Impact performance of the marine rotating machinery coupled with floating raft-airbag-limiter under heaving motion

Abstract

A dynamic model of an asymmetric system is developed with limiter under heave conditions to explore the impact characteristics of a marine rotating machinery coupled airbag-floating raft-limiter systems. The steady-state response of the system is numerically solved using the Runge-Kutta theory. Nonlinear dynamic analysis methods, including energy trajectories and time-averaged energy diagrams, have been introduced to examine the influence of rotor speed, heaving amplitude, limiter clearance, and contact stiffness on the dynamic characteristics of the system. The results show that the amplitude of the heave increases, the system collides with the limiter, resulting in a significant reduction in amplitude. Meanwhile, the dynamic behavior of system transitions from quasi-periodic motion to chaotic state. After the impact, the energy of the system begins to concentrate in the region of impact, and the higher the time-averaged energy value, the more severe the impact of the system. The method proposed in this paper can provides an optimization scheme and theoretical reference for the impact problem of this system.

Keywords

Heaving motion rotating machinery floating raft-airbag-limiter impact

Introduction

In rough seas, ships undergo dynamic movements such as heaving, pitching, and rolling.¹ The ship hull experiences lateral displacement in response to wave undulations, with the frequency of this displacement influenced by the frequency of the waves. While the heaving frequency is lower than the rotating frequency of the mechanical systems within a marine pontoon raft-airbag assembly, the inertial forces produced during heaving significantly impact the normal functioning of these mechanical systems. Therefore, the study of the dynamic properties of the floating raft-airbag system under heave conditions can offer more precise theoretical insights for refining the parameters of the floating raft-airbag.

The floating raft-airbag isolation system serves as a crucial mechanism for mitigating and isolating vibrations, with wide-ranging applications in the maritime industry, offshore platforms, and various structures.^2,3 This system effectively reduces the impact of external forces such as heave, thereby enhancing ship stability and minimizing the risk of structural fatigue and damage. Eminent scholars have conducted extensive investigations into the dynamic behavior of the floating raft-airbag vibration isolation system. Their research indicates that this system exhibits complex dynamic behaviors similar to those of rotating mechanical systems, including phenomena such as single-cycle, multi-cycle, quasi-periodic, and chaotic dynamics.^4–6 For example, Lv et al. improved the low-frequency isolation effect by incorporating an airbag isolator into a floating raft isolation device.⁷ This determination of optimal airbag pressure not only maintained raft stability but also provided theoretical support for vibration control and system optimization.

A limiter acts as a mechanism engineered to regulate and demarcate the allowable operational boundaries within which machinery or equipment can equipment.⁸ Its primary objective is to stop or redirect the movement of the machinery upon reaching a pre-defined position, thereby averting damage or accidents that might arise from surpassing the motion range. In the domain of marine rotating machinery, recent studies have demonstrated that opting for a smaller limiter gap and combining it with suitable stiffness can prove advantageous in reducing system resonance.^9,10 Ship operations in demanding sea conditions pose a substantial challenge to the rotational machinery of mechanical systems on ships. The utilization of limiters boosts the speed and efficiency of ship rotational machinery. Nevertheless, incorrect design or installation of limiters can lead to equipment damage or malfunction. Hence, undertaking parameter optimization and design research for limiters is indispensable.

Contact impact is a prevalent mechanical phenomenon in engineering, classified as fully elastic, non-fully elastic, or plastic based on the contact conditions of the impact.^11–13 The existence of friction during impact further differentiates it into smooth or non-smooth types. Scholars has been dedicated to experimental and theoretical studies on the collision problem. For instance, Vasilev et al. developed a variable recovery coefficient model considering material yield strength and initial impact velocity, demonstrating its efficacy through simulation and testing.¹⁴ Sun et al. have adopted a combination of the multi-harmonic balance method and the alternating frequency domain technique. This study investigates the steady-state responses and stability of a dual-rotor system with friction impact. This approach is utilized to calculate the precise amplitudes of each harmonic component.¹⁵ Zhang et al. employed the methods of energy vibration and power flow analysis to explore the patterns of rotor energy variation, the energy characteristics produced by various faults, as well as the energy response and transmission properties in the presence of nonlinear phenomena triggered by faults.^16–19 Xiao applied power flow analysis to study rolling mill vibrations, demonstrating its superiority over traditional methods.²⁰ Additionally, Zhu explored vibration power flow characteristics in cracked structures, proposing a crack identification method based on structural power flow.²¹ Additionally, scholars have investigated mechanical vibration systems with clearances subjected to harmonic excitation. Through comprehensive simulations considering multiple parameters and performance metrics, these studies have elucidated the relationship between dynamic performance and system parameters, as well as the principles of parameter matching.^22–24

The aforementioned studies have explored the vibration isolation characteristics of floating raft-airbag systems, the parameter optimization of limiters, and the dynamic behavior of mechanical collisions. Nevertheless, it is essential to investigate the dynamic behavior of rotating mechanical systems with floating raft-airbag structures under entrained motion and the parameter optimization of limiters when colliding with them. Consequently, this paper mainly focuses on the nonlinear dynamic behavior of marine rotating machinery coupled with a floating raft-airbag system under heaving motion and its interaction with limiters.

Dynamic model

In this section, a theoretical model of the impact between marine rotating machinery and a floating raft-airbag-limiter system is established, which is schematically represented in three dimensions in Figure 1. Based on the principles of analytical mechanics, the governing differential equations of the system are derived and subsequently dimensionless.

Figure 1.

Schematic three-dimensional diagram of the floating raft-airbag-limiter system.

Motion equations

Figure 2 presents a schematic illustration of marine rotating machinery coupled with floating raft-airbag-limiter under heaving motion. To streamline the analysis, several assumed conditions are proposed

In actual operational conditions, the floating raft-airbag system is not aligned with the ship centerline. To simplify calculations, it is assumed that the system is centrally installed.

The floating raft-airbag and rotating machinery are considered a unified entity with a total mass of $m$ . $m_{e}$ and $e$ are the rotational eccentric mass and eccentricity distance, respectively. $b$ and $a$ are the floating raft length and width, respectively. Where $x_{0}$ and $θ$ are the vertical displacement and the angular deflection, respectively. The floating raft-airbag exhibits cubic nonlinear elastic force with a magnitude of $k_{1} x_{0} + k_{12} x_{0}^{3}$ . $k_{2}$ and $c$ are the linear stiffness of the limiter and the vertical linear damping, respectively.

x_H is the displacement induced by heaving motion is idealized as the translational displacement of the base.

The contact point between the impact surfaces is considered, assuming that no lateral friction is present. The impact is modeled as inelastic, which requires a change in the velocity of the system both before and after the impact.

Figure 2.

