Analysis and design of the power law damping based on the nonlinear vessel isolation system

Abstract

In this study, the applications of the cubic power law damping in vessel isolation systems are investigated. The isolation performance is assessed using the force transmissibility of the vessel isolation system, which is simplified as a multiple-degree-of-freedom system with two parallel freedoms. The force transmissibilities of different working conditions faced in practice are discussed by applying the cubic power law damping on different positions of the vessel isolation system. Numerical results indicate that by adding the cubic power law damping to an appropriate position, the isolation system can not only suppress the force transmissibility over the resonant frequency region but also keep the force transmissibility unaffected at the nonresonant frequency region. Moreover, the design of the nonlinear vessel isolation system is discussed by finding the optimal nonlinear damping of the isolation system.

Keywords

Power law damping nonlinear analysis vessel force transmissibility vibration isolation

Introduction

Vessels have been widely applied in the survey of marine resources. The vibration and noise of the vessel system are curial to its stability and comfort in engineering practice.^1,2

In the vessels’ vibration isolation systems, linear damping is often applied by many literatures to reduce the output amplitude and the force transmitted to the base, which is known as the transmissibility of the system.^3,4 However, by using linear damping isolators, the transmissibility of the resonant region is suppressed, but the transmissibility of the nonresonant region can be increased.⁵ In order to address this problem, many nonlinear vibration isolation approaches, such as the quasi-zero stiffness (QZS),⁶ $H_{\infty}$ controller design approach,⁷ the viscous damping,⁸ and the linear–quadratic–Gaussian (LQG) method,⁹ have been investigated.

For example, Yang et al.¹⁰ presented an active–passive vibration isolation approach by establishing a mathematical model of the floating raft system, and the $H_{\infty}$ controller was designed so as to suppress the vibration and noise. Song and Sun¹¹ analyzed the dynamic characteristics of a multiple-degree-of-freedom (MDOF) isolation system and achieved significant isolation effects. Li et al.¹² investigated the dynamic characteristics and periodic vibration of the floating raft vibration isolation system by adopting the average approach. In Li and Xu,¹³ the QZS isolator was applied in the underwater vehicles, and the nonlinear model of the QZS system was conducted using a series of theoretical and numerical analysis.

Most recently, Lang and colleagues^14–16 introduced the power law nonlinear damping and nonlinear stiffness into the vibration isolation system. Ho et al.¹⁵ investigated the effects of spring nonlinearity on the power transmissibility. In Lang et al.,¹⁷ the effects of nonlinear viscous damping on single-degree-of-freedom (SDOF) structures were analyzed. The output frequency response function (OFRF) was employed to determine the relationships between the force transmissibility and the system characteristic parameters, such as to systematically facilitate the analysis and design of nonlinear system.^18,19 Peng et al.^20,21 investigated the force transmissibility of series MDOF structures with a cubic nonlinear damping, demonstrating that a cubic power law viscous damping can achieve a better vibration isolation performance and overcame the disadvantages of linear damping. Mofidian et al.²² presented the theory and experimental study about the combination of magnetic springs and viscous and magnetic damping.

It is worth pointing out that, most nonlinear damping theory investigations of the vibration isolation system focused on the SDOF and series MDOF structures. However, for vessel isolation systems, the structures are complex and usually of unchain type, which can provide more abundant isolation phenomenon than the SDOF and series MDOF structures. In this study, the MDOF structure with parallel freedoms is applied to represent the vessel isolation system. Moreover, the application of the cubic power law damping is, for the first time, discussed based on the parallel MDOF system under different working conditions. Finally, the optimal design of the vessel isolation system is conducted, and the design requirements and process are summarized.

Nonlinear damping–based vessel isolation system

Mechanical structure of the isolation system

The vessel isolation system simplified from laboratory equipment is depicted in Figure 1(a), where the two diesel engines are the vibration sources. Diesel engines and other equipments are installed on the middle support together. Four dampers are set on the bottom of the each diesel engine, shown as the blue circle in Figure 1(b). Under the middle support, there are six uniformly distributed springs and dampers, shown as the triangle in Figure 1(b).

