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Abstract

A flexible rotor supported by two journal bearings under nonlinear suspension and the lubricating oil is assumed to be temperature dependent in this study. Analytical tools were inclusive of bifurcation diagrams, dynamic trajectories, power spectra, Poincaré map, fractal dimension, and Lyapunov exponent are used to verify the dynamic characteristics of the rotor-bearing system, and abundant periodic, subharmonic, quasi-periodic, and even chaotic motions are found in this study. From the simulation results, the higher operating temperature will also cause the dynamic trajectory of the system to become non-periodic vibration. The rotor-bearing system operating in a non-periodic motion state may lead to possible broadband vibration with large amplitude and increase the possibility of fatigue or failure of the system. Such simulation analysis will help engineers avoid unnecessary trial and error when designing and applying rotor-bearing systems.

Keywords

Temperature-dependent viscosity bifurcation rotor-bearing

Introduction

Turbo-machineries play a vital role in modern industrial applications, whether in vehicles, machine tools, or other industrial technology applications. Therefore, the stable operating and long service life of turbo-machineries will enable the continuous development of these industrial applications. It is well known that as the operating speed of rotating machinery increases, the system’s operating temperature will also increase. However, the viscosity of the lubricating fluid in the turbo-machineries is quite sensitive to the temperature. When the temperature is too low, the turbo-machineries may fail to rotate, and when the temperature is too high, the lubrication may fail. Therefore, to gain a more comprehensive understanding of the dynamic characteristics of the systems, it will be helpful to consider temperature effects in the dynamic characteristic analysis of turbo-machineries. Jain et al.¹ analyze and compare the performance of the laminar and turbulent regimes in circular bearings considering viscosity variation. The pressure–viscosity coefficients are the focus of their analysis. Clarke et al.² propose a model to simulate the floating ring-bearing system considering the thermal effect. The steady-state model is established to analyze in their study. San et al.³ introduced a thermo-hydrodynamic model to analyze the performance characteristics of cryogenic liquid annular seals in the turbulent flow regime. They prove that the temperature effect is undoubtedly significant in the study of the performance analysis of those systems, especially at very high rotating speeds. Taylor⁴ present the short-bearing approximation model, including lubricant shear thinning, that is, the non-Newtonian model. He also mentions that modern automotive lubricants are multigrade oils containing polymeric additives at treatment rates of several percent, and it will lead to the lubricant viscosity having a significant dependence on the shear rate. San et al.⁵ provide a helpful physical model and fast computational programs to improve the design and performance of rotodynamic. The simulation results agree well with the measurements, and the dynamic stability analysis is also studied. Bukovnik et al.⁶ propose a modified thermal-elasto-hydrodynamic lubrication model for journal bearing, considering shear rate-dependent viscosity. Kadam et al.⁷ perform an analysis of thermal hydrodynamic analysis of plain journal bearing with modified viscosity–temperature. They specify that thermal effects should be considered when the bearing is highly loaded. Torgal et al.⁸ present a modified model to analyze the thermo-hydrodynamic of journal bearing to find out the equivalent temperature. Bagul et al.⁹ use CFD software to simulate the performance of the journal bearing considering with temperature effect. They validate the situation in which the temperature rise will cause instability in the system’s operation.

It is said that the viscosity coefficient in the lubricating fluid is not only a function of pressure but also a function of temperature. However, in the general study of the dynamic characteristics of the rotor-bearing system, they assume that the viscosity coefficient is a constant value, so there is a possibility of underestimating the effect of the viscosity coefficient. This study takes that a rigid rotor supported by two journal bearings under nonlinear suspension, nonlinear oil film force, and lubricating oil is temperature dependent. Analytical tools, including bifurcation diagrams, dynamic trajectories, power spectra, Poincaré map, fractal dimension, and Lyapunov exponent, are used to verify the dynamic characteristics of the rotor-bearing system.

