# Nonlinear Coupled Dynamics of a Rod Fastening Rotor under Rub-Impact and Initial Permanent Deflection

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Modeling of a Rub-Impact Rod Fastening Rotor System

_{b1}and m

_{b2}, m

_{1}and m

_{2}are the lumped mass of two disks, e

_{1}and e

_{2}are the eccentric distance of two disks, and φ is the angle between unbalance mass of the two disks. The stiffness of shaft is k, c

_{1}and c

_{2}are the damping coefficient in the journal bearing and disks, and the damping coefficient in contact layer is c

_{3}, δ

_{0}is initial deflection. The dynamic model of the rod fastening rotor under rub-impact and initial permanent deflection is simplified with following assumptions: (1) the effects of rods and contacts on the system response are modeled as a flexural spring with nonlinear stiffness; (2) the shaft connecting the bearing and disk is flexible massless, and only lateral vibration is considered in the model; (3) the rod fastening rotor is supported by identical oil-film bearings at both sides, and the nonlinear oil-film force of the journal bearing is satisfied with the theory of short bearing; and (4) the rub-impact between disk 1 and stator is elastic, the radial deformation is elastic deformation, and the tangential friction force meets the Coulomb friction law.

#### 2.1. Rub-Impact Force

_{0}between disk 1 and stator. O

_{s}is the stator center, O

_{r}

_{1}′ is the geometric center of disk 1, O

_{r}

_{1}is the initial geometric center of disk 1.

_{N}can be expressed following linear elastic deformation theory. The tangential rub force P

_{T}can be represented following Coulomb law [9]:

_{c}is the radial stiffness of the stator, and η is the friction coefficient.

_{N}and the tangential rub force P

_{T}can be written in x-y coordinates as

_{1}= x

_{1}/c, Y

_{1}= y

_{1}/c, c is radial clearance of the bearing. Equation (2) can be rewritten as:

#### 2.2. Nonlinear Oil-Film Force

#### 2.3. The Governing Equations of Motion

_{x}and F

_{y}are the nonlinear oil-film force in the x-direction and y-direction, P

_{x}and P

_{y}are the rub-impact force in the x-direction and y-direction, φ is the angle between mass eccentricity of the two disks, and β is the phase angle between the mass eccentricity and the initial permanent deflection. F

_{cx}

_{1}, F

_{cy}

_{1}, F

_{cx}

_{2}, F

_{cy}

_{2}are the nonlinear restoring forces of contact layer, the expressions are as the following:

_{1}is the linear contact stiffness, and k

_{1}′ is the nonlinear contact stiffness.

_{b}

_{1}= x

_{b}

_{1}/c, X

_{b}

_{2}= x

_{b}

_{2}/c, Y

_{b}

_{1}= y

_{b}

_{1}/c, Y

_{b}

_{2}= y

_{b}

_{2}/c, X

_{1}= x

_{1}/c, X

_{2}= x

_{2}/c, Y

_{1}= y

_{1}/c, and Y

_{2}= y

_{2}/c.

## 3. Numerical Results and Discussion

_{b}

_{1}= 4 kg, m

_{b}

_{2}= 4 kg, m

_{1}= 32.1 kg, m

_{2}= 32.1 kg, c

_{1}= 1050 N·s/m, c

_{2}= 2100 N·s/m, c

_{3}= 2100 N·s/m, k = 2.5 × 10

^{7}N/m, k

_{1}= 2.5 × 10

^{7}N/m, k

_{1}′ = 2.5 × 10

^{7}N/m, k

_{c}= 1 × 10

^{7}N/m, r

_{0}= 0.18 mm, e

_{1}= 0.05 mm, e

_{2}= 0.05 mm, δ

_{0}= 0.01 mm, β = π/4, η = 0.1, φ = 0, R = 25 mm, L = 12 mm, c = 0.11 mm, μ = 0.018 Pa·s, and g = 9.81 m/s

^{2}. Assuming that the rod fastening rotor has rigid support, the natural frequency ω

_{c}= (k/m

_{1})

^{1/2}= 882 rad/s. In fact, the rotor cannot have complete rigid support, and the critical speed should be less than 882 rad/s. The coupled nonlinear dynamic analysis of a rod fastening rotor under rub-impact and initial permanent deflection is carried out by the fourth-order Runge–Kutta method. The period of the dimensionless system is 2π; the integral step length of each period is 1/100; the calculation is 200 periods; we choose the last 100 periods as the effective analysis data; and the number of the effective data points is 10,000. Bifurcation diagram, vibration waveform, frequency spectrum, shaft orbit and Poincaré map are presented to illustrate the nonlinear dynamic phenomena of system as follows.