Figure 1.

Structural characteristics of the vibration isolation system: (a) front view and (b) top view.

In the proposed vessel isolation system, $m_{1} = 1000 kg$ is the mass of the middle support and $m_{3} = 1700 kg$ is the mass of the diesel engine, where the masses of the two diesel engines are identical. $m_{2}$ , $m_{4}$ and $m_{5}$ are the masses of other equipments located on the middle support with $m_{2} = 4800 kg$ , $m_{4} = 1120 kg$ , and $m_{5} = 680 kg$ , respectively.

Under the middle support, the stiffness values of the left, middle, and right springs are $k_{1} = 6.2 \times 10^{6} N / m$ , $k_{2} = 2.7 \times 10^{6} N / m$ , and $k_{3} = 9 \times 10^{5} N / m$ , respectively, and the stiffness values of springs under the diesel engine are the same, given as $k_{4} = 8 \times 10^{5} N / m$ .

Assuming the middle support is rigid, the vessel isolation system in Figure 1 can further be simplified as a 3-DOF system with two parallel freedoms, shown in Figure 2.

Figure 2.

Simplified structure of a vibration isolation system with a cubic power law damping.

Symbols in Figure 2 are defined as follows:

$f_{in 1} (t)$ and $f_{in 2} (t)$ are the input forces of the diesel engines $D_{1}$ and $D_{2}$ , respectively; $f_{out} (t)$ is the output force transmitted to the base; $x_{1}$ , $x_{21}$ , and $x_{22}$ are the output displacements of the middle support and the diesel engines $D_{1}$ and $D_{2}$ , respectively.

$m_{21}$ and $m_{22}$ are the masses of the diesel engines $D_{1}$ and $D_{2}$ , respectively, with $m_{21} = m_{22} = m_{3} = 1700 kg$ . $m_{11}$ is the equivalent mass and can be obtained as

m_{11} = m_{1} + m_{2} + 2 \times m_{4} + m_{5} = 8720 kg

(1)

$k_{21}$ and $k_{22}$ are the equivalent support stiffness of the diesel engines $D_{1}$ and $D_{2}$ , respectively, and $k_{11}$ is the stiffness of the equivalent support, which are computed as

{\begin{matrix} k_{21} = k_{22} = 4 \times k_{4} = 3.2 \times 10^{6} N / m \\ k_{11} = 2 (k_{1} + k_{2} + k_{3}) = 1.96 \times 10^{7} N / m \end{matrix}

(2)

$c_{21}$ and $c_{22}$ are equivalent linear damping of the diesel engines $D_{1}$ and $D_{2}$ with $c_{21} = c_{22} = 4 c_{2}$ , respectively. $r_{11}, r_{21}, and r_{22}$ are the cubic power law damping proposed to be added in the isolation system. The analysis and design of the linear and nonlinear damping on the force transmissibility are conducted in the following discussion.

Transmissibility of the isolation system

The differential equation of the isolation system can be written as

M \overset{\cdot\cdot}{x} + C \overset{\cdot}{x} + Kx + F_{N} = F

(3)

where x is the vector of the displacement, $x = [x_{21}, x_{22}, x_{1}]^{T}$ ; $F_{N}$ is the vector of the cubic nonlinear damping force, $F_{N} = {[r_{21} {({\overset{\cdot}{x}}_{21} - {\overset{\cdot}{x}}_{1})}^{3}, r_{22} {({\overset{\cdot}{x}}_{22} - {\overset{\cdot}{x}}_{1})}^{3}, r_{11} {\overset{\cdot}{x}}_{1}^{3} + r_{21} {({\overset{\cdot}{x}}_{1} - {\overset{\cdot}{x}}_{21})}^{3} + r_{22} {({\overset{\cdot}{x}}_{1} - {\overset{\cdot}{x}}_{22})}^{3}]}^{T}$ ; F is the input force vector with $F = {[f_{in, 1} (t), f_{in, 2} (t), 0]}^{T}$ ; and $M = (\begin{matrix} m_{21} & 0 & 0 \\ 0 & m_{22} & 0 \\ 0 & 0 & m_{11} \end{matrix})$ , $C = (\begin{matrix} c_{21} & 0 & - c_{21} \\ 0 & c_{22} & - c_{22} \\ - c_{21} & - c_{22} & c_{21} + c_{22} + c_{11} \end{matrix})$ , and $K = (\begin{matrix} k_{21} & 0 & - k_{21} \\ 0 & k_{22} & - k_{22} \\ - k_{21} & - k_{22} & k_{21} + k_{22} + k_{11} \end{matrix})$ are the mass, damping, and stiffness matrices, respectively.

