Nonlinear Dynamics of Marine Rotor System Coupled with Air Bag-Floating Raft Subjected to the Basement Excitations in Lateral Directions

Abstract

This paper presents an investigation into the nonlinear dynamic behaviors of the mechanical isolation system coupled with air-bag and floating-raft subject to basement excitation in lateral directions. First, the coupling effects between the excitation source and isolation system are considered. Also, the mechanical isolation model under basic excitation and its motion equation are deduced, and then the dynamic responses are mainly investigated by using the techniques of displacement response, frequency spectrum, rotor orbit, Poincaré maps, and the bifurcation diagram. Last, the bifurcations of the mechanical isolation system with different parameters are analyzed through numerical methods, especially the effect of excitation frequency and amplitude. The result predicts that period-5 is mainly performed, with the increase of rotor speed, and the system moves into quasi-bifurcation. However, the system stays in chaos state at high rotor speed, and the vibration amplitude rises rapidly until against bearing bush. Furthermore, the effects of basement excitation on the mechanical isolation system are mainly concentrated on the stage of lower rotor speed, but with the increasing speed, the effects become weak and at the same time the vibration amplitude reduces significantly. The points projected on the Poincaré section are five, three, or two solitary attractors, in which the system stays in periodic motion. Above all, the dynamic characteristics can provide the theoretic supporting for the dynamic, vibration control and its parametric optimization of the marine mechanical isolation system coupled with air-bag and floating-raft.

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

The external environment is relatively complex because of work requirements on the rotating machinery. Usually, base excitation of rotating machinery could occur when its surroundings are equipped with a plurality of mechanical power equipment such as vessels and submarines on ground that is prone to seismic motion or other similar environments. Foundation excitations are the most common source of harmonic excitation of rotating machinery because any form of excitation can be converted into harmonic excitation by fast Fourier transformation (FFT). A rotating machine under motion base could be excited at its base by another harmonic excitation that in turn could arise from, for example, vibration from basement motion on the ground or ship. The reduction of undesired high vibration amplitudes is of paramount importance. It is essential for a designer to have at their disposal dynamic models to predict the vibrational behavior of rotating machines operating under such multifrequency excitations at the design stage to avoid expensive design modifications or in-field catastrophic failure.

There are many researches on the basic excitation. Wilson and Brebbia investigated the dynamic behavior of steel foundations for turbo-alternators by the finite element displacement method [1]; Gasch studied the vibration of large turbo-rotors in fluid-film bearings on an elastic foundation by the finite element analysis [2]. Hori reviewed 19 works mainly on the foundation excitation effect on the dynamics/stability of rotor-fluid film bearings systems under basement excitation [3]. Rajan and Davies considered the random primary response of a duffing oscillator subject to narrowband excitation by using the method of multiple scales and stochastic averaging [4]. Nayfeh and Serhan analyzed the stationary mean and mean square responses and its local stability of a Duffing–Rayleigh oscillator excited by the sum of a deterministic harmonic component and a random component by using a second-order closure method [5]; Chen studied the nonlinear dynamical behavior of a single-axis rate gyro with harmonic excitation mounted on a space vehicle [6]; American scientists researched a method for determining the relationship between the structural type of a manipulator and its susceptibility to motion-induced vibrational excitation [7]; Feng and Hu developed a set of nonlinear differential equations by using Kane’s method for the planar oscillation of slender beams subject to a parametric excitation of the base movement [8, 9]. Fawzi developed a formulation for the dynamic analysis of the rigid rotor subject to base excitations plus mass imbalance. Han and Chu acquired the parametric instability regions induced by the periodic base angular motions based on the discrete state transition matrix method [10, 11]. Dakel et al. investigated the dynamics of two different rotor configurations under various support motion through the numerical method [12]. Saeed and Kamel investigated the vibration control of a horizontally suspended Jeffcott-rotor system. A nonlinear restoring force and the rotor weight are considered in the system model [13]. Lei et al. focused on the nonlinear responses of a cracked rotor-ball bearing system caused by aircraft flight maneuvers [14]. Saeed and El-Ganaini deal with the vibration analysis of a Jeffcott-rotor system under various supports. Finally, a comparison with the papers that published previously is included [15–18]. Chen and Wang developed a general model for the flexible rotor system subjected to time-varying base excitation and studied the direct effects of angular base motion on the dynamic behaviors of a simple rotor [19].