Schematic diagram of ship board rotating machinery coupling floating raft-airbag under heaving motion: (a) schematic diagram of a rotating machinery coupled floating raft-airbag-limiter system and (b) deformation diagram of a rotating machinery coupled floating raft-airbag-limiter system.

When a floating raft-airbag collides with a limiter, the displacements of the four pivotal points where the impact occurs are interconnected as follows

{\begin{matrix} x_{1} = x_{0} + \frac{a}{2} \cos θ + \frac{b}{2} \sin θ - \frac{a}{2} \\ x_{2} = x_{0} + \frac{a}{2} \cos θ - \frac{b}{2} \sin θ - \frac{a}{2} \\ x_{3} = x_{0} - \frac{a}{2} \cos θ - \frac{b}{2} \sin θ + \frac{a}{2} \\ x_{4} = x_{0} - \frac{a}{2} \cos θ + \frac{b}{2} \sin θ + \frac{a}{2} \end{matrix}

(1)

the total kinetic energy of the system can be expressed as

T = T_{m} + T_{m_{e}}

(2)

Where

T_{m} = \frac{1}{2} (m - m_{e}) {x'_{z}}^{2} + \frac{1}{2} J_{0} θ'^{2}

(3)

T_{m_{e}} = \frac{1}{2} m_{e} {v_{x}}^{2} + \frac{1}{2} m_{e} {v_{y}}^{2} + \frac{1}{2} J_{1} ω^{2}

(4)

In equation (3), $x_{z} = x_{0} + x_{H}$ represents the absolute displacement of the system, $J_{0}$ signifies the moment of inertia of the floating raft, $J_{1}$ denotes the moment of inertia of the rotating machinery, while $v_{x}$ and $v_{y}$ denote the horizontal and vertical velocities of the eccentric mass $m_{e}$ , respectively, and can be expressed as

{\begin{matrix} v_{x} = \frac{d}{d t} (x_{z} + e \sin (ω t + θ)) \\ = {x^{'}}_{z} + e (ω + θ^{'}) \cos (ω t + θ) \\ v_{y} = \frac{d}{d t} (e \cos (ω t + θ)) \\ = - e (ω + θ^{'}) \sin (ω t + θ) \end{matrix}

(5)

Thus, the total kinetic energy is as follows

\begin{matrix} T = T_{m} + T_{m_{e}} = \frac{1}{2} mx'_{z}^{2} + \frac{1}{2} J_{0} {θ'}^{2} + \frac{1}{2} J_{1} ω^{2} + \\ \frac{1}{2} m_{e} e^{2} {(ω + θ')}^{2} + m_{e} e {x'}_{z} (ω + θ') \cos (ω t + θ) \end{matrix}

(6)

The Rayleigh dissipation function of the system is given by

R = \sum_{i = 1}^{n} \frac{1}{2} c_{i} r'_{i}^{2} = \frac{1}{2} c ({x'}_{1}^{2} + {x'}_{2}^{2})

(7)

Due to adverse maritime conditions and extraneous variables, the system may experience substantial oscillations, potentially leading to collisions with the limiter. Therefore, it is suggested that the impact between the system and the limiter occurs in a frictionless manner, with energy loss confined to the intervals immediately before and after the impact. These collisions are characterized as vertical in nature. The magnitudes of the elastic forces within the limiters are as follows

f_{i} = k_{2} σ_{i} (i = 1, 2, 3, 4)

(8)

In equation (8), $σ_{i}$ represents the deformation of the limiter, which can be expressed as

σ_{i} = {\begin{matrix} 0 & | x_{i} - x_{10} | < δ \\ | x_{i} - x_{10} | - δ & | x_{i} - x_{10} | \geq δ \end{matrix}

(9)

In equation (9), represents the static equilibrium displacement of rotating mechanically coupled floating raft-airbag system, which satisfies the following relationship

k_{1} x_{10} + k_{12} x_{10}^{3} = mg

(10)

Then the limiter spring force of the system is

f = - f_{1} - f_{2} + f_{3} + f_{4}

(11)

The force moment generated by the impact of the system with limiter can be expressed as

M_{i} = f_{i} \frac{b}{2} (i = 1, 2, 3, 4)

(12)

At the moment of impact, that is, at $x_{i} - x_{10} > δ (i = 1, 2)$ or $x_{i} - x_{10} < - δ (i = 3, 4)$ , the velocity before and after the impact can be represented as

x'_{0}^{+} = μ_{i} x'_{0}^{-} (i = 1, 2)

(13)

In equation (13), symbols $x'_{0}^{+}$ and $x'_{0}^{-}$ denote instantaneous velocities before and after impact, respectively. The impact recovery coefficient denoted as $μ_{i}$ , collision recovery coefficient signifies overall kinetic energy of the system rigid body motion that is redistributed due to collision, and it represents the extent of energy loss during the collision process.²⁵ Let $μ_{1}$ denote impact recovery coefficient for impacts with upper limiter, and $μ_{2}$ symbolize impact recovery coefficient for impacts with lower limiter.

The generalized force vector $Q_{j}$ in the Lagrange function encompasses gravitational forces, limiter elasticity and force moments, can be expressed as

Q_{j} = {\begin{matrix} mg \cos θ + Θ f \\ Θ M - m_{e} ge \cos (ω t + θ) \end{matrix}}

(14)

In equation (14), $Θ$ is a $Heaviside$ function, which is specifically expressed as

Θ = {\begin{matrix} 0 | x_{i} - x_{10} | < δ \\ 1 | x_{i} - x_{10} | \geq δ \end{matrix} (i = 1, 2, 3, 4)

(15)

According to the II type Lagrange equation

\frac{d}{dt} \frac{\partial T}{\partial {q'}_{j}} - \frac{\partial T}{\partial q_{j}} + \frac{\partial u}{\partial q_{j}} = Q_{j} - \frac{\partial R}{\partial {q'}_{j}} (j = 1, 2, 3, \dots, k)

(16)

$q_{j}$ is generalized coordinate, $T$ is kinetic energy of the system, $Q_{j}$ is generalized force, and $u$ is elastic potential energy, as shown in equation (17). The differential equation of motion for the system can be represented as

{\begin{matrix} m {x^{″}}_{0} = m g \cos θ + Θ f - m {x^{″}}_{H} - m_{e} e θ^{″} \cos \\ (ω t + θ) + m_{e} e {(ω + θ^{'})}^{2} \sin (ω t + θ) - \\ \frac{1}{2} c {x^{'}}_{1} - \frac{1}{2} c {x^{'}}_{2} - k_{1} (x_{0} - \frac{a}{2} + \frac{a \cos θ}{2}) \\ - \frac{1}{2} k_{12} x_{1}^{3} - \frac{1}{2} k_{12} x_{2}^{3} \\ J_{0} θ^{″} = Θ M - m_{e} e^{2} θ^{″} - m_{e} e {x^{″}}_{0} \cos (ω t + θ) \\ - m_{e} e {x^{″}}_{H} \cos (ω t + θ) - m_{e} e g \cos (ω t + θ) \\ - (\frac{b}{4} \cos θ - \frac{a}{4} \sin θ) (k_{1} x_{1} + k_{12} x_{1}^{3} + c {x^{'}}_{1}) \\ + (\frac{b}{4} \cos θ + \frac{a}{4} \sin θ) (k_{1} x_{2} + k_{12} x_{2}^{3} + c {x^{'}}_{2}) \end{matrix}