Considering that input force generated by the diesel engine is harmonic yields

f_{in, i} (t) = F_{i} \sin (Ω t), i = 1, 2

(4)

where $Ω$ is the input frequency; and $F_{i}$ is the magnitude, where the subscript i denotes the diesel engines $D_{1}$ and $D_{2}$ .

Consider only one diesel engine works, the force transmissibility of the isolation system is defined as the ratio of the output force spectrum with respect to the input force spectrum,²³ which can be expressed as

T_{i, j} (Ω) = \frac{| F_{out, j} (j Ω) |}{| F_{in, i} (j Ω) |} = \frac{| F_{out, j} (j Ω) |}{F_{i}}, {\begin{matrix} i = 1, 2 \\ j = 0, 1, 2 \end{matrix}

(5)

where $T_{i, 0}$ , $T_{i, 1}$ , and $T_{i, 2}$ denote the ratio of the output force transmitted to the ground, the equivalent support mass $m_{11}$ , and the other diesel engine that does not work with respect to the input force $F_{i}$ , respectively. The output force spectra $F_{out, j} (j Ω)$ can be obtained by computing the Fourier transform of the output forces $f_{out, j} (t)$

{\begin{matrix} f_{out, 0} (t) = k_{11} x_{1} + c_{11} {\dot{x}}_{1} + r_{11} {\dot{x}}_{1}^{3} \\ f_{out, 1} (t) = k_{21} x_{21} + c_{21} {\dot{x}}_{21} + r_{21} {\dot{x}}_{21}^{3} \\ f_{out, 2} (t) = k_{22} x_{22} + c_{22} {\dot{x}}_{22} + r_{22} {\dot{x}}_{22}^{3} \end{matrix}

(6)

where $f_{out}, 0 (t)$ is the output force transmitted to the ground, $f_{out}, 1 (t)$ is the output force transmitted to $m_{11}$ , and $f_{out}, 2 (t)$ is the output force transmitted to the unworked diesel engine.

When two diesel engines synchronously work, the transmissibility is defined as

T_{0} (Ω) = \frac{| F_{out, 0} (j Ω) |}{| F_{in} (j Ω) |} = \frac{| F_{out, 0} (j Ω) |}{F_{1} + F_{2}}

(7)

where $F_{in} (j Ω)$ is the Fourier transform of $f_{in, 1} (t) + f_{in, 2} (t)$ .

In the following studies, the transmissibilities of the ground and the unworked engine are considered, where the transmissibility analysis for $j = 1$ will be omitted since they are usually not concerned in practice.

Numerical analysis of the nonlinear vessel isolation system

In order to investigate the vibration transmissibility characteristics of an isolation system and the effects of a cubic nonlinear damping characteristic on the force transmissibility, numerical analysis was conducted on a vessel isolation system to calculate the force transmissibility. It is worth noting that, many numerical methods, such as the Newmark method²⁴ and the Runge–Kutta method,²⁰ can also be used to compute the system output responses. In this study, the Runge–Kutta method and the Fourier transform are applied to analyze the vibration transmissibility of the vessel isolation system.

For the isolation system in Figure 2, the previous three orders’ natural frequencies of the system can be numerically obtained as listed in Table 1.

Table 1.

Natural frequencies of the isolation system.