However, it is worth mentioned that researchers’ works mainly concentrated on the effect of base motions while the mechanical isolation system coupled with air bag-floating raft under base motions is not referred. In addition, the excitation resource from their researches is always in single direction, and the definition of base excitation was not accurate enough to describe the real movement of the base.

In this paper, an investigation into the nonlinear dynamic behaviors of the mechanical isolation system coupled with air-bag and floating-raft under basement excitation in lateral directions is presented. First, the coupling effects between the excitation source and isolation system are considered, and the excitation on the longitudinal and transverse directions is also taken into account. Based on this, the mechanical isolation model subjected to basic excitation and its motion equation are developed, and then its dynamic responses are mainly investigated by using the techniques of displacement response, frequency spectrum, rotor orbit, Poincaré maps, and the bifurcation diagram. Last, the bifurcations of the mechanical isolation system with different parameters are analyzed through numerical methods, especially the effect of excitation frequency and amplitude. In particular, this work aims to present design guidelines for predicting dynamics of the mechanical isolation system under various basement excitation.

2. Dynamic Model under Basement Excitation

Due to the low frequency vibration of ship hull and other power machinery equipment, mainly motor or diesel engine and other power equipment, the form of its vibrations is mainly harmonic vibrations [20].

2.1. Dynamic Model

Figure 1 depicts the rotating mechanical isolation system coupled with the air bag-floating raft device subjected to basement excitation, in which the floating-raft and the rotor are considered as a rigid body and the air-bag is regarded as a spring with vertical and horizontal deformations. In general, the air-bag and floating-raft isolation device is a kind of a mixed complex dynamic isolation system which includes air-bags, floating-raft, and rotor-bearings. Particularly, all the system should be fixed on the ground such as the ship and vessel that can be prone to seismic motion.

[figure omitted; refer to PDF]

It is well known that any form of excitation can be converted into harmonic excitation by the FFT technique; thus, basement motions in this paper are considered as harmonic motion S(t) driven by the angular frequency Ω₁ and Ω₂ in the longitudinal and transverse directions, respectively, and by considering the coupling effects between the excitation source and isolation system, the mechanical model of rotating machinery under base excitation is established.

Also, based on the dynamics theory, the differential equations of motion [21] under base excitation motion are derived as follows:$$\begin{array}{}\text{(1)}& \left\{\begin{array}{c}{m}_1{\ddot{x}}_1=F_x\left[\left(x_1-x_2\right),\left(y_1-y_2\right),\left({\dot{x}}_1-{\dot{x}}_2\right),\left({\dot{y}}_1-{\dot{y}}_2\right)\right]+m_{1}e{\Omega }^2\cos \Omega t+m_{1}g,\\ m_1{\ddot{y}}_1=F_y\left[\left(x_1-x_2\right),\left(y_1-y_2\right),\left({\dot{x}}_1-{\dot{x}}_2\right),\left({\dot{y}}_1-{\dot{y}}_2\right)\right]+m_{1}e{\Omega }^2\sin \Omega t,\\ m_2{\ddot{x}}_2=-F_x\left[\left(x_1-x_2\right),\left(y_1-y_2\right),\left({\dot{x}}_1-{\dot{x}}_2\right),\left({\dot{y}}_1-{\dot{y}}_2\right)\right]-k_1\left(x_2-s_1\right)-d_1\left({\dot{x}}_2-{\dot{s}}_1\right)+m_{2}g,\\ m_2{\ddot{y}}_2=-F_y\left[\left(x_1-x_2\right),\left(y_1-y_2\right),\left({\dot{x}}_1-{\dot{x}}_2\right),\left({\dot{y}}_1-{\dot{y}}_2\right)\right]-k_2\left(y_2-s_2\right)-d_2\left({\dot{y}}_2-{\dot{s}}_2\right),\end{array}\right.\end{array}$$where m₁, m₂, e, k, d, Ω, F_x, and F_y are the mass of disc, the mass of floating-raft, eccentricity, stiffness, damping, rotor speed, nonlinear oil film in x and y directions, respectively; s₁(t) and s₂(t) are the longitudinal and transverse basement excitation/driven-base excitation/foundation excitation, respectively, and its motion parametric settings can be expressed as $s_1\left(t\right)={\Phi }_a\sin {\Omega }_{1}t$ and $s_2\left(t\right)={\Phi }_b\sin {\Omega }_{2}t$;the parameters Ω₁ and Ω₂ are the angular frequency in the longitudinal and transverse directions, respectively; and ${\Phi }_a$ and ${\Phi }_b$ are the excitation amplitudes of longitudinal and transverse directions, respectively.

