Study on dynamic characteristics of bio-inspired vibration isolation platform

Abstract

Inspired by the body movement of the kangaroo, a multi-degree-of-freedom vibration isolation platform containing three units, that is, a protected object, a nonlinear energy sink, and an X-shaped structure, has been modeled, and the differential equations of the system have been given in the form of uniform relative coordinates. Furthermore, the displacement transmissibility analysis and numerical calculation are supported by the method of harmonic balance and Runge–Kutta algorithm, which shows that (a) there are nonlinear behaviors and resonant phenomenon in the time–frequency response and (b) quasi-periodic motion may be a predictor of periodic steady-state response or strong resonance, and the displacement evolution before quasi-periodic motion may be used to distinguish the two phenomena. In addition, based on the numerical method, the system energy changes in a selected frequency are discussed. Finally, the correctness of the theoretical analysis is verified by simulation data in Adams. Taken together, these results demonstrate that the dynamic characteristics are adjustable and designable of structural parameters in a specific frequency band and can provide a useful way to reduce the amplitude of resonant peaks and improve the vibration isolation performance for practical engineering applications.

Keywords

Nonlinear dynamics vibration isolation platform harmonic balance method numerical simulation bionic design

1. Introduction

The dynamical features and the vibration reduction of nonlinear engineering structure, as is well known, are widespread problems [Kamesh et al., 2012; Thanh and Kyoung, 2011; Robertson et al., 2009]. Great efforts, from various perspectives, were made to improve the performance of vibration isolation structure, such as installing metal rubber [Yu et al., 2017] to achieve vibration isolation, using a versatile n-layer scissorlike structure [Sun et al., 2014], attaching a mass to provide an inertial force [Chen et al., 2019], and using active components matching various control strategies (a disturbance feedforward control strategy, PID control, and an active tuned inerter damper with $ℋ_{\infty}$ optimization criterion) [Michiel et al., 2018; Syed et al., 2018; Zhao et al., 2019], which brings new ideas and promotion to engineering vibration reduction. However, we all know that it is most economical and practical to use simple spring and damper elements to achieve vibration isolation under passive conditions.

In recent years, bionic vibration isolation design has attracted a lot of attention [Feng and Jing, 2019; Feng et al., 2019; Jing et al., 2019; Zang et al., 2018]. Inspired by the vibration isolation idea of the bird skeleton, Wu Z-J [Wu et al., 2015] designed a nonlinear mechanism to realize the zero stiffness and negative stiffness. Feng X [Feng et al., 2019] proposed an anti-vibration structure considering the swing of the human arm (abstracting the swing of the human arm as an inertial mass). In his subsequent research [Feng and Jing, 2019], he changed the arrangement of a spring–damper element and continued to discuss the dynamic characteristics of bionic human structure. Inspired by a bird limb skeleton, Jing X-J [Jing et al., 2019 ] proposed a bio-inspired anti-vibration exoskeleton, and the dynamic analysis and experiments demonstrate that the novel structure has significant engineering application. These works show that the bionic structure can provide satisfactory performance to the vibration isolation application.

Kangaroo, an ancient creature, can jump in various ground conditions and even cliffs. Except for the movement of its legs, the swing of the tail is always found during movement. Based on these behavioral characteristics, after an overall observation of kangaroo body, it can be inferred that its movement and body structure are consistent with requirements of vibration reduction. Furthermore, from the kangaroo anatomy diagram, it can be seen that the length ratio of thigh bone, calf bone, and foot bone is 1:2:1 as shown in Figure 1(a), which is consistent with the results obtained by dissecting 52 kangaroos [Hopwood and Butterfield, 1990]. Therefore, inspired by bionic thinking and the existing bionic vibration isolation mechanism [Wu et al., 2015; Jing et al., 2019; Feng and Jing, 2019; Feng et al., 2019], a multi-degree-of-freedom vibration isolation platform coupled with an inertial mass shown in Figure 1(b) is modeled and systematically studied. In Figure 1(b), the X-shaped structure is used to equivalent the kangaroo’s leg bones, and the inertial mass is used to equivalent the vibrating of the tail. And the spring–dampers in the X-shaped structure and the protected object are linear elements, and the inertial mass is linked by a nonlinear spring and a linear damper, known as the nonlinear energy sink (NES). Based on some similar references [Chouvion, 2019; Chen et al., 2020; Chen et al., 2020; Chen et al., 2019; Yang et al., 2017; Zang et al., 2018; Zang et al., 2019], the proposed design will be explored in this study.

In many dynamic response evaluations, the displacement transmissibility is often used for strong nonlinear systems [Feng and Jing, 2019; Feng et al., 2019; Zang et al., 2019]. In addition, the responses in the time domain [Zang et al., 2019; Zang et al., 2018; Huang et al., 2019] and the changes of system energy [Zang et al., 2019; Tarcisio et al., 2018; Liu et al., 2017] can also disclose the dynamic behaviors in the specific frequency, which brings clear physical meaning and with computational convenience. Therefore, in the present work, the dynamical analysis in time and frequency domains will be completed to illustrate the effect of different parameters on vibration reduction. In addition, the effect of parameter changes on system energy dissipation is also discussed, which reveals a more comprehensive parameter influence.