Order	1	2	3
Natural frequencies (Hz)	5.36	6.91	9.72

In the following studies, two working conditions will be discussed for the isolation system, including single diesel engine and double diesel engine rotating synchronously. The effects of both linear and nonlinear damping are discussed under these working conditions, such as to illustrate the advantages of nonlinear damping in the vessel isolation systems.

Single diesel engine rotating

Considering the isolation system works with single diesel engine rotating, for example, the diesel engine $D_{1}$ keeps rotating and the diesel engine $D_{2}$ maintains static.

The input force $f_{in, 1} (t)$ is a sinusoidal force in equation (4) with $F_{1} = 1 N$ and the excitation frequency $Ω$ sweeps in the range of $0 \leq Ω \leq 15 (Hz)$ according to the resonant frequencies shown in Table 1.

Therefore, the force transmissibility in equation (5) can be simplified as

T_{1, j} (Ω) = | F_{out, j} (j Ω) |; j = 0, 2

(8)

and the effects of both linear and nonlinear damping on the force transmissibility $T_{1, 0}$ and $T_{1, 2}$ will be investigated under the single diesel engine rotating condition as follows.

Effects of linear damping

The values of linear damping are assumed to be proportional to the stiffness values $k_{21} = k_{22} = 3.2 \times 10^{6} N / m$ and $k_{11} = 1.96 \times 10^{7} N / m$ , which can be denoted as²⁰

c_{11} = C_{1} k_{11}

(9)

c_{21} = c_{22} = C_{2} k_{21} = C_{2} k_{22}

(10)

where $C_{1}$ and $C_{2}$ are linear damping ratios.

The linear and nonlinear damping ratios can be determined based on the study of Peng et al.²⁰ and contained the light and strong damping values. Furthermore, based on the vibration isolation system, the damping ratios are optimized according to the numerical analysis and ensured in the available range. Given the linear damping ratio $C_{2} = 0.002$ and the cubic power law damping is $r_{11} = r_{21} = r_{22} = 0$ . The force transmissibilities $T_{1, 0}$ and $T_{1, 2}$ of the vibration isolation system are evaluated under different linear damping values of $C_{1} = 0.0001$ , $C_{1} = 0.004$ , and $C_{1} = 0.02$ , respectively, and the results are depicted in Figure 3. Similar process is also conducted by fixing $C_{1} = 0.004$ , and the transmissibilities of $C_{2} = 0.0005$ , $C_{2} = 0.002$ , and $C_{2} = 0.01$ are also calculated and shown in Figure 4.

Figure 3.

Force transmissibility analysis with linear damping parameter $C_{1}$ . The force transmissibilities (a) $T_{1, 0}$ and (b) $T_{1, 2}$ .

Figure 4.

Force transmissibility with the linear damping parameter $C_{2}$ . The force transmissibilities (a) $T_{1, 0}$ and (b) $T_{1, 2}$ .

As indicated in Figures 3(a), 4(a), and 4(b), the force transmissibility $T_{1, 0}$ is greatly reduced with the increase of linear damping $C_{1}$ and $C_{2}$ over the resonant frequency region. However, in the vibration isolation range, the increase of linear damping $C_{1}$ and $C_{2}$ leads to the increase in the force transmissibilities. Moreover, resonant frequencies of the isolation system are shifted due to a strong linear damping. For example, the resonant frequency is shifted from 5.36 to 5.88 Hz, as shown in Figure 3(a). Figures 3 and 4 were obtained under linear damping, and the cubic power law damping was not introduced into the force transmissibility analysis. Therefore, the phenomenon was generated due to the very basic linear damping effects.

However, in Figure 3(b), only the second-order resonant frequency remains when the linear damping $C_{1}$ is large, and the transmissibility $T_{1, 2}$ decreases in higher frequency range. This is very different from the well-known linear damping effects on SDOF systems and series MDOF systems,¹⁷ such as Figures 3(a) and 4, where the transmissibility will increase in the high frequency range.