2.2. Oil Film Force

Figure 2 illustrates the schematic diagram of journal bearing, in which $F_r$ and $F_{\tau }$ are the nonlinear oil forces in the radial and tangential directions, respectively. In this article, the long bearing theory is introduced; accordingly, the Reynolds equation for the oil film pressure p yields [22, 23]$$\begin{array}{}\text{(2)}& \frac{1}{R_b^2}\frac{\partial }{\partial \phi }\left(\frac{h^3}{u}\frac{\partial p}{\partial \phi }\right)=6\left(\Omega -2\dot{\theta }\right)\frac{\partial h}{\partial \phi }+12\dot{e}\cos \phi ,\end{array}$$where $h=c+e\cos \theta =c\left(1+\epsilon \cos \theta \right).$

[figure omitted; refer to PDF]

After considering the semi-Sommerfeld condition $\theta =0$, $p=0$; $\theta =\pi$, $p=0$, the oil film forces in radial and tangent directions are obtained as follows:$$\begin{array}{}\text{(3)}& \left\{\begin{array}{c}{F}_r=6uBR{\left(\frac{R}{c}\right)}^2\left[\left(\Omega -2\frac{\text{d}\theta }{\text{d}t}\right)E_1+2\frac{\text{d}\epsilon }{\text{d}t}{E}_2\right],\\ F_{\tau }=6uBR{\left(\frac{R}{c}\right)}^2\left[\left(\Omega -2\frac{\text{d}\theta }{\text{d}t}\right)E_3+2\frac{\text{d}\epsilon }{\text{d}t}{E}_4\right],\end{array}\right.\end{array}$$where$$\begin{array}{c}\text{(4)}& E_1=\frac{2{\alpha }^2}{\left(1-{\alpha }^2\right)\left(2+{\alpha }^2\right)},E_2=\frac{1}{{\left(1-{\alpha }^2\right)}^{3/2}}\left[\frac{\pi }{2}-\frac{8}{\pi \left(2+{\alpha }^2\right)}\right],\\ E_3=\frac{\pi \alpha }{{\left(1-{\alpha }^2\right)}^{1/2}\left(2+{\alpha }^2\right)},\\ E_4=\frac{2\alpha }{\left(1-{\alpha }^2\right)\left(2+{\alpha }^2\right)}.\end{array}$$

The oil film forces in equation (3) can be expressed as$$\begin{array}{}\text{(5)}& \left\{\begin{array}{c}{F}_x\left(x_1-x_2,y_1-y_2,{\dot{x}}_1-{\dot{x}}_2,{\dot{y}}_1-{\dot{y}}_2\right)=-F_r\cos \phi -F_t\sin \phi ,\\ F_y\left(x_1-x_2,y_1-y_2,{\dot{x}}_1-{\dot{x}}_2,{\dot{y}}_1-{\dot{y}}_2\right)=-F_r\sin \phi +F_t\cos \phi ,\end{array}\right.\end{array}$$where F_x and F_y are the nonlinear oil film forces components of journal bearing in the x and y direction, respectively.

2.3. Nondimensional Motion Equation

For the simplification and wide applicability of the following analysis, the nondimensional variables are obtained on the basis of the bearing clearance, weight of disc, and time as shown in Table 1.

Table 1

The dimensionless expressions of the parameters.