These dynamic characteristics mentioned above can be revealed and discussed by analyzing or numerically solving the established mathematical model. There are a lot of methods that can be used to solve the differential equations. The usual methods are multiple scales method [Foroutan et al., 2019; Liu et al., 2017; Zulli and Luongo, 2016], harmonic balance method [Zang, et al., 2018; Zang, et al., 2019], the average method [Barkham and Soudack, 1969; Liu et al., 2017], complexification averaging technique [Huang et al., 2019], and some alternating frequency–time techniques [Fares et al., 2019; Zang, et al., 2018]. These proposed and applied methods well reveal the dynamic response in the time domain or frequency domain. So, in this work, the primary dynamic behavior of the system, including displacement transmissibility, the response in the time domain, and energy changes, is analytically and numerically explored by the harmonic balance method, Newton–Raphson iteration, and the Runge–Kutta numerical technique.

The article is organized as follows. In Section 2, the modeling process and the corresponding derivation is finished. Section 3 gives the displacement transmissibility analysis of the system under the periodic excitation. Section 4 reveals a series of numerical results in the specific frequency, such as amplitude spectrum, time history, and phase portrait under some different parameter groups. In Section 5, the change of the system energy is analyzed. Section 6 lists the conclusion.

2. Modeling and derivation tasks

2.1. Physical modeling

For convenience in modeling and analyzing, the vibration isolation device is designed as shown in Figure 2. From Figure 2, it is obvious that there are three units: the X-shaped unit which strictly imitates the ratio of kangaroo limb bones; the NES unit which is used to simulate the effect of the kangaroo’s tail on the vertical vibration reduction; and the protected object $m_{1}$ linked by a linear spring $k_{1}$ and a linear damper $c_{1}$ , which symbolizes the body and protected part of the kangaroo, where $α$ is the angle between the X-shaped unit and the horizontal reference, $k_{2}$ and $c_{2}$ are the nonlinear stiffness and linear damper, respectively, $k_{h}$ and $c_{h}$ are the horizontal spring–damper element, and $k_{v}$ and $c_{v}$ are the vertical spring–damper element.

2.2. Mathematical modeling, simplification, and dimensionless

The multi-degree-of-freedom dynamic equations can be obtained by the Lagrange principle

m_{0} \overset{..}{y_{0}} + k_{1} (y_{0} - y_{1}) + k_{2} {(y_{0} - y_{2})}^{3} + k_{h} (2 x) 2 \frac{\partial x}{\partial y} + k_{v} (y_{3} - z) \frac{\partial (y_{3} - z)}{\partial y} + c_{v} (\dot{y_{3}} - \dot{z}) \frac{\partial (\dot{y_{3}} - \dot{z})}{\partial \dot{y}} + c_{h} (2 \dot{x}) \frac{\partial 2 \dot{x}}{\partial \dot{y}} + c_{1} (\dot{y_{0}} - \dot{y_{1}}) + c_{2} (\dot{y_{0}} - \dot{y_{2}}) = 0

(1a)

m_{1} \overset{..}{y_{1}} + k_{1} (y_{1} - y_{0}) + c_{1} (\dot{y_{1}} - \dot{y_{0}}) = 0

(1b)

m_{2} \overset{..}{y_{2}} + k_{2} {(y_{2} - y_{0})}^{3} + c_{2} (\dot{y_{2}} - \dot{y_{0}}) = 0

(1c)

where

y_{i}

are the vertical displacements of

m_{i}

(i = 0,1,2)

y_{3}

is the vertical displacement of the middle node of the X-shaped unit,

x

is the corresponding tensile length of a spring

k_{h}

in the horizontal direction, and

z

is the external excitation.

Now, the geometric relations are listed as follows to be used to simplify the complex dynamic equations. When the structure is disturbed, the shape of the X-shaped unit will change accordingly. The geometric analysis is shown in Figure 3

\begin{array}{l} y_{3} - z = \frac{L_{1}}{L_{1} + L_{2}} (y_{0} - z) = η_{1} Y \\ x = L_{2} cos (α) - \sqrt{L_{2}^{2} - {(L_{2} sin (α) + \frac{1}{2} η_{2} Y)}^{2}} \end{array}

(2)

where

η_{i} = L_{i} / L_{1} + L_{2}

(i = 1,2)

is the dimensionless coefficients. And

Y = y_{0} - z

is a relative displacement in the dynamic process.

