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Abstract

In this work, a revised Bingham model for magneto-rheological damper is used to investigate the primary resonance reduction in the double-layer semi-active isolation system of marine auxiliary machinery. An analytical solution for the auxiliary double-layer semi-active isolation system’s primary resonance is obtained with an averaging method, and this is verified numerically using the Maple software. The effect of model parameters of magneto-rheological damper on the system’s vibration transmissibility is studied. The research results show that the damping of magneto-rheological damper and the control force have a significant effect on the vibration transmissibility in the resonance region. The vibration transmissibility of the double-layer semi-active isolation system decreases with increase in damping of the magneto-rheological damper and control force. Yet the zero force velocity contributes very little to changes in vibration transmissibility.

Keywords

Double-layer system vibration isolation marine auxiliary machinery MR damper averaging method

Introduction

With the development of vibration control techniques and increasing strict requirements for vibration isolation in ships, classical single-layer isolation systems exhibit poor performances.^1–3 To achieve more efficient vibration cancellation, double-layer isolation systems have received increased research attention in recent years. Double-layer can isolate vibration of marine auxiliary machinery and reduce the structural noise of ships and submarines subjected to external loads and sudden shocks. Passive vibration isolation systems employing rubber isolators have some advantages such as design simplicity and cost effectiveness. However, as the design of passive vibration isolation systems is fixed, its performance could be limited when the operating conditions change, due to variation of rotating speed of the marine auxiliary machinery. In contrast, an active vibration isolation system utilizing hydraulic or pneumatic actuators can provide high control performance. However, it requires high power and sophisticated control implementation.⁴ Besides, the active systems could also be destabilized as mechanical power could be imported into the systems. Therefore, semi-active vibration isolation system utilizing semi-active dampers is proposed to overcome the aforementioned shortcomings in order to achieve high performance with low power while the marine auxiliary machinery is stable and fail-safe.

As far as semi-active dampers are considered, it has been found that magneto-rheological (MR) fluid devices are quite promising for vibration reduction applications.^5,6 MR fluids are one kind of controllable fluids that respond to a magnetic field with dramatic changes in rheological behaviours. The essential feature of MR fluid is its ability for reversible change from free flowing liquid to semisolid with controllable yield strength in milliseconds^7,8 when exposed to a magnetic field. Comparing with conventional semi-active devices such as variable orifice dampers, MR dampers have the advantages that they are fast responding, have no moving parts, and make them simple and reliable. These have been the primary factors for motivating the development of such devices.⁹ Various researchers have studied the application of MR dampers in vehicle suspension systems, buildings, and so on.^10,11

The Bingham plastic model is often used to characterize MR fluid properties.¹² Based on the Bingham plastic constitutive equation, a quasi-steady, field-controllable damper model is developed to predict the damper’s performance. In this model, a control force is added to the linear damping force, however, there is no hysteresis element included.

The averaging method which is also named as the KB method (Krylov–Bogoliubov method) is in common for obtaining the analytical solution of non-linear systems. This method is equivalent to the method of multiple scales. But the derivation process of averaging method is simpler than the method of multiple scales.¹³ Vahdati devised a criterion for oscillation stability using the averaging method. The proposed criterion is applied to the stability analysis of negative resistance diode oscillators and Colpitts oscillator.¹⁴ Haberman¹⁵ used the averaging method to analyse perturbations to strongly non-linear partial differential equations with oscillatory solutions.

In this study, the double-layer semi-active vibration isolation systems of marine auxiliary machinery are investigated, using a revised Bingham model for MR dampers. A simple relation is introduced to represent the behaviour of MR dampers. The analytical solution for the double-layer semi-active vibration isolation systems is obtained using an averaging method. The physical conditions necessary for primary resonance to occur are determined to assist in the dynamic systems evaluation and in control strategy development. The response of the double-layer semi-active vibration isolation systems is obtained using Maple and finally, the analytical solution for dynamic systems response is validated with numerical simulations.

Model of MR forces

The Bingham plastic model is often used to characterize MR fluid properties.⁵ Based on the Bingham plastic constitutive equation, a quasi-steady, field-controllable damper model was developed to predict the damper’s performance. In this model, a control force is added to the linear damping force, however, there is no hysteresis element included. The Bingham plastic model can be expressed as

F_{MRD} = c_{1} V + F_{y} sign (V)

(1)

where F is the damping force of the MR damper. F_y is the control force of MR damper. V is the relative velocity between piston and cylinder of the MR damper, and c₁ is the viscous damping coefficient.