Mathematical modeling

Figure 1 shows a rigid rotor supported by journal bearing under nonlinear suspension and the lubricating oil is assumed to be temperature dependent in this study. According to Newton’s second law, the dynamic equations considering with nonlinear suspension (i.e., hard spring) for the bearing geometric center O_b can be presented

{\begin{cases} M_{b} \ddot{X_{b}} + C_{b} \dot{X_{b}} + K_{1} X_{b} + K_{2} {X_{b}}^{3} - F_{x} = 0 \\ M_{b} \ddot{Y_{b}} + C_{b} \dot{Y_{b}} + K_{1} Y_{b} + K_{2} {Y_{b}}^{3} - F_{y} = - M_{b} g \end{cases}

(1)

where

M_{b}

is the mass of the bearing and

C_{b}

is the damping coefficient of the bearing.

K_{1}

and

K_{2}

are the first- and third-order stiffness, respectively, to present nonlinear suspension effect.

Figure 1.

Rotor-bearing system supported by nonlinear suspension and considering temperature-dependent viscosity.

Dynamic equations used to illustrate the geometric center of the rotor O_r can be performed

{\begin{cases} M_{r} \ddot{X_{r}} + C_{r} \dot{X_{r}} + K_{s} (X_{r} - X_{j}) = M_{r} ρ ω^{2} \cos ϕ \\ M_{r} \ddot{Y_{r}} + C_{r} \dot{Y_{r}} + K_{s} (Y_{r} - Y_{j}) = M_{r} ρ ω^{2} \sin ϕ - M_{r} g \end{cases}

(2)

where

M_{r}

is the mass of the rotor,

C_{r}

is the damping coefficient of the shaft,

K_{s}

is the stiffness of the shaft,

ρ

is the mass eccentricity of the rotor,

ω

is the rotational speed of the rotor, and

ϕ

is rotational angle (ϕ = ωt).

O_j is defined as the journal geometric center, and then the nonlinear oil film forces in the horizontal and vertical directions can be expressed from the force equilibrium in each direction

{\begin{cases} F_{x} = f_{e} \cos φ - f_{φ} \sin φ = \frac{K_{s} (X_{r} - X_{j})}{2} \\ F_{y} = f_{e} \sin φ + f_{φ} \cos φ = \frac{K_{s} (Y_{r} - Y_{j})}{2} \end{cases}

(3)

The temperature-dependent viscosity could be performed as⁴

μ = μ_{\infty} + \frac{μ_{0} - μ_{\infty}}{1 + \frac{γ}{γ_{c}}} = r μ_{0} + \frac{μ_{0} - r μ_{0}}{1 + \frac{γ}{γ_{c}}}

(4)

where

\log_{10} γ_{c} = A + B T

\frac{μ_{\infty}}{μ_{0}} = r

(the ratio r is defined in the region of

0.5 < r < 1.0

μ_{0} = ϑ \exp (\frac{T_{1}}{T_{2} + T})

h = c (1 + ε \cos θ)

, and

γ = \frac{U}{h} = \frac{R ω}{c (1 + ε \cos θ)}

. For SAE-15w/40,

r = 0.8

A = 2.5

B = 0.0225

ϑ = 0.0388

T_{1} = 1372.5 ℃

, and

T_{2} = 134.8 ℃

\log_{10} γ_{c} = 2.5 + 0.0225 T

γ_{c} = 10^{2.5 + 0.0225 T}

and

\frac{μ_{\infty}}{μ_{0}} = 0.8

. Thus

μ_{0} (T) = 0.0388 \exp (\frac{1372.5}{134.8 + T})

(5)

The temperature-dependent viscosity can be introduced by getting equation (5) into equation (4), as shown below

μ (T, θ) = 0.8 μ_{0} (T) + \frac{0.2 μ_{0} (T)}{1 + \frac{\frac{R ω}{c (1 + ε \cos θ)}}{10^{2.5 + 0.0225 T}}}

(6)

Apparently, we could find that the viscosity is not only a function of temperature but also a function of $θ$ .