Combining the above geometric relations and Taylor series theory, equation (1a) can be expressed as

m_{0} \overset{..}{y_{0}} + k_{1} (y_{0} - y_{1}) + k_{2} {(y_{0} - y_{2})}^{3} + k_{h} (μ_{1} Y + μ_{2} Y^{2} + μ_{3} Y^{3} + μ_{4} Y^{4}) + η_{1}^{2} k_{v} (y - z) + η_{1}^{2} c_{v} (\dot{y} - \dot{z}) + c_{h} (ν_{0} + ν_{1} Y + ν_{2} Y^{2} + ν_{3} Y^{3} + ν_{4} Y^{4}) \dot{Y} + c_{1} (\dot{y_{0}} - \dot{y_{1}}) + c_{2} {(\dot{y_{0}} - \dot{y_{2}})}^{3} = 0

(3)

where

\begin{array}{l} k_{h} (2 x) 2 \frac{\partial x}{\partial y} = k_{h} (μ_{0} + μ_{1} Y + μ_{2} Y^{2} + μ_{3} Y^{3} + μ_{4} Y^{4} + O (Y^{5})) \\ c_{h} (2 \dot{x}) \frac{\partial 2 \dot{x}}{\partial \dot{y}} = c_{h} (ν_{0} + ν_{1} Y + ν_{2} Y^{2} + ν_{3} Y^{3} + ν_{4} Y^{4} + O (Y^{5})) \dot{Y} \\ k_{v} (y_{3} - z) \frac{\partial (y_{3} - z)}{\partial y} = η_{1}^{2} k_{v} Y \\ c_{v} (\dot{y_{3}} - \dot{z}) \frac{\partial (\dot{y_{3}} - \dot{z})}{\partial \dot{y}} = η_{1}^{2} c_{v} \dot{Y} \end{array}

(4)

where

μ_{i}

and

ν_{i}

(i = 0,1,2,3,4)

are the corresponding polynomial shown in Appendix A.

Furthermore, combining equation (3), equation (1b), and equation (1c), equation (1) reduces to the dimensionless form

\begin{array}{l} ε_{0} y_{0}^{″} + (y_{0} - y_{1}) + β_{2} {(y_{0} - y_{2})}^{3} \\ + β_{h} (μ_{1} Y + μ_{2} Y^{2} + μ_{3} Y^{3} + μ_{4} Y^{4}) \\ + η_{1}^{2} β_{v} (y - z) + η_{1}^{2} ς_{v} (y^{'} - z^{'}) \\ + ς_{h} (ν_{0} + ν_{1} Y + ν_{2} Y^{2} + ν_{3} Y^{3} + ν_{4} Y^{4}) Y^{'} \\ + ς_{1} (y_{0}^{'} - y_{1}^{'}) + ς_{2} (y_{0}^{'} - y_{2}^{'}) = 0 \end{array}

(5a)

y_{1}^{″} + (y_{1} - y_{0}) + ς_{1} (y_{1}^{'} - y_{0}^{'}) = 0

(5b)

ε_{2} y_{2}^{″} + β_{2} {(y_{2} - y_{0})}^{3} + ς_{2} (y_{2}^{'} - y_{0}^{'}) = 0

(5c)

where

{(*)}^{'} = \frac{d (*)}{d τ}, τ = ω_{n} t, Ω = \frac{ω}{ω_{n}}, ω_{n}^{2} = \frac{k_{1}}{m_{1}}, ε_{i} = \frac{m_{i}}{m_{1}}, (i = 0, 2), β_{l} = \frac{k_{l}}{k_{1}}, ς_{l} = \frac{c_{l}}{m_{1} ω_{n}}, l = 1, 2, v, h

(6)

To display all variables in a unified form, equation (7a) can be obtained by subtracting $ε_{0} Z^{''}$ from both sides of equation (5a); and equation (7b) and (7c) can be obtained by subtracting equation (5a) from equations (5b) and (5c), respectively. Equation (7) is given as follows

\begin{array}{l} ε_{0} (y_{0}^{″} - z^{″}) + (y_{0} - y_{1}) + β_{2} {(y_{0} - y_{2})}^{3} + β_{h} (μ_{1} Y + μ_{2} Y^{2} + μ_{3} Y^{3} + μ_{4} Y^{4}) + η_{1}^{2} β_{v} (y - z) \\ + η_{1}^{2} ς_{v} (y^{'} - z^{'}) + ς_{h} (ν_{0} + ν_{1} Y + ν_{2} Y^{2} + ν_{3} Y^{3} + ν_{4} Y^{4}) Y^{'} + ς_{1} (y_{0}^{'} - y_{1}^{'}) + ς_{2} (y_{0}^{'} - y_{2}^{'}) = - ε_{0} z^{'} \end{array}