(17)

The ship hull displacement due to heave follows a sinusoidal relationship as described in equations $x_{H} = a_{H} \sin ω_{H} t$ ,^26,27 where $a_{H}$ and $ω_{H}$ are amplitude of heave and heaving frequency, respectively. As shown in equation (18). Hence, the expression for heaving acceleration of ship under heaving-induced oscillations is expressed as

{x ″}_{H} = - a_{H} ω_{H}^{2} \sin ω_{H} t

(18)

Let $ω_{n}^{2} = k_{1} / m$ , $c = 2 m ζ ω_{n}$ , $λ = \sqrt{k_{1} / e^{2} k_{12}}$ . Where $ζ$ represents the damping ratio, $ω_{n}$ denotes natural frequency of floating raft-airbag system, and $λ$ is characteristic length. Formula (18) can be organized as follows

{\begin{matrix} {x ″}_{0} = g \cos θ + \frac{Θ f}{m} + a_{H} ω_{H}^{2} \sin ω_{H} t - \frac{m_{e}}{m} e θ ″ \cos (ω t + θ) \\ + \frac{m_{e}}{m} e {(ω + θ')}^{2} \sin (ω t + θ) - ζ ω_{n} {x'}_{1} - ζ ω_{n} {x'}_{2} - \\ ω_{n}^{2} (x_{0} - \frac{a}{2} + \frac{a \cos θ}{2}) - (\frac{ω_{n}^{2}}{2 λ^{2} e^{2}}) x_{1}^{3} - (\frac{ω_{n}^{2}}{2 λ^{2} e^{2}}) x_{2}^{3} \\ θ ″ = \frac{Θ M}{J_{0}} - \frac{m_{e} e^{2} θ ″}{J_{0}} - \frac{m_{e} e {x ″}_{0}}{J_{0}} \cos (ω t + θ) \\ + \frac{m_{e} e a_{H} ω_{H}^{2} \sin (ω_{H} t)}{J_{0}} \cos (ω t + θ) \\ - \frac{m_{e} eg}{J_{0}} \cos (ω t + θ) - \end{matrix}

{\begin{matrix} (\frac{b}{4} \cos θ - \frac{a}{4} \sin θ) (\frac{ω_{n}^{2} m x_{1}}{J_{0}} + \frac{ω_{n}^{2} m x_{1}^{3}}{J_{0} λ^{2} e^{2}} + \frac{2 m ζ ω_{n} {x^{'}}_{1}}{J_{0}}) + \\ (\frac{b}{4} \cos θ + \frac{a}{4} \sin θ) (\frac{ω_{n}^{2} m x_{2}}{J_{0}} + \frac{ω_{n}^{2} m x_{2}^{3}}{J_{0} λ^{2} e^{2}} + \frac{2 m ζ ω_{n} {x^{'}}_{1}}{J_{0}}) \end{matrix}

(19)

Dimensionless equations

In order to mitigate the influence of various physical quantity units and streamline subsequent analytical procedures, the introduction of characteristic length eccentricity and the transformation of differential equations into dimensionless form are necessary. The dimensionless parameters can be found in Table 1. In equation (20), the dimensionless quantization of the elastic force of the limiter can be expressed as

F_{i} = φ \frac{Ω_{n}^{2}}{Ω^{2}} σ'_{i} (i = 1, 2, 3, 4)

(20)

where $φ = k_{2} / k_{1}$ ,where $φ$ and $σ'_{i}$ are the proportionality coefficient and dimensionless deformation of the limiters, respectively. $σ'_{i}$ can be dimensionless as

σ'_{i} = {\begin{matrix} 0 | X_{i} - X_{10} | < δ' \\ | X_{i} - X_{10} | - δ' | X_{i} - X_{10} | \geq δ' \end{matrix}

(21)

Table 1.

Dimensionless parameter expressions.

Parameter name	Parameter	Expression
Dimensionless displacement	$x, X_{1}, X_{2}$	$x_{0} / e, x_{1} / e, x_{2} / e$
Dimensionless limiter gap	$α$	$δ / e$
Dimensionless amplitude of the heave	$ε$	$a_{H} / e$
Mass ratio	$n$	$m_{e} / m$
Dimensionless rotor speed	$Ω$	$ω \sqrt{e / g}$
Dimensionless heaving frequency	$Ω_{H}$	$ω_{H} \sqrt{e / g}$
Floating raft-airbag dimensionless vertical intrinsic frequency	$Ω_{n}$	$ω_{n} \sqrt{e / g}$
Dimensionless raft-airbag moment of inertia	$J$	$J_{0} / m e^{2}$
Limiter dimensionless torque	$\bar{M}$	$M / mge$
Raft dimensionless length, width	$γ, β$	$b / e, a / e$
Dimensionless time	$τ$	$ω t$

By the way, denote $x'_{0} = d x_{0} / dt$ , $x'_{0} = d x_{0} / dt$ , $X'_{1} = d x_{1} / d τ$ , and $X'_{2} = d x_{2} / d τ$ . Subsequently, the dimensionless differential equation governing motion of the system is formulated as

{\begin{matrix} x^{″} = \frac{\cos θ}{Ω^{2}} + Θ F + ε \frac{Ω_{H}^{2}}{Ω^{2}} \sin \frac{Ω_{H}}{Ω} τ - n θ^{″} \cos (τ + θ) \\ + n {(1 + θ^{'})}^{2} \sin (τ + θ) - ζ \frac{Ω_{n}}{Ω} {X^{'}}_{1} - ζ \frac{Ω_{n}}{Ω} {X^{'}}_{2} \\ - \frac{Ω_{n}^{2}}{Ω^{2}} (x - \frac{β}{2} + \frac{β \cos θ}{2}) - \frac{Ω_{n}^{2}}{2 λ^{2} Ω^{2}} X_{1}^{3} - \frac{Ω_{n}^{2}}{2 λ^{2} Ω^{2}} X_{2}^{3} \\ θ^{″} = \frac{Θ M^{'}}{J Ω^{2}} - \frac{n θ^{″}}{J} - \frac{n x^{″}}{J} \cos (τ + θ) \\ + \frac{n ε Ω_{H}^{2}}{J Ω^{2}} \sin (\frac{Ω_{H}}{Ω} τ) \cos (τ + θ) - \frac{n}{J Ω^{2}} \cos (τ + θ) \\ - (\frac{γ}{4} \cos θ - \frac{β}{4} \sin θ) \\ (\frac{Ω_{n}^{2} X_{1}}{J Ω^{2}} + \frac{Ω_{n}^{2} X_{1}^{3}}{J λ^{2} Ω^{2}} + \frac{2 ζ Ω_{n} {X^{'}}_{1}}{J Ω^{2}}) \\ + (\frac{γ}{4} \cos θ + \frac{β}{4} \sin θ) \\ (\frac{Ω_{n}^{2} X_{2}}{J Ω^{2}} + \frac{Ω_{n}^{2} X_{2}^{3}}{J λ^{2} Ω^{2}} + \frac{2 ζ Ω_{n} {X^{'}}_{2}}{J Ω^{2}}) \end{matrix}