The modified Reynolds equation with the short-bearing approximation (i.e., L/D < 0.25、∂p/∂θ << ∂p/∂z, and then we can set ∂p/∂θ = 0.) could be performed as

\frac{\partial^{2} p}{\partial z^{2}} = - \frac{6 μ (T, θ) c}{h^{3}} [(ω - 2 \dot{φ}) ε \sin θ - 2 \dot{ε} \cos θ]

(7)

The pressure distribution with the boundary conditions ${\begin{cases} \frac{\partial p}{\partial z} = 0, z = 0 \\ p = 0, z = \pm \frac{L}{2} \end{cases}$ will be

p (θ, z, t) = - \frac{3 c}{h^{3}} μ (T, θ) [(ω - 2 \dot{φ}) ε \sin θ - 2 \dot{ε} \cos θ] (z^{2} - \frac{L^{2}}{4})

(8)

The oil film force in the radial and tangential direction could be presented by integrating the function $p (θ, z, t)$ both with $θ$ and z. i.e.

{\begin{cases} f_{e} = \frac{L^{3} R}{{2 c}^{2}} \int_{0}^{π} \frac{μ (T, θ) [(ω - 2 \dot{φ}) ε \sin θ - 2 \dot{ε} \cos θ]}{{(1 + ε \cos θ)}^{3}} \cos θ d θ \\ f_{φ} = - \frac{L^{3} R}{{2 c}^{2}} \int_{0}^{π} \frac{μ (T, θ) [(ω - 2 \dot{φ}) ε \sin θ - 2 \dot{ε} \cos θ]}{{(1 + ε \cos θ)}^{3}} \sin θ d θ \end{cases}

(9)

Let

Π_{0} = - \frac{L^{3} R}{{2 c}^{2}}

Π_{1} = - \int_{0}^{π} \frac{μ (T, θ) ω ε \sin θ \cos θ}{{(1 + ε \cos θ)}^{3}} d θ

Π_{2} = \int_{0}^{π} \frac{2 μ (T, θ) ε \sin θ \cos θ}{{(1 + ε \cos θ)}^{3}} d θ

Π_{3} = \int_{0}^{π} \frac{2 μ (T, θ) \cos θ \cos θ}{{(1 + ε \cos θ)}^{3}} d θ

Π_{4} = \int_{0}^{π} \frac{μ (T, θ) ω ε \sin θ \sin θ}{{(1 + ε \cos θ)}^{3}} d θ

Π_{5} = \int_{0}^{π} \frac{- 2 μ (T, θ) ε \sin θ \sin θ}{{(1 + ε \cos θ)}^{3}} d θ

, and

Π_{6} = \int_{0}^{π} \frac{- 2 μ (T, θ) \cos θ \sin θ}{{(1 + ε \cos θ)}^{3}} d θ

and equation (6) could be simplified as follows

{\begin{cases} f_{e} = Π_{0} (Π_{1} + Π_{2} \dot{φ} + Π_{3} \dot{ε}) \\ f_{φ} = Π_{0} (Π_{4} + Π_{5} \dot{φ} + Π_{6} \dot{ε}) \end{cases}

(10)

Substituting equation (10) into equation (3) and rearranging equations for the oil film forces by considering temperature effect in the horizontal and vertical direction. Meanwhile, let $Π_{7} = \frac{K_{s} (X_{r} - X_{b} - e c o s φ)}{2} \cos φ$ + $\frac{K_{s} (Y_{r} - Y_{b} - e s i n φ)}{2} \sin φ$ , $Π_{8} = \frac{K_{s} (Y_{r} - Y_{b} - e s i n φ)}{2} \cos φ$ - $\frac{K_{s} (X_{r} - X_{b} - e c o s φ)}{2} \sin φ$ and then

{\begin{cases} Π_{0} Π_{2} \dot{φ} + Π_{0} Π_{3} \dot{ε} = Π_{7} - {Π_{0} Π}_{1} \\ {Π_{0} Π}_{5} \dot{φ} + Π_{0} Π_{6} \dot{ε} = Π_{8} - Π_{0} Π_{4} \end{cases}

