© AFM, EDP Sciences 2020

1 Introduction

It is well known that high speed automotive turbochargers are sources of high vibration level on vehicles. Moreover, the bearing behavior nonlinearities makes their spectrum very complex including synchronous vibration, self-excited oil whirl and oil whip phenomena, subharmonics, superharmonics, combination frequencies and jump phenomena [4]. On [2] Gjika et al. did a critical analysis of different turbocharger vibro-acoustics mechanisms and sources as per Garrett's experience. There are outlined 9 noise types, which cover a full audible frequency range from 0 to 20000 Hz. Their sources can be rotordynamics or aerodynamics, and the transfer paths structural or gaseous. The turbocharger rotor-bearing system concept & design, assembling & balancing processes, housings and vehicle vibro-acoustics management are becoming a real challenge for both turbocharger manufacturers and OEMs.

Many researches have investigated the vibration interaction between bearing system rotordynamics and housings dynamics [1–13].

Foundation dynamics characteristics can be very well represented by frequency response functions (FRF), which can be obtained by finite element (FE) analysis or test; their quality is a fundamental step on mechanical vibration investigation.

Cavalca et al. [5] identified the influence of support flexibility on rotordynamics unbalance response; both rotor-bearing system and foundation have been modeled by finite elements. Dakel et al. [6] predicted the rotordynamics performances of a flexible rotor (symmetric and asymmetric) under masse unbalance combined with rigid base excitations; the stability chart, Campbell diagrams, steady-state responses as well as orbits of the rotor are analyzed. On [7] Ewins described how to obtain high quality of experimental FRFs and Yamaguchi et al. [8] made the best use of Fast Fourier Transformation (FFTs) approach on measured FRFs. Nicholas et al. [9] developed a concept of equivalent bearing coefficients based on the combination of experimental foundation FRFs with bearing force coefficients and used that on a rotordynamics code for predicting the critical speed. Xu and Vance [10] improved the model by refining the foundation modal damping from test data. Both studies demonstrated the impact of the flexible foundation on rotor-bearing system critical speed, validated by test data. Vazquez et al. [11,12] analyzed the rotordynamics performance of a three-disc rotor supported by two fluid film bearings on flexible anisotropic supports; the experimental FRFs were used to identify the characteristics of the foundations, an equivalent stiffness matrix was developed, and the damping was considered to be zero; the prediction of the first two critical speeds and of the threshold speed of instability correlates well with the test data. Shaposhnikov et al. [13] suggested a hybrid rotor-foundation model for a 2MW gas turbine engine, which has a very complex support structure. The rotordynamics has been modeled by finite elements and the foundation characteristics (direct, cross-coupling and cross talk dynamic stiffness and damping) obtained experimentally have been integrated into that. It shows better accuracy prediction of the critical speeds and support structural resonances.

A finite element analysis of a turbomachine which integrates in a single model the bearing system rotordynamics and the support dynamics can also be used for vibration performance prediction, but it seems to be costly and time consuming for foundation structures that are complex or have nonlinear behavior [9].

This paper describes a prediction model and validation aspects for the synchronous forced vibration of ball bearing turbocharger housings, but which can be applied to any other ball bearing turbomachine.

2 Automotive turbocharger description

Garrett-Advancing Motion considers the bearing system as the heart of the turbocharger. As part of continuous improvement and OEMs needs they have developed different technologies such as fluid bearings and ball bearings [1]. Advanced concepts such air foil bearing [14,15] are now implemented on Two-Stage Electric Compressor and the application on turbocharging technology is being validated.

Figure 1 shows a fully-floating fluid bearing turbocharger. It is the original design, relatively low cost and still in production for commercial vehicle (CV) applications. That incorporates two oil films: the inner film between the shaft and the ring, and the outer one between the ring and the housing. Both behave as hydrodynamic films under unbalance forces. The axial load is supported by a separated hydrodynamic thrust bearing. Such a concept is known for its high damping, which reduces the shaft motion and vibration transmission on the housings. Gas stand shaft motion test is a good predictor of on-engine shaft motion behavior, but the high number of self-excitation frequencies (3 frequencies − inner film whirl, outer film whirl & ring speed) causes the design to be more susceptible to rotordynamics instability. It was found to be difficult to qualify shaft motion for small frame sizes.

Figure 2 presents Garrett's semi-floating fluid bearing with integrated thrust bearing that is successfully implemented on light vehicle (LV) turbochargers. Both bearings, compressor side and turbine side, are part of a single bushing, which is prevented from rotation by a pin; the inner oil film acts as hydrodynamic bearing and the outer film is behaving as squeeze film damper. The axial grooves on the bearing inside diameter serving oil on the thrust bearings while improving rotordynamics stability. It is a demonstrated design for very good rotordynamics stability due to the squeeze film damper, low radial power losses due to the small journal diameter and low vibration transmission due to the slender shaft design.

