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Abstract

The dynamic vibration absorbers have been applied to attenuate the critical or unbalanced vibration but may create the fluid-induced vibration instability in the rotor/seal system. The major purpose of this study is devoted to the effects of the dynamic vibration absorber on the nonlinear dynamic behavior and stability of the fluid-induced vibration in the rotor/seal system. The dynamic vibration absorber is attached on the shaft in the perpendicular directions. The model of the rotor/seal-dynamic vibration absorber system is established as the modified Jeffcott rotor system, and Muszynska nonlinear seal force is applied. The numerical method is used for the dynamic behavior analysis. The effects of the natural frequency and damping ratio of the dynamic vibration absorber on the dynamic behavior are discussed. The stability of the rotor/seal-dynamic vibration absorber system is judged by the eigenvalue theory. The variations of the instability threshold with the parameters of the dynamic vibration absorber are obtained. The results show that the instability threshold and instability vibration frequency are changed by the dynamic vibration absorber. The parameters of the dynamic vibration absorber must be selected carefully to avoid reducing the instability threshold and causing the instability vibration to occur in advance when the dynamic vibration absorber is applied to attenuate the critical or unbalanced vibration of the rotor/seal system.

Keywords

Rotordynamics nonlinear dynamics dynamic vibration absorber fluid-induced vibration stability

Introduction

The excessive vibration in the rotating machinery is a serious problem. Especially, it leads to the rotor vibration faults, even causes the rotor to shut down quickly. The unacceptable vibration must be controlled to ensure the stable operation of the rotating machinery.^1,2

Two kinds of methods, including adjusting the parameters and applying the external forces, are applied to attenuate the rotor vibration. The mass, stiffness, damping, and clearance can be adjusted to attenuate the excessive vibration. The rotor dynamic balancing is a common method to reduce the critical vibration when the rotor passes the critical speed.³ Wang et al.⁴ proposed the measurement point vector method to identify rotor unbalance while operating. The results from the dynamic analysis and numerical experiments provided that the method was efficient for the rotor balancing without test runs and external excitations. The shape memory alloy is commonly used to attenuate the rotor vibration by adjusting the stiffness.⁵ Majewska et al.⁶ presented the smart bearing support with the magnetic shape memory actuators to control the rotor vibration by the experimental work. Ribeiro et al.⁷ performed the viscoelastic supports for the vibration control in the rotating machines. In addition, various types of dampers are used for the rotor vibration attenuation, for example, the magnetorheological squeeze film damper was utilized to reduce the rotor vibration by Hemmatian and Ohadi.⁸ The bearing parameters can be self-adjusted by the tilting-pad journal bearing, and it is suitable to suppress the oil film whirl and whip occurring in the rotor system; for example, Chasalevris and Dohnal,^9,10 Santos and colleagues,^11–13 and Queiroz et al.¹⁴ investigated the active tilting-pad journal bearing and the active fluid-film bearing to extend the stability margins for the rotor system. Another example of the parameter adjusting is the seal parameter adjusting. The rotating labyrinth gas turbine seal can disrupt the fluid flow in the seal,¹⁵ and the parameters of the seal force were adjusted to suppress the fluid-induced vibration in the rotor/seal system. The external force applying is another method, for example, Fan and Pan¹⁶ proposed the electromagnetic force generated by the active electromagnetic actuator to eliminate the oil and dry whips in the rotor system. The ultrasonic force was very suitable for small rotor vibration attenuation as described by Yao et al.¹⁷ Chen et al.¹⁸ presented the active magnetic bearings of the adaptive frequency estimator for the rotor unbalance vibration suppression.

The vibration absorbing method using the dynamic vibration absorber (DVA) has been applied to attenuate the structure vibration, such as the pipe conveying fluid,¹⁹ the linear beam,²⁰ and the wing model.²¹ Recently, the DVAs have also been applied to attenuate the rotor vibration.²² One kind of the DVA is attached on the bearing²³ or the shaft by the rolling bearing,²⁴ so it does not rotate with the rotor. Another kind of the absorber is attached on the rotor by the rod²⁵ or the centrifugal pendulum,²⁶ and it rotates with the rotor. The DVA can be divided into two by the stiffness coefficient: the linear and nonlinear DVA. Generally, the stiffness of the DVA is linear. For the linear DVA, the vibration responses and attenuation effects are studied. Campos and Nicoletti²⁷ proposed the rotating dynamic absorbers to attenuate the vibration for the vertical washing machine. Hu and He²⁴ presented the rotor DVAs to online control the critical speed vibration of the single-span rotor. Yao et al.²² presented the negative stiffness DVAs to attenuate the critical vibration for the rotor system, which is the wide-frequency DVAs. The nonlinear energy sink (NES) is a kind of the nonlinear DVA which has a linear damping and an essentially nonlinear stiffness. Bab et al.²⁸ investigated the vibration reduction of the unbalanced rotor using the multi-smooth NESs, and the parameters’ conditions of the occurrence of the strongly modulated responses were obtained analytically by the multiple-scale-harmonic balance method. Then, they discussed the vibration reduction of the unbalanced rotor supported by the journal bearings with the nonlinear suspensions and the unbalanced rotor–blisk–journal bearing system by the multi-smooth NESs.^23,29 Yao et al.³⁰ proposed the magnetic NESs for the vibration attenuation of the unbalanced rotor. Tehrani and Dardel³¹ utilized the DVA and NES to prevent contact between the rotor and stator. Taghipour et al.³² investigated the vibration reduction of the rotor system under the nonlinear restoring forces using the linear DVA, the NES, and combined energy sinks DVA-NES.