#### 3.1. Effect of Speed

_{0}= 0, k

_{c}= 1 × 10

^{7}N/m. The system response has a big difference in high rotating speed compared with Figure 4. When ω < 507 rad/s, the system state is synchronous periodic-1 motion. It can be seen from Figure 6 that there is one isolated point in the Poincaré map and one-peak amplitude in the frequency spectrum, and the rub-impact does not occur under this condition. The system turns into periodic-2 motion at ω = 507 rad/s. With the increase of rotating speed, the period doubling bifurcation occurs. The periodic-2 motion, periodic-4 motion, periodic-8 motion and chaotic motion are observed one by one. Figure 7 shows the periodic-4 motion at ω = 670 rad/s, the Poincaré map contains four isolated points, the amplitude of half fundamental frequency exceeds the amplitude of fundamental frequency, and the oil whirl occurs at this rotating speed. When ω > 810 rad/s, inverse period doubling bifurcation occurs. The quasi-periodic motion can be observed at the interval of 1038 rad/s < ω < 1615 rad/s, as shown in Figure 8, the Poincaré map of the system presents a closed loop, and the frequency spectrum contains incommensurate frequency components. Meanwhile, the oil whip frequency is less than half of the fundamental frequency, and the amplitude of oil whip frequency is bigger than the amplitude of fundamental frequency. The oil whirl develops into oil whip at this speed, and the shaft orbit exceeds rubbing boundary. When ω > 1615 rad/s, the system experiences periodic-3 motion, and with the increase of rotating speed, the system finally enters into quasi-periodic motion. Figure 9 shows the periodic-3 motion at ω = 1750 rad/s, and the Poincaré map performs three isolated points.

_{0}= 0.01 mm, β = π/4. Figure 10 shows the bifurcation diagram of a disk 1 center in a horizontal direction considering the initial permanent deflection in the system motion equations.

#### 3.2. Effect of Initial Permanent Deflection

_{c}= 1 × 10

^{7}N/m, r

_{0}= 0.18 mm, e

_{1}= 0.05 mm, e

_{2}= 0.05 mm, and β = π/4, Figure 15 shows the bifurcation diagrams of the disk 1 center in a horizontal direction under different initial permanent deflection lengths δ

_{0}= 0.01 mm, δ

_{0}= 0.02 mm, δ

_{0}= 0.03 mm, and δ

_{0}= 0.04 mm. As shown in Figure 15a, the system bifurcates into chaotic motion at ω = 608 rad/s and leaves chaotic motion at ω = 898 rad/s, with the increase of rotating speed, the system undergoes an inverse period doubling bifurcation. At the interval of 1639 rad/s < ω < 1991 rad/s, the system keeps periodic-3 motion. When δ

_{0}= 0.02 mm, the system enters into chaotic motion at ω = 716 rad/s and leaves chaotic motion at ω = 936 rad/s, similar to the bifurcation diagram at δ

_{0}= 0.01 mm, with the increase of rotating speed, the system experiences periodic-16 motion,periodic-8 motion,periodic-4 motion,periodic-2 motion, quasi-periodic motion and keeps periodic-3 motion at 1655 rad/s < ω < 1992 rad/s. Figure 15c shows the bifurcation diagram at δ

_{0}= 0.03 mm, and the system bifurcates into chaotic at ω = 782 rad/s. When the rotating speed reaches 962 rad/s, the system undergoes an inverse period doubling bifurcation and leaves chaotic motion. At ω = 1192 rad/s, the system becomes quasi-periodic motion from periodic-2 motion and keeps periodic-3 motion at 1663 rad/s < ω < 1991 rad/s with the increase of rotating speed. As indicated from Figure 15d, the system bifurcates into periodic-2 motion at ω = 826 rad/s and leaves chaotic motion at ω = 948 rad/s. The system presents similar characteristics with the increase of rotating speed compared with δ

_{0}= 0.03 mm.

_{0}do not present an obvious differences in high-speed regions. In a low-speed region, the system keeps periodic-1 motion under different initial permanent deflection lengths. The instability speed of the system gradually rises with the increase of initial permanent deflection length. Meanwhile, the chaotic motion region becomes smaller and smaller. Although the instability speed of the system increased with the value of initial permanent deflection length δ

_{0}, the initial permanent deflection is not expected.