The Bingham plastic model is often used in predicting MR damper response, however, this model does not include hysteretic element. A revised Bingham model with hysteretic loop is proposed, as follows

F_{MRD} = {c_{1} V + F_{y} sign (V - V_{0}) V > 0 c_{1} V + F_{y} sign (V + V_{0}) V < 0

(2)

where V is the relative velocity between piston and cylinder wall of the MR damper. c₁ is the viscous damping coefficient, and V₀ is the zero force velocity of the MR damper. This model is schematically shown in Figure 1. The figure is composed of two branches, A and B, which come from the two equations of the above model, respectively.

Figure 1.

The revised Bingham model.

The proposed model requires only three parameters that need to be determined empirically. The theoretical results are shown in Figure 1. The MR damper RD-8040 of LORD Co. Ltd is tested. The experimental setup is shown in Figure 2. A typical experimental result showing the force-velocity relationship of a prototype MR damper is depicted in Figure 3. It can be seen that the model can capture the hysteretic behaviour of an MR damper.

Figure 2.

The experiment setup of MR damper.

Figure 3.

Force versus velocity model of MR damper.

Model of vibration isolation systems

The double-layer semi-active vibration isolation system of marine auxiliary machinery considered in this work is shown in Figure 4. The dynamic equations of the systems are

{m_{1} {\overset{··}{y}}_{1} + k_{1} y_{1} + k_{2} (y_{1} - y_{2}) + c_{2} ({\overset{\cdot}{y}}_{1} - {\overset{\cdot}{y}}_{2}) + F_{MRD} = 0 m_{2} {\overset{··}{y}}_{2} + k_{2} (y_{2} - y_{1}) + c_{2} ({\overset{\cdot}{y}}_{2} - {\overset{\cdot}{y}}_{1}) + F = 0

(3)

where y₁ and y₂ are displacements of the medium mass and marine auxiliary machinery, m₁ and m₂ represent the masses of the medium and marine auxiliary machinery, k₁ and k₂ are the upper stiffness and the lower stiffness,

F = F_{0} \cos (ω t)

is the excitation force of marine auxiliary machinery,

F_{MRD} = c_{1} {\overset{\cdot}{y}}_{1} + F_{y} sign ({\overset{\cdot}{y}}_{1} \mp V_{0})

is the force of MR damper, K is the spring stiffness, F_y is the control force of MR damper, V₀ is the zero force velocity of the MR damper, c₁ is viscous damping coefficient.

Figure 4.

The double-layer semi-active vibration isolation system of marine auxiliary machinery.

For the sake of brevity, simplifying equation (3) with dimensionless displacement $x_{1} = y_{1} k_{2} / F_{0}$ , $x_{2} = y_{2} k_{2} / F_{0}$ , and dimensionless time $τ = ω t$ , the dimensionless equation of motion is obtained

{{\overset{··}{x}}_{1} + B_{11} (x_{1} - x_{2}) + B_{12} ({\overset{\cdot}{x}}_{1} - {\overset{\cdot}{x}}_{2}) + B_{13} x_{1} + B_{14} {\overset{\cdot}{x}}_{1} + B_{15} sign ({\overset{\cdot}{x}}_{1} \mp {\overset{\cdot}{x}}_{10}) = 0 {\overset{··}{x}}_{2} + B_{21} (x_{2} - x_{1} + sin τ) + B_{22} ({\overset{\cdot}{x}}_{2} - {\overset{\cdot}{x}}_{1}) = 0

(4)

where, the dots stand for differential with respect to dimensionless time τ, and

B_{11} = \frac{k_{2}}{m_{1} ω^{2}}

B_{12} = \frac{c_{2}}{m_{1} ω}

B_{13} = \frac{k_{1}}{m_{1} ω^{2}}

B_{14} = \frac{c_{1}}{m_{1} ω}

B_{15} = \frac{F_{y}}{m_{1} ω}

{\overset{\cdot}{x}}_{10} = \frac{V_{0} k_{2}}{ω F_{0}}

B_{21} = \frac{k_{2}}{m_{2} ω^{2}}

B_{22} = \frac{c_{2}}{m_{2} ω}

In this case, the parameter B₁₅ is small compared with other parameters. Thus, the equation is a weakly non-linear oscillation systems. The analytical solution of equation (4) is obtained through the averaging method as follows. Consider the following solution for equation (4)