Parameters	Expression
α	α = e/c
$\tau$	$\tau =\Omega t$
$\overline{\sigma }$	$\overline{\sigma }=6uB{R}^3/c^2$
n	n = m2/m1
$\omega$	$\omega =\Omega \sqrt{c/g}$
D	$D=dc\Omega /m_{1}g$
${\overline{x}}_i$	${\overline{x}}_i=x_i/c$
${\overline{y}}_i$	${\overline{y}}_i=y_i/c$
f j	$f_j=F_j/m_{1}g$
$\sigma$	$\sigma =\overline{\sigma }/m_1\sqrt{gc}$
$\lambda$	$\lambda =B/2R$
K	$K=\text{kc}/m_{1}g$
${\eta }_1$	${\eta }_1={\Omega }_1/\Omega$
${\eta }_2$	${\eta }_2={\Omega }_2/\Omega$
${\varphi }_a$	${\varphi }_a={\Phi }_a/c$
${\varphi }_b$	${\varphi }_b={\Phi }_b/c$
$f_x$	$f_x=-f_r\cos \phi -f_t\sin \phi$
$f_y$	$f_x=-f_r\cos \phi -f_t\sin \phi$

By the way, denote $\text{d}x/\text{d}t=\dot{x}\text{\hspace{0.17em}and\hspace{0.17em}}\text{d}x/\text{d}\tau ={\overline{x}}^{\prime }$. The meaning of the parameters in the above equations is shown in nomenclature, and equation (3) can be put into a convenient dimensionless form as follows:$$\begin{array}{}\text{(6)}& \left\{\begin{array}{c}{f}_r=\sigma \omega \left[\left(1-2{\phi }^{\prime }\right)E_1+2e^{\prime }{E}_2\right],\\ f_{\tau }=\sigma \omega \left[\left(1-2{\phi }^{\prime }\right)E_3+2e^{\prime }{E}_4\right].\end{array}\right.\end{array}$$

Thereby, we obtain the following set of first-order and second-order differential relation:$$\begin{array}{}\text{(7)}& \left\{\begin{array}{c}\frac{\text{d}x}{\text{d}t}=\frac{\text{d}x}{\text{d}\tau }\frac{\text{d}\tau }{\text{d}t}=\Omega \frac{\text{d}x}{\text{d}\tau },\\ \frac{{\text{d}}^{2}x}{\text{d}{t}^2}=\frac{\text{d}}{\text{d}t}\left(\Omega \frac{\text{d}x}{\text{d}\tau }\right)=\frac{\text{d}}{\text{d}\tau }\frac{\text{d}\tau }{\text{d}t}\left(\Omega \frac{\text{d}x}{\text{d}\tau }\right)={\Omega }^2\frac{{\text{d}}^{2}x}{\text{d}{\tau }^2}.\end{array}\right.\end{array}$$

We rewrite $\text{d}x/\text{d}t\text{\hspace{0.17em}and\hspace{0.17em}}{\text{d}}^{2}x/\text{d}{t}^2$ as follows:$$\begin{array}{c}\text{(8)}& \frac{\text{d}x}{\text{d}t}=\Omega {\overline{x}}^{\prime },\frac{{\text{d}}^{2}x}{\text{d}t}=\Omega {\overline{x}}^{″}.\end{array}$$

In the same way, we get$$\begin{array}{c}\text{(9)}& {\ddot{x}}_1={\Omega }^2{\overline{x}}_1^{″},{\ddot{y}}_1={\Omega }^2{\overline{y}}_1^{″},\\ {\ddot{x}}_2={\Omega }^2{\overline{x}}_2^{″},\\ {\ddot{y}}_2={\Omega }^2{\overline{y}}_2^{″}.\end{array}$$