#### 3.3. Effect of Radial Stiffness of the Stator

_{0}= 0.11 mm, δ

_{0}= 0.01 mm.

_{c}= [0–1.71 × 10

^{7}] N/m. As shown in Figure 17, the orbit is irregular, the Poincaré map presents a strange attractor, and the frequency spectrum contains continuous frequency bands. Meanwhile, there is a half fundamental frequency component in the frequency spectrum, and oil whirl occurs at this condition. With the increase of radial stiffness of stator k

_{c}, the system experiences periodic-8 motion, periodic-4 motion, and periodic-2 motion. Figure 18 shows the periodic-4 motion at k

_{c}= 1.9 × 10

^{7}N/m, and there are four isolated points in Poincaré map. At k

_{c}= [2.19 × 10

^{7}–2.73 × 10

^{7}] N/m, the system undergoes multi-periodic motions and returns back to periodic-2 motion at k

_{c}= 2.73 × 10

^{7}N/m (see in Figure 19). When k

_{c}> 2.97 × 10

^{7}N/m, the system displays a synchronous periodic-1 motion, and the oil whirl disappears under this condition (see in Figure 20).

## 4. Conclusions

- The dynamic responses of the rod fastening rotor bearing system under rub-impact and initial permanent deflection exhibit a rich nonlinear dynamic diversity, synchronous periodic-1 motion, multi-periodic motion, chaotic motion and quasi-periodic motion can be observed through the analysis.
- Initial permanent deflection length has a great effect on the dynamic response of the system in the low-speed regions. With the increase of initial permanent deflection length, the instability speed of the system gradually rises, and the chaotic motion region becomes smaller and smaller.
- With the increase of radial stiffness of the stator, the system response becomes simpler under certain conditions. Meanwhile, the oil whirl is weaker or even disappears at a certain rotating speed.
- It is unsuitable to take the rod fastening rotor as an integral rotor in analyzing the coupled nonlinear dynamic responses of the system under rub-impact and initial permanent deflection.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Symbols

c | Radial clearance of bearing |

μ | Oil viscosity |

L | Bearing length |

R | Bearing radius |

δ | Sommerfeld correction coefficient |

h | Thickness of oil-film |

p | Dimensionless pressure of oil-film |

F_{x}, F_{y} | Nonlinear oil-film force in x-direction and y-direction |

f_{x}, f_{y} | Dimensionless nonlinear film force in x-direction and y-direction |

P_{T}, P_{N} | Rub-impact force in radial and tangential direction |

P_{x}, P_{y} | Rub-impact in x-direction and y-direction |

η | Friction coefficient |

r_{0} | Initial clearance |

δ_{0} | Initial permanent deflection |

k_{c} | Radial stiffness of the stator |

F_{cx}, F_{cy} | Restoring force of contact layer in x-direction and y-direction |

m_{b1}, m_{b2} | Lumped mass of bearings |

m_{1}, m_{2} | Lumped mass of disks |

e_{1}, e_{2} | Eccentric distance of disks |

φ | Angle between mass eccentricity of the two disks |

β | Angle between mass eccentricity and initial permanent deflection |

k | Shaft stiffness |

k_{1} | Linear contact stiffness |

k_{1}′ | Nonlinear contact stiffness |

c_{1} | Damping of bearing |

c_{2} | Damping of disk |

c_{3} | Damping of contact layer |

x_{i}, y_{i} (i = 1, 2) | Displacements of disks in x-direction and y-direction |

x_{bi}, y_{bi} (i = 1, 2) | Displacements of bearings in x-direction and y-direction |

X_{i}, Y_{i} (i = 1, 2) | Dimensionless displacements of disks in x-direction and y-direction |

X_{bi}, Y_{bi} (i = 1, 2) | Dimensionless displacements of bearings in x-direction and y-direction |

ω | Rotating speed |

g | Gravitational acceleration |

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**Figure 4.**Bifurcation diagram of integral rotor disk center in the x-direction at k

_{c}= 1 × 10

^{7}N/m, δ

_{0}= 0.

**Figure 5.**Bifurcation diagram of rod fastening rotor disk 1 center in the x-direction at k

_{c}= 1 × 10

^{7}N/m, δ

_{0}= 0.