Besides, the centrifugal pendulum vibration absorbers (CPVAs) are used to attenuate the torsional vibration in the rotating and reciprocating machines.³³ Shi and Parker³⁴ developed the analytical models of the CPVAs systems with the equally spaced and identical absorbers. The analytical models were used to investigate the vibration properties and stability of the CPVAs’ systems. Then, they studied the symmetry breaking effects on the vibration-mode structure of the CPVAs systems³⁵ and the vibration reduction in the rigid rotor with the tilting, rotational, and translational motions by the CPVAs.³⁶ Nishimura et al.³⁷ investigated the dynamic behavior and torsional vibration suppression of the rotor with the multiple CPVAs.

The effects of the DVA have been verified by lot of researches. However, the research on the DVAs for the rotor system vibration reduction is mainly focused on the forced vibration (the unbalanced and torsional vibration). To the authors’ knowledge, few papers are devoted to the effects of the DVA on the fluid-induced vibration in the rotor/seal system. The dynamic characteristics and stability of the rotor/seal system may be changed by the DVA. So, the DVA-based nonlinear dynamic behavior and stability of the fluid-induced vibration in the rotor/seal system is proposed in this article. Section “Dynamic model of the rotor/seal system with the DVA” describes the dynamic model of the rotor/seal system with the DVA. Section “Nonlinear dynamic characteristics of the rotor/seal-DVA system” compares the nonlinear dynamic behaviors of the rotor/seal system without and with the DVA. Then, the effects of the natural frequency and damping ratio of the DVA on the dynamic behaviors of the rotor/seal-DVA system are discussed in section “Influences of the parameters of the DVA on the nonlinear dynamic characteristics of the rotor/seal system.” Section “Stability of the rotor/seal-DVA system” compares the stability of the rotor/seal system without and with the DVA. The conclusions are in section “Conclusion.”

Dynamic model of the rotor/seal system with the DVA

The schematic diagram of the structure of the rotor/seal-DVA system is shown in Figure 1. A single-disk rotor is supported rigidly and the seal force is equivalently acting on the disk. Then, the DVA is mounted on the shaft by the rolling bearing with the frame in the horizontal and vertical directions, respectively.^22,24,30 So, it does not rotate with the rotor.

Figure 1.

(a) The structure of the whole test rig (left) and (b) the structure of the DVA (right).

The dynamic model of the rotor/seal-DVA system is shown in Figure 2. The rotor/seal system can be modeled as the modified Jeffcott rotor. The 2-degree-of-freedom isotropic rotor model with the seal force in the clearance between the rotor and seal by Muszynska² is proposed. The dynamic equations of the rotor/seal system are established as shown in equations (1)–(3) from the classical vibration theory.^2,38 Then, the DVA is attached on the shaft. They act equivalently on the disk. A 4-degree-of-freedom isotropic rotor/seal-DVA system model²² is proposed. The dynamic equations of the rotor/seal-DVA system are established as shown in equation (4)^22,24,30

\begin{array}{l} m \overset{\cdot\cdot}{X} + d \overset{\cdot}{X} + k X + f_{X} = m_{e} ω^{2} r cos ω t \\ m \overset{\cdot\cdot}{Y} + d \overset{\cdot}{Y} + k Y + f_{Y} = m_{e} ω^{2} r sin ω t \end{array}

(1)

\begin{matrix} f_{X} = m_{f} \overset{\cdot\cdot}{X} + d_{f} \overset{\cdot}{X} + 2 m_{f} ω λ \overset{\cdot}{Y} + (k_{f} - m_{f} ω^{2} λ^{2}) X \\ + ω d_{f} λ Y \\ f_{Y} = m_{f} \overset{\cdot\cdot}{Y} - 2 m_{f} ω λ \overset{\cdot}{X} + d_{f} \overset{\cdot}{Y} - ω d_{f} λ X \\ + (k_{f} - m_{f} ω^{2} λ^{2}) Y \end{matrix}

(2)

\begin{matrix} d_{f} = d_{0} {(1 - U^{2})}^{- n}, k_{f} = k_{0} {(1 - U^{2})}^{- n}, \\ λ = λ_{0} {(1 - U)}^{b}, U = \frac{\sqrt{X^{2} + Y^{2}}}{r_{f}} \end{matrix}

(3)

\begin{matrix} m \overset{\cdot\cdot}{X} + d \overset{\cdot}{X} + kX + f_{X} + d_{a} (\overset{\cdot}{X} - {\overset{\cdot}{X}}_{a}) + k_{a} (X - X_{a}) \\ = m_{e} ω^{2} r \cos ω t \\ m \overset{\cdot\cdot}{Y} + d \overset{\cdot}{Y} + kY + f_{Y} + d_{a} (\overset{\cdot}{Y} - {\overset{\cdot}{Y}}_{a}) + k_{a} (Y - Y_{a}) \\ = m_{e} ω^{2} r \sin ω t \\ m_{a} {\overset{\cdot\cdot}{X}}_{a} + d_{a} ({\overset{\cdot}{X}}_{a} - \overset{\cdot}{X}) + k_{a} (X_{a} - X) = 0 \\ m_{a} {\overset{\cdot\cdot}{Y}}_{a} + d_{a} ({\overset{\cdot}{Y}}_{a} - \overset{\cdot}{Y}) + k_{a} (Y_{a} - Y) = 0 \end{matrix}

