# Nonlinear Dynamic Response of an Unbalanced Flexible Rotor Supported by Elastic Bearings Lubricated with Piezo-Viscous Polar Fluids

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{i}and body couples noted ℓ

_{i}in addition to the body forces and surface forces Figure 1.

**Figure 1.**Balance of forces and couples acting on elementary volume according to V. K. Stokes theory.

^{−8}(m

^{2}/N) for paraffinic oils, and a value between 2.5–3.5 × 10

^{−8}(m

^{2}/N) for aromatic oils according to Klamann [36].

^{−1}), and ${\mu}_{0}$ is the atmospheric dynamic viscosity in (cP) or (mPa·s). There are also other equations for the calculation of the pressure-viscosity coefficient available in the literature. Some of these equations are accurate for certain fluids and inaccurate for others. One of the problems associated with available formulae is that they only allow the accurate calculation of pressure-viscosity coefficients at low shear rates. An accurate value of this coefficient can be determined experimentally.

_{r}can be expressed as

_{r}is the Young’s modulus of rotor, ${I}_{Gy}={\int}_{S}^{}{x}^{2}ds=\frac{\pi {D}^{4}}{64}$ is the second moment of area of the rotor (shaft) cross-section, and L

_{r}denotes the span of the rotor (shaft).

## 2. Governing Equations

#### 2.1. Momentum Equations of the Polar or Couple Stress Fluid

^{−1}T

^{−1}) whereas the dimensions of $\mathrm{\eta}$ is that of momentum (MLT

^{−1}). The ratio $\mathrm{\eta}/\mathrm{\mu}$ has dimension of length square and, in the following, we denote this material constant by $\ell $, where $\ell ={\left(\raisebox{1ex}{$\eta $}\!\left/ \!\raisebox{-1ex}{$\mu $}\right.\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$. Some experiments for determining the material constants $\mathrm{\mu}$ and $\mathrm{\eta}$ for incompressible fluids are given in Equation (1). Physically, the quantity $\ell $ can be regarded as a characteristic length of the additives which are added to the base oil and which can be polymers or co-polymers.

#### 2.2. Modified Transient Nonlinear Piezo-Viscous Reynolds’ Equation and Boundary Conditions

_{j}= ɷR is the linear velocity of the journal surface.

#### 2.3. Rotor-Dynamic Equations

- -
- the weight of the rotor 2W
_{0}= 2 mg; - -
- the dynamic load components ${W}_{X}(t)$ and ${W}_{Y}(t)$ due an unbalance mass characterized by its eccentricity e;
- -
- the hydrodynamic forces F
_{X}and F_{Y}due to the presence of the lubricating oil film.

_{r}, and rotor stiffness of 2k

_{r}.

_{0}= mg, and a synchronous dynamic excitation due to an unbalance mass $\left|\overrightarrow{W}(t)\right|=me{\omega}^{2}$.

_{X}and F

_{Y}are the hydrodynamic forces in X and Y directions that are nonlinear functions of the displacement components (X,Y) and the velocities $\left(\dot{X},\dot{Y}\right)$ of the journal center O

_{j}. They are calculated by integrating the hydrodynamic pressure over the bearing surface. This latter is obtained by solving the transient modified Reynolds’ equation by finite element method.

## 3. Computation Procedure of the Journal Bearing Nonlinear Dynamic Response

_{j}for balanced and unbalanced shafts is determined Figure 4 (See Appendix A for more details).

## 4. Finite Element Treatment of the Steady-State Modified Reynolds’ Equation

#### 4.1. Weak Integral Formulation and Finite Element Discretization

_{i}(Lagrangian polynomials) are expressed as

#### 4.2. Method of Solution of Steady-State Nonlinear Modified REYNOLDS’ Equation

_{0}is an under-relaxation factor which ensures and accelerates the convergence of the iterative process.