{x_{i} = a_{i} \cos ϕ_{i} {\overset{\cdot}{x}}_{i} = - a_{i} \sin ϕ_{i} i = 1, 2

(5)

where

ϕ_{i} = τ - θ_{i}

. Differentiating equation (5) with respect to τ, and using equation (4), yields the following

{{\overset{\cdot}{a}}_{i} \cos ϕ_{i} + a_{i} {\overset{\cdot}{θ}}_{i} \sin ϕ_{i} = 0 {\overset{\cdot}{a}}_{i} \sin ϕ_{i} - a_{i} {\overset{\cdot}{θ}}_{i} \cos ϕ_{i} = f_{i}

(6)

where

f_{1} = (B_{11} + B_{13} - 1) a_{1} \cos ϕ_{1} - B_{11} a_{2} \cos ϕ_{2} - (B_{12} + B_{14}) a_{1} \sin ϕ_{1} + B_{12} a_{2} \sin ϕ_{2} + B_{15} sign (- a_{1} \sin ϕ_{1} \mp {\overset{\cdot}{x}}_{10}) f_{2} = (B_{21} - 1) a_{2} \cos ϕ_{2} - B_{21} a_{1} \cos ϕ_{1} + B_{21} \sin τ - B_{22} a_{2} \sin ϕ_{2} - B_{22} a_{1} \sin ϕ_{1}

From equation (6), two equations can be obtained as follows

{{\overset{\cdot}{a}}_{i} = f_{i} \sin ϕ_{i} a_{i} {\overset{\cdot}{θ}}_{i} = - f_{i} \cos ϕ_{i}

(7)

Performing a K-B transformation to equation (7), and using $ϕ_{1} - ϕ_{2} = θ_{2} - θ_{1}$ , the following equations can be obtained

{{\overset{\cdot}{a}}_{1} = P_{1} {\overset{\cdot}{θ}}_{1} = R_{1} / a_{1} {\overset{\cdot}{a}}_{2} = P_{2} {\overset{\cdot}{θ}}_{2} = R_{2} / a_{2}

(8)

where

P_{1} = \frac{1}{2 π} \int_{0}^{2 π} f_{1} \sin ϕ_{1} d ϕ_{1} = \frac{B_{11} a_{2}}{2} \sin (θ_{2} - θ_{1}) - \frac{(B_{12} + B_{14}) a_{1}}{2} + \frac{B_{12} a_{2}}{2} \cos (θ_{2} - θ_{1}) + \frac{1}{2 π} \int_{0}^{2 π} B_{15} sign (- a_{1} \sin ϕ_{1} \mp {\overset{\cdot}{x}}_{10}) \sin ϕ_{1} d ϕ_{1} = \frac{B_{11} a_{2}}{2} \sin (θ_{2} - θ_{1}) - \frac{(B_{12} + B_{14}) a_{1}}{2} + \frac{B_{12} a_{2}}{2} \cos (θ_{2} - θ_{1}) - \frac{2 B_{15} \sqrt{a_{1}^{2} - {\overset{\cdot}{x}}_{10}^{2}}}{π a_{1}} R_{1} = - \frac{1}{2 π} \int_{0}^{2 π} f_{1} \cos ϕ_{1} d ϕ_{1} = - \frac{(B_{11} + B_{13} - 1) a_{1}}{2} + \frac{B_{11} a_{2}}{2} \cos (θ_{2} - θ_{1}) - \frac{B_{12} a_{2}}{2} \sin (θ_{2} - θ_{1}) - \frac{1}{2 π} \int_{0}^{2 π} B_{15} sign (- a_{1} \sin ϕ_{1} \mp {\overset{\cdot}{x}}_{10}) \cos ϕ_{1} d ϕ_{1} = - \frac{(B_{11} + B_{13} - 1) a_{1}}{2} + \frac{B_{11} a_{2}}{2} \cos (θ_{2} - θ_{1}) - \frac{B_{12} a_{2}}{2} \sin (θ_{2} - θ_{1}) - \frac{2 B_{15} {\overset{\cdot}{x}}_{10}}{π a_{1}} P_{2} = \frac{1}{2 π} \int_{0}^{2 π} f_{2} \sin ϕ_{2} d ϕ_{2} = - \frac{B_{21} a_{1}}{2} \sin (θ_{2} - θ_{1}) - \frac{B_{22} a_{2}}{2} + \frac{B_{21}}{2} \cos θ_{2} - \frac{B_{22} a_{1}}{2} \cos (θ_{2} - θ_{1}) R_{2} = - \frac{1}{2 π} \int_{0}^{2 π} f_{2} \cos ϕ_{2} d ϕ_{2} = \frac{B_{21} a_{1}}{2} \cos (θ_{2} - θ_{1}) - \frac{(B_{21} - 1) a_{2}}{2} - \frac{B_{21}}{2} \sin θ_{2} - \frac{B_{22} a_{1}}{2} \sin (θ_{2} - θ_{1})