(4)

where m, d, and k are the rotor-generalized (modal) mass, lateral external damping, and lateral isotropic stiffness coefficients, respectively. m_e and r are the eccentric mass and eccentricity of the rotor, respectively. ω is the rotational speed. r_f is the radial clearance of the seal. m_f, d_f, and k_f are the fluid inertia, fluid film radial damping, and stiffness coefficients in the seal, respectively. λ is the fluid circumferential average velocity ratio. d₀, k₀, λ₀, n, and b are the parameters of the nonlinear seal force. The parameters of the fluid force are described in detail by Muszynska.²m_a, d_a, and k_a are the mass, damping, and stiffness coefficients of the DVA, respectively. X, Y, X_a, and Y_a are the displacements of the rotor and DVA, respectively. The rotor stiffness coefficient k contains the elements of the shaft stiffness k_s and the support stiffness k_b

k = \frac{1}{1 / k_{s} + 1 / k_{b}}

(5)

Figure 2.

(a) The seal force model (left) and (b) the rotor/seal system with the DVA model (right).

The nondimensional transform is as follows

τ = ω t, \frac{d}{d t} = ω \frac{d}{d τ}, \frac{d^{2}}{d t^{2}} = ω^{2} \frac{d^{2}}{d τ^{2}}, x = \frac{X}{r_{f}}, y = \frac{Y}{r_{f}}, x_{a} = \frac{X_{a}}{r_{f}}, y_{a} = \frac{Y_{a}}{r_{f}}

(6)

\begin{matrix} \begin{matrix} x' = \frac{d}{d τ} x \\ y' = \frac{d}{d τ} y \end{matrix}, & \begin{matrix} x ″ = \frac{d^{2}}{d τ^{2}} x \\ y ″ = \frac{d^{2}}{d τ^{2}} y \end{matrix}, & \begin{matrix} {x'}_{a} = \frac{d}{d τ} x_{a} \\ {y'}_{a} = \frac{d}{d τ} y_{a} \end{matrix}, & \begin{matrix} {x ″}_{a} = \frac{d^{2}}{d τ^{2}} x_{a} \\ {y ″}_{a} = \frac{d^{2}}{d τ^{2}} y_{a} \end{matrix} \end{matrix}

(7)

Substituting equations (6) and (7) into equations (1)–(4), the original equations become the nondimensional equations

\begin{matrix} x ″ + 2 ζ_{r} {\bar{ω}}_{r} x' + {\bar{ω}}_{r}^{2} x + f_{x} + 2 ε ζ_{a} {\bar{ω}}_{a} (x' - {x'}_{a}) + ε {\bar{ω}}_{a}^{2} (x - x_{a}) \\ = ε_{e} \cos τ \\ y ″ + 2 ζ_{r} {\bar{ω}}_{r} y' + {\bar{ω}}_{r}^{2} y + f_{y} + 2 ε ζ_{a} {\bar{ω}}_{a} (y' - {y'}_{a}) + ε {\bar{ω}}_{a}^{2} (y - y_{a}) \\ = ε_{e} \sin τ \\ {x ″}_{a} + 2 ζ_{a} {\bar{ω}}_{a} ({x'}_{a} - x') + {\bar{ω}}_{a}^{2} (x_{a} - x) = 0 \\ {y ″}_{a} + 2 ζ_{a} {\bar{ω}}_{a} ({y'}_{a} - y') + {\bar{ω}}_{a}^{2} (y_{a} - y) = 0 \end{matrix}

(8)

\begin{matrix} f_{x} = ε_{f} x ″ + μ_{d} x' + 2 ε_{f} λ y' + (μ_{k} - ε_{f} λ^{2}) x + μ_{d} λ y \\ f_{y} = ε_{f} y ″ - 2 ε_{f} λ x' + μ_{d} y' - μ_{d} λ x + (μ_{k} - ε_{f} λ^{2}) y \end{matrix}

(9)

\begin{matrix} d_{f} = d_{0} {(1 - u^{2})}^{- n}, k_{f} = k_{0} {(1 - u^{2})}^{- n}, \\ λ = λ_{0} {(1 - u)}^{b}, u = \sqrt{x^{2} + y^{2}} \end{matrix}

(10)

where

\begin{matrix} ω_{r} = \sqrt{k / m}, ω_{a} = \sqrt{k_{a} / m_{a}}, ζ_{r} = \frac{d}{(2 m ω_{r})}, \\ ζ_{a} = \frac{d_{a}}{(2 m_{a} ω_{a})}, ε = \frac{m_{a}}{m}, ε_{e} = \frac{m_{e} r}{(m r_{f})}, \\ {\bar{ω}}_{r} = \frac{ω_{r}}{ω}, {\bar{ω}}_{a} = \frac{ω_{a}}{ω}, μ_{d} = \frac{d_{f}}{(m ω)}, \\ μ_{k} = \frac{k_{f}}{(m ω^{2})}, ε_{f} = \frac{m_{f}}{m} \end{matrix}

Nonlinear dynamic characteristics of the rotor/seal-DVA system

Newmark time integration method³⁹ is used to solute the complicated nonlinear equations. The algorithm is unconditionally convergent as the values of parameters β ≥ 1/4 and γ ≥ 1/2.⁴⁰ The bifurcation diagrams, waterfall diagrams, frequency spectrums, rotor orbits, and Poincaré maps are used to illustrate the nonlinear dynamic behaviors of the rotor/seal-DVA system.