#### 4.3. Iterative Research of the Steady-State Equilibrium Position

_{0},Y

_{0}) due to the application of static load ${\mathit{W}}_{0}=\left({W}_{X0},{W}_{Y0}\right)$ is determined when the integration of the resulting steady-state hydrodynamic pressure obtained from Equation (28a) balances with the applied load components $\left({W}_{X0},{W}_{Y0}\right)$, i.e.,

## 5. Finite Element Treatment of the Transient Modified Reynolds’ Equation

#### 5.1. Weak Integral Formulation and Finite Element Discretization

#### 5.2. Method of Solution of Transient Nonlinear Modified Reynolds’ Equation

## 6. Results and Discussions

#### 6.1. Transient Solution vs. Steady-State Solution

_{0}= 340 kN (S = 0.18, ${W}_{0}^{*}$ = 1.8), α

^{*}= 0, ℓ

^{*}= 0, C

_{d}= 0, ε = 0.

_{0}= 2720 kN (S = 0.0225, ${W}_{0}^{*}$ = 14), α = 17 × 10

^{−9}Pa

^{−1}, ε = 0, ℓ

^{*}= 0.3, t

_{h}= 10 × 10

^{−3}m, E = 0.9 GPa, σ = 0.35.

Parameter | Symbol | Unit | Value |
---|---|---|---|

Bearing length | L | m | 0.320 |

Journal diameter | D = 2R | m | 0.500 |

Radial clearance | C | m | 3.5 × 10^{−4} |

Bearing-liner thickness | ${t}_{h}$ | m | 10^{−2} |

Young’s modulus of the bearing-liner (polyethylene high density at 20 °C) [41] | E | Pa | 0.9 × 10^{9} |

Poisson’s ratio of the bearing-liner at 20 °C [41] | σ | - | 0.35 |

Dynamic viscosity of lubricant at atmospheric pressure | ${\mu}_{0}$ | Pa·s | 15 × 10^{−3} |

Density of lubricant | ρ | kg·m^{−3} | 870 |

Pressure-viscosity coefficients | α | Pa^{−1} | 0 |

17 × 10^{−9} | |||

42.5 × 10^{−9} | |||

212.5 × 10^{−9} | |||

Rotor speed | $N$ | rpm | 3 × 10^{3} |

Rotor mass (disc) | 2m | kg | 68 × 10^{3} |

Rotor length | L_{r} | m | 10 |

Young’s modulus of the rotor (steel) at 20 °C | E_{r} | Pa | 210 × 10^{9} |

Rotor stiffness | k_{r} | N/m | 5 × 10^{6} |

Rotor damping | b_{r} | N·s/m | 0 |

Mass unbalance eccentricities | e | m | 0 |

70 × 10^{−6} | |||

280 × 10^{−6} | |||

Unbalance dynamic loads | meω^{2} | kN | 0 235 940 |

Static load applied per bearing | W_{0} = mg | kN | 340 |

^{*}= 0.439241, Y

^{*}= 0.501445).

**Figure 6.**Journal center trajectory and final position for the case 1 (W

_{0}= 340 kN (${W}_{0}^{*}$ = 1.8), S = 0.18, α

^{*}= 0, ℓ

^{*}= 0, C

_{d}= 0, ε = 0).

**Figure 7.**Comparison of steady-state oil-film pressure and film thickness curves calculated from steady-state and transient analysis. (W

_{0}= 340 kN (${W}_{0}^{*}$ = 1.8), S = 0.18, α

^{*}= 0, ℓ

^{*}= 0 C

_{d}= 0, ε = 0).

^{*}= 0.3272, Y

^{*}= 0.3136). As in the case 1, the discrepancy between the two results is very small.

**Figure 8.**Journal center trajectory and final position for the case 2. (W

_{0}= 2720 kN (${W}_{0}^{*}$ = 14.14), S = 0.0225, α = 17 × 10

^{−9}Pa

^{−1}, ε = 0, ℓ

^{*}= 0.3, t

_{h}= 10 × 10

^{−3}m, E = 0.9 GPa, σ = 0.35).

**Figure 9.**Comparison of nonlinear unbalanced shaft center trajectories for flexible rotor and large unbalance mass (ε = 0.80).

