated system natural frequency and modal damping have potential implications for rotordynamic analyses of flexible rotors.

Tilting Pad Journal Bearings: A Discussion on Stability Calculation, Frequency Dependence, and Pad and Pivot Flexibility

2012 ◽  
Author(s):  
Jason C. Wilkes ◽  
Dara W. Childs
Keyword(s):  
Degrees Of Freedom ◽  
Journal Bearing ◽  
Operating Temperature ◽  
Excitation Frequency ◽  
Full Model ◽  
Stability Calculation ◽  
Damping Coefficients ◽  
Tilting Pad ◽  
Stiffness And Damping ◽  
Tilting Pad Journal Bearing

For several years, researchers have presented predictions showing that using a full tilting-pad journal bearing (TPJB) model (retaining all of the pad degrees of freedom) is necessary to accurately perform stability calculations for a shaft operating on TPJBs. This paper will discuss this issue, discuss the importance of pad and pivot flexibility in predicting impedance coefficients for the tilting-pad journal bearing, present measured changes in bearing clearance with operating temperature, and summarize the differences between measured and predicted frequency dependence of dynamic impedance coefficients. The current work presents recent test data for a 100 mm (4 in) five-pad TPJB tested in load on pad (LOP) configuration. Measured results include bearing clearance as a function of operating temperature, pad clearance and radial displacement of the loaded pad (the pad having the static load vector directed through its pivot), and frequency dependent stiffness and damping. Measured hot bearing clearances are approximately 30% smaller than measured cold bearing clearances and are inversely proportional to pad surface temperature; predicting bearing impedances with a rigid pad and pivot model using these reduced clearances results in overpredicted stiffness and damping coefficients that are several times larger than previous comparisons. The effect of employing a full bearing model versus a reduced bearing model (where only journal degrees of freedom are retained) in a stability calculation for a realistic rotor-bearing system is assessed. For the bearing tested, the bearing coefficients reduced at the frequency of the unstable eigenvalue (subsynchronously reduced) predicted a destabilizing cross-coupled stiffness coefficient at the onset of instability within 1% of the full model, while synchronously reduced coefficients for the lightly loaded bearing required 25% more destabilizing cross-coupled stiffness than the full model to cause system instability. The same stability calculation was performed using measured stiffness and damping coefficients at synchronous and subsynchronous frequencies. These predictions showed that both the synchronously measured stiffness and damping and predictions using the full bearing model were more conservative than the model using subsynchronously measured stiffness and damping, an outcome that is completely opposite from conclusions reached by comparing different prediction models. This contrasting outcome results from a predicted increase in damping with increasing excitation frequency at all speeds and loads; however, this increase in damping with increasing excitation frequency was only measured at the most heavily loaded conditions.

An Analytical Complete Model of Tilting-Pad Journal Bearing Considering Pivot Stiffness and Damping

2010 ◽  
Vol 133 (1) ◽  
Author(s):  
Zhiyong Yan ◽  
Yi Lu ◽  
Tiesheng Zheng
Keyword(s):  
Degrees Of Freedom ◽  
Journal Bearing ◽  
Local Coordinate System ◽  
Dynamical Behavior ◽  
Complete Model ◽  
Damping Coefficients ◽  
Tilting Pad ◽  
Generalized Force ◽  
Stiffness And Damping ◽  
Tilting Pad Journal Bearing

Considering the freedom of pad tilting and pad translation along preload orientation, an analytical complete model, as well as mathematical method, which contains 2n+2 degrees of freedom, is presented for calculating the dynamical characteristics of tilting-pad journal bearing. Based on the motion relationship of shaft and pad, the local coordinate system, the generalized displacement, and the generalized force vector are chosen. The concise transformation of generalized displacement, generalized force, and its Jacobian matrix between the local and global coordinate systems are built up in matrix form. A fast algorithm using the Newton–Raphson method for calculating the equilibrium position of journal and pads is proposed. The eight reduced stiffness and damping coefficients can be obtained assuming that the journal and all pads are subject to harmonic vibration. Numerical results show that the reduced damping coefficients and the threshold speed can be effectively enhanced by giving suitable pad pivot stiffness and damping simultaneously, and this analytical method can be applied to analyze dynamical behavior of the tilting-pad journal bearing rotor system.