The parameters of the rotor/seal system are as follows. The rotor mass m = 1 kg, the natural frequency ω_r = 200 rad/s, and the damping ratio ζ_r = 0.025. The radial clearance of the seal r_f = 10⁻³ m and the eccentricity ratio ε_e = 0.01. The fluid inertia ratio ε_f = 1×10⁻⁶ and other parameters d₀ = 100 N m/s, k₀ = 1 N/m, λ₀ = 0.48, n = 2, and b = 0.5. It is known that the “cross-stiffness” terms (±ωλd_f) in equation (2) are the most important factors affecting the rotor nonlinear behaviors and stability. The nonlinear behaviors of the real rotor/seal system can be described by the above-mentioned parameters.

The parameters of the DVA are considered as follows. Picking the mass ratio ε = 0.01 since the DVA mass ratio is limited for the practical considerations. It is known that the fluid-induced vibration frequency is almost equal to the natural frequency of the rotor, so picking the natural frequency of the DVA ω_a = ω_r. Normally, the damping ratio is relatively small, so picking the damping ratio ζ_a = 0.01. The influences of the natural frequency and damping ratio are discussed in the next section.

The bifurcation and waterfall diagrams in the x-coordinate of the rotor/seal system without the DVA are established and the results are shown in Figure 3. The rotational speed ω as a bifurcation parameter is considered. f is the rotor vibration response frequency. Assuming the nondimensional rotational speed s = ω/ω_r, Figure 3(a) shows that the response is the synchronous vibration when s < 2.31, and the instability vibration due to the nonlinear seal force appears when s ≥ 2.31. s = 2.31 is the instability threshold of the rotor/seal system. The instability vibration frequency is close to the natural frequency ω_r of the rotor, and the amplitude of the instability vibration is much higher than that of the synchronous vibration as shown in Figure 3(b). The rotational frequency ω and the combination frequency ω – 2ω_r exist in the waterfall diagram besides the instability frequency ω_r, and the amplitude of the combination frequency ω – 2ω_r is much smaller than that of the rotational frequency ω. The combination frequency appears with the instability frequency.

Figure 3.

(a) The bifurcation (left) and (b) waterfall (right) diagrams of the rotor/seal system without the DVA in the x-coordinate.

Figures 4 and 5 show the synchronous and instability vibrations for s = 1.5 and s = 2.605, respectively. Figure 4 shows that the rotational frequency ω only exists in the frequency spectrum, the orbit of the rotor is a regular circle, and an isolated point is in the Poincaré section. Figure 5 represents that the rotational frequency ω, the natural frequency ω_r of the rotor, and the combination frequency ω – 2ω_r exist in the frequency spectrum; the orbit of the rotor is irregular; and a closed curve formed by the infinite points is in the Poincaré section. It is a quasi-periodic response.

Figure 4.

The synchronous vibration of the rotor/seal system at s = 1.5. (a) The frequency spectrum of the x-coordinate (left),(b) the orbit of the rotor (middle) and (c) the Poincaré map of the x-coordinate (right).

Figure 5.

The instability vibration of the rotor/seal system at s = 2.605. (a) The frequency spectrum of the x-coordinate (left),(b) the orbit of the rotor (middle) and (c) the Poincaré map of the x-coordinate (right).

The bifurcation and waterfall diagrams in the x-coordinate of the rotor/seal system with the DVA are established and the results are shown in Figure 6. Figure 6(a) shows that the response is the synchronous vibration when s < 2.265, and the instability vibration due to the nonlinear seal force appears when s ≥ 2.265. s = 2.265 is the instability threshold of the rotor/seal-DVA system. Figure 6(b) and (c) shows that the instability vibration frequency is close to the first-order natural frequencies ω_n₁ of the rotor-DVA system as the rotational speed s is between 2.265 and 3.795, and the amplitude of the instability vibration is much higher than that of the synchronous vibration. In this case, the rotational frequency ω and the combination frequency ω – 2ω_n₁ exist in the waterfall diagram besides the instability frequency ω_n₁, and the amplitude of the combination frequency ω – 2ω_n₁ is much smaller than that of the rotational frequency ω. The combination frequency ω – 2ω_n₁ appears with the instability frequency ω_n₁. The instability vibration frequency is close to the second-order natural frequencies ω_n₂ of the rotor-DVA system as the rotational speed s exceeds 3.795, and the amplitude of the instability vibration is much higher than that of the synchronous vibration. In this case, the rotational frequency ω and the combination frequency ω – 2ω_n₂ exist in the waterfall diagram besides the instability frequency ω_n₂ and the amplitude of the combination frequency ω – 2ω_n₂ is much smaller than that of the rotational frequency ω. The combination frequency ω – 2ω_n₂ appears with the instability frequency ω_n₂.

Figure 6.

(a) The bifurcation diagram (upper left), (b) the waterfall diagram at view 1 (upper right), and (c) waterfall diagram at view 2 (lower middle) of the rotor/seal system with the DVA in the x-coordinate.