**Figure 10.**Comparison of nonlinear unbalanced shaft center trajectories for flexible rotor and small unbalance mass (ε = 0.20).

#### 6.2. Parametric Study

#### 6.2.1. Flexible Rotor with Large Unbalance Mass

_{0}= 340 kN, t

_{h}= 10

^{−2}m, E = 0.9 GPa, σ = 0.35, b

_{r}= 0, k

_{r}= 5 MN/m, N = 3000 rpm or ω = 100 × π rad/s, and ε = 0.80. Note that for the rigid bearing-liner, the Young’s modulus of liner E tends to infinity and the deformation coefficient ${C}_{d}$ is then equal to zero because it is inversely proportional to the elasticity modulus.

^{2}= 940 kN at synchronous frequency (ν/ω = 1). For this operating condition, the value of W is greater than the static load W

_{0}(W/W

_{0}≈ 3) and is representative of some emergency conditions in turbomachinery when a blade is lost for example.

^{*}= 0.0 (Newtonian case) and l

^{*}= 0.0 (non-Newtonian couple stress fluid), and different values of the piezo-viscosity coefficient ranging from 0.0 (iso-viscous case) to 0.50. In these operating conditions, the rotor has a very large amplitude of circular or pseudo-circular motion and the nonlinear dynamic behavior appears clearly. This is due to the fact that the dynamic loading due to a large unbalance mass is very important compared to the static one as afore-mentioned above.

^{*}= 0 that increasing the pressure coefficient α

^{*}shorten the shaft trajectories. This is due to the pressure rise leading to a higher load carrying capacity which reacts and reduces the trajectories size. We can notice that the operating eccentricity of the journal bearing can be greater than the radial clearance in this case where the unbalance mass is large. Furthermore, the orbits described by the shaft center are widely modified by the bearing compliance especially for the non-Newtonian case as clearly illustrated in Figure 9a. They exactly follow the shape of the deformed bearing.

^{*}on the orbits is weak in both Newtonian and non-Newtonian cases. That means in this case the distortion plays a major role in the bearing response. Moreover, the non-Newtonian orbits are smaller than those obtained in the Newtonian case, i.e., when the couple stress parameter l

**of the lubricating fluid increases.**

^{*}#### 6.2.2. Flexible Rotor with Small Unbalance Mass

^{2}= 235 kN at synchronous frequency (ν/ω= 1) which corresponds to 0.70 times the static load. This defect may be attributed to a large residual unbalance which could exist in the shaft.

^{*}and the piezo-viscosity coefficient. As expected, the shaft center moves around the equilibrium position in both cases since the dynamic load is smaller than the static one. Moreover, the co-ordinates of the equilibrium position change when increasing the piezo-viscosity coefficient especially for higher values of this coefficient in both Newtonian and non-Newtonian cases. So, it results a shift of orbits towards the bearing center as depicted in the figure.

Parameter | Symbol | Value |
---|---|---|

Number of elements in the circumferential direction of bearing | N_{θ} | 30 |

Number of elements in the axial direction of half-bearing | N_{z} | 10 |

Number of Gauss integration points in ξ-direction | - | 2 |

Number of Gauss integration points in η-direction | 2 | |

Under-relaxation factor for the substitution iterative method | Ω_{0} | 0.10 |

Convergence criterion of the substitution iterative method | ε_{p} | 10^{−5} |

Dimensionless time increment | Δ${t}^{*}=\omega \u2206t$ | $2\pi /50$ |

Time limit | ${{t}^{*}}_{max}$ | 40π ^{†} |

^{†}This value corresponds to 20 revolutions of the shaft.