Substituting equations (3) to (9) into equation (1), we get the nondimensional formulation of motion:$$\begin{array}{}\text{(10)}& \left\{\begin{array}{c}{\overline{x}}_1^{″}=\frac{1}{{\omega }^2}+\frac{f_x}{{\omega }^2}+\alpha \cos \tau ,\\ {\overline{y}}_1^{″}=\frac{f_y}{{\omega }^2}+\alpha \sin \tau ,\\ {\overline{x}}_2^{″}=\frac{1}{{\omega }^2}-\frac{f_x}{n{\omega }^2}-\frac{K_1}{n{\omega }^2}{\overline{x}}_2-\frac{D_1}{n{\omega }^2}{\overline{x}}_2^{\prime }+\frac{K_1}{n{\omega }^2}+\frac{D_1}{n{\omega }^2}{\varphi }_{\text{a}}{\eta }_1\cos \left({\eta }_1\tau \right),\\ {\overline{y}}_2^{″}=-\frac{f_y}{n{\omega }^2}-\frac{K_2}{n{\omega }^2}{\overline{y}}_2-\frac{D_2}{n{\omega }^2}{\overline{y}}_2^{\prime }+\frac{K_2}{n{\omega }^2}{\varphi }_{\text{b}}\sin \left({\eta }_2\tau \right)+\frac{D_2}{n{\omega }^2}{\varphi }_{\text{b}}{\eta }_2\cos \left({\eta }_2\tau \right).\end{array}\right.\end{array}$$

Equation (10) is a two order nonlinear differential equation group controlled by 11 varied parameters such as α, ω, n, η₁, η₂, ϕ_a, ϕ_b, D₁, D₂, K₁, and K₂. The motion styles of system under the unbalanced excitation and basement excitation are complex, where η₁ and η₂ are the dimensionless vibration frequency in the vertical and horizontal direction and ϕ_a and ϕ_b are the dimensionless excitation amplitude, respectively. Hence, we introduce $s_1\left(\tau \right)$ and $s_2\left(\tau \right)$ as the unbalanced excitation and basement excitation, and we have$$\begin{array}{c}\text{(11)}& \left\{\begin{array}{c}{s}_1\left(\tau \right)={\varphi }_a\sin \left({\eta }_1\tau \right),\\ {\dot{s}}_1\left(\tau \right)={\varphi }_a{\eta }_1\cos \left({\eta }_1\tau \right),\end{array}\right.\left\{\begin{array}{c}{s}_2\left(\tau \right)={\varphi }_b\sin \left({\eta }_2\tau \right),\\ {\dot{s}}_2\left(\tau \right)={\varphi }_b{\eta }_2\cos \left({\eta }_2\tau \right).\end{array}\right.\end{array}$$

Plugging equation (11) into equation (10), we get$$\begin{array}{}\text{(12)}& \left\{\begin{array}{c}{\dot{Z}}_1\left(\tau \right)={\dot{\overline{x}}}_1\left(\tau \right)=Z_2\left(\tau \right),\\ {\dot{Z}}_2\left(\tau \right)={\ddot{\overline{x}}}_1\left(\tau \right)=\frac{1}{{\omega }^2}+\frac{f_x}{{\omega }^2}+\alpha \cos \tau ,\\ {\dot{Z}}_3\left(\tau \right)={\dot{\overline{y}}}_1\left(\tau \right)=Z_4\left(\tau \right),\\ {\dot{Z}}_4\left(\tau \right)={\ddot{\overline{y}}}_1\left(\tau \right)=\frac{f_y}{{\omega }^2}+\alpha \sin \tau ,\\ {\dot{Z}}_5\left(\tau \right)={\dot{\overline{x}}}_2\left(\tau \right)=Z_6\left(\tau \right),\\ {\dot{Z}}_6\left(\tau \right)={\ddot{\overline{x}}}_2\left(\tau \right)=\frac{1}{{\omega }^2}-\frac{f_x}{n{\omega }^2}-\frac{K_1}{n{\omega }^2}{Z}_5\left(\tau \right)-\frac{D_1}{n{\omega }^2}{Z}_6\left(\tau \right)+\frac{K_1}{n{\omega }^2}{s}_1\left(\tau \right)+\frac{D_1}{n{\omega }^2}{\dot{s}}_1\left(\tau \right),\\ {\dot{Z}}_7\left(\tau \right)={\dot{\overline{y}}}_2\left(\tau \right)=Z_8\left(\tau \right),\\ {\dot{Z}}_8\left(\tau \right)={\ddot{\overline{y}}}_2\left(\tau \right)=-\frac{f_y}{n{\omega }^2}-\frac{K_2}{n{\omega }^2}{Z}_7\left(\tau \right)-\frac{D_2}{n{\omega }^2}{Z}_8\left(\tau \right)+\frac{K_2}{n{\omega }^2}{s}_2\left(\tau \right)+\frac{D_2}{n{\omega }^2}{\dot{s}}_2\left(\tau \right).\end{array}\right.\end{array}$$