Figures 7 –9 show the synchronous and instability vibrations for s = 1.5, s = 3.5, and s = 4.11, respectively. Figure 7 shows that the rotational frequency ω only exists in the frequency spectrum, the orbit of the rotor is a regular circle, and an isolated point is in the Poincaré section. Figure 8 represents that the rotational frequency ω, the first-order natural frequency ω_n₁ of the rotor-DVA system, and the combination frequency ω – 2ω_n₁ exist in the frequency spectrum; the orbit of the rotor is irregular; and a closed curve formed by the infinite points is in the Poincaré section. It is also a quasi-periodic response. Figure 9 represents that the rotational frequency ω, the second-order natural frequency ω_n₂ of the rotor-DVA system, and the combination frequency ω – 2ω_n₂ exist in the frequency spectrum; the orbit of the rotor is irregular; and a closed curve formed by the infinite points is in the Poincaré section. It is also a quasi-periodic response.

Figure 7.

The synchronous vibration of the rotor/seal-DVA system at s = 1.5. (a) The frequency spectrum of the x-coordinate (left), (b) the orbit of the rotor (middle), and (c) the Poincaré map of the x-coordinate (right).

Figure 8.

The instability vibration of the rotor/seal-DVA system at s = 3.5. (a) The frequency spectrum of the x-coordinate (left), (b) the orbit of the rotor (middle), and (c) the Poincaré map of the x-coordinate (right).

Figure 9.

The instability vibration of the rotor/seal-DVA system at s = 4.11. (a) The frequency spectrum of the x-coordinate (left), (b) the orbit of the rotor (middle), and (c) the Poincaré map of the x-coordinate (right).

Compared with the nonlinear dynamic characteristics between the rotor/seal system without and with the DVA, it can be seen that (1) the instability threshold of the rotor/seal system becomes smaller by attaching the DVA as shown in the bifurcation diagrams in Figures 3(a) and 6(a), that is, the instability threshold of the rotor/seal system can be changed by the DVA. (2) Only one instability frequency is in the waterfall diagram in Figure 3(b) in the rotor/seal system. However, two instability frequencies are in the waterfall diagrams in Figure 6(b) and (c) in the rotor/seal system with the DVA. The two frequencies do not appear at the same time. The first-order natural frequency appears first, and then, the second-order natural frequency appears, while the first-order natural frequency disappears. The combination frequencies appear or disappear with the instability frequencies. (3) The type of the vibration does not change by the DVA as shown in the frequency spectrums, rotor orbits, and Poincaré maps in Figures 4, 5, and 7 –9. The synchronous and instability vibrations still exist in the rotor/seal system with the DVA. The effects of the parameters of the DVA on the nonlinear dynamic characteristics of the rotor/seal system are discussed as follows.

Influences of the parameters of the DVA on the nonlinear dynamic characteristics of the rotor/seal system

The DVA consists of the mass, stiffness, and damping elements. The mass ratio is limited for the practical considerations, so the natural frequency (stiffness element) and damping ratio (damping element) are analyzed for the dynamic behavior of the rotor/seal-DVA system in this section.

The effects of the natural frequency of the DVA on the nonlinear dynamic characteristics in the rotor/seal system with the DVA are presented as follows. Assume the nondimensional natural frequency σ_a =ω_a/ω_r. Figure 10 indicates the bifurcation and waterfall diagrams in the x-coordinate of the rotor/seal system with the DVA when σ_a =0.8. Figure 10(a) shows that the response is the synchronous vibration when s < 2.335, and the instability vibration due to the nonlinear seal force appears when s ≥ 2.335. s = 2.335 is the instability threshold of the rotor/seal-DVA system. The instability vibration frequency is close to the second-order natural frequency ω_n₂ of the rotor-DVA system, and the amplitude of the instability vibration is much higher than that of the synchronous vibration as shown in Figure 10(b). The rotational frequency ω and the combination frequency ω – 2ω_n₂ exist in the waterfall diagram besides the instability frequency ω_n₂, and the amplitude of the combination frequency ω – 2ω_n₂ is much smaller than that of the rotational frequency ω. The combination frequency ω – 2ω_n₂ appears with the instability frequency ω_n₂.

Figure 10.

(a) The bifurcation (left) and (b) waterfall (right) diagrams of the rotor/seal system with the DVA in the x-coordinate when σ_a = 0.8.

Figure 11 indicates the bifurcation and waterfall diagrams in the x-coordinate of the rotor/seal system with the DVA when σ_a = 0.9. The variations in the nonlinear dynamic characteristics in Figure 11 are the same as in Figure 10. Only the value of the instability threshold is different. s = 2.365 is the instability threshold of the system.

Figure 11.

(a) The bifurcation (left) and (b) waterfall (right) diagrams of the rotor/seal system with the DVA in the x-coordinate when σ_a = 0.9.

Figure 12 indicates the bifurcation and waterfall diagrams in the x-coordinate of the rotor/seal system with the DVA when σ_a =1.1. Compared with Figure 11, the instability vibration frequency becomes close to the first-order natural frequency ω_n₁ instead of the second-order natural frequency ω_n₂ of the rotor-DVA system. Correspondingly, the combination frequency becomes ω – 2ω_n₁. s = 2.265 is the instability threshold of the rotor/seal-DVA system.

Figure 12.

(a) The bifurcation (left) and (b) waterfall (right) diagrams of the rotor/seal system with the DVA in the x-coordinate when σ_a = 1.1.