(22)

In equation (23), the dimensionless equations for the four impact points are

{\begin{matrix} X_{1} = x + \frac{β}{2} \cos θ + \frac{γ}{2} \sin θ - \frac{β}{2} \\ X_{2} = x + \frac{β}{2} \cos θ - \frac{γ}{2} \sin θ - \frac{β}{2} \\ X_{3} = x - \frac{β}{2} \cos θ - \frac{γ}{2} \sin θ + \frac{β}{2} \\ X_{4} = x - \frac{β}{2} \cos θ + \frac{γ}{2} \sin θ + \frac{β}{2} \end{matrix}

(23)

In equation (12), dimensionless equation for the moment of force $M'$ is

M'_{i} = F_{i} \frac{γ}{2} (i = 1, 2, 3, 4)

(24)

Parameter analysis

Taking into account cubic nonlinear properties of airbag and segmented elastic force function of limiter, dimensionless vibration differential equations of the system are resolved employing 45th-order Runge-Kutta method. Dimensionless parameters of the raft-airbag coupled marine rotating machinery system are as follows: $β = 400,$ $n = 0.01,$ $Ω_{n} = 0.5$ , $ζ = 0.1$ , $λ = 20$ , $φ = 20$ , $α = 1$ , and $J = 8.67 \times 10^{5}$ . A time step of 1/100th of system motion period is chosen for calculation, and the initial conditions of system are set to $x = 4.0$ , $x' = 0$ , $θ = 0.0001$ , and $θ' = 0$ . The impact recovery factor of the system is $μ_{1} = μ_{2} = 1$ , which corresponds to scenario where there is no impact between airbag floating raft and limiter, and $μ_{1} = 0.74, μ_{2} = 0.8$ as in the event of impact. The scope of marine heaving amplitude and frequency during heaving motion is delineated in Table 2.²⁸

Table 2.

Range of values for the ship heaving parameters.

Parameter name	Parameter	Retrieve value
Amplitude of heave	$ε$	0–5000
Frequency of heave	$Ω_{H}$	0–0.03

In Figure 3, the impact of rotational speed on the system behavior is illustrated across different heaving amplitudes and frequencies, along with the steady-state response of the system at various heaving amplitudes. Specifically, Figure 3(a) highlights how heaving amplitude affects the system at diverse rotational speeds with a fixed limiter clearance and heaving frequency $Ω_{H} = 0.01$ . Notably, in absence of a limiter, displacement of the system increases linearly with the heaving amplitude. With a limiter in place and at a heaving amplitude of $ε < 2700$ , there is no impact between floating raft-airbag and limiter. However, at heaving amplitude $2700 \leq ε < 2900$ , there is unilateral contact between floating raft-airbag and limiter resulting in a noticeable decrease in slope of amplitude for this situation. Additionally, at low rotational speeds, the floating raft-airbag exhibits a linear increase in amplitude with the enhancement of heaving amplitude. Conversely, at high rotational speeds, the curve of the floating raft-airbag exhibits fluctuations. The findings indicate that the bilateral limiter have significant effect on the amplitude of the system. At $ε \geq 2900$ , the findings indicate that bilateral limiters have significant effect on system amplitude. At certain levels collision between floating raft-airbag system and the limiters induces oscillatory trajectories near these limits indicating maximum influence from wave excitation while influence from rotation starts manifesting.

Figure 3.

Steady state response of the system with different influences: (a) effect of speed on the system with different heaving amplitudes, (b) the effect of speed on the system with heaving frequencies varied, and (c) steady state response of the system with different heaving amplitudes.

Figure 3(b) illustrates impact of heaving frequency on the system at different speeds, while maintaining a fixed limiter gap $α = 1$ and heaving amplitude $ε = 2000$ . As speeds varies, the amplitude of the system increases nonlinearly with heaving frequency at $Ω_{H} < 0.012$ , and the slope becomes progressively steeper. When $Ω_{H} \geq 0.012$ , the system impacts the unilateral limiter at rotor speed $Ω = 1.12$ , there is a decrease in the system response amplitude. Additionally, beyond this threshold rotor speed, the rate of growth in amplitude slow down, indicating that the limiter has a more significant influence on the system at lower rotor speeds. The highest system occurs at heaving frequency $0.017 \leq Ω_{H} \leq 0.026$ and rotor speed $Ω = 2.45$ , suggesting that this combination makes the system more sensitive to the effect of the limiter.

Figure 3(c) illustrates the steady-state response of the different heaving amplitudes, while maintaining constant limiter clearance $α = 1$ , heaving frequency $Ω_{H} = 0.01$ , and rotor speed $Ω = 1.12$ . When the heaving amplitude is $ε = 0$ , periodic motion occurs due to the rotor eccentricity-induced excitation force. As the heaving amplitude increases up to 2000, there is a significant increase in system amplitude, indicating that heave excitation has a substantial impact on the system. However, when the heaving amplitude reaches $ε = 2700$ , collisions with limiters result in reduced motion amplitude and altered dynamics. Further increasing leads to additional collisions with bilateral limits.

Influence of rotor speed on system dynamics with different heaving amplitudes

Figure 4 exhibits vibrational across various heaving amplitudes with constant parameters: heaving frequency $Ω_{H} = 0.01$ , limiter clearance $α = 1$ , and rotor speed $Ω = 1.12$ . Figure 4(a) shows vibrational response at a heaving amplitude of $ε = 2000$ . In the time-domain response, the heave motion initiates a stable macro periodic vibration within the system. The energy response curve manifests a smooth “mountain-shaped” distribution. Notably, spectral analysis reveals the predominance of heaving frequencies $f_{H}$ and 2 $f_{H}$ , in addition to the rotating frequency $f_{1}$ , suggesting that the heaving action is the primary influence on the motion state of the system. The phase diagram of the system exhibits a symmetric “ring” structure, indicating a quasi-periodic motion state.