## 7. Conclusions

## Acknowledgments

## Author Contributions

## Nomenclature

$C$ | bearing radial clearance, m |

$E$ | Young’s modulus of the bearing-liner, Pa |

e | unbalance eccentricity, m |

${F}_{X},\hspace{0.17em}{F}_{Y}$ | lift force components, N |

$h$ | fluid-film thickness, m |

${h}^{*}$ | dimensionless fluid-film thickness $=\frac{h}{C}$, |

${h}_{0}$ | static fluid-film thickness, m |

${h}_{0}^{*}$ | dimensionless static fluid-film thickness $=\frac{{h}_{0}}{C}$, |

${k}_{r}$ | rotor stiffness, ${\scriptscriptstyle \raisebox{1ex}{$N$}\!\left/ \!\raisebox{-1ex}{$m$}\right.}$ |

${b}_{r}$ | rotor damping, ${\scriptscriptstyle \raisebox{1ex}{$N.s$}\!\left/ \!\raisebox{-1ex}{$m$}\right.}$ |

$L$ | length of bearing, m |

$\ell $ | couple-stress parameter=$\sqrt{{\scriptscriptstyle \raisebox{1ex}{$\eta $}\!\left/ \!\raisebox{-1ex}{${\mu}_{0}$}\right.}}$, m |

${\ell}^{*}$ | dimensionless couple-stress parameter=${\scriptscriptstyle \raisebox{1ex}{$\ell $}\!\left/ \!\raisebox{-1ex}{$C$}\right.}$, |

$\complement $ | scalar compliance operator, ${\scriptscriptstyle \raisebox{1ex}{${m}^{3}$}\!\left/ \!\raisebox{-1ex}{$N$}\right.}$ |

$m$ | mass of rotor per bearing, kg |

N | rotation velocity of the rotor, rpm |

$p$ | fluid-film pressure, Pa |

${p}^{*}$ | normalized film pressure $=\frac{p}{{\mu}_{0}\omega {\left({\scriptscriptstyle \raisebox{1ex}{$R$}\!\left/ \!\raisebox{-1ex}{$C$}\right.}\right)}^{2}}$, |

${p}_{0}$ | steady-state pressure, Pa |

$R$ | journal radius, m |

S | Sommerfeld number,$=\frac{{\mu}_{0}\omega RL{\left({\scriptscriptstyle \raisebox{1ex}{$R$}\!\left/ \!\raisebox{-1ex}{$C$}\right.}\right)}^{2}}{\pi {W}_{0}}$ |

$t$ | time, s |

${t}^{*}$ | dimensionless time$=\omega t$, |

${t}_{h}$ | thickness of bearing-liner, m |

${W}_{0}=mg$ | static load applied on the journal bearing |

X,Y | displacement components of the journal centre, m |

${X}^{*},\hspace{0.17em}{Y}^{*}$ | dimensionless displacements$=\frac{\left(X,Y\right)}{C}$, |

$z$ | axial co-ordinate measured from middle section plane of the bearing, m |

${z}^{*}$ | non-dimensional axial co-ordinate$=\frac{z}{L}$, |

$\alpha $ | pressure-viscosity coefficient, Pa ^{−1} |

${\alpha}^{*}$ | non-dimensionless pressure-viscosity coefficient$={\mu}_{0}\omega {\left({\scriptscriptstyle \raisebox{1ex}{$R$}\!\left/ \!\raisebox{-1ex}{$C$}\right.}\right)}^{2}\alpha $, |

$\epsilon $ | unbalance eccentricity ratio, $\epsilon =\frac{e}{C}$ |

$\eta $ | material constant responsible for couple-stresses, kg·m·s ^{−1} |

$\mu $ | absolute viscosity of lubricant, Pa·s |

${\mu}_{0}$ | absolute viscosity at atmospheric pressure, Pa·s |

$\sigma $ | Poisson’s ratio of the bearing-liner |

$\mathrm{\theta}$ | bearing angle, rad |

$\omega $ | angular velocity of the rotor (shaft)=${\scriptscriptstyle \raisebox{1ex}{$2\pi N$}\!\left/ \!\raisebox{-1ex}{$60$}\right.}$, rad/s |