Tilting Pad Journal Bearings—A Discussion on Stability Calculation, Frequency Dependence, and Pad and Pivot

2012 ◽  
Vol 134 (12) ◽  
Author(s):  
Jason C. Wilkes ◽  
Dara W. Childs
Keyword(s):  
Degrees Of Freedom ◽  
Journal Bearing ◽  
Operating Temperature ◽  
Excitation Frequency ◽  
Full Model ◽  
Stability Calculation ◽  
Damping Coefficients ◽  
Tilting Pad ◽  
Stiffness And Damping ◽  
Tilting Pad Journal Bearing

For several years, researchers have presented predictions showing that using a full tilting-pad journal bearing (TPJB) model (retaining all of the pad degrees of freedom) is necessary to accurately perform stability calculations for a shaft operating on TPJBs. This paper will discuss this issue, discuss the importance of pad and pivot flexibility in predicting impedance coefficients for the tilting-pad journal bearing, present measured changes in bearing clearance with operating temperature, and summarize the differences between measured and predicted frequency dependence of dynamic impedance coefficients. The current work presents recent test data for a 100 mm (4 in.) five-pad TPJB tested in load on pad (LOP) configuration. Measured results include bearing clearance as a function of operating temperature, pad clearance and radial displacement of the loaded pad (the pad having the static load vector directed through its pivot), and frequency-dependent stiffness and damping. Measured hot-bearing clearances are approximately 30% smaller than measured cold-bearing clearances and are inversely proportional to pad surface temperature; predicting bearing impedances with a rigid pad and pivot model using these reduced clearances results in overpredicted stiffness and damping coefficients that are several times larger than previous comparisons. The effect of employing a full bearing model versus a reduced bearing model (where only journal degrees of freedom are retained) in a stability calculation for a realistic rotor-bearing system is assessed. For the bearing tested, the bearing coefficients reduced at the frequency of the unstable eigenvalue (subsynchronously reduced) predicted a destabilizing cross-coupled stiffness coefficient at the onset of instability within 1% of the full model, while synchronously reduced coefficients for the lightly loaded bearing required 25% more destabilizing cross-coupled stiffness than the full model to cause system instability. The same stability calculation was performed using measured stiffness and damping coefficients at synchronous and subsynchronous frequencies. These predictions showed that both the synchronously measured stiffness and damping and predictions using the full bearing model were more conservative than the model using subsynchronously measured stiffness and damping, an outcome that is completely opposite from conclusions reached by comparing different prediction models. This contrasting outcome results from a predicted increase in damping with increasing excitation frequency at all speeds and loads; however, this increase in damping with increasing excitation frequency was only measured at the most heavily loaded conditions.

Ball bearing turbocharger vibration management: application on high speed balancer

2020 ◽  
Vol 21 (6) ◽  
pp. 619
Author(s):  
Kostandin Gjika ◽  
Antoine Costeux ◽  
Gerry LaRue ◽  
John Wilson
Keyword(s):  
Dynamic Behavior ◽  
High Speed ◽  
Internal Combustion Engines ◽  
Ball Bearing ◽  
Squeeze Film ◽  
Design And Technology ◽  
Squeeze Film Damper ◽  
Damping Coefficients ◽  
Bearing Loads ◽  
Stiffness And Damping

Today's modern internal combustion engines are increasingly focused on downsizing, high fuel efficiency and low emissions, which requires appropriate design and technology of turbocharger bearing systems. Automotive turbochargers operate faster and with strong engine excitation; vibration management is becoming a challenge and manufacturers are increasingly focusing on the design of low vibration and high-performance balancing technology. This paper discusses the synchronous vibration management of the ball bearing cartridge turbocharger on high-speed balancer and it is a continuation of papers [1–3]. In a first step, the synchronous rotordynamics behavior is identified. A prediction code is developed to calculate the static and dynamic performance of “ball bearing cartridge-squeeze film damper”. The dynamic behavior of balls is modeled by a spring with stiffness calculated from Tedric Harris formulas and the damping is considered null. The squeeze film damper model is derived from the Osborne Reynolds equation for incompressible and synchronous fluid loading; the stiffness and damping coefficients are calculated assuming that the bearing is infinitely short, and the oil film pressure is modeled as a cavitated π film model. The stiffness and damping coefficients are integrated on a rotordynamics code and the bearing loads are calculated by converging with the bearing eccentricity ratio. In a second step, a finite element structural dynamics model is built for the system “turbocharger housing-high speed balancer fixture” and validated by experimental frequency response functions. In the last step, the rotating dynamic bearing loads on the squeeze film damper are coupled with transfer functions and the vibration on the housings is predicted. The vibration response under single and multi-plane unbalances correlates very well with test data from turbocharger unbalance masters. The prediction model allows a thorough understanding of ball bearing turbocharger vibration on a high speed balancer, thus optimizing the dynamic behavior of the “turbocharger-high speed balancer” structural system for better rotordynamics performance identification and selection of the appropriate balancing process at the development stage of the turbocharger.