(7a)

\begin{array}{l} ε_{0} (y_{1}^{″} - y_{0}^{″}) + (1 + ε_{0}) (y_{1} - y_{0}) + (1 + ε_{0}) ς_{1} (y_{1}^{'} - y_{0}^{'}) + β_{2} {(y_{2} - y_{0})}^{3} \\ + ς_{2} (y_{2}^{'} - y_{0}^{'}) - β_{h} (μ_{1} Y + μ_{2} Y^{2} + μ_{3} Y^{3} + μ_{4} Y^{4}) - η_{1}^{2} β_{v} (y - z) \\ - η_{1}^{2} ς_{v} (y^{'} - z^{'}) - ς_{h} (ν_{0} + ν_{1} Y + ν_{2} Y^{2} + ν_{3} Y^{3} + ν_{4} Y^{4}) Y^{'} = 0 \end{array}

(7b)

\begin{array}{l} ε_{2} ε_{0} (y_{2}^{″} - y_{0}^{″}) + (ε_{0} + ε_{2}) β_{2} {(y_{2} - y_{0})}^{3} + (ε_{0} + ε_{2}) ς_{2} (y_{2}^{'} - y_{0}^{'}) \\ + ε_{2} (y_{1} - y_{0}) + ε_{2} ς_{1} (y_{1}^{'} - y_{0}^{'}) - ε_{2} β_{h} (μ_{1} Y + μ_{2} Y^{2} + μ_{3} Y^{3} + μ_{4} Y^{4}) \\ - ε_{2} η_{1}^{2} β_{v} (y - z) - ε_{2} η_{1}^{2} ς_{v} (y^{'} - z^{'}) \\ - ε_{2} ς_{h} (ν_{0} + ν_{1} Y + ν_{2} Y^{2} + ν_{3} Y^{3} + ν_{4} Y^{4}) Y^{'} = 0 \end{array}

(7c)

Introduction of new relative dimensionless displacements as equation (8)

\begin{array}{l} u_{0} = \frac{1}{L_{2}} (y_{0} - z) \\ u_{1} = \frac{1}{L_{2}} (y_{1} - y_{0}) \\ u_{2} = \frac{1}{L_{2}} (y_{2} - y_{0}) \end{array}

(8)

And one can transform equation (7) into the dimensionless form

ε_{0} u_{0}^{″} - u_{1} - β_{n} u_{2}^{3} + β_{h} (ρ_{1} u_{0} + ρ_{2} u_{0}^{2} + ρ_{3} u_{0}^{3} + ρ_{4} u_{0}^{4}) + β_{V} u_{0} + ς_{V} u_{0}^{'} + ς_{h} (δ_{0} + δ_{1} u_{0} + δ_{2} u_{0}^{2} + δ_{3} u_{0}^{3} + δ_{4} u_{0}^{4}) u_{0}^{'} ς_{1} u_{1}^{'} - ς_{2} u_{2}^{'} = - ε_{0} Z^{″}

(9a)

\begin{array}{l} ε_{0} u_{1}^{″} + (1 + ε_{0}) u_{1} \\ + (1 + ε_{0}) ς_{1} u_{1}^{'} + β_{n} u_{2}^{3} + ς_{2} u_{2}^{'} \\ - β_{h} (ρ_{1} u_{0} + ρ_{2} u_{0}^{2} + ρ_{3} u_{0}^{3} + ρ_{4} u_{0}^{4}) \\ - β_{V} u_{0} - ς_{V} u_{0}^{'} - ς_{h} (δ_{0} + δ_{1} u_{0} + δ_{2} u_{0}^{2} + δ_{3} u_{0}^{3} + δ_{4} u_{0}^{4}) u_{0}^{'} = 0 \end{array}

(9b)

\begin{array}{l} ε_{0} ε_{2} u_{2}^{″} + (ε_{0} + ε_{2}) β_{n} u_{2}^{3} \\ + (ε_{0} + ε_{2}) ς_{2} u_{2}^{'} + ε_{2} u_{1} + ε_{2} ς_{1} u_{1}^{'} \\ - ε_{2} β_{h} (ρ_{1} u_{0} + ρ_{2} u_{0}^{2} + ρ_{3} u_{0}^{3} + ρ_{4} u_{0}^{4}) \\ - ε_{2} β_{V} u_{0} - ε_{2} ς_{V} u_{0}^{'} - ε_{2} ς_{h} (δ_{0} + δ_{1} u_{0} + δ_{2} u_{0}^{2} + δ_{3} u_{0}^{3} + δ_{4} u_{0}^{4}) u_{0}^{'} = 0 \end{array}

(9c)

where

Z = \frac{z}{L_{2}}, ρ_{j} = μ_{j} L_{2}^{j - 1}, (j = 1, 2, 3, 4), δ_{k} = ν_{k} L_{2}^{k}, (k = 0, 1, 2, 3, 4), β_{V} = η_{1}^{2} β_{v}, ς_{V} = η_{1}^{2} ς_{v}

(10)