(11)

{\begin{cases} \dot{φ} = \frac{(Π_{7} - {Π_{0} Π}_{1}) Π_{0} Π_{6} - (Π_{8} - Π_{0} Π_{4}) Π_{0} Π_{3}}{{Π_{0}}^{2} Π_{2} Π_{6} - {Π_{0}}^{2} Π_{3} Π_{5}} \\ \dot{ε} = \frac{(Π_{8} - Π_{0} Π_{4}) Π_{0} Π_{2} - (Π_{7} - {Π_{0} Π}_{1}) Π_{0} Π_{5}}{{Π_{0}}^{2} Π_{2} Π_{6} - {Π_{0}}^{2} Π_{3} Π_{5}} \end{cases}

(12)

Equations (1), (2), and (12) are the nonlinear dynamic equations used to simulate the dynamic responses of the rotor-bearing system considering temperature-dependent viscosity in this study and then integrating them to be the following dimensionless equations

{\begin{cases} x_{b}^{″} + \frac{2 c_{1}}{a_{1}} x_{b}^{} + \frac{1}{{a_{1}}^{2}} x_{b} + \frac{α}{a^{2}} {x_{b}}^{3} - \frac{1}{2 r_{b r} a^{2}} (x_{r} - x_{b} - ε \cos φ) = 0 \\ y_{b}^{″} + \frac{2 c_{1}}{a_{1}} y_{b}^{} + \frac{1}{{a_{1}}^{2}} y_{b} + \frac{α}{a^{2}} {y_{b}}^{3} - \frac{1}{2 r_{b r} a^{2}} (y_{r} - y_{b} - ε \sin φ) + \frac{f_{b}}{a^{2}} = 0 \end{cases}

(13)

{\begin{cases} x_{r}^{″} + \frac{2 c_{2}}{a} x_{r}^{} + \frac{1}{a^{2}} (x_{r} - x_{b} - ε \cos φ) = β \cos ϕ \\ y_{r}^{″} + \frac{2 c_{2}}{a} y_{r}^{} + \frac{1}{a^{2}} (y_{r} - y_{b} - ε \sin φ) = β \sin ϕ - \frac{f_{r}}{a^{2}} = 0 \end{cases}

(14)

{\begin{cases} φ^{'} = \frac{(Π_{7} - {Π_{0} Π}_{1}) Π_{0} Π_{6} - (Π_{8} - Π_{0} Π_{4}) Π_{0} Π_{3}}{{Π_{0}}^{2} Π_{2} Π_{6} - {Π_{0}}^{2} Π_{3} Π_{5}} \\ ε^{'} = \frac{(Π_{8} - Π_{0} Π_{4}) Π_{0} Π_{2} - (Π_{7} - {Π_{0} Π}_{1}) Π_{0} Π_{5}}{{Π_{0}}^{2} Π_{2} Π_{6} - {Π_{0}}^{2} Π_{3} Π_{5}} \end{cases}

(15)