This is because, the mass, stiffness, and damping values of the parallel diesel engines $D_{1}$ and $D_{2}$ are completely identical, and most energies are directly transmitted from $D_{1}$ to $D_{2}$ by passing the support mass $m_{11}$ instead of going through $C_{1}$ and $k_{11}$ . In this case, the transmissibility of the second-order resonant frequency will not be affected by changing $C_{1}$ , which is related to the diesel engine $D_{2}$ .

In the next section, the nonlinear damping technique will be applied to improve these unexpected isolation performance for a parallel MDOF isolation system.

Effects of nonlinear damping

Two cases will be discussed in this section, where nonlinear damping is applied under the equivalent mass $m_{11}$ and both diesel engines, respectively.

Considering that a cubic power law damping $r_{11}$ is applied under the diesel engine $D_{1}$ , the linear damping ratios are $C_{1} = 0.004$ and $C_{2} = 0.002$ in equation (10) and the nonlinear damping parameter is $r_{21} = r_{22} = 0$ . Transmissibilities are not affected by the weak nonlinear damping, but too strong nonlinear damping will deduce the transmissibility increase in the nonresonant region. The suitable nonlinear damping values can be determined by the numerical analysis. Transmissibilities $T_{1, 0}$ and $T_{1, 2}$ are shown in Figure 5(a) and (b) with the power law damping $r_{11} = 0$ , $r_{11} = 2 \times 10^{9} N s^{3} / m^{3}$ and $r_{11} = 4 \times 10^{10} N s^{3} / m^{3}$ , respectively. Similarly, by giving $r_{11} = 0$ , transmissibilities $T_{1, 0}$ and $T_{1, 2}$ with different nonlinear damping of $R = r_{21} = r_{22} = 0$ , $R = 4 \times 10^{8} N s^{3} / m^{3}$ , and $R = 2 \times 10^{9} N s^{3} / m^{3}$ are shown in Figure 6(a) and (b), respectively.

Figure 5.

Force transmissibility with different nonlinear damping $r_{11}$ . The force transmissibility (a) $T_{1, 0}$ and (b) $T_{1, 2}$ .

Figure 6.

Transmissibility analysis with different nonlinear damping R. The force transmissibility (a) $T_{1, 0}$ and (b) $T_{1, 2}$ .

In Figures 5 and 6, the results indicate that the power law damping can not only suppress the force transmissibilities $T_{1, 0}$ and $T_{1, 2}$ over the resonant frequency region but also keep the transmissibility of nonresonant frequencies unaffected. Figures 5(b) and 6(b) show that the transmissibilities at different resonant frequencies can be decreased by applying the nonlinear damping at different positions. For example, the nonlinear damping $r_{11}$ can decrease the transmissibilities of the first-order resonant frequency, but the transmissibility of the second- and third-order resonant frequencies is not affected, whereas R can decrease the transmissibilities of the first- and second-order resonant frequencies, but the transmissibility of the third-order resonant frequency is not affected.

These results indicate that the isolation performance of the vessel isolation system can be optimized by applying nonlinear damping. Before conducting the optimized design of the nonlinear damping based on the vessel isolation system, another operating condition, where the double diesel engines rotate synchronously, is briefly discussed as follows, indicating the optimization is also needed in this case with a nonlinear damping.

Synchronous rotation of double diesel engines

When the two diesel engines rotate synchronously, the isolation system in Figure 2 can readily be simplified as a 2-DOF system as shown in Figure 7.

Figure 7.

2-DOF vibration isolation system.

The equivalent input force $f_{in} (t)$ of the whole system can be described as

f_{in} (t) = F_{0, 1} (t) + F_{0, 2} (t) = 2 F_{0} \sin (Ω t)

(11)

Both power law damping $r_{11}$ and R are introduced into the isolation system. In this case, the linear parameters are given as ${C_{1}}^{'} = C_{1} = 0.004$ and ${C_{2}}^{'} = 2 C_{2} = 0.004$ , and the transmissibility $T_{0}$ introduced in equation (7) is shown in Figure 8 with different pairs of nonlinear damping.