3. Vibration amplitude response evaluation in frequency domain

Those strong nonlinear dynamical equations can be reduced to obtain steady-state response using the HBM. Many studies have proved that the harmonic balance method can give high accuracy compared with the other numerical method [Zang and Chen, 2017; Zang et al., 2019]. Therefore, to achieve a faster convergence of the numerical result, the external excitation and the responses can be assumed to be harmonic as equation (11)

\begin{array}{l} Z (τ) = F \sin (Ω τ) \\ u_{0} (τ) = r_{0} + a_{0} \cos (Ω τ) + b_{0} \sin (Ω τ) \\ u_{1} (τ) = r_{1} + a_{1} \cos (Ω τ) + b_{1} \sin (Ω τ) \\ u_{2} (τ) = r_{2} + a_{2} \cos (Ω τ) + b_{2} \sin (Ω τ) \end{array}

(11)

where

r_{i}

(i = 0,1,2)

is the bias term,

a_{i}

and

b_{i}

(i = 0,1,2)

are the amplitudes of corresponding harmonic components, and

F = f / L_{2}

is the dimensionless amplitude of the external excitation.

Substituting equations (11) into (9), and equating the coefficients of each harmonic term and constant term of the both hands of resulting equations, a set of nonlinear algebraic equations are obtained as shown in Appendix B. For given choice parameters, the nonlinear algebraic equations can be numerically solved by the Newton–Raphson iteration. To evaluate the vibration isolation performance of the bio-inspired structure, the displacement transmissibility can be written as

T_{d} = \frac{| (a_{0} cos (Ω τ) + b_{0} sin (Ω τ)) + (a_{1} cos (Ω τ) + b_{1} sin (Ω τ)) + F sin (Ω τ) |}{| F sin (Ω τ) |} = \frac{\sqrt{a_{0}^{2} + 2 a_{0} a_{1} + a_{1}^{2} + {(F + b_{0} + b_{1})}^{2}}}{| F |}

(12)

where

| * |

means to take the norm.

The parameters are set as $m_{0} = 200$ kg, $m_{1} = 40$ kg, $m_{2} = 5$ kg, $k_{1} = 2.2 \times 10^{6}$ N/m, $k_{2} = 3.15 \times 10^{7}$ N/m³, $k_{h} = 3.78 \times 10^{6}$ N/m, $k_{v} = 2.0 \times 10^{6}$ N/m, $c_{1} = 500$ Ns/m, $c_{2} = 300$ Ns/m, $c_{h} = 2500$ Ns/m, and $c_{v} = 2100$ Ns/m. The input excitation amplitude is supposed to be fixed as $f = 0.05$ m. The X-shaped structural parameters $L_{1} = 0.35$ m, $L_{2} = 0.35$ m, and $α = 35$ ° are the main support for the vibration isolation system. Then, the different results of displacement transmissibility under different structural parameters can be obtained in Figures 4–6.

From the observation in Figure 4, for different $m_{0}$ , $m_{1}$ , $k_{1}$ , and $c_{1}$ , the displacement transmissibility $T_{d}$ is demonstrated, respectively, in Figures 4(a)–(d). It is obvious that there are two resonant peaks at the frequency $Ω \in [0.1, 0.3]$ and $Ω \in [1, 1.2]$ . In Figure 4(a), with the increase of the mass $m_{0}$ , the resonant frequency and resonant peak in the second-order resonant frequency are reduced simultaneously. And the resonant frequency is decreased with increasing the mass $m_{0}$ , but there is almost no change in the resonance peak at the first-order resonance frequency. From Figure 4(b), it can be seen that the change of the peak value in the first-order resonant frequency is not obvious. However, the resonant frequency in the first-order and second-order is reduced by decreasing the protected object mass $m_{1}$ , and the resonant peak value at second-order resonant frequency is also reduced.

In Figure 4(c), the resonant frequency is decreased with the increase of the stiffness $k_{1}$ , and the change of the resonant peak value is not obvious in the first-order resonant frequency. Moreover, the resonant peak is decreased with increasing the stiffness $k_{1}$ at the second-order resonant frequency, but the resonant frequency here does not show significant change.

The transmissibility curves for different damping $c_{1}$ are shown in Figure 4(d). It is obvious that the peak value around the second-order resonant frequency $(Ω \in [1,1.2])$ is increased with the decrease of the damping $c_{1}$ , and the displacement transmissibility does not display significant change around other frequency bands.

With other parameters determined as before, the effect of different NES parameters on frequency domain response is shown in Figure 5.

In Figure 5(a), it can be seen that the large mass $m_{2}$ can reduce the peak value at the first resonant frequency, and the change of the transmissibility is not obvious at other frequency bands (as the yellow enlarged diagram in Figure 5(a)). Figure 5(b) shows that the peak value decreases at the first resonant frequency with decreasing the nonlinear stiffness $k_{2}$ (as the yellow enlarged diagram in Figure 5(b)), and the amplitude has no change at other resonant frequency bands. Moreover, there is sudden change (may be bifurcation phenomenon) in the frequency range $(0.3, 0.6)$ as shown in the blue enlarged diagram in Figure 5(b). Form Figure 5(c), it can be seen that the decrease of the NES damping $c_{2}$ can obviously increase the resonant amplitude around the first and second resonant frequencies as shown in the yellow and blue enlarged diagrams, respectively.