Equation (10) belongs to high-dimensional, strong constrained, and nonlinear nonautonomous problems with various parameters; generally, in order to facilitate calculation and solution for convenience. Equation (10) could be discussed on the state space, and hence the four second-order equations can be easily converted into eight first-order ones and the equation of motion governing the behavior of the mechanical isolation system can be written as equation (12).

3. Nonlinear Dynamic Analysis

For the strongly nonlinear differential equation (12), no analytical method exists, which is solved by applying the Runge–Kutta integration routine with a variable step in the present work. The calculation parameters of the system are as follows [24, 25]:

n = 10, α = 0.1, σ = 3, K₁ = K₂ = 12, D₁ = D₂ = 0.1, $x_1=0.2$, ${\dot{x}}_1=0.0$, $y_1=0.1$, ${\dot{y}}_1=0.0$, $x_2=0.1$, ${\dot{x}}_2=0.0$, $y_2=0.1$, and ${\dot{y}}_2=0.0$.

Equation (12) is eight one-order nonlinear differential equations controlled by 11 varied parameters. Solving the analytical solution or the approximate analytical solution of the system is much more difficult to address than we imagine. Thus, numerical methods are widely used to address the nonlinear problems. In order to simulate easily, the initial conditions are given as following: η₁ = η₂ = η = 0.125; more importantly, the dynamic characteristic of the basement excitation isolation system under given conditions can be analyzed in detail.

3.1. Bifurcation Diagram

Because equation (12) is a nonlinear nonautonomous system with multidimensional rotor-bearing system, which contains the nonlinear oil film, nonlinear functions, and excitation function about $\sin \varphi$ and $\cos \varphi$, consequently, the analytic solution or the approximate one cannot be obtained easily. A theoretical analysis has been conducted for the mechanical isolation system. Toward this, (12) is integrated over a long period of time for bifurcation of initial condition by using the Rung–Kutta technique to be introduced to calculate dynamic response of the rotating mechanical isolation system coupled with air bag-floating raft under basement excitation.

In the non-linear analysis, for the purpose, the bifurcation diagram is constructed via numerical calculations. Construction of a bifurcation diagram is a standard approach used to analyze various nonlinear systems. In the bifurcation diagram, the dynamical behaviors may be viewed globally over a range of parameter values and allow us to compare simultaneously different types of motion. Therefore, the bifurcation diagram provides a summary of essential dynamics and is a useful tool for acquiring an overview.

Figure 3 illustrates the bifurcation diagram based on the above initial condition and shows the theoretically calculated bifurcation diagram of the mechanical isolation system with respect to the key motion parameter ω, in which ω varied from 0.60 to 3.50. It can be summarized from the bifurcation diagram (a) that the main influences by basic excitation on the mechanical isolation system are mainly concentrated in the low rotating speed stage, where the system amplitude is large in comparation with the prior research [26] and the response shows a periodic motion. With the rapid increments of rotor speed, the system vibration decreases gradually in the vertical direction. Figure 3(b) shows bifurcation of the same system for the displacement response y₁ with respect to the bifurcation parameter ω. The results indicate it has the same dynamics at lower speed, but with the increase of rotor speed, the vibration starts to rise and the dynamic performance demonstrates period-5, and when the system stays at higher speed, strong vibration and chaos will appear.

[figures omitted; refer to PDF]

3.2. Dynamic Steady-State Response

Figures 4 and 5 illustrate the steady response and its frequency spectrum, rotor obit, and its Poincaré map when the rotor speed is ω = 1.20 and 2.25, respectively. The numerical results show that the dynamic response of the isolation system contains the rotor frequency and the basement excitation frequency. The basement excitation affects the external force of the motion differential equation; therefore, the trajectory of the system is more complex and the nonlinear factor on the mechanical isolation system has weak effects; the repeatability of the rotor orbit is better expressed clearly, and the same time it is worth mentioning here that points on the Poincaré section reflect five isolated points, which means the dynamic characteristics of the system shows period-5 motion.