Figure 3 describes a conventional ball bearing cartridge turbocharger for LV and CV applications. The cartridge inner race is press fitted on the shaft and the outer race is prevented from rotating by a pin; the outer oil film behaves as squeeze film damper. This design shows very low power loss for improved fuel economy and transient performance, as well as excellent cold start behavior; Garrett-Advancing Motion identified no shaft motion dynamic instability issues due to missing hydrodynamic oil film. High speed balancing performance is affected by high rotordynamics bearing loads related to the rotating group stiffness and initial unbalance due to the assembly process.

Fig. 1 Fully-floating fluid-bearing system.

Fig. 2 Semi-floating fluid bearing system.

Fig. 3 Ball bearing system.

3 Synchronous vibration prediction method for high speed ball bearing turbomachinery

Three steps are integrated on the method.

A “ball bearing-squeeze film damper” model allows to predict the isotropic stiffness and damping coefficients as function of the eccentricity; they are integrated on a rotordynamics code for bearing loads prediction under unbalances; the eccentricity is updated by a converging iterative process.

The foundation transfer functions are obtained from a commercial FE code.

The forced vibration on the housing are calculated by coupling rotating bearing load with transfer functions.

3.1 “Ball bearing-squeeze film damper” modelling

The dynamic behavior of balls, Figure 4, is modeled by springs; the isotropic stiffness is calculated from Tedric Harris formulas [16], equation (1), and the damping is considered null.(1)with:

k, ball stiffness (N/m)

z, number of balls

d, ball diameter (mm)

Δ, radial deflection (mm)

α, contact angle (radian)

The deflection is calculated by the equation (2).(2)where F_o, is the load acting on the outer squeeze film damper.

The squeeze film damper model is derived from the Osborne Reynolds equation for incompressible and synchronous fluid loading [1]. Both the balls and the squeeze film damper are assumed to support the same load (i.e. the cartridge outer race is massless) Thermal model is considered adiabatic and viscosity/temperature is described by Walther's formula [17]. The bearing is assumed infinitely short and the oil film pressure is modeled as a cavitated π film model. The stiffness and damping coefficients can be calculated respectively by equations (3) and (4).(3) (4)where:

μ, fluid viscosity

R, journal radius

L, bearing length

c_r, radial bearing clearance

ε, eccentricity ratio

N_s, shaft speed (rpm)

N_eff = |2N_load − N_journal|, effective speed [18,19]

For a non-rotating bearing (case of the squeeze film damper), N_journal = 0 and for unbalance loading, N_load = N_s, then N_eff/N_s = 2N_s/N_s = 2.

Fig. 4 Ball bearing squeeze film damper modeling.

3.2 Rotordynamics and bearing load modelling

The fundamentals of rotordynamics and the associated finite element code are presented at [20]; the equations (5), which govern the synchronous phenomena are detailed in [1].(5)where {F_unb (Ωt)} is the mass unbalance vector.

The rotating bearing loads can be presented by the formulas (6).(6)where:

F_brg, bearing load magnitude

Ω, rotor speed (rad/s)

Depending on the distribution of unbalances on the rotating group, the load vectors on each of the bearings can be different in magnitude and phase. In case of two bearings they can be presented with the equations (7) and (8) where Φ is the phase angle between load vectors on the bearings.(7) (8)

3.3 Housings vibration modelling

The structural dynamics of housings can be modeled by any commercial FE prediction code. In this study ANSYS code [21] is used, and the fundamentals of modelling are presented on [1].

For modelling purposes, the OD (squeeze film damper clearance) bearing loads are applied to two dummy nodes on each bearing locations, featuring two lump mass elements with negligible mass, which are connected to the housing bore by constraint equations. The housing transfer functions are predicted by applying a rotating bearing load unit on each of dummy nodes. The coupling of the dynamic bearing loads with transfer functions allows to predict vibration on the housings.

4 Case study: ball bearing turbocharger vibration management on high speed balancer

Figure 5 shows a high speed balancer (HSB) for automotive turbocharger. A ball bearing CHRA (center housing rotating assembly) unbalance master, which allows the implementation of unbalances by using small metallic screws, is clamped on the mechanical tooling. The vibrations of the “CHRA-HSB” assembly are measured by an accelerometer implemented on the HSB fixture.

Fig. 5 Turbocharger unbalance master on HSB.

4.1 Turbocharger rotordynamics

The structural rotordynamics model of a ball bearing turbocharger is presented in Figure 6. It includes 44 beam finite elements for the rotating group and 8 for the bearing outer race; the wheel inertia characteristics are attached on their center of inertia; the balls and squeeze film damper are respectively modeled by springs and oil stiffness & damping coefficients.

Fig. 6 Turbocharger rotordynamics model.

4.2 Free-free rotordynamics

Figure 7 shows how the experimental transfer functions (γ/F) of free-free rotating group are obtained; Figure 8 demonstrates good test/prediction correlation of natural frequencies that occur up to 7000 Hz.

Fig. 7 Turbocharger rotating group on light elastic bands.

Fig. 8 Rotating group free-free natural frequencies and associated mode shapes. Test/prediction.