Figure 8.

Effects of power law damping $r_{11}$ and R.

The results discussed above indicate that, the vessel isolation system with two engines working synchronously can be simplified as a 2-DOF system, which has been well studied for the analysis and design problems.²⁰

In addition, the finite element simulation and experiments will be conducted in the further investigation. In the next section, an optimal design of the vessel isolation system is studied with only one diesel engine working.

Design of the vessel isolation system

The design restrictions of the vessel isolation system are depicted in Figure 9, where the force transmissibility curve cannot be presented in the forbidden area.¹⁵ For the working condition of single diesel engine rotating, the design requirements are described as

{\begin{matrix} T (ω) < T_{a}, ω \leq ω_{b} \\ \begin{matrix} T (ω) < T_{b}, ω_{b} < ω \leq ω_{c} \\ T (ω) < T_{c}, ω > ω_{c} \end{matrix} \end{matrix}

(12)

where $ω_{b} = \frac{ω_{1} + ω_{2}}{2}$ , and $ω_{c}$ is the cut-off frequency and takes the value of $\sqrt{2} ω_{2}$ .

Figure 9.

Design restrictions of the vessel isolation system.

Equation (12) limits the force transmissibility of the nonresonant region by adopting a less significant linear damping, as well as restricts the resonant peak values of the force transmissibility by increasing the nonlinear damping parameters. The ideal transmissibility curve of a vessel isolation system must lie in the white region by designing the linear and nonlinear damping.

The design issue can be described as an optimal design problem as follows:

Find

η_{0} = {C_{1}, C_{2}, r_{11}, R}

(13)

Solve the optimization problem

{\begin{matrix} min_{{C_{1}, C_{2}, r_{11}, R}} | T_{a} (ω_{1}) | \\ min_{{C_{1}, C_{2}, r_{11}, R}} | T_{b} (ω_{2}) | \end{matrix}

(14)

Under the constraint of

T (ω) \leq {\begin{matrix} T_{a} (ω_{1}) & 0 \leq ω \leq ω_{b} \\ T_{b} (ω_{2}) & ω_{a} \leq ω \leq ω_{c} \\ - 12.39 dB & ω \geq ω_{c} \end{matrix}

(15)

Design steps

Assuming that mass and stiffness parameters of the vibration isolation system are fixed, the damping parameters can be determined in three steps to achieve the design requirements. First, considering that the linear damping will increase the force transmissibility in the isolation range of $ω > ω_{c}$ , a smaller value $T_{c}$ can be designed by adopting a less significant linear damping. Then, in order to obtain smaller values of the resonant peak transmissibilities $T (ω_{1}), T (ω_{2})$ and ensure the transmissibility $T_{c}$ of nonresonant region is not increased, the design of nonlinear damping $r_{11}$ and R is conducted to satisfy the design equation (12). Finally, compare the results and requirements.

Step 1. Determination of linear damping characteristics $C_{1}$ and $C_{2}$

Assuming the vibration isolation system only has linear damping forces, $ω_{c}$ can be calculated as 13.7 Hz according to Table 1. A less strong linear damping should be adopted so as to obtain the smaller transmissibility over the nonresonant range. Furthermore, combining with Figure 3(a) and (b), $T_{c} (Ω_{1})$ can be obtained as −12.39 dB with linear damping characteristics $C_{1} = 0.004$ and $C_{2} = 0.002$ . The bigger linear damping will shift the peak values, and the force transmissibility of the vibration isolation region will remain at the higher level. Therefore, the linear damping can be determined by satisfying equation (12), which is the force transmissibility of the nonresonant region.