Figure 13 indicates the bifurcation and waterfall diagrams in the x-coordinate of the rotor/seal system with the DVA when σ_a = 1.2. The variations in the nonlinear dynamic characteristics in Figure 13 are the same as in Figure 12. Only the value of the instability threshold is different. s = 2.28 is the instability threshold of the system.

Figure 13.

(a) The bifurcation (left) and (b) waterfall (right) diagrams of the rotor/seal system with the DVA in the x-coordinate when σ_a = 1.2.

Compared with the bifurcation diagrams in Figures 6(a), 10(a), 11(a), 12(a), and 13(a), it can be seen that the natural frequency of the DVA has an effect on the instability threshold of the rotor/seal system. The instability threshold increases first, then decreases, and increases again (2.335 → 2.365 → 2.265 → 2.265 → 2.28) as the natural frequency of the DVA increases (0.8 → 0.9 → 1 → 1.1 → 1.2). The instability threshold is maximum when σ_a = 0.9. Compared with the waterfall diagrams in Figures 6(b) and (c), 10(b), 11(b), 12(b), and 13(b), it can be seen that the natural frequency of the DVA has an effect on the instability frequency. The instability frequency is the first-order natural frequency of the rotor-DVA system when σ_a = 1.1 and σ_a = 1.2. However, the instability frequency is the second-order natural frequency of the rotor-DVA system when σ_a = 0.8 and σ_a = 0.9. The instability frequency is the first-order natural frequency of the rotor-DVA system when the instability vibration appears, and then, the instability frequency becomes the second-order natural frequency of the rotor-DVA system as the rotational speed increases when σ_a = 1. The combination frequency changes accordingly with the instability frequency.

The effects of the damping ratio of the DVA on the nonlinear dynamic characteristics in the rotor/seal system with DVA are considered as follows. Figure 14 indicates the bifurcation and waterfall diagrams in the x-coordinate of the rotor/seal system with the DVA when ζ_a = 0.02. The variations in the nonlinear dynamic characteristics in Figure 14 are the same as in Figure 6. Only the values of the instability threshold and the rotational speed at which the instability frequencies occur are different. s = 2.34 is the instability threshold of the system. The instability vibration frequency is close to the first-order natural frequency ω_n₁ of the rotor-DVA system as the rotational speed s is between 2.34 and 3.69. The instability vibration frequency is close to the second-order natural frequency ω_n₂ of the rotor-DVA system as the rotational speed s exceeds 3.69.

Figure 14.

Figure 15 indicates the bifurcation and waterfall diagrams in the x-coordinate of the rotor/seal system with the DVA when ζ_a = 0.03. The variations in the nonlinear dynamic characteristics in Figure 15 are the same as in Figure 14. Only the values of the instability threshold and the rotational speed at which the instability frequencies occur are different. s = 2.425 is the instability threshold of the system. The instability vibration frequency is close to the first-order natural frequency ω_n₁ of the rotor-DVA system as the rotational speed s is between 2.425 and 3.92. The instability vibration frequency is close to the second-order natural frequency ω_n₂ of the rotor-DVA system as the rotational speed s exceeds 3.92.

Figure 15.

Figure 16 indicates the bifurcation and waterfall diagrams in the x-coordinate of the rotor/seal system with the DVA when ζ_a = 0.04. Compared with Figure 15, the instability vibration frequency becomes close to the first-order natural frequency ω_n₁ instead of the first- and second-order natural frequencies ω_n₁ and ω_n₂ of the rotor-DVA system. Correspondingly, the combination frequency becomes ω – 2ω_n₁. s = 2.505 is the instability threshold of the rotor/seal-DVA system.

Figure 16.

(a) The bifurcation (left) and (b) waterfall (right) diagrams of the rotor/seal system with the DVA in the x-coordinate when ζ_a = 0.04.

Figure 17 indicates the bifurcation and waterfall diagrams in the x-coordinate of the rotor/seal system with the DVA when ζ_a = 0.05. Compared with Figure 16, the instability vibration frequency becomes close to the natural frequency ω_r of the rotor instead of the first-order natural frequency ω_n₁ of the rotor-DVA system. Correspondingly, the combination frequency becomes ω–2ω_r. s = 2.58 is the instability threshold of the rotor/seal-DVA system.

Figure 17.

(a) The bifurcation (left) and (b) waterfall (right) diagrams of the rotor/seal system with the DVA in the x-coordinate when ζ_a = 0.05.

Figure 18 indicates the bifurcation and waterfall diagrams in the x-coordinate of the rotor/seal system with the DVA when ζ_a = 0.06. The variations in the nonlinear dynamic characteristics in Figure 18 are the same as in Figure 17. Only the values of the instability threshold and instability frequency are different. s = 2.605 is the instability threshold of the system.

Figure 18.

(a) The bifurcation (left) and (b) waterfall (right) diagrams of the rotor/seal system with the DVA in the x-coordinate when ζ_a = 0.06.