$[\u2022]$ | square matrix |

$\langle \u2022\rangle $ | line vector, =${\{\u2022\}}^{T}$ |

$\{\u2022\}$ | column vector |

${(\u2022)}^{*}$ | dimensionless quantity |

$\overrightarrow{\nabla}\cdot $ | divergence operator |

## Appendix A

#### Computation Procedure of the Journal Bearing Nonlinear Dynamic Response

**Step 1**: At time ${t}^{*}=0$ , for an initial position of the shaft center $\left\{{U}_{0}\right\}=\left\{\begin{array}{c}{X}_{0}^{*}\\ {Y}_{0}^{*}\end{array}\right\}$ and initial velocities $\left\{U{\prime}_{0}\right\}=\left\{\begin{array}{c}{X}_{0}^{*\prime}\\ {Y}_{0}^{*\prime}\end{array}\right\}=\left\{\begin{array}{c}0\\ 0\end{array}\right\}$, we solve the set of nonlinear algebraic equations resulting from discretization of the normalized steady-state modified Reynolds Equation (21) (i.e., without the transient term) by the finite element method using the relaxed substitution iterative method in order to obtain the hydrodynamic pressure field ${p}_{0}^{*}$ and the film thickness distribution ${h}_{0}^{*}$, and by integration of this latter over the bearing surface, we get the oil film force components:

**Step 2**: The acceleration components of the shaft center $\left\{U\prime {\prime}_{0}\right\}=\left\{\begin{array}{c}{X}_{0}^{*\prime \prime}\\ {Y}_{0}^{*\prime \prime}\end{array}\right\}$ are then determined from dynamic Equation (27).

**Step 3**: The new position together with the velocity components of the journal center are predicted at dimensionless time $\left(\tilde{t}+\u2206\tilde{t}\right)$ from the explicit Euler’s scheme, that is:

**Step 4**: The new position and velocities allow us to calculate new pressure and film thickness distributions ${p}_{{t}^{*}+\u2206{t}^{*}}^{*}$ and ${h}_{{t}^{*}+\u2206{t}^{*}}^{*}$ by solving the first order differential equations system resulting from the discretization of the transient modified Reynolds’ equation by means of the implicit Euler’s scheme (β = 1) and the substitution iterative method. Thus, new components of the hydrodynamic (lift) force$\left\{\begin{array}{c}{F}_{X}^{*}\left({t}^{*}+\u2206{t}^{*}\right)\\ {F}_{Y}^{*}\left({t}^{*}+\u2206{t}^{*}\right)\end{array}\right\}$ can be calculated through integration of ${p}_{{t}^{*}+\u2206{t}^{*}}^{*}$.

**Step 5**: The acceleration components at time $\left({t}^{*}+\u2206{t}^{*}\right)$ are then calculated from dynamic Equations (27) considering the new values of the lift force components ${F}_{X}^{*}\left({t}^{*}+\u2206{t}^{*}\right)$ and ${F}_{Y}^{*}\left({t}^{*}+\u2206{t}^{*}\right)$.

**Step 6**: The next time interval is then considered while ${t}^{*}<{t}_{max}^{*}$, the values obtained earlier play the role of $\left\{\begin{array}{c}{X}_{0}^{*}\\ {Y}_{0}^{*}\end{array}\right\}$ and $\left\{\begin{array}{c}{X}_{0}^{*\prime}\\ {Y}_{0}^{*\prime}\end{array}\right\}$ in

**Step 3**.

## Appendix B

#### Method of Solution for Steady-State Nonlinear Modified Reynolds’ Equation

**Step 1**: Select the input parameters of the problem ${X}_{0}^{*}{Y}_{0}^{*},\lambda ,\hspace{0.17em}\alpha ,{t}_{h}^{*},{C}_{d}^{*},\hspace{0.17em}\sigma ,\hspace{0.17em}{\ell}^{*}$, under-relaxation factor ${\mathrm{\Omega}}_{0}$whose the value ranges from 0 to 1, convergence criterion ${\epsilon}_{p}$ and maximum number of iterations, ${k}_{\mathrm{max}}$ for the steady-state pressure solution.

**Step 2**: Initialize the iteration number k to 0, the norm $\Vert n\Vert $to 1, and the global vector containing nodal dimensionless steady-state pressures $\left\{{P}_{0}^{\left(k\right)}\right\}=0$.