The effect of the X-shaped unit parameters on the displacement transmissibility is shown in Figure 6. In Figure 6(a), the resonant peak is decreased with the increasing of stiffness $k_{h}$ at the first-order resonant frequency, but the resonant frequency here is increased. Besides, the resonant peak is increased with the increase of $k_{h}$ at the second-order resonant frequency, but the resonant frequency here does not show obvious change. Figure 6(b) shows, in the first and second resonant frequencies, the peak value is increased with the increase of the damping $c_{h}$ , but the change of the resonant frequency is not obvious (as the yellow and blue enlarged diagrams in Figure 6(b)).

The effect of the stiffness $k_{v}$ and the damping $c_{v}$ is similar to the effect of the stiffness $k_{h}$ and the damping $c_{h}$ shown in Figures 6(c) and (d), but at the first-order resonant frequency, the effect of the stiffness $k_{v}$ on the peak value of $T_{d}$ is opposite to the effect of the stiffness $k_{h}$ . It is clear that with the decrease of the stiffness $k_{v}$ , the resonant frequency and the resonant peak are decreased at the first-order resonant frequency, whereas for the second-order resonant frequency, the peak value is increased with increasing the stiffness $k_{v}$ , and the resonant frequency change is not obvious. Figure 6(d) shows that the peak value is increased with increasing the damping $c_{v}$ in the first and second resonant frequencies (as the yellow and blue enlarged diagrams in Figure 6(d)), but the change of the resonant frequency is not obvious.

The effects of the rod length $L_{i}$ $(i = 1,2)$ and the angle $α$ are shown in Figures 6(e) and (f). It can be seen that peak value is decreased with smaller rod length in the first-order resonant frequency, and $T_{d}$ is not changed at other frequency bands. Figure 6(f) shows a lower resonant peak is achieved with larger angle $α$ , but the resonant frequency is increased at the first-order resonant frequency. Moreover, at the second-order resonant frequency, the resonant peak is increased with increasing the angle $α$ , but the change of the resonant frequency is not obvious.

In a summary, (1) the system has two resonant frequencies over the entire frequency band, which are distributed around $Ω \in [0.1, 0.3]$ and $Ω \in [1, 1.2]$ and (2) from the enlarged diagrams, there are bifurcations around $Ω \in [0.3, 0.6]$ under some certain parameter groups.

4. Numerical analysis

4.1 Derivation of numerical calculation

Introducing some new variables by

u_{0} = w_{1}, u_{0}^{'} = w_{2}, u_{1} = w_{3}, u_{1}^{'} = w_{4}, u_{2} = w_{5}, u_{2}^{'} = w_{6}

(13)

Equation (9a)–(9c) can be rewritten as

\begin{array}{l} w_{1}^{'} = w_{2} \\ w_{2}^{'} = (w_{3} + β_{n} w_{5}^{3} - β_{h} (ρ_{1} w_{1} + ρ_{2} w_{1}^{2} + ρ_{3} w_{1}^{3} + ρ_{4} w_{1}^{4}) - β_{V} w_{1} - ς_{V} w_{2} - ς_{h} (δ_{0} + δ_{1} w_{1} + δ_{2} w_{1}^{2} + δ_{3} w_{1}^{3} + δ_{4} w_{1}^{4}) w_{2} + ς_{1} w_{4} + ς_{2} w_{6} + ε_{0} F Ω^{2} sin (τ Ω)) \frac{1}{ε_{0}} \\ w_{3}^{'} = w_{4} \\ w_{4}^{'} = (- (1 + ε_{0}) w_{3} - (1 + ε_{0}) ς_{1} w_{4} - β_{n} w_{5}^{3} - ς_{2} w_{6} + β_{h} (ρ_{1} w_{1} + ρ_{2} w_{1}^{2} + ρ_{3} w_{1}^{3} + ρ_{4} w_{1}^{4}) + β_{V} w_{1} + ς_{V} w_{2} + ς_{h} (δ_{0} + δ_{1} w_{1} + δ_{2} w_{1}^{2} + δ_{3} w_{1}^{3} + δ_{4} w_{1}^{4}) w_{2}) \frac{1}{ε_{0}} \\ w_{5}^{'} = w_{6} \\ w_{6} = (- (ε_{0} + ε_{2}) β_{n} w_{5}^{3} - (ε_{0} + ε_{2}) ς_{2} w_{6} - ε_{2} w_{3} - ε_{2} ς_{1} w_{4} + ε_{2} β_{h} (ρ_{1} w_{1} + ρ_{2} w_{1}^{2} + ρ_{3} w_{1}^{3} + ρ_{4} w_{1}^{4}) + ε_{2} β_{V} w_{1} + ε_{2} ς_{V} w_{2} + ε_{2} ς_{h} (δ_{0} + δ_{1} w_{1} + δ_{2} w_{1}^{2} + δ_{3} w_{1}^{3} + δ_{4} w_{1}^{4}) w_{2}) \frac{1}{ε_{0} ε_{2}} \end{array}