Figure 4.

The steady-state response of the system, energy response curve, phase diagram, and spectrogram are displayed with $Ω = 1.12$ : (a) $ε = 2000$ , (b) $ε = 2700$ , and (c) $ε = 3500$ .

In Figure 4(b), the vibrational response of the system at a heaving amplitude of $ε = 2700$ is depicted. The amplitude of the system increases in the time-domain response, leading to impact with the lower limiter. This asymmetry in the system, influenced by gravity and the joint action of the floating raft-airbag, results in initial impact with lower limiter. The trajectory of impact region in the energy response curve experiences sudden change, with energy surging to four times its original value. This indicates that there is energy input into system during collision process and collision process exhibits nonlinear characteristics. In phase diagram, abrupt changes in phase trajectory within collision region indicate transition of system motion state from stability to instability. Dominant feature in the frequency domain response is heaving frequency $f_{H}$ , with a significant proportion, accompanied by 2 $f_{H}$ and the rotating frequency $f_{1}$ , indicating that motion state of the system has not changed. Motion state of the system is quasi-periodic.

Figure 4(c) shows the vibration response of the system at a heaving amplitude of $ε = 3500$ . In time-domain response, the amplitude experiences further increment and encounters both upper and lower limiters. The energy response curve reveals that the energy is predominantly concentrated in collision region, and energy has increased to approximately five times its original value. The evidence indicates that the degree of collision within the system is more severe. Nevertheless, the augmentation in collision severity leads to rapid decrease in system energy curve, which is attributed to the fact that during collision process, damper might cause a swift dissipation of energy within a short period of time, thereby resulting in the rapid decline of the energy curve. In spectral diagram, heaving frequency $f_{H}$ accounts for the largest proportion, 2 $f_{H}$ disappears, being replaced by frequencies of 3 $f_{H}$ and 5 $f_{H}$ , along with some unidentified frequencies. Moreover, continuous spectrum emerges in the vicinity of the rotating frequency, signifying a shift in the motion status of the system. The motion state of system exhibits chaotic behavior.

Figure 5 illustrates the steady-state response of the system across varying amplitudes of heave at rotor speed $Ω = 1.65$ . The time-domain response depicted in Figure 5(a) indicates that the amplitude of the system remains constant despite an increase in rotor speed, with the trajectory of the system in subsequent cycle aligning closely with that of preceding cycles. The energy response curve demonstrates a decrease in the maximum value as rotational speed increases, suggesting a corresponding reduction in energy evaluation at this specific moment. This observation implies that the system exhibits enhanced stability compared to conditions where $Ω = 1.12$ . The phase diagram reveals that the system of motion within the system remains constant and further suggests that heave amplitude is a primary factor influencing dynamic behavior. In addition, the spectrogram shows only heaving frequency $f_{H}$ and rotating frequency $f_{1}$ , indicating that the motion state of the system is quasi-periodic.

Figure 5.

The steady-state response of the system, energy response curve, phase diagram, and spectrogram are displayed with $Ω = 1.65$ : (a) $ε = 2000$ , (b) $ε = 2700$ , and (c) $ε = 3500$ .

Figure 5(b) presents a time-domain response plot which reveals minor collisions with limiters, leading to a cessation of periodic variations in system amplitude. In contrast, post-collision alterations become more pronounced within the energy response curve. Specifically, there is an observed doubling of energy trajectory at lower rotational speeds while energy values within non-collision regions decline. This indicates that collisions facilitate an energy transfer from non-collision areas to collision zones. The increase in energy trajectory suggests heightened input from limiters and signifies an intensification of collision scenarios. Consequently, it can be inferred that due to these collisions, there is significant migration of energy from non-collision regions into collision zones. Furthermore, analysis through phase diagrams highlights abrupt transitions within trajectories occurring specifically within collision regions. The spectrogram indicates that solely heaving frequency $f_{H}$ and rotating frequency $f_{1}$ are present, indicating that the system is merely grazing limiter, which exerts minimal influence on system motion.

Figure 5(c), illustrates the time-domain response plot, which demonstrates that the system experiences simultaneous collisions with both the upper and lower limiters due to a progressive increase in heaving amplitude. The impact between the system and the limiters become more severe. In the energy response curve, these collisions result in abrupt changes that extend from the center of impact toward both sides. Notably, there is a pronounced increase in the maximum value of energy transition, indicating an extended duration of contact between the system and the stoppers, thereby leading to an increased energy input into the system. Abrupt changes are observed on both sides of the phase diagram trajectory. Due to inherent asymmetry within the system, alterations on the left side of this diagram are more significant than those on its right side; this suggests that collisions occurring on the left side are more intense. This observation aligns with findings from the energy response curve. Furthermore, analysis through spectrogram reveals a continuous spectrum emerging alongside six times the heaving frequency $f_{H}$ as well as an unidentified frequency near both heaving and rotational frequencies, indicative of a transformation in motion state within this system. The emergence of harmonics underscores nonlinearity within it while also suggesting that at this juncture, there is a transition from a pseudo-periodic state toward chaotic behavior.

Figure 6 illustrates steady-state responses across varying heaving amplitudes at constant rotor speed $Ω = 2.45$ . As depicted in Figure 6(a), low heaving amplitudes yield periodic motion where rotational speed exerts minimal influence over overall motion states within this context. Conversely, while depicting an uneven “mountain” shape for energy response curves indicates some level of impact by rotational speed upon system dynamics at this stage. In the phase diagram, oscillatory behavior of the phase trajectories is observed in specific regions. The spectrum indicates the presence of only the heaving frequency $f_{H}$ and the rotating frequency $f_{1}$ , suggesting that the system operates in a quasi-periodic state.

Figure 6.

The steady-state response of the system, energy response curve, phase diagram, and spectrogram are displayed with $Ω = 2.45$ : (a) $ε = 2000$ , (b) $ε = 2700$ , and (c) $ε = 3500$ .

In Figure 6(b), the time-domain response reveals that as the heaving amplitude increases, minor collisions with the lower limiter occur. However, compared to conditions at rotational speed $Ω = 1.65$ , these collisions are significantly reduced. The energy response curve further emphasizes this phenomenon, within the collision zone, energy levels consistently remain lower than peak energy levels observed in unaffected regions. Nevertheless, there is an overall increase in energy level attributed to the elevated rotor speed. Within the phase diagram, trajectory alterations are minimal within the collision region. Both heaving frequency $f_{H}$ and rotating frequency $f_{1}$ appear in the frequency domain response, with heaving frequency $f_{H}$ predominating, indicating that it is the primary responsible for influencing system motion.