where

x_{i} = \frac{X_{i}}{c} (i = b, r & j),

c is the clearance of oil film,

ϕ = ω t

\frac{d}{d t} = ω \frac{d}{d ϕ}

Ω^{2} = \frac{K_{s}}{M_{r}}, a^{2} = \frac{ω^{2}}{Ω^{2}}

β = \frac{M_{r} ρ ω^{2}}{M_{r} c ω^{2}}

f_{b} = \frac{M_{r} g}{K_{s} δ}

c_{1} = \frac{C_{b}}{2 \sqrt{K_{1} M_{b}}}

c_{2} = \frac{C_{r}}{2 \sqrt{K_{s} M_{r}}}

R_{m} = \frac{M_{b}}{M_{r}}

R_{k} = \frac{K_{s}}{K_{1}}

{a_{1}}^{2} = R_{m} R_{k} a^{2}

, and

α = \frac{K_{2} c^{2}}{K_{s} R_{m}}

Results and discussions

The fourth-order Runge–Kutta method is a valuable tool for solving autonomous differential equations, and we use it to simulate dimensionless nonlinear dynamic equations for this reason. The time step for the calculation is π/300, and the error tolerance is less than 0.0001. The time series data of the first 800 revolutions are not used for dynamic behavior investigation to guarantee that the data used are for a steady state. Also, 30000 data points from the bearing and rotor geometric center displacement time series are used to generate the attractors in embedding space. These data generate the dynamic orbit, power spectrum, Poincaré map, and bifurcation diagram. The bifurcation analysis including supercritical or subcritical bifurcations are well studied by some researchers. Some excellent studies such as Li et al.¹⁰ studied coherence resonance and stochastic bifurcation behaviors of simplified standing-wave thermoacoustic systems. Sun et al.¹¹ analyzed lean premixed swirling combustion systems. Due to the fluid flow-acoustics-combustion interaction in this system, such combustion systems are more susceptible to nonlinear combustion instability. Li and his team also presented experimental and theoretical bifurcation study of a nonlinear standing-wave thermoacoustic system.¹² Therefore, related supercritical or subcritical bifurcation analysis and application may refer the above literatures. The fractal dimension and maximum Lyapunov exponent are also used to identify nonlinear emotional responses, and according to Nayfeh,¹³ the optimum delay time was found by the autocorrelation function to be about one-third of a revolution of the rotor. Thus, the fractal dimension of the system can be determined by using the plot of the log value of the correlation function versus the log value of the radius of an N-dimensional hyper-sphere, that is, (log c(r)) versus (log r).¹⁴ As for the fractal dimension, Smith¹⁵ defined the greatest integer M, which is less than the fractal dimension of the attracting set, and the number of points used to estimate the size should be less than 42^M. In this study, we take the dimensionless rotational speed ratio s, unbalanced parameter β, and the dimensionless damping ratio c₁ as the bifurcation control parameters. The following values are the non-dimensional parameters used in this study: $β = 0.45$ , $f_{b} = 0.0972$ , $c_{1} = 0.0525$ , $c_{2} = 0.026$ , $R_{m} = 0.2$ , $R_{k} = 2.0$ , ${a_{1}}^{2} = 0.4 a^{2}$ , and $α = 0.1$ . The initial values of the rotor-bearing system are defined as follows: $x_{b} (0) = 0.2, x_{b}^{,} (0) = 0.00000001$ , $y_{b} (0) = 0.4, x_{b}^{,} (0) = 0.00000001$ , $x_{r} (0) = 0.5, x_{b}^{,} (0) = 0.00000001$ , and $y_{b} (0) = 0.2, x_{b}^{,} (0) = 0.00000001$ .