Step 2. Designing nonlinear damping R and $r_{11}$

In order to satisfy equation (12), the strong nonlinear damping $r_{11}$ and $R = r_{21} = r_{22}$ can be introduced to decrease the resonant peak values in the vibration isolation region. The ideal nonlinear damping $r_{11}$ and $R = r_{21} = r_{22}$ can be designed by the numerical analysis. The transmissibility analysis result is depicted in Figure 10 with the different nonlinear damping. For the forbidden area, $T (Ω_{1}) = 23.7 dB$ and $T (ω_{2}) = 10.4 dB$ can be determined according to the transmissibility with linear damping values $C_{1} = 0.004$ and $C_{2} = 0.002$ . In this numerical example, the dotted dashed line indicates that too large nonlinear damping will lead to the transmissibility increasing over nonresonant region, and the resonant frequency was shifted. Furthermore, it is clear that the dotted dashed line lies in the forbidden area. In addition, the transmissibility of the second-order resonant frequency is lightly affected by the nonlinear damping. If the force transmissibility of the second-order resonant frequency is not satisfied as the minimum value of equation (12), linear damping $C_{2}$ can be increased to achieve the ideal vibration isolation performance and satisfy the requirement according to Figure 4.

Figure 10.

Force transmissibility with double nonlinear damping where $r_{21} = 0, R = r_{21} = r_{22}$ , solid; $r_{21} = 4 \times 10^{10}, R = r_{21} = r_{22} = 2 \times 10^{9}$ , dashed; and $r_{21} = 4 \times 10^{10}, R = r_{21} = r_{22} = 2 \times 10^{11}$ , dotted dashed.

For the vibration isolation system, the nonlinear damping values can be determined as $r_{21} = 4 \times 10^{10}, R = r_{21} = r_{22} = 2 \times 10^{9}$ .

Step 3. Compare the results and requirements

If $T (ω) < T_{c}$ for $ω > ω_{c}$ is violated, reducing linear damping can achieve smaller values for nonresonant region.

If the force transmissibility is violated for $ω \leq ω_{c}$ , the nonlinear damping can be increased to obtain a smaller peak value.

If the force transmissibility of the second-order resonant frequency is not satisfied with equation (12), the linear damping $C_{2}$ can be increased to obtain a smaller value. Then, repeat the design process of nonlinear damping values.

The optimization process is summarized in Figure 11.

Figure 11.

Optimization process of the damping parameters.

Conclusion

In this article, the power law nonlinear damping was studied based on a parallel 3-DOF nonlinear system simplified from a vessel isolation system in laboratory. Two working conditions, including the single diesel engine rotating condition and the double diesel engine rotating condition, were investigated, where the effects of the power law damping on the force transmissibility of the system were investigated. Furthermore, the design of the power law damping was discussed. The detailed conclusions are given as follows:

Nonlinear damping has a better performance than linear damping in isolating the vessels’ vibration. The force transmissibilities over all frequency ranges are relatively low under different working conditions.

With the single diesel engine rotating, three resonant peaks can be observed in the plot of the force transmissibility $T_{1, 2}$ , but the second-order resonant frequency (6.91 Hz) disappeared in the plot of the force transmissibility $T_{1, 0}$ since the isolation system has two identical parallel DOFs.

With the single diesel engine rotating, the force transmissibility $T_{1, 2}$ of the second-order resonant frequency (6.91 Hz) was little affected by the linear damping ratio $C_{1}$ and nonlinear damping $r_{11}$ , and the force transmissibility $T_{1, 0}$ and $T_{1, 2}$ of the third-order resonant frequency (9.72 Hz) was not affected by the nonlinear damping $r_{11}$ and R.

The design process of the power law damping was presented by determining the linear and nonlinear damping characteristics under a given forbidden area. Strong nonlinear damping can increase the transmissibility over the nonresonant region.

Footnotes

Appendix 1

Handling Editor: Yunn-Lin Hwang

Declaration of conflicting interests

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

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study was financially supported by the Youth Innovation Promotion Association Foundation of the Chinese Academy of Sciences (grant no. 2016187) and the Key Research Program of the Chinese Academy of Sciences (grant no. KGFZD-135-16-006).

ORCID iD

You Wang

References

Almotiri

Elleithy

Retinal vessels segmentation techniques and algorithms: a survey. Appl Sci 2018; 8: 155.

