A scheme of the turbocharger is shown in Fig.1a. The turbocharger in Fig.1a consists of a turbine and a compressor attached to the shaft, supported by two thrust bearings. This turbocharger can be modeled by the equivalent system in Fig.1b, considering that there is only relative motion between the collar and the thrust bearings, increasing or decreasing the oilfilm thickness. It is assumed that the thrust bearings are clamped and the collar moves only in the axial direction (defined as the xcoordinate). The thrust bearings support the external forcesΔF to maintain the system working properly. The springs and viscous dampers are the equivalent
Fig. 1 aScheme of turbochargerbequivalent system of the thrust bearing
Experimental Estimation of Equivalent Damping Coefficient… 19
coefficients of the oil film between the bearings and the collar. The geometric dimensions of the thrust bearings are shown in Table1, through the ratios of the main design parameters. The main variables of the thrust bearings, also shown schematically in Fig.2, are the innerðriÞand outerðroÞradius, the angular length of the pad ramp ðθrampÞ, the angular length of the pad ðθ0Þ, the minimumðh0Þ and maximumðhmaxÞoil thickness and the number of padsðnpadÞof the bearing.
The turbocharger axial vibration is modeled as a one degree of freedom (DOF) system [2], admitting that its entire massMis concentrated in the collar of the shaft. The springs and dampers of Fig.1b are in parallel to each other, i.e., the equivalent stiffness and damping coefficients are the sum of those coefficients, Kxx=Kxx1+Kxx2 and Cxx=Cxx1+Cxx2. Therefore, the equation of motion of the system is
MẍðtÞ+CxxẋðtÞ+KxxxðtÞ=FðtÞ ð1Þ which can be numerically evaluated, by integrating Eq. (1) using the state space model defined as [14]:
ẋðtÞ ẍðtÞ
= 0 1
−KMxx −CMxx
xðtÞ ẋðtÞ
+ 0
FðtÞ M
ð2Þ Table 1 Geometric ratios of
the bearings [2] Variable Bearing 1 Bearing 2
ro ̸ri 1.5 1.7
ri ̸h0 350 340
hmax ̸h0 2.4 2.0
θramp ̸θ0−θramp
5.0 4.0
npad 3 3
Fig. 2 aBearing pad andbfluid thickness profile (adapted from [2])
20 T. F. Peixoto et al.
In the test condition, the turbine is driven by compressed air, while the com- pressor is open to the atmosphere. An impact hammer, in the shaft end, axially excites the system. The axial displacement of the system is measured, as well as the force applied by the hammer, in time domain, as illustrated schematically in Fig.3.
The aim of this work is to estimate the equivalent coefficients of the bearings, so the simulated response is compatible with the experimental results obtained, due to an impulsive excitation force. The stiffness coefficient can be estimated as described by [13], which consists in solving the Reynolds’Equation utilizing a THD model, by the Finite Volume Method (FVM), to obtain the pressure distribution of the oil film circulating in the thrust bearings, accounting for parameters like the geometry of the bearings and the temperature distribution in the oilfilm.
The Reynolds’Equation is the governing equation for pressure distribution in the oilfilm. To account for the temperature variation of the oilfilm, which changes the oil viscosity along the bearing, the generalized Reynolds’ Equation must be utilized. This equation was introduced by [7] and is written in cylindrical coordi- nates as
1 r
∂
∂θ F2∂p
∂θ
+ ∂
∂r rF2∂p
∂r
=Ωr ∂
∂θ F1
F0
+r∂h
∂t ð3Þ
in whichΩis the rotational speed of the shaft and the functionsF0,F1 andF2are introduced to account for the temperature distribution in the oilfilm, which changes the oil viscosityðμÞalong the mesh. These functions are defined as
F0= Zh
0
1
μdx, F1= Zh
0
x
μdx, F2= Zh
0
x
μ x− F1
F0
dx ð4Þ
Fig. 3 Scheme of experimental setup of the turbocharger (adapted from [2])
Experimental Estimation of Equivalent Damping Coefficient… 21
These integrals are responsible for the coupling between the viscosity variation along the oil film and the pressure to be calculated by the Reynolds’ Equation (Eq.3).
Daniel et al. [13] approached the problem to obtain the equivalent stiffness coefficient of the thrust bearing as suggested by Lund [3], by solving the Reynolds’ Equation applying small perturbations around the previously calculated equilibrium position, disregarding thefluidfilm thickness variation with the time ð∂h ̸∂t= 0Þ.
However, to estimate this coefficient, it is necessary to observe that the circulating oil in the thrust bearing enters the turbocharger through journal bearings, which causes an increase in its temperature. A THD model of the journal bearing isfirst solved to estimate the temperature of the oil leaving the journal bearing, assumed as the temperature of the oil inlet in the thrust bearing.
The results obtained by [13] are checked using the one DOF equation (Eq.1) and a finite element method (FEM) to discretize the turbocharger, following the method suggested by [15,16]. Peixoto et al. [15] compared the natural frequency of the system obtained by the one DOF system, by a FEM solution and the equation of longitudinal vibration of uniform bars with discontinuities of concentrated masses and springs. Peixoto [16] calculated the oil thickness of the fluid in the thrust bearing, using a step force and the equivalent stiffness coefficient calculated in the simulations of [13], comparing the values with measured, empirical results. The stiffness coefficient is estimated from the experimental results by dividing the measured force by the measured axial displacement of the system.
The oil thickness was calculated according to Fig.4. The oil thickness of each thrust bearing is originallyh01andh02, for thrust bearings 1 and 2, respectively. The collar attached to the shaft changes its equilibrium position byΔxafter an external excitation, modeled as a step excitation, acts on the system. The collar changes its static equilibrium position due to the step excitation, which causes the oil thickness to change an amount equal toh0±Δx. The minimum estimated oil thickness of the bearing ish0−j jΔx and this value is compared to the experimental measurements.
To check the oil thickness change of the model means to verify if the estimated stiffness coefficient is in agreement with experimental results. However, the stiff- ness coefficient changes only the static equilibrium position of the system, but gives little information on its dynamic response. To fully add theflexibility of the oilfilm
Fig. 4 Simplified scheme of the assembly bearings and collar (adapted from [16])
22 T. F. Peixoto et al.
in the dynamic response, it is also necessary to obtain the equivalent damping coefficient. This coefficient is estimated by optimizing this parameter, setting as the objective function the difference between the simulated and the experimental response, during the transient response of the system.
The optimization problem is constructed admitting that the objective function to minimize is the maximum absolute difference between the experimental response
xexp
and the simulated responsefxsimg:
min
cmin≤Cxx≤cmax
max fxsimg− xexp
ð5Þ The vector xexp
is obtained from experimental measured results [2], along with the time vectorftg, while the simulated responsefxsimgis obtained from the numerical integration of the system (Eq.2), which is a function of the damping coefficient Cxx. The input force for the simulation is the force measured in the impact hammer, during the experiment. It is important to notice that the max function returns the largest element of the array,j j⋅ is the element-wise absolute value function and the minð⋅Þ function is the min operator, which returns the minimum value of the function.
The system is numerically integrated, with initial conditions x0= 0, i.e., the system starts from rest, andẋ0=v0, i.e., with an initial velocity given by
v0=dx
dt ≅x½2−x½1
t½2−t½1 ð6Þ
so, the initial velocity of the system is estimated by the forward difference approximation, utilizing the measured values of displacement and time. The opti- mization problem is solved using thefmincon function from MATLAB®, using the interior point algorithm [17]. The range admitted for the damping coefficient, in the case studied here, is 0 <Cxx< 106 N s/m.