In Figure 2, the bifurcation diagram is for the bearing geometric center with dimensionless rotational speed ratio s (s=0.01∼6.00) at temperature T=25°C. We found that the bearing geometric center performs aperiodic motions with large amplitude at low rotating speeds, that is, $s \leq 0.3$ . Still, the dynamic behaviors become periodic with the increasing of rotating states also pick two dimensionless rotational speed ratios, for example, $s = 0.2$ and $s = 0.3,$ to show the orbit and Poincaré map. Its dynamic behavior is unstable in the orbit and closed circle, which is also found in the Poincaré map at s = 0.2, so it’s obviously quasi-periodic. The dynamic behavior is periodic for $s = 0.3$ because a circle and a single return point can be found in the orbit and Poincaré map, respectively. The bifurcation diagram of the rotor geometric center using dimensionless rotational speed ratio s (s = 0.01∼6.00) for temperature T = 25°C is shown in the Figure 3. The dynamic orbit behaves almost the same as the bearing case, except for the amplitude is much smaller at low rotating speeds. The range of the disorder of the dynamic behaviors gradually expands as the temperature rises, such as the Figure 4. The dynamic behaviors perform non-periodic motions before the rotating speed $s \leq 2.6$ and become periodic and order at high rotating speeds. The bifurcation diagram shown in Figure 4 is for the bearing geometric center with dimensionless rotational speed ratio s at temperature T = 90°C. In order to more clearly illustrate the dynamic characteristics of the system at high and low speeds, we especially choose the dynamic orbit, Poincaré map, and power spectrum of the bearing geometric at dimensionless rotational speed ratio s = 1.0, 1.2, 1.8, 2.4, 2.6, and 3.0 to describe the difference among them as shown in the Figure 5. They are obviously quasi-periodic responses at s = 1.0, 1.2, 1.8, and 2.6, but it is chaotic at s = 2.4 for the reason of the return points in the Poincaré map is a geometric structure. We also use the fractal dimension and maximum Lyapunov exponent to reinforce the specification and verify chaotic motions, that is, Figure 6. The fractal dimension is 1.17 and the maximum Lyapunov exponent is positive, so we prove again that the dynamic response is a chaotic motion at s = 2.4. Then let us observe the Figure 7 which is the bifurcation diagram of the rotor geometric center using dimensionless rotational speed ratio s as control parameter for temperature T = 90°C. We can also find that the dynamic responses are quasi-periodic or chaotic motions with relatively low rotating speeds, that is, $s \leq 2.6$ . And soon afterward, it returns to periodic ones with the increasing rotating speeds. A geometric structure is also found in the Poincaré map at s = 2.4. The results prove that the bearing and rotor geometric center dynamic responses synchronously perform chaotic motions for s = 2.4 and T = 90°C. The disorder area is further expanded to $0.01 < s \leq 2.8$ for the temperature rise to 100°C by observing bifurcation diagrams shown in Figures 8 and 9. Until the temperature rises to 120°C, the dynamic responses are almost disorder in all of the range of rotating speeds (see Figures 10 and 11). Figures 12 and 13 are bifurcation diagrams of the bearing geometric center and rotor geometric center using unbalanced parameter β as the bifurcation control parameter (β = 0.01∼0.88) for s = 2.4 and temperature T = 90°C. It behaves in a chaotic motion for the bearing and rotor geometric center for s = 2.4 and temperature T = 90°C from the above simulation results, so it means that the unbalanced parameters must be greater to suppress the aperiodic motions. Figures 12 and 13 both prove that the dynamic behaviors are aperiodic motions and then they are periodic ones when $β \geq 0.36$ . Apart from dimensionless rotational speed ratio s and unbalanced parameters β are significant bifurcation control parameters, the dimensionless damping ratio is also a critical and useful bifurcation control parameter to analyze dynamic responses of the rotor-bearing system. Therefore, the bifurcation diagrams of the bearing and rotor geometric center using dimensionless damping ratio c₁ as the bifurcation control parameter (c₁ = 0.01∼1.00) for s = 0.2 and temperature T = 25°C are introduced in the Figures 14 and 15. We can find abundant quasi-periodic or even chaotic motions at low rotating speed s = 0.2 and low temperature T = 25°C with dimensionless damping ratio c₁ nearly approximate 0.9. The result shows that the dynamic responses hardly escape aperiodic motions with low rotating speed even for low working temperature. Figures 16 and 17 are the diagrams using damping ratio as bifurcation control parameter at rotating speed s = 0.2 and higher temperature T = 80°C. The vibration amplitude shown in the figure is greater than in low-temperature cases and the dynamic response cannot escape non-periodic motions even at higher damping ratio. Therefore, it is obvious from the results that the dynamic response of the system is deeply affected by temperature.

Figure 2.

Bifurcation diagram of the bearing geometric center using dimensionless rotational speed ratio s (s = 0.01∼6.00) for temperature T = 25°C.

Figure 3.

Bifurcation diagram of the rotor geometric center using dimensionless rotational speed ratio s (s = 0.01∼6.00) for temperature T = 25°C

Figure 4.

Bifurcation diagram of the bearing geometric center using dimensionless rotational speed ratio s (s = 0.01∼5.50) for temperature T = 90°C.