Compared with the bifurcation diagrams in Figures 6(a), 14(a) 15(a), 16(a), 17(a), and 18(a), it can be seen that the damping ratio of the DVA has an effect on the instability threshold of the rotor/seal system. The instability threshold increases (2.265 → 2.34 → 2.425 → 2.505 → 2.58 → 2.605) as the damping ratio of the DVA increases (0.01 → 0.02 → 0.03 → 0.04 → 0.05 → 0.06). The instability threshold is maximum when ζ_a = 0.06. Compared with the waterfall diagrams in Figures 6(b) and (c), 14(b) and (c), 15(b) and (c), 16(b), 17(b), and 18(b), it can be seen that the damping ratio of the DVA has an effect on the value of the instability frequency. Two instability frequencies ω_n₁ and ω_n₂ are in the waterfall diagrams in the rotor/seal system with the DVA when ζ_a = 0.01, ζ_a = 0.02, and ζ_a = 0.03. However, only one instability frequency ω_n₁ is in the waterfall diagrams in the rotor/seal system when ζ_a = 0.04. Then, as the damping ratio continues to increase, the instability frequency becomes close to the rotor natural frequency ω_r when ζ_a = 0.05 and ζ_a = 0.06.

The effects of the DVA on the nonlinear dynamic characteristics in the rotor/seal system are listed as shown in Table 1. It can be seen that the instability frequency is the first-order natural frequency of the rotor-DVA system when the natural frequency of the DVA is larger than that of the rotor. The instability frequency is the second-order natural frequency of the rotor-DVA system when the natural frequency of the DVA is smaller than that of the rotor. In the case of the rotor natural frequency is equal to the DVA natural frequency, the instability frequency is the first-order natural frequency of the rotor-DVA system, and then, it becomes the second-order natural frequency as the rotational speed increases when the damping ratio is smaller. The instability frequency becomes the rotor natural frequency again as the damping ratio continues to increase. The variations in the instability threshold are discussed in detail in the following section.

Table 1.

The effects of the DVA on the nonlinear dynamic characteristics in the rotor/seal system.

Case	σ_a	ζ_a	Instability threshold	Instability frequency
Rotor without DVA	–	–	2.31	ω_r
Case 1	1	0.01	2.265	ω_n ₁ and ω_n₂
Case 2	0.8	0.01	2.335	ω_n ₂
Case 3	0.9	0.01	2.365	ω_n ₂
Case 4	1.1	0.01	2.265	ω_n ₁
Case 5	1.2	0.01	2.28	ω_n ₁
Case 6	1	0.02	2.34	ω_n ₁ and ω_n₂
Case 7	1	0.03	2.425	ω_n ₁ and ω_n₂
Case 8	1	0.04	2.505	ω_n ₁
Case 9	1	0.05	2.58	ω_r
Case 10	1	0.06	2.605	ω_r

DVA: dynamic vibration absorber.

Stability of the rotor/seal-DVA system

The natural frequency and damping ratio of the DVA can influence the instability threshold of the rotor/seal system based on the above analysis. Therefore, the stability region of the rotor/seal system will be extended to higher rotational speeds possibly, that is, the appearance of the instability vibration will be postponed possibly using the DVA. In this section, the effects of the DVA on the stability of the rotor/seal system are discussed.

Equations (1)–(4) are linearized for the stability analysis by the eigenvalue theory. The nonlinear terms and the fluid inertia coefficient in equations (1)–(4) are neglected as they are minimal and the unbalance force is assumed to be zero.^2,16 The nonlinear model becomes a classical linear model

\begin{matrix} m \overset{\cdot\cdot}{z} + (d + d_{0}) \overset{\cdot}{z} + (k + k_{0}) z - i d_{0} λ_{0} ω z \\ + d_{a} (\overset{\cdot}{z} - {\overset{\cdot}{z}}_{a}) + k_{a} (z - z_{a}) = 0 \\ m_{a} {\overset{\cdot\cdot}{z}}_{a} + d_{a} ({\overset{\cdot}{z}}_{a} - \overset{\cdot}{z}) + k_{a} (z_{a} - z) = 0 \end{matrix}

(11)

where

\begin{matrix} z = X + i Y \\ z_{a} = X_{a} + i Y_{a}, i = \sqrt{- 1} \end{matrix}

(12)

The solutions for equation (11) are as follows

\begin{matrix} z = A e^{μ t} \\ z_{a} = A_{a} e^{μ t} \end{matrix}

(13)

where A and A_a are the integration constants and μ is the complex eigenvalue.

Substituting solutions (13) into equation (11), the complex characteristic equation is provided

\begin{matrix} m m_{a} μ^{4} + [d_{a} m + (d + d_{a} + d_{0}) m_{a}] μ^{3} \\ + [(d + d_{0}) d_{a} + k_{a} m + (k + k_{a} + k_{0} - i d_{0} λ_{0} ω) m_{a}] μ^{2} \\ + [(k + k_{0} - i d_{0} λ_{0} ω) d_{a} + (d + d_{0}) k_{a}] μ \\ + (k + k_{0} - i d_{0} λ_{0} ω) k_{a} = 0 \end{matrix}

(14)

The complex eigenvalue can be obtained by equation (14). The response of the system is stable when the eigenvalue has negative real part. The response is at the threshold of instability when the real part of the eigenvalue equals to zero. The vibration is unstable when the eigenvalue has positive real part.