[figures omitted; refer to PDF]

[figures omitted; refer to PDF]

Figure 6 shows the oscillations at ω = 3.412, in which the nonlinear dynamic behaviors are the quasi-period motion and the displacement response (Figure 6(a)) highlights steady vibration. The 1/2 frequency components mainly dominate on the frequency spectrum (Figure 6(a)), the steady-state response of this system is confined to an annular region, and the rotor makes complex movements as shown in (Figure 6(c)). Points on the Poincaré maps are an approximate closed curve, the dynamic characteristic of the quasi-period motion at this rotor speed.

[figures omitted; refer to PDF]

Figures 7–9 illustrate the displacement response and its frequency spectrum, rotor obit, and its Poincaré map when ω = 3.35, 3.50, and 3.58, respectively. The dynamic behaviors of the system mainly show chaotic motion, in which the displacement response shows asymmetry and a continuous frequency is not expressed in the frequency spectrum clearly, but trajectory is a mess network structure and its Poincaré section shows some break, scattered points that are not closed.

[figures omitted; refer to PDF]

[figures omitted; refer to PDF]

[figures omitted; refer to PDF]

4. Parametric Analysis

4.1. The Effect of Excitation Frequency

Figure 10 illustrates the bifurcation diagram of floating-raft forced by the displacement response x₁ and y₁ with respect to the ratio of excitation frequency η, varying from 0.1 to 0.5 in certain parameters for ω = 1.20 based on the initial condition as $\sigma =3$, $\alpha =0.01$, $n=30,$ $K_1=100$, $K_2=40$, $D_1=D_2=0.1$, $\lambda =0.2$, and $\varphi =0.0175$. It is worth mentioning here that the external force of the motion differential equation is influenced by the basement excitation; thus, the trajectory is no longer a circle or ellipse, but a complex shape with the η increasing. The vertical displacement x₁ of the mechanical isolation system at low frequency shows the high amplitude due to the effect of basic excitation frequency, but the amplitude starts to drop gradually with the increase of frequency (however the y₂ had a opposite trend); throughout the whole period, the dynamic responses are dominated by period-5, quasi-period, period-2, period-3, and chaos.

[figures omitted; refer to PDF]

Figures 11–15 illustrate the displacement response and its frequency spectrum, rotor obit, and its Poincaré map at ω = 1.20 when η = 0.125, 0.167, 0.25, 0.5, and 0.65, respectively.

[figures omitted; refer to PDF]

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From the spectrum, it can be found that f/2 in the frequency response component is mainly dominated as shown. At this point, the frequency whirl vibration of the system is obvious; meanwhile, the unbalance magnitude of the system plays a dominant role during the whole process of operation.

It can be seen from the Poincaré maps that five isolated points project on the Poincaré section when η = 0.125, which presents period-5 motion. When η = 0.167 and 0.25, the dynamic characters of the system demonstrate period-3 motion because of the three unrelated and isolated attractors shown and the steady response have a steady beat vibration and trajectories are simple. However, with the increase of η, the trajectories start to become complex; furthermore, an unclosed curve is shown in Figure 16, and the Poincaré section also proves a chaos motion.

[figures omitted; refer to PDF]

Generally, after periodic motion at single frequency disturbing, the system will move into steady state, and its trajectories will acquire a disjoint closed form. However, in the system under the basement external stimuli η = 0.5 (Figure 17), the phase plane will extend, unbalanced force plays a dominant role, the closure of “8” trajectory appears as shown in Figure 17, and its motion trajectory is periodic; two isolated points projected on the Poincaré maps create attractors, that is to say that the system move in period-two motion.

[figures omitted; refer to PDF]

4.2. The Effect of Excitation Amplitude

Figure 18 illustrates the bifurcation diagram at ω = 1.20 with respect to the ratio of excitation amplitude ϕ, in which ϕ varied from 0.01 to 0.10. The basement excitation affects the external force of the motion differential equation; therefore, the trajectory of the system is more complex. The mechanical isolation system is weakly affected by the excitation amplitude at minor excitation amplitude. However, its vibration amplitude will increase with the excitation amplitude. The dynamic behavior of the system stayed at state of period-5 throughout the whole period.