Vertical vibration modelling and vibration response analysis of Chinese high-speed train passengers at different locations of a high-speed train

2020 ◽  
Vol 235 (1) ◽  
pp. 35-46 ◽  
Author(s):  
Chun-jun Chen ◽  
Chao Fang ◽  
Guo-qing Qu ◽  
Zhi-ying He
Keyword(s):  
High Speed ◽  
Degrees Of Freedom ◽  
Vibration Signal ◽  
Vertical Vibration ◽  
Response Analysis ◽  
High Speed Train ◽  
Vehicle Model ◽  
Internal Organs ◽  
Damping Coefficients ◽  
Track Irregularity

To study the vibration of a passenger's head and internal organs at different locations of a high-speed train, a 9-degrees-of-freedom (DOF) model of seated passengers is proposed in this paper, and its parameters of the damping coefficients and stiffnesses are identified. Next, the response of the head and internal organs is simulated by applying the vibrational stimulation generated by a 27-DOF vehicle model under track irregularity. Moreover, by applying the measured vibration signal, the following conclusions can be drawn: (1) the weakest response is detected at the centre of the compartment of the wagon, and a stronger response is detected at the centre of the bogie, with the rolling motion having a greater effect 1 m away from the centre of the bogie; (2) the response of the human internal organs is stronger than that of the head under stimulation with a lower frequency of less than 3 Hz, and a similar conclusion can be drawn in the range of 5 to 8 Hz. However, if the frequency is in the range between 8 and 15 Hz, the situation is entirely different. The responses of both the head and internal organs are reduced at frequencies over 20 Hz; (3) from the real application, it can be inferred that the greatest response can be detected at approximately 3 Hz for internal organs and at 8 Hz or higher for the head.

Three-Dimensional Thermo-Elasto-Hydrodynamic Computational Fluid Dynamics Model of a Tilting Pad Journal Bearing—Part II: Dynamic Response

2019 ◽  
Vol 141 (6) ◽  
Author(s):  
Jongin Yang ◽  
Alan Palazzolo
Keyword(s):  
Fluid Dynamics ◽  
Computational Fluid Dynamics ◽  
Dynamic Response ◽  
Journal Bearing ◽  
Three Dimensional ◽  
Static Response ◽  
Damping Coefficients ◽  
Tilting Pad ◽  
Stiffness And Damping ◽  
Tilting Pad Journal Bearing

Part II presents a novel approach for predicting dynamic coefficients for a tilting pad journal bearing (TPJB) using computational fluid dynamics (CFD) and finite element method (FEM), including fully coupled elastic deflection, heat transfer, and fluid dynamics. Part I presented a similarly novel, high fidelity approach for TPJB static response prediction which is a prerequisite for the dynamic characteristic determination. The static response establishes the equilibrium operating point values for eccentricity, attitude angle, deflections, temperatures, pressures, etc. The stiffness and damping coefficients are obtained by perturbing the pad and journal motions about this operating point to determine changes in forces and moments. The stiffness and damping coefficients are presented in “synchronously reduced form” as required by American Petroleum Institute (API) vibration standards. Similar to Part I, an advanced three-dimensional thermal—Reynolds equation code validates the CFD code for the special case when flow Between Pad (BP) regions is ignored, and the CFD and Reynolds pad boundary conditions are made identical. The results show excellent agreement for this validation case. Similar to the static response case, the dynamic characteristics from the Reynolds model show large discrepancies compared with the CFD results, depending on the Reynolds mixing coefficient (MC). The discrepancies are a concern given the key role that stiffness and damping coefficients serve instability and response predictions in rotordynamics software. The uncertainty of the MC and its significant influence on static and dynamic response predictions emphasizes a need to utilize the CFD approach for TPJB simulation in critical machines.

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