(14)

In fact, equation (14) is a matrix equation that can be calculated by the Runge–Kutta method. In this section, the dynamic characteristics of the system in these special frequency bands mentioned above under some parameter groups are explored. And the initial parameters are set as $m_{0} = 200$ kg, $m_{1} = 40$ kg, $m_{2} = 5$ kg, $k_{1} = 2.2 \times 10^{6}$ N/m, $k_{2} = 3.15 \times 10^{7}$ N/m³, $c_{1} = 500$ Ns/m, and $c_{2} = 300$ Ns/m. The input excitation amplitude is fixed as $f = 0.05$ m. The X-shaped structure is the main support of the vibration isolation system, and its parameters are determined as $k_{h} = 3.78 \times 10^{6}$ N/m, $k_{v} = 2.0 \times 10^{6}$ N/m, $c_{h} = 2500$ Ns/m, $c_{v} = 2100$ Ns/m, $L_{1} = 0.35$ m, $L_{2} = 0.35$ m, and $α = 40$ °. Some representative parameters are changed for numerical calculation to further clarify the dynamic behavior of the system. The blue text in the white textbox is the stable displacement amplitude of the mass $m_{1}$ in Figures 7 and 8.

Figure 7 shows the amplitude spectrum, as well as the time history and phase diagram under initial parameters. It can be seen that the amplitude spectrum has three kinds of unusual regions, that is, region I, region II, and region III as shown in Figure 7(a). Then, it is obvious that the frequency bands of these three unusual regions are consistent with those special frequency bands in the frequency domain analysis obtained by the harmonic balance method in Section 3. Next, the time history and phase diagram are analyzed by selecting the corresponding frequency from those three regions shown in Figure 7(b)–(i).

In Figure 7, picking $Ω = 0.3, 0.6, 1.1$ represents the three unusual regions, respectively, and $Ω = 0.8$ represents the other region. Then, it is clear that chaotic motion occurs at $Ω = 0.3, 0.6$ . Besides, the displacement is rapidly decreased, and the motion of the protected object changes into quasi-periodic motion in the frequency $Ω = 0.8$ , finally. On the contrary, the displacement is increased, and the motion of the protected object tends to quasi-periodic motion at $Ω = 1.1$ , gradually.

From Figure 8, it can be seen that with the decreasing of horizontal spring stiffness $k_{h}$ , the displacement amplitude across the entire frequency bands begins to decrease, which can be obviously seen in Figure 8(b), (d), (f), and (h). And the chaotic motion of the protected body is transformed into quasi-periodic motion at $Ω = 0.6$ as shown in Figure 8(d) and (e).

From the analysis and diagrams above, the following points can be summarized.

When the damping parameter of the protected object is strong, it has an important suppression effect on the high-frequency resonant peak.

The NES parameter $k_{2}$ has a significant effect on the vibration attenuation of higher frequency bands.

The bifurcation phenomenon at the frequency $Ω = 0.6$ is related to the chaotic motion of the system, which can be eliminated by adjusting the X-shaped unit parameters, such as horizontal stiffness $k_{h}$ .

When the displacement of the protected object gradually decreases and converges to a quasi-periodic motion, a steady-state motion may occur; when the displacement gradually increases and converges to a quasi-periodic motion, and in this case, a resonance phenomenon may occur.

5. Energy analysis

The change of the system energy can be used to evaluate dynamic characteristics and parameter effect [Zang et al., 2019; Tarcisio et al., 2018; Li et al., 2017a, 2017b]. The kinetic energy and potential energy of the system are given

T_{s} = \frac{1}{2} ε_{0} w_{2}^{2} + \frac{1}{2} ε_{1} w_{4}^{2} + \frac{1}{2} ε_{2} w_{6}^{2}

(15)

V_{s} = \frac{1}{2} β_{h} {(2 x (τ))}^{2} + \frac{1}{2} β_{v} Y_{3} {(τ)}^{2} + \frac{1}{2} β_{1} {(w_{3} - w_{1})}^{2} + \frac{1}{2} β_{2} {(w_{5} - w_{1})}^{2}

(16)

Because the expression of $x$ is a nonlinear term given by structural parameters, a polynomial is obtained by using series theory

x {(τ)}^{2} = l_{0} + l_{1} w_{1} + l_{2} w_{1}^{2} + l_{3} w_{1}^{3} + l_{4} w_{1}^{4} + O (w_{1}^{5})

(17)

where

l_{i}

(i = 0,1,2,3,4)

are the expansion coefficient of series listed in Appendix C.