In Figure 6(c), severe collisions between the system and bilateral limiters are evident from time-domain responses. The motion trajectory reveals that non-collision regions is time-domain trajectories are affected by these collisions, resulting in oscillatory phenomena. The energy response curve indicates a more intense collision scenario compared to speed $Ω = 1.65$ , showing a twofold increase in collision energy values. Furthermore, following each collision event, there is a rapid decrease in energy response curves – suggesting that increased instantaneous energy input into the system leads to accelerated dissipation of energy through damping mechanisms. In summary, within the phase diagram context, an escalation of rotational speed results in abrupt changes to trajectory patterns. In the frequency domain response, a sixfold increase in heaving frequency $f_{H}$ is observed, attributed to the nonlinear stiffness of the floating raft-airbag system. Additionally, a continuous spectrum and unknown frequencies emerge, indicating that the system is in a chaotic state.

Figure 7 illustrates the steady-state response of the system across varying heaving amplitudes at a constant rotor speed $Ω = 3.18$ . At a heaving amplitude of $ε = 2000$ , as depicted in Figure 7(a), the system exhibits periodic motion in time-domain response. In the energy response curve, surface roughness becomes more pronounced along the energy trajectory, suggesting that an increase in rotational speed leads to an unstable motion state for the system. The spectral response reveals both heaving frequency $f_{H}$ and rotating frequency $f_{1}$ are observed. However, within certain regions of the phase diagram, local oscillations in velocity arise primarily due to increase rotational speed. These phenomena indicate that at low heaving amplitudes, significant impacts on system stability are caused by rising rotational speed.

Figure 7.

The steady-state response of the system, energy response curve, phase diagram, and spectrogram are displayed with $Ω = 3.18$ : (a) $ε = 2000$ , (b) $ε = 2700$ , and (c) $ε = 3500$ .

At a heaving amplitude of $ε = 2700$ , as shown in Figure 7(b), there is evidence of slight collision between the system and one side limiter within the time-domain response graph. This collision region influences motion states elsewhere within unaffected areas. In comparison to previous rotational speed, an increase is noted in energy responses during collision conditions on this curve. This indicates that energy input from limiters into the system exceeds levels seen at prior speeds while also suggesting heightened collision condition for said systems. Furthermore, increases in rotor speed results in abrupt change within non-collision area energy response curves, indicating sensitive of these curves to variations in rotor speed. In the phase diagram, the trajectories on the left side exhibit abrupt transitions due to collisions, with those within the collision zone being larger than those associated with the rotational speed $Ω = 1.65$ . The unaffected trajectories in proximity to the collision zone experience an amplification in heaving amplitude as a result of collision influence, which is corroborated by the energy response curve. In the frequency domain response, both heaving frequency $f_{H}$ and rotating frequency $f_{1}$ are present.

The heaving amplitude of $ε = 3500$ is illustrated in Figure 7(c). In the time domain response plot, significant collisions occur with both bilateral limiters, leading to a more pronounced impact on the unaffected region. The energy response curve indicates that the collision region expands, resulting in a marked energy mutation. Concurrently, there is a decrease in energy value within the unaffected region, indicating that energy is transferred from this area to those affected by collisions. This phenomenon also highlights the highly nonlinear nature of system interactions during collision. The rapid transformation and release of energy result in abrupt changes in the nonlinear dynamics of the system under specific conditions. The rapid transformation and release of energy lead to abrupt changes in nonlinear dynamics under specific condition. Within the frequency domain response, rotating frequency $f_{1} / 2$ emerges alongside a continuous spectrum appearance, this signifies that the system motion state has transitioned from a simulated cycle into a chaotic state.

Effect of limiter gap on system dynamics at different heaving amplitudes

System vibration arises from internal energy transfer and conversion processes that induce variations in system displacement. In steady state conditions, internal energy remains balanced. However, external forces disrupting this balance create an imbalance of energy and consequently shift its state. This section introduces time-averaged energy as a means to quantify average energy levels over specific timeframes. It is calculated by aggregating transient energies over a defined duration and averaging them within a specified time interval. The introduction of the concept of time-averaged energy allows for an efficient, accurate, and rapid reflection of changes in collision severity resulting from variations in system parameters. This research provides theoretical support for optimizing system parameters and diagnosing collision severity. The dimensionless time-averaged energy value $E'$ takes following specific form

E' = \frac{1}{T} \int_{τ_{0}}^{τ_{0} + T} E d τ

(25)

Where $E$ represents the transient energy of the system following dimensionless transformation.

Figure 8 illustrates the relationship between the displacement amplitude of the floating raft-airbag and the limiter gap for varying heaving amplitudes, along with corresponding time-averaged energy profiles for differing limiter clearances. In Figure 8(a), at a heaving amplitude of $ε = 2000$ , it can be observed that as the limiter gap expands, the displacement amplitude of the system increases linearly until it encounters unilateral contact with a limiter. Following this point, further widening of the limiter gap results in stabilization of system amplitude. A transition occurs in the nature of collisions between the limiter and system when the limiter gap ranges from 0.71 to 0.75. During this interval, engagement is exclusively with the lower limiter, and upon the increasing the limiter gap to 0.75, maximum system amplitude reaches 1.46. The time-averaged energy diagram indicates that when the gap is $δ = 0.65$ , the system collides with both sides of the limiter, resulting in an energy value of 0.134. Conversely, when the gap is $δ = 0.73$ , the system interacts with only one side of the limiter, yielding an energy value of 0.0762. Finally, when the gap $δ > 0.73$ , the system experiences no contact with the limiter, leading to an energy value of 0.0592. As the gap between the limiter increases, there is a corresponding decrease in energy levels, this suggests a reduction in system collisions. In other words, lower time-averaged energy values correlate with fewer collisions within the system.

Figure 8.

Displays the plot of the system amplitude versus limiter gap for various heaving amplitudes at $Ω = 1.56$ : (a) $ε = 2000$ ,(b) $ε = 2700$ , and (c) $ε = 3500$ , along with the time-averaged energy for different limiter gaps.

Figure 8(b) depicts the relationship between system amplitude and limiter gap across varying heaving amplitudes. Notably, as heaving amplitude $ε = 2700$ , system amplitude rises nearly linearly alongside widening gaps in the limiter. When this gap reaches 0.95, significant changes occur in impact characteristics, unilateral impacts are observed along with a maximum system amplitude of 1.97 units achieved thereafter. As heaving amplitude continues to increase, beyond this point and after colliding unilaterally with limiters, growth rates for system amplitude begin to decelerate while simultaneously expanding ranges for unilateral impacts indicate that higher heaving amplitudes lead to more pronounced interactions between systems and limiters, resulting in greater suppression effects on overall system amplitudes. When limiter gap increases to 1, the maximum system amplitude is 1.97, and further increments do not significantly influence overall system amplitude, thus it can be concluded that limiters exhibiting gaps exceeding this measurement exert negligible effects on performance metrics. In the time-averaged energy diagram, as the limiter gap increases, the time-averaged energy value gradually decreases but at a reduced rate, indicating transitions within systemic impacts shift from severe encounters to milder interactions. A comparison of the time-averaged energy values with the heaving amplitude $ε = 2000$ (Figure 7(a)) indicates that these values remain relatively stable during the impact phase. This stability is attributed to the limiter capacity to suppress system amplitude escalation. Conversely, in the absences of impacts, system energy increases in response to an augmented heaving amplitude, despite the presence of the limiter.