[figures omitted; refer to PDF]

Figures 19–22 illustrate the dynamic steady response at ϕ = 0.020, 0.055, 0.075, and 0.100, respectively. The numerical simulation results indicate that the region value of system vibration, with the increase of excitation amplitude, grows rapidly and vibration amplitude also rises. The beat vibration and 1f are mainly shown on displacement response and frequency, which is the interaction of both system’s unbalance magnitude and basement excitation; thus, oil film oscillation appears. Furthermore, the other frequency components cannot be easily actuated under these specific values compared with the first mode frequency which indicates that the unbalance force and nonlinear oil force of the system play a dominant role during the whole process of operation.

[figures omitted; refer to PDF]

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Due to the basement excitation, the external force of equations is affected, as a result the rotor orbit has changed and clearly shows several single-closed curves. Besides, we note that the trajectory will expand outward in y direction as the excitation amplitude increases gradually when we compare the trajectories under various excitation values. More importantly, the points on the Poincaré map could be numbered and five isolated simple attractors appear apparently on its section, which demonstrates incontestably that the system moves into period-5 motion.

5. Conclusions

In this paper, an investigation into the nonlinear dynamic behavior of the mechanical isolation system coupled with air-bag and floating-raft under lateral basement excitation is presented. First, the coupling effects between the excitation source and isolation system are considered, the mechanical isolation model subjected to the basic excitation is built, and its motion equation is developed and dynamic responses are mainly investigated by using the displacement response, frequency spectrum, rotor orbit, Poincaré maps, and the bifurcation diagram. The result predicts that period-5 is mainly performed, and with the increase of rotor speed, the system moves into quasi-bifurcation. However, the system stays in chaos state at high rotor speed, and the vibration amplitude rises rapidly until against bearing bush. Furthermore, the effects of basement excitation on the mechanical isolation system are mainly concentrated on the stage of lower rotor speed, but with the increasing speed, the effects become weak and at the same time, the vibration amplitude reduces significantly. The points projected on the Poincaré section are five, three, or two solitary attractors, in which the system stays in periodic motion. Above all, the dynamic characteristics can provide theoretic supporting for the dynamic, vibration control and its parametric optimization of the marine mechanical isolation system coupled with air-bag and floating-raft on the motion base. Besides, it is of importance for a designer to have at their disposal dynamic models to predict the vibrational behavior of rotating machines operating under base excitations at the design stage to avoid expensive design modifications or in-field catastrophic failure.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (grant no. 11972282) and the Natural Science Foundation of Shaanxi Province, China (grant no. 2018JZ1001). The authors would like to express their deep thanks for the people who supported them during writing this paper. Many thanks go to the author’s advisor Prof. M. Li, parents, and brother. This support is gratefully acknowledged.

Glossary

Nomenclature

c:Bearing clearance

e:Eccentricity position of rotor center line in journal bearing

d:Dimensionless damping

D:Dimensionless damping

f_x, f_y:Nondimensional oil film forces of journal bearing

F_xF_y:Oil film forces of journal bearing

$g$:Acceleration of gravity

h:Thickness of oil film

k:Dimensionless stiffness

K:Dimensional stiffness

m₁:Mass of disc

m₂:Mass of the floating-raft

n:Ratio of mass

p:Oil film pressure

t:Time

u:Vector variable

x, y:Coordinates of the centers of disc

z:Coordinates in width and circumference directions

α:Dimensionless eccentricity

ε:Bearing eccentricity

φ:Angular position of rotor center line in journal bearing

ϕ_a, ϕ_b:Dimensionless excitation amplitude

η₁, η₂:Dimensionless vibration frequency

σ:Sommerfeld of bearing

λ:Ratio of length diameter

τ:Nondimensional time

Ω:Rotating speed of rotor

ω:Nondimensional rotor speed

1, 2:Disc (or rotor-bearing) and floating-raft.

References

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