From ${\overset{\cdot}{a}}_{i} = 0, {\overset{\cdot}{θ}}_{i} = 0, i = 1, 2$ , equation (9) can be obtained as follows

{\frac{B_{11} a_{2}}{2} \sin (θ_{2} - θ_{1}) - \frac{(B_{12} + B_{14}) a_{1}}{2} + \frac{B_{12} a_{2}}{2} \cos (θ_{2} - θ_{1}) - \frac{2 B_{15} \sqrt{a_{1}^{2} - {\overset{\cdot}{x}}_{10}^{2}}}{π a_{1}} = 0 - \frac{(B_{11} + B_{13} - 1) a_{1}}{2} + \frac{B_{11} a_{2}}{2} \cos (θ_{2} - θ_{1}) - \frac{B_{12} a_{2}}{2} \sin (θ_{2} - θ_{1}) - \frac{2 B_{15} {\overset{\cdot}{x}}_{10}}{π a_{1}} = 0 - \frac{B_{21} a_{1}}{2} \sin (θ_{2} - θ_{1}) - \frac{B_{22} a_{2}}{2} + \frac{B_{21}}{2} \cos θ_{2} - \frac{B_{22} a_{1}}{2} \cos (θ_{2} - θ_{1}) = 0 \frac{B_{21} a_{1}}{2} \cos (θ_{2} - θ_{1}) - \frac{(B_{21} - 1) a_{2}}{2} - \frac{B_{21}}{2} \sin θ_{2} - \frac{B_{22} a_{1}}{2} \sin (θ_{2} - θ_{1}) = 0

(9)

From equation (9), the following two equations can be obtained

(B_{11}^{2} + B_{12}^{2}) a_{1}^{2} a_{2}^{2} π^{2} = [(B_{12} + B_{14}) a_{1}^{2} π + 4 B_{15} \sqrt{a_{1}^{2} - {\overset{\cdot}{x}}_{10}^{2}}]^{2} + [(B_{11} + B_{13} - 1) a_{1}^{2} π + 4 B_{15} {\overset{\cdot}{x}}_{10}]^{2}

(10)

2 (B_{22}^{2} - B_{21}^{2} + B_{21}) ((B_{12}^{2} + B_{12} B_{14} + B_{11}^{2} + B_{11} B_{13} - B_{11}) \times a_{1}^{2} π + 4 B_{12} B_{15} \sqrt{a_{1}^{2} - {\overset{\cdot}{x}}_{10}^{2}} + 4 B_{11} B_{15} {\overset{\cdot}{x}}_{10}) + 2 B_{22} a_{1} a_{2} (2 B_{21} - 1) ((B_{11} B_{14} - B_{12} B_{13} + B_{12}) \times a_{1}^{2} π + 4 B_{11} B_{15} \sqrt{a_{1}^{2} - {\overset{\cdot}{x}}_{10}^{2}} - 4 B_{12} B_{15} {\overset{\cdot}{x}}_{10} + ((B_{21}^{2} + B_{22}^{2}) a_{1}^{2} + (B_{22}^{2} + B_{21}^{2} - 2 B_{21} + 1) \times a_{2}^{2} - B_{21}^{2}) (B_{11}^{2} + B_{12}^{2}) a_{2} = 0

(11)