**Figure 6.**Numerical analysis results at ω = 500 rad/s, δ

_{0}= 0 mm. (

**a**) Time domain waveform; (

**b**) Shaft orbit; (

**c**) Frequency spectrum; (

**d**) Poincaré map.

**Figure 7.**Numerical analysis results at ω = 670 rad/s, δ

_{0}= 0 mm. (

**a**) Time domain waveform; (

**b**) Shaft orbit; (

**c**) Frequency spectrum; (

**d**) Poincaré map.

**Figure 8.**Numerical analysis results at ω = 1400 rad/s, δ

_{0}= 0 mm. (

**a**) Time domain waveform; (

**b**) Shaft orbit; (

**c**) Frequency spectrum; (

**d**) Poincaré map.

**Figure 9.**Numerical analysis results at ω = 1750 rad/s, δ

_{0}= 0 mm. (

**a**) Time domain waveform; (

**b**) Shaft orbit; (

**c**) Frequency spectrum; (

**d**) Poincaré map.

**Figure 10.**Bifurcation diagram of disk 1 center in a horizontal direction at k

_{c}= 1 ×10

^{7}N/m, δ

_{0}= 0.01 mm.

**Figure 11.**Numerical analysis results at ω = 500 rad/s, δ

_{0}= 0.01 mm. (

**a**) Time domain waveform; (

**b**) Shaft orbit; (

**c**) Frequency spectrum; (

**d**) Poincaré map.

**Figure 12.**Numerical analysis results at ω = 670 rad/s, δ

_{0}= 0.01 mm. (

**a**) Time domain waveform; (

**b**) Shaft orbit; (

**c**) Frequency spectrum; (

**d**) Poincaré map.

**Figure 13.**Numerical analysis results at ω = 1400 rad/s, δ

_{0}= 0.01 mm. (

**a**) Time domain waveform; (

**b**) Shaft orbit; (

**c**) Frequency spectrum; (

**d**) Poincaré map.

**Figure 14.**Numerical analysis results at ω = 1750 rad/s, δ

_{0}= 0.01 mm. (

**a**) Time domain waveform; (

**b**) Shaft orbit; (

**c**) Frequency spectrum; (

**d**) Poincaré map.

**Figure 15.**Bifurcation diagram of disk 1 center in a horizontal direction at k

_{c}= 1 × 10

^{7}N/m. (

**a**) δ

_{0}= 0.01 mm; (

**b**) δ

_{0}= 0.02 mm; (

**c**) δ

_{0}= 0.03 mm; (

**d**) δ

_{0}= 0.04 mm.

**Figure 17.**Numerical analysis results at k

_{c}= 1.2 × 10

^{7}N/m. (

**a**) Time domain waveform; (

**b**) Shaft orbit; (

**c**) Frequency spectrum; (

**d**) Poincaré map.

**Figure 18.**Numerical analysis results at k

_{c}= 1.9 × 10

^{7}N/m. (

**a**) Time domain waveform; (

**b**) Shaft orbit; (

**c**) Frequency spectrum; (

**d**) Poincaré map.

**Figure 19.**Numerical analysis results at k

_{c}= 2.85 × 10

^{7}N/m. (

**a**) Time domain waveform; (

**b**) Shaft orbit; (

**c**) Frequency spectrum; (

**d**) Poincaré map.

**Figure 20.**Numerical analysis results at k

_{c}= 3.2 × 10

^{7}N/m. (

**a**) Time domain waveform; (

**b**) Shaft orbit; (

**c**) Frequency spectrum; (

**d**) Poincaré map.

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## Share and Cite

**MDPI and ACS Style**

Hu, L.; Liu, Y.; Teng, W.; Zhou, C.
Nonlinear Coupled Dynamics of a Rod Fastening Rotor under Rub-Impact and Initial Permanent Deflection. *Energies* **2016**, *9*, 883.
https://doi.org/10.3390/en9110883

**AMA Style**

Hu L, Liu Y, Teng W, Zhou C.
Nonlinear Coupled Dynamics of a Rod Fastening Rotor under Rub-Impact and Initial Permanent Deflection. *Energies*. 2016; 9(11):883.
https://doi.org/10.3390/en9110883

**Chicago/Turabian Style**

Hu, Liang, Yibing Liu, Wei Teng, and Chao Zhou.
2016. "Nonlinear Coupled Dynamics of a Rod Fastening Rotor under Rub-Impact and Initial Permanent Deflection" *Energies* 9, no. 11: 883.
https://doi.org/10.3390/en9110883