Figure 8(c) illustrates how system amplitude progresses linearly with an increase in the limiter gap under a heaving amplitude $ε = 3500$ . As the limiter gap $α$ extends to 1.2, there is a transformation in system impact dynamics with respect to the limiter, leading to unilateral impacts. Upon reaching a gap of 1.3, interaction between the system and limiter ceases altogether, culminating in a maximum amplitude of 2.54. Compared to lower heaving amplitudes, this results in a slower rate of increase for system amplitude, this phenomenon is particularly pronounced during unilateral impacts involving the limiter. This suggests that an increase in heaving amplitude leads to more severe interactions between the system and limiters, thereby enhancing their inhibitory effect on overall system amplitude. In examining time-averaged energy values during bilateral impacts within this context, they reached 0.19557 while those for unilateral collisions amounted to 0.09294. In contrast with lower heaving amplitudes, there was a significant increase in time-averaged energy values, increased heaving amplitudes exerted greater influence on systemic behavior than did any inhibitory effects from limiters on overall amplitude. Furthermore, as depicted within time-averaged energy diagrams where increasing gaps are considered, it becomes evident that energy values generally exhibit a downward trend at diminishing rates over time intervals examined here (Figure 8(b)). Notably compared against specific levels of heaving amplitudes present earlier mentioned figures indicate substantial increases among averaged energies suggesting heightened severity regarding impact dynamics occurring between systems and limiters at these particular settings.

The effect of limiter stiffness on system dynamics at different heaving amplitudes

Figure 9 illustrates the steady state responses of the system under varying contact stiffnesses at a rotational speed of $Ω = 1$ , heaving amplitude of $ε = 3500$ , and limiter gap of $α = 1$ . In Figure 9(a), with a contact stiffness of $φ = 10$ , the time-domain response indicates that the limiter effectively reduces the system heaving amplitude, resulting in vibrations occurring near the limiter. The localized amplification graph shows that the vibration amplitude gradually diminishes before stabilizing. Due to relatively low contact stiffness, significant vibration primarily occurs below the limiter, this is mainly attributed to heaving action leading to impacts within the system and subsequent alterations in its motion state. In the phase diagram, oscillations manifest on both sides of the phase trajectory, expanding to twice its original size. The frequency spectrum reveals that while the primary frequency corresponds to heaving frequency. However, rotating frequency components remain minimal alongside some unidentified frequencies, indicating that the system has entered a chaotic state.

Figure 9.

Shows the time-domain response, localized magnification diagram, phase diagram, and spectrum of the system for different contact stiffnesses, (a) $φ = 10$ , (b) $φ = 30$ , and (c) $φ = 50$ , with $Ω = 1$ .

By maintaining a constant limiter gap and increasing contact stiffness, as depicted in Figure 9(b) with a contact stiffness of $φ = 30$ , the time domain response initially exhibits severe impacts with the limiter initially against the limiter which leads to an increase in heaving amplitude. As these impacts continue, however, there is a noticeable reduction in oscillation amplitude over time. Under this higher contact stiffness condition, waveform, characteristics as well as trajectory and frequency components closely resemble those observed for lower contact stiffness (Figure 9(a)). The localized amplification graph indicates that system vibration are now closer to reaching their limiters while also showing an increase in heaving frequency. This suggests that even with reduced contact stiffness levels, both vibration characteristics and impact behaviors largely remain unchanged.

As the contact stiffness is further increased, as depicted in Figure 9(c), the contact stiffness $φ = 50$ , the system experiences more pronounced impacts with the lower limiter during the initial stage of contact. This phenomenon arises from the asymmetric characteristics of the system, which lead to varying impact patterns between the upper and lower limiters. In the localized amplification graph, a noticeable increase in oscillation amplitude is observed within the impact region, accompanied by secondary rebounds characterized by distinct oscillatory forms. These secondary rebounds contribute to a rapid amplification of the system oscillation trajectory, presenting considerable challenges to its stable operation. The phase diagram reveals a larger velocity discontinuity for the lower limiter compared to that of the upper limiter, with an observed fivefold increase in velocity. These findings imply that an increase in contact stiffness exacerbates the severity of impacts between the system and its lower limiter, consistent with observations made in the time-domain diagram. In the frequency spectrum, it is noted that while heaving frequency remains dominant, it is followed by harmonic frequencies at three times and five times this fundamental frequency, conversely, rotating frequency exhibits lesser prominence. This suggests that the motion state of the system persists within a chaotic regime.

To more accurately describe the state of the system during the impact process, this paper analyzes the elastic potential energy of the floating raft-airbag system, which effectively detects the impact of rotating machinery on ships. This has significant implications for the vibration control of the system. Consequently, a third-order non-linear elastic potential energy of the airbag is introduced to characterize the changes in the system state during the impact process. The expression is as follows

\begin{matrix} u = \frac{1}{2} \int_{0}^{x_{1}} k_{1} x + k_{12} x^{3} dx + \frac{1}{2} \int_{0}^{x_{2}} k_{1} x + k_{12} x^{3} dx \\ = \frac{1}{4} k_{1} x_{1}^{2} + \frac{1}{8} k_{12} x_{1}^{4} + \frac{1}{4} k_{1} x_{2}^{2} + \frac{1}{8} k_{12} x_{2}^{4} \end{matrix}

(26)

After dimensionless of the introduced third-order non-linear airbag elastic potential energy, its expression is as follows

U = \frac{1}{4} Ω_{n}^{2} x_{1}^{2} + \frac{1}{8} \frac{Ω_{n}^{2}}{λ^{2}} x_{1}^{4} + \frac{1}{4} Ω_{n}^{2} x_{2}^{2} + \frac{1}{8} \frac{Ω_{n}^{2}}{λ^{2}} x_{2}^{4}

(27)

To mitigate the impact of rotor speed $Ω = 1$ on the system motion state, this study aims to investigate the motion dynamics and energy variations of the floating raft-airbag system upon impact with limiters under varying contact stiffness and heaving amplitudes.