The amplitudes a₁ and a₂ of the steady state resonance solution of the systems can be obtained from equations (10) and (11). The phase shifts $θ_{1}$ and $θ_{2}$ of the steady state resonance solution of the systems can be obtained from the last two equations of equation (9) and the amplitudes a₁, a₂. Therefore, the resonance solution of double-layer semi-active vibration isolation systems is obtained as follows

x_{1} = a_{1} \cos (τ - θ_{1}), x_{2} = a_{2} \cos (τ - θ_{2})

(12)

The vibration transmissibility of double-layer semi-active vibration isolation systems is

η = \frac{F_{1}}{F_{0}} = \frac{k_{1} y_{1} + F_{MRD}}{F_{0}} = \frac{k_{1}}{k_{2}} x_{1} + \frac{c_{1} ω}{k_{2}} {\overset{\cdot}{x}}_{1} + F_{y} sign (\frac{ω}{k_{2}} {\overset{\cdot}{x}}_{1} \mp \frac{V_{0}}{F_{0}}) = \frac{k_{1} a_{1}}{k_{2}} \cos (τ - θ_{1}) - \frac{c_{1} ω a_{1}}{k_{2}} \sin (τ - θ_{1}) + F_{y} sign (- \frac{ω a_{1}}{k_{2}} \sin (τ - θ_{1}) \mp \frac{V_{0}}{F_{0}})

(13)

where F₀ is the excitation force of marine auxiliary machinery, F₁ is the transmission force of base.

Numerical results

In order to verify the theoretical results of this work, equation (4) was integrated numerically with the Maple software. The system parameters used were as follows: m₂ = 1000 kg, m₁ = 620 kg, V₀ = 0.1 m/s, k₁ = 3.2 × 10⁶N/m, k₂ = 4.0 × 10⁵N/m, c₁ = 1500 Ns/m, c₂ = 3000 Ns/m, F_y = 60 N, F₀ = 500 N, $x_{1} = 0, {\overset{\cdot}{x}}_{1} = 0$ . The numerical simulation results are shown in Figure 5. It can be seen that the theoretical results agree well with the simulation result.

Figure 5.

Theoretical and numerical solutions of vibration isolation system.

Increasing the off-state viscous damping of the MR damper reduces the vibration transmissibility of the systems, as shown in Figure 6. As c₁ makes little contribution to the systems equation, the resonance frequency changes inconspicuously.

Figure 6.

The effect of c₁ on the vibration transmissibility.

The MR force F_y has a significant impact on the vibration transmissibility of the systems, as shown in Figure 7. It can be seen that as F_y increases, the vibration transmissibility decreases and the systems becomes more stable. As F_y makes little contribution to the systems equation, the resonance frequency changes inconspicuously.

Figure 7.

The effect of F_y on the vibration transmissibility.

The value of V₀, zero-force velocity of the MR damper, depends on the compressibility of the MR fluid and the accumulator. The effect of V₀ on the vibration transmissibility of the double-layer semi-active isolation systems in the resonance region is small, as shown in Figure 8. It can also be seen that V₀ reduces the response of the systems in the resonance region.

Figure 8.

The effect of V₀ on the vibration transmissibility.

Conclusions

A revised Bingham model for MR damper is used to describe the non-linear damping force of marine auxiliary machinery. The influence of system parameters on vibration transmissibility of double-layer semi-active isolation systems controlled MR damper has been investigated. The factors include the control force F_y, the viscous damping c₁ and the zero force V₀ of the MR damper, among which the last three factors are related to the compressibility of the MR damper. Increasing of the compressibility leads to a larger value of F_y, c₁ and V₀. By increasing the control force of the MR damper, the threshold value for the system resonance will increase, and hence, the system would be more stable. In the resonance region, increasing viscous damping c₁ or the control force of the MR damper F_y will restrain the primary resonance greatly. However, in the non-resonance region, the effect of them is very minor. The zero force velocity V₀ has less effect on the vibration transmissibility of the systems in the resonance region. Increasing the zero force velocity V₀ would reduce the vibration transmissibility of the systems.

Footnotes

Declaration of conflicting interests

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

Funding

This study is supported in part by the National Science Foundation of China (11302088) and the Natural Science Foundation of Jiangsu Province (BK2012278).

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Non-linear dynamic analysis of double-layer semi-active vibration isolation systems using revised Bingham model

Abstract

Keywords

Introduction

Model of MR forces

Model of vibration isolation systems

Numerical results

Conclusions

Footnotes

Declaration of conflicting interests

Funding

References