The instability threshold of the rotor/seal system with the DVA is presented in Figures 19 and 20. The red and black solid lines represent the instability threshold of the rotor/seal system without and with the DVA, respectively. Figure 19 illustrates the effect of the natural frequency of the DVA on the instability threshold. The instability threshold of the rotor/seal system with the DVA is smaller first, then larger, and again smaller than that of the rotor/seal system without the DVA as σ_a increases. Figure 19(a) shows that the instability threshold of the rotor/seal system with the DVA is smaller than that of the rotor/seal system without the DVA in the range of σ_a (0.01, 0.75) and (0.96, 2), respectively. The instability threshold of the rotor/seal system with the DVA is larger than that of the rotor/seal system without the DVA in the range of σ_a (0.75, 0.96). s = 2.37 is the maximum of the instability threshold at σ_a =0.93 when ζ_a = 0.01. The variations in the instability threshold in Figures 19(b)–(h) are the same as in Figure 19(a). Only the values of the instability threshold are different. The range of σ_a that the instability threshold of the rotor/seal system with the DVA is larger than that of the rotor/seal system without the DVA varies with the damping ratio ζ_a. The upper limit of the range increases (0.96 → 1.01 → 1.04 → 1.07 → 1.08 → 1.1 → 1.11 → 1.13) and the lower limit of the range decreases (0.75 → 0.73 → 0.71 → 0.69 → 0.68 → 0.66 → 0.64 → 0.63) as ζ_a increases (0.01 → 0.02 → … → 0.08). However, the maximum of the instability threshold increases first and then decreases (2.37 → 2.46 → 2.51 → 2.65 → 2.67 → 2.63 → 2.58 → 2.54) as ζ_a increases. s = 2.67 is the maximum of the instability threshold at σ_a = 0.99 when ζ_a = 0.05.

Figure 19.

The stability diagrams of the rotor/seal system with the DVA when the natural frequency of the DVA changes.

Figure 20.

The stability diagrams of the rotor/seal system with the DVA when the damping ratio of the DVA changes.

Figure 20 illustrates the effect of the damping ratio of the DVA on the instability threshold. Figure 20(a)–(e) shows that the instability threshold of the rotor/seal system with the DVA is smaller first, then larger, and again smaller than that of the rotor/seal system without the DVA as ζ_a increases. Figure 20(a) shows that the instability threshold of the rotor/seal system with the DVA is smaller than that of the rotor/seal system without the DVA in the range of ζ_a (0.01, 0.03) and (0.98, 1), respectively. The instability threshold of the rotor/seal system with the DVA is larger than that of the rotor/seal system without the DVA in the range of σ_a (0.03, 0.98). s = 2.33 is the maximum of the instability threshold at ζ_a = 0.28 when σ_a =0.7. The variations in the instability threshold in Figure 20(b)–(e) are the same as in Figure 20(a). Only the values of the instability threshold are different. Figure 20(a–f) shows that the range of ζ_a that the instability threshold of the rotor/seal system with the DVA is larger than that of the rotor/seal system without the DVA varies with the natural frequency ratio σ_a. The upper limit of the range decreases until 0.26 (0.98 → 0.87 → 0.78 → 0.68 → 0.57 → 0.26) and the lower limit of the range decreases first and then increases until 0.26 (0.03 → 0⁺ → 0⁺ → 0.02 → 0.06 → 0.26) as σ_a increases (0.7 → 0.8 → … → 1.2). The maximum of the instability threshold increases first and then decreases (2.33 → 2.35 → 2.41 → 2.58 → 2.34 → 2.31) as σ_a increases. s = 2.58 is the maximum of the instability threshold at ζ_a = 0.05 when σ_a =1. Figure 20(f) shows that the maximum of the instability threshold of the rotor/seal system with the DVA is equal to that of the rotor/seal system without the DVA, and others of the rotor/seal system with the DVA are smaller than that of the rotor/seal system without the DVA when σ_a = 1.2. Figure 20(g) and (h) shows that the instability threshold of the rotor/seal system with the DVA is smaller than that of the rotor/seal system without the DVA when σ_a =1.3 and σ_a = 1.4.

Conclusion

The effects of the DVA on the nonlinear dynamic behavior and stability of the fluid-induced vibration instability in the rotor/seal system are investigated in this article. The 4-degree-of-freedom isotropic rotor/seal system with the DVA model is established. Newmark time integration method is used to solve the differential equations. The effects of the natural frequency and damping ratio of the DVA on the dynamic characteristics of the fluid-induced vibration in rotor/seal system are discussed. The complex characteristic equation is provided for the stability analysis. The effects of the DVA on the stability region of the fluid-induced vibration in the rotor/seal system are obtained. Some conclusions are as follows:

The type of the dynamic response of the rotor/seal system is not changed by the DVA. The synchronous and instability vibrations still exist in the rotor/seal system with the DVA.

The instability vibration frequency of the rotor/seal system is changed by the DVA. The instability frequency is changed from the rotor natural frequency to the first- or second-order natural frequency of the rotor-DVA system.

The instability threshold of the rotor/seal system is changed by the DVA. The instability threshold is decreased by the DVA in some parameter ranges, and the instability vibration will occur at a lower rotational speed when the DVA is used to attenuate the critical or unbalanced vibration of the rotor. This result must be avoided.

Footnotes

Handling Editor: Elsa de Sa Caetano

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) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study was financially supported by the National Natural Science Foundation of China (grant nos 51475085 and U1708257) and 2017 Higher Education Institution Basic Research Project of Liaoning Province (grant no. LQGD2017019).

ORCID iD

Qi Xu

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Nonlinear dynamic behavior and stability of a rotor/seal system with the dynamic vibration absorber

Abstract

Keywords

Introduction

Dynamic model of the rotor/seal system with the DVA

Nonlinear dynamic characteristics of the rotor/seal-DVA system

Influences of the parameters of the DVA on the nonlinear dynamic characteristics of the rotor/seal system

Stability of the rotor/seal-DVA system

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References