Figure 10(a) to (c) illustrate the three-dimensional energy trajectories of the floating raft-airbag cubic non-linear elastic potential energy for different limiter stiffnesses. As contact stiffness increases, the energy trajectory associated with interaction between the system and the upper limiter begins to decrease in a vertical direction, while that for the lower limiter starts to increase. Additionally, there is a reduction in energy trajectory within non-impact region. This observation indicates that as contact stiffness rises, the severity of impact with the lower limiter gradually exceeds that with the upper limiter, resulting in an increased concentration of system energy near these limiters and intensifying impact conditions. With changes in contact stiffness, the energy trajectory of the system in the $x$ direction projects onto $xoz$ plane as a curve of constant magnitude. This suggests that limiters effectively suppress displacement in the $x$ direction. However, the limiters cannot restrict the system movement in the $θ$ direction, leading to a chaotic energy trajectory in the $θ$ direction and affecting the system stable operation.

Figure 10.

Nonlinear elastic potential energy of floating raft-airbag with different contact stiffness (a) $φ = 10$ , (b) $φ = 30$ , (c) $φ = 50$ , at $Ω = 1$ , (d) variation of system amplitude with heaving amplitude with different limiter gaps, and (e) time-averaged energy diagram of airbag floating raft with different limiter gaps.

Figure 10(d) illustrates the relationship between system amplitude variations and different contact stiffnesses as the heaving amplitude ranges from 2900 to 5000. As the contact stiffness increases from 10 to 30, a significant decrease in system amplitude is observed, indicating that an increase in contact stiffness effectively reduces the heaving amplitude of the system. However, when the contact stiffness reaches 50, further reductions in amplitude become minimal, suggesting that additional changes in contact stiffness have a negligible impact on system amplitude. The system consistently oscillates around a mean value as the heaving amplitude increases, which is attributable to limiters effectively dampening the system vibration amplitude. Moreover, variations in contact stiffness also exert a certain influence on vibration amplitude.

The time-averaged energy results for rotational speed $Ω = 1$ are depicted in Figure 10(e). From these energy values variations, it is evident that as contact stiffness increases from 10 to 30, the time-averaged energy of the floating raft-airbag system increases by 0.29. This indicates that higher contact stiffness allows for more effective absorption of energy from heaving by the floating raft-airbag, resulting in an increase in overall vibration amplitude within the system. However, when contact stiffness further increases to 50, time-averaged energy remains unchanged, indicates that alteration beyond this point do not significantly affect either system energy or its associated amplitude. These findings align with observations in Figure 10(e), while also indicates that the impact dynamics of the system become more complex.

Conclusions

This paper focuses on the impact characteristics of a marine rotating machinery coupled floating raft-airbag-limiter system under heaving motion. Taking in account the influence of heaving motion, limiter clearance, and contact stiffness on the coupled floating raft-airbag system, a dynamic model of the asymmetric system with clearance under heave effects was established. The equations have been rendered non-dimensional, and numerical methods were employed to obtain the steady-state response of the system, to analyze the impact characteristics of the system under heave effects. The study mainly revealed the followings:

Under the effect of heaving motion, the motion state of marine rotating machinery coupled floating raft-airbag-limiter system transitions from quasi-periodicity to chaos as the heaving amplitude increases. Compared with the case without limiter, the limiter effectively suppresses the increase in the vibration amplitude of the system.

As the contact stiffness of the limiter increases, the elastic potential energy trajectories of the floating raft-airbag shift from the upper limiter toward the lower limiter under the influence of heaving motion. The energy trajectories concentrate toward the impact region, and collisions lead to significant changes in the energy trajectories in the angular direction.

Considering the influence of heaving motion, the marine rotating machinery coupled floating raft-airbag-limiter system experiences a linear increase in motion amplitude as the limiter gap widens. Once the vibration amplitude of the system reaches a certain level, the rate of increase in amplitude significantly decreases. The more severe the collisions experienced in the system, the higher the time-averaged energy values. Increasing contact stiffness within a specific range exerts a restraining effect on the system amplitude. When the system experiences severe collisions with the limiters, reducing the clearance between the system and the limiters can effectively control the amplitude of system vibrations without altering the state of motion of the system.

Footnotes

Appendix

Notation

$a$	width of floating raft
$a_{H}$	amplitude of pendulum swing
$b$	length of floating raft
$c$	friction
$e$	rotor eccentricity
$E$	instantaneous energy value
$E'$	hourly average energy value
$f_{i}$	limiters elasticity $(i = 1, 2, 3, 4)$
$F_{i}$	dimensionless limiter spring $(i = 1, 2, 3, 4)$
$g$	acceleration of gravity
$J$	dimensionless raft-airbag moment of inertia
$J_{0}$	airbag-raft moment of inertia
$J_{1}$	moment of inertia of a rotating machine
$k_{1}$	floating raft-airbag stiffness
$k_{12}$	airbag-flotation deck nonlinear stiffness
$k_{2}$	limiter stiffness
$m$	mass of the flotation raft and rotating machinery
$m_{e}$	rotor mass
$M_{i}$	limiters torque $(i = 1, 2, 3, 4)$
$M'_{i}$	limiter dimensionless torque $(i = 1, 2, 3, 4)$
$n$	mass ratio
$q_{j}$	generalized coordinate
$Q_{j}$	generalized force vector
$t$	time
$T$	systemic kinetic energy
$u$	elastic potential energy of the airbag raft
$U$	elastic potential energy of dimensionless airbag raft
$v_{x}$	vertical velocity of rotating machinery
$v_{y}$	horizontal speed of rotating machinery
$x$	dimensionless displacement of the system
$x_{10}$	static displacement of airbag rafts
$x_{0}$	vertical displacement of floating raft
$x_{i}$	collision point displacements $(i = 1, 2, 3, 4)$
$x_{H}$	vertical displacement of a heave
$x_{z}$	absolute displacement
$x'_{0}^{+}$	instantaneous pre-collision speed
$x'_{0}^{-}$	instantaneous post-collision speed
$X_{i}$	dimensionless displacement of the collision point $(i = 1, 2, 3, 4)$
$α$	dimensionless limiter gap
$β$	dimensionless width of airbag-float raft
$γ$	dimensionless length of airbag-float raft
$ε$	dimensionless vibration amplitude
$ω$	rotor speed
$ω_{H}$	heaving frequency
$ω_{n}$	intrinsic frequency of an air-bagged raft
$φ$	scale factor
$θ$	deflection angle
$ζ$	damping ratio
$λ$	feature length
$τ$	dimensionless time
$σ_{i}$	limiter contraction amount $(i = 1, 2, 3, 4)$
$σ'_{i}$	dimensionless limiter contraction amount
$μ_{1}$	collision factor with upper limiters
$μ_{2}$	collision factor with lower limiters
$Ω$	dimensionless rotor speed
$Ω_{H}$	dimensionless heaving frequency
$Ω_{n}$	dimensionless intrinsic frequency

Handling Editor: Jianjun Feng

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 work was supported by the National Natural Science Foundation of China (Grant Nos. 11972282 and 12002267).

ORCID iD

Ming Li

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