Gan

et al . New insights into airfoil sail selection for sail-assisted vessel with computational fluid dynamics simulation. Adv Mech Eng. Epub ahead of print 18 April 2018. DOI: 10.1177/1687814018771254.

Ribeiro

AMR

Silva

JMM

Maia

NMM.

On the generalisation of the transmissibility concept. Mech Syst Signal Pr 2000; 14: 29–35.

Wang

Huang

Zhou

MG.

Transfer matrix approach of vibration isolation analysis of periodic composite structure. Arch Appl Mech 2007; 77: 461–471.

Ruzicka

Derby

TF.

Influence of damping in vibration isolation. Washington, DC: Department of Defense, Shock and Vibration Information Center, 1971.

Carrella

Brennan

Kovacic

et al . On the force transmissibility of a vibration isolator with quasi-zero-stiffness. J Sound Vib 2009; 322: 707–717.

Yang

Zhang

Modeling and control of active-passive vibration isolation for floating raft system. Math Probl Eng 2017; 2017: 5361702.

Terenzi

Dynamics of SDOF systems with nonlinear viscous damping. J Eng Mech 1999; 125: 956–963.

Petit

Conan

Kulcsar

et al . First laboratory validation of vibration filtering with LQG control law for adaptive optics. Opt Express 2008; 16: 87–97.

10.

Yang

Vicario

et al . Improvements of magnetic suspension active vibration isolation for floating raft system. Int J Appl Electrom 2017; 53: 193–209.

11.

Song

Sun

Modeling and dynamics of a MDOF isolation system. Appl Sci 2017; 7: 393.

12.

et al . Nonlinear dynamic analysis of 2-DOF nonlinear vibration isolation floating raft systems with feedback control. Chaos Soliton Fract 2012; 45: 1092–1099.

13.

DL.

Spectrum reconstruction of quasi-zero stiffness floating raft systems. Chaos Soliton Fract 2016; 93: 123–129.

14.

Lang

Guo

Takewaki

Output frequency response function based design of additional nonlinear viscous dampers for vibration control of multi-degree-of-freedom systems. J Sound Vib 2013; 332: 4461–4481.

15.

Lang

Billings

SA.

Design of vibration isolators by exploiting the beneficial effects of stiffness and damping nonlinearities. J Sound Vib 2014; 333: 2489–2504.

16.

Guo

Lang

Peng

ZK.

Analysis and design of the force and displacement transmissibility of nonlinear viscous damper based vibration isolation systems. Nonlinear Dynam 2012; 67: 2671–2687.

17.

Lang

Jing

Billings

et al . Theoretical study of the effects of nonlinear viscous damping on vibration isolation of SDOF systems. J Sound Vib 2009; 323: 352–365.

18.

Zhu

Lang

ZQ.

Design of nonlinear systems in the frequency domain: an output frequency response function-based approach. IEEE T Contr Syst T 2018; 26: 1358–1371.

19.

Zhu

Lang

. Analysis of output response of nonlinear systems using nonlinear output frequency response functions. In: Proceedings of the UKACC 11th international conference on control, Belfast, 31 August–2 September 2016, pp.1–6. New York: IEEE.

20.

Peng

Lang

Zhao

et al . The force transmissibility of MDOF structures with a non-linear viscous damping device. Int J Nonlinear Mech 2011; 46: 1305–1314.

21.

Peng

Lang

Billings

SA.

Crack detection using nonlinear output frequency response functions. J Sound Vib 2007; 301: 777–788.

22.

Mofidian

SMM

Bardaweel

. Theoretical study and experimental identification of elastic-magnetic vibration isolation system. J Intell Mater Syst Struct 2018; 29: 3550–3561.

23.

Zhu

Lang

ZQ.

The effects of linear and nonlinear characteristic parameters on the output frequency responses of nonlinear systems: the associated output frequency response function. Automatica 2018; 93: 422–427.

24.

Yao

Zhang

et al . Nonlinear dynamic response analysis of cylindrical composite stiffened laminates based on the weak form quadrature element method. Compos Struct 2018; 203: 446–457.