Figure 5.

Dynamic orbit, Poincaré map, and power spectrum of the bearing geometric at dimensionless rotational speed ratio s = 1.0, 1.2, 1.8, 2.4, 2.6, and 3.0 for temperature T = 90°C.

Figure 6.

Fractal dimension and Maximum Lyapunov Exponent of the bearing geometric at dimensionless rotational speed ratio s = 2.4 for temperature T = 90°C.

Figure 7.

Bifurcation diagram of the rotor geometric center using dimensionless rotational speed ratio s (s=0.01∼5.50) for temperature T=90°C.

Figure 8.

Bifurcation diagram of the bearing geometric center using dimensionless rotational speed ratio s (s=0.01∼6.00) for temperature T=100°C.

Figure 9.

Bifurcation diagram of the rotor geometric center using dimensionless rotational speed ratio s (s = 0.01∼6.00) for temperature T = 100°C.

Figure 10.

Bifurcation diagram of the bearing geometric center using dimensionless rotational speed ratio s (s = 0.01∼6.00) for temperature T = 120°C.

Figure 11.

Bifurcation diagram of the rotor geometric center using dimensionless rotational speed ratio s (s = 0.01∼6.00) for temperature T = 120°C.

Figure 12.

Bifurcation diagram of the bearing geometric center using unbalanced parameter β as the bifurcation control parameter (β = 0.01∼0.88) for s = 2.4 and temperature T = 90°C.

Figure 13.

Bifurcation diagram of the rotor geometric center using unbalanced parameter β as the bifurcation control parameter (β = 0.01∼0.88) for s = 2.4 and temperature T = 90°C.

Figure 14.

Bifurcation diagram of the bearing geometric center using dimensionless damping ratio c₁ as the bifurcation control parameter (c₁ = 0.01∼1.00) for s = 0.2 and temperature T = 25°C.

Figure 15.

Bifurcation diagram of the rotor geometric center using dimensionless damping ratio c₁ as the bifurcation control parameter (c₁ = 0.01∼1.00) for s = 0.2 and temperature T = 25°C.

Figure 16.

Bifurcation diagram of the bearing geometric center using dimensionless damping ratio c₁ as the bifurcation control parameter (c₁ = 0.01∼1.00) for s = 0.2 and temperature T = 80°C.

Figure 17.

Bifurcation diagram of the rotor geometric center using dimensionless damping ratio c₁ as the bifurcation control parameter (c₁ = 0.01∼1.00) for s = 0.2 and temperature T = 80°C.

Discussion and conclusions

As we all know, the viscosity coefficient in lubricating fluid is not only a function of pressure but also a function of temperature. However, in the general study of the dynamic characteristics of the rotor-bearing system, they assume that the viscosity coefficient is a constant value, so there is a possibility of underestimating the effect of the viscosity coefficient. In this study, a rigid rotor supported by two journal bearings under nonlinear suspension and the lubricating oil is assumed to be temperature dependent in this study. Analytical tools inclusive of bifurcation diagrams, dynamic trajectories, power spectra, Poincaré map, fractal dimension, and Lyapunov exponent are used to verify the dynamic characteristics of the rotor-bearing system, and abundant periodic, subharmonic, quasi-periodic, and even chaotic motions are found in this study.

We hope that the rotor-bearing system operates in a predictable periodic motion state, because when the rotor-bearing system operates in a non-periodic motion state, the possible broadband vibration with large amplitude will increase the possibility of the fatigue or failure of the system. However, as the speed increases, the operating temperature of the rotor-bearing system will also increase, which may cause lubrication failure. From the simulation results, the higher operating temperature will also cause the dynamic trajectory of the system to become non-periodic vibration.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Cai-Wan Chang-Jian

Data Availability Statement

The generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

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Bifurcation analysis of a rotor-bearing system with temperature-dependent viscosity

Abstract

Keywords

Introduction

Mathematical modeling

Results and discussions

Discussion and conclusions

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

Data Availability Statement

References