4.3 Rotordynamics on HSB operation condition

Figures 9 and 10 summarize respectively the prediction of the Campbell diagram and OD bearing loads on the speed range up to 210000 rpm; bearing stiffness and damping coefficients are calculated for the converged eccentricity and the housings support is considered rigid. The first bending critical speed occurs at 186000 rpm.

Fig. 9 Campell diagram.

Fig. 10 Rotodynamics od bearing loads.

4.4 “CHRA housing - HSB fixture” modal analysis

Figure 11 presents the finite element model of “CHRA housing − HBS fixture”, which is validated by test/prediction transfer functions up to 4000 Hz (Figs. 12 and 13). At high speed range the turbocharger vibration response can be affected by the natural frequency at 2913 Hz.

Fig. 11 Finite element model of “CHRA-HSB” housings.

Fig. 12 Experimental vibration identification of “CHRA-HSB” structure.

Fig. 13 “CHRA-HSB” modal analysis. (a) Transfer function. (b) Mode shapes.

4.5 Turbocharger vibration on HSB

CHRAs unbalance masters, which is shown in Figure 14, with ODmin, ODmax and ODnom squeeze film damper clearances were run on the high-speed balancer. Test masses have been implemented on different rotor planes (CWnose, CWface, TWface and TWnose) by using small screws and the vibration responses under different unbalance configurations have been collected by the accelerometer on the HSB fixture (see Fig. 4). Figure 15 summarizes 6 cases of test/prediction dimensionless responses; it shows good correlation. The vibration prediction model is validated.

Fig. 14 Turbocharger unbalance master.

Fig. 15 Turbocharger vibration response: test/prediction (dimensionless).

5 Conclusion

The paper describes a prediction method for synchronous vibration management of ball bearing turbomachines; it is a continuation of the papers [1–3].

A model and the associated prediction code for the static and dynamic behavior performance of “ball bearing cartridge-squeeze film damper” are developed. The dynamic behavior of the balls is modeled with Tedric Harris formulas and the squeeze film damper characteristics derive from the Osborne Reynolds equation; the stiffness and damping coefficients are used on a rotordynamics code to predict the bearing loads by converging with the bearing system eccentricity ratio; the OD rotating bearing loads are coupled with foundation transfer functions predicted by FE analysis and the vibration on the housing are predicted.

The approach is validated with numerous test data from a Garrett's ball bearing turbocharger on the high-speed balancer, but it is generic and can be applied to any turbomachinery.

The next step consists in the validation of the method on vehicle operating conditions.

Nomenclature

B : Squeeze film damping

[C] : “Shaft-bearing” damping matrix

c_r : Radial bearing clearance

CHRA : Center housing rotating assembly

CV : Commercial vehicle

d : Ball diameter

Δ : Ball radial deflection

F_brg : Bearing load

FE : Finite element

FRF : Frequency response function

{F_unb(Ωt)} : Mass unbalance force vector

[G] : Gyroscopic matrix

HSB : High speed balancer

i : Imaginary unit

k : Ball stiffness

K : Squeeze film stiffness coefficient

[K] : “Shaft-bearing” stiffness matrix

L : Bearing length

LV : Light vehicle

[M] : “Shaft-discs-bearing” inertia matrix

N_eff : Effective speed

N_L : Load speed

N_s : Shaft speed

OD : Squeeze film damper clearance

R : Journal radius

t : Time

z : Number of balls

: Global DOF vectors (displacement, velocity and acceleration)

α : Contact angle

ε : Eccentricity ratio

Φ : Phase difference between compressor side and turbine side bearings load vectors

μ : Fluid viscosity

Ω : Rotational speed

Subscripts

brg : Bearing (bearing load)

eff : Effective (effective speed)

min, max : Minimum, maximum (bearing clearance)

unb : Mass unbalance

r : Radial (radial clearance)

Acknowledgments

The authors are indebted to Garrett Advancing Motion for permission to publish this work.

References

All Figures

	Fig. 1 Fully-floating fluid-bearing system.
In the text

	Fig. 2 Semi-floating fluid bearing system.
In the text

	Fig. 3 Ball bearing system.
In the text

	Fig. 4 Ball bearing squeeze film damper modeling.
In the text

	Fig. 5 Turbocharger unbalance master on HSB.
In the text

	Fig. 6 Turbocharger rotordynamics model.
In the text

	Fig. 7 Turbocharger rotating group on light elastic bands.
In the text

	Fig. 8 Rotating group free-free natural frequencies and associated mode shapes. Test/prediction.
In the text

	Fig. 9 Campell diagram.
In the text

	Fig. 10 Rotodynamics od bearing loads.
In the text

	Fig. 11 Finite element model of “CHRA-HSB” housings.
In the text

	Fig. 12 Experimental vibration identification of “CHRA-HSB” structure.
In the text

	Fig. 13 “CHRA-HSB” modal analysis. (a) Transfer function. (b) Mode shapes.
In the text

	Fig. 14 Turbocharger unbalance master.
In the text

	Fig. 15 Turbocharger vibration response: test/prediction (dimensionless).
In the text