**Step 3**:

**Step 4**: Apply the Gümbel’s rupture film conditions by vanishing all the negative terms of calculated pressure.

**Step 5**: Calculate the steady-state lift force components using Equation (35).

## Appendix C

#### Iterative Research of the Steady-State Equilibrium Position

**Step 1**. Select the input parameters of the procedure which are:

**Step 2**. Solve either the steady-state HD lubrication problem governed by Equation (31a) for ${p}_{0}^{*}$, or the steady-state EHD lubrication problem described by the coupled Equations (31a) and (31b) by iterations for ${h}_{0}^{*}\mathrm{and}{p}_{0}^{*}$.

**Step 3**. Calculate the steady-state hydrodynamic lift components:$\left\{\begin{array}{c}{F}_{{X}_{0}}^{*}\\ {F}_{{Y}_{0}}^{*}\end{array}\right\}={\displaystyle \underset{-{\scriptscriptstyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{\overset{{\scriptscriptstyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{\int}}{\displaystyle \underset{0}{\overset{2\pi}{\int}}{p}_{0}^{*}\left(\theta ,{z}^{*}\right)\left\{\begin{array}{c}\mathrm{cos}\theta \\ \mathrm{sin}\theta \end{array}\right\}d\theta d{z}^{*}}}$

**Step 4**. Calculate the residual components: ${W}_{X}^{*}and{W}_{Y}^{*}$ Equation (C1).

**Step 5**. Evaluate Jacobean matrix coefficients by numerical differentiation:

^{*}= 10

^{−6}.

**Step 6**. Calculate the corrections $\left(\delta {X}_{k}^{*},\delta {Y}_{k}^{*}\right)$ by using Equation (C5).

**Step 7**. Calculate the new approximations of the nonlinear system:$\left\{\begin{array}{c}{X}_{k+1}^{*}\\ {Y}_{k+1}^{*}\end{array}\right\}=\left\{\begin{array}{c}{X}_{k}^{*}\\ {Y}_{k}^{*}\end{array}\right\}+\mathrm{\Omega}\left\{\begin{array}{c}\delta {X}_{k}^{*}\\ \delta {Y}_{k}^{*}\end{array}\right\}$

**Step 8**. If $\sqrt{\langle \begin{array}{cc}{W}_{X}^{{*}^{\left(k+1\right)}}& {W}_{Y}^{{*}^{\left(k+1\right)}}\end{array}\rangle \left\{\begin{array}{c}{W}_{X}^{{*}^{\left(k+1\right)}}\\ {W}_{Y}^{{*}^{\left(k+1\right)}}\end{array}\right\}}\le {\epsilon}_{1}and\hspace{1em}k{k}_{\mathrm{max}}$, the convergence is reached, i.e., the values of $\left({X}_{k+1}^{*},{Y}_{k+1}^{*}\right)$ correspond to the co-ordinates of equilibrium position $\left({X}_{0}^{*},{Y}_{0}^{*}\right)$ which results from the applied load W

_{0}.

## Conflicts of Interest

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Lahmar, M.; Bou-Saïd, B. Nonlinear Dynamic Response of an Unbalanced Flexible Rotor Supported by Elastic Bearings Lubricated with Piezo-Viscous Polar Fluids. *Lubricants* **2015**, *3*, 281-310.
https://doi.org/10.3390/lubricants3020281

**AMA Style**

Lahmar M, Bou-Saïd B. Nonlinear Dynamic Response of an Unbalanced Flexible Rotor Supported by Elastic Bearings Lubricated with Piezo-Viscous Polar Fluids. *Lubricants*. 2015; 3(2):281-310.
https://doi.org/10.3390/lubricants3020281

**Chicago/Turabian Style**

Lahmar, Mustapha, and Benyebka Bou-Saïd. 2015. "Nonlinear Dynamic Response of an Unbalanced Flexible Rotor Supported by Elastic Bearings Lubricated with Piezo-Viscous Polar Fluids" *Lubricants* 3, no. 2: 281-310.
https://doi.org/10.3390/lubricants3020281