Substituting equations (17) into (16), the potential energy can be obtained

V_{s} = 2 β_{h} (l_{0} + l_{1} w_{1} + l_{2} w_{1}^{2} + l_{3} w_{1}^{3} + l_{4} w_{1}^{4}) + \frac{1}{2} β_{v} η_{1}^{2} w_{1}^{2} + \frac{1}{2} β_{1} {(w_{3} - w_{1})}^{2} + \frac{1}{2} β_{2} {(w_{5} - w_{1})}^{2}

(18)

Then, the system energy is given in the following

E_{s} = T_{s} + V_{s}

(19)

and the change in the sum of kinetic energy and potential energy is used to assess the dissipation of system energy

E_{s}

5.1. Effect on system energy of the protected object parameters

The change of system energy between the initial state and varying protected object parameters are shown below. Some of the results of energy changes are shown in Figure 9, and finally, all the results are shown in Table 1.

Table 1.

Effect of parameters on the changing of system energy.

Symbol	Mean	Initial value	End value	Rate of energy change
$m_{1}$	$kg$	$40$	$55$	$52.495 %$ ↓
$k_{1}$	$N / m$	$2.2 \times 10^{6}$	$3.2 \times 10^{6}$	$8.122 %$ ↓
$c_{1}$	$Ns / m$	$500$	$900$	$5.762 %$ ↓
$m_{2}$	$kg$	$5$	$10$	$1.018 %$ ↓
$k_{2}$	$N / m$	$3.15 \times 10^{7}$	$3.15 \times 10^{8}$	$52.495 %$ ↓
$c_{2}$	$Ns / m$	$300$	$600$	$4.576 %$ ↓
$k_{h}$	$N / m$	$3.78 \times 10^{6}$	$1.78 \times 10^{6}$	$16.951 %$ ↓
$c_{h}$	$Ns / m$	$2500$	$3500$	$2.378 %$ ↓
$k_{v}$	$N / m$	$2.0 \times 10^{6}$	$1.0 \times 10^{6}$	$11.015 %$ ↓
$c_{v}$	$Ns / m$	$2100$	$3100$	$6.822 %$ ↓
$L_{i} (i = 1,2)$	$m$	$0.35$	$0.4$	$34.573 %$ ↓
$α$	$deg$	$40$	$35$	$13.561 %$ ↓

Figure 9 and Table 1 reveal that a larger protected object mass $m_{1}$ makes the system have a smaller energy at $τ = 0$ , and finally, the system energy tends to be stable and reduces by 52.495%. The strong damper and large stiffness connected to the protected object have little effect on the system energy dissipation, and finally, the system energy reduces by 8.122% and 5.762%, respectively. Consequently, when the mass of the protected object is large, the system energy can still be kept at a low level, which shows the stability and robustness of vibration isolation of the proposed structure.

Compared with the initial system energy, a larger nonlinear stiffness $k_{2}$ and the strong damping $c_{2}$ make the system energy tend to reduce by 52.495% and 4.576%. However, with the increase of NES mass $m_{2}$ , the system energy is increased by 1.018% as in Table 1.

As for the evaluation of the influence of X-shaped unit parameters on energy dissipation, the system energy is reduced by 16.951% relative to the energy of initial state with decreasing horizontal spring $k_{h}$ . As the horizontal damper $c_{h}$ increases, the system energy reduces by 2.378%.

Besides, the system energy decreases with the decreasing of vertical spring stiffness $k_{v}$ , which indicates that the energy dissipation is reduced by 11.015%. The influence of increasing vertical damper is investigated, and it shows that the system energy decreases by 6.822%.

Finally, it can be seen that with larger rod length, the initial energy of the system is reduced by 34.573%. As the angle $α$ decreases, the energy of the system decreases continuously by 13.561%.

From the analysis about the system energy change, the vibration isolation performance caused by the parameters change is more comprehensively understood. In detail, although the displacement transmissibility becomes larger at the entire frequency domain for the larger mass of the protected object $m_{1}$ , the energy dissipation of the entire system becomes better. Moreover, the smaller NES mass, combined with the larger stiffness, may play a positive role in promoting the vibration isolation performance.

5.2. Simulation verification

To validate the theoretical results, a simulation model as shown in Figure 10 is established in Adams. In this simulation test, the parameters of the simulation model are the same as the initial parameters in Section 3. The verification results are shown in Figure 11.

From the comparison between the simulation data and theoretical result (Figure 11), it can be clearly seen that the resonant frequency and the peak value of the displacement transmissibility curve of the theoretical result are in good agreement with the Adams simulation data, which indicates that the theoretical derivation is reliable.

5.3. Comparison discussions

Three systems are compared and discussed as shown in Figure 12. Figure 12(a) shows a linear system coupled with NES, where $k = k_{h} + k_{v}$ is the linear stiffness and $c = c_{h} + c_{v}$ is the linear damping. Figure 12(b) represents an X-shaped system, and Figure 12(c) is the proposed system.

The kinetic energy, the potential energy, and the dissipated energy of the system in Figure 12(a) are

\begin{array}{l} T = \frac{1}{2} m_{0} {\dot{y_{0}}}^{2} + \frac{1}{2} m_{1} {\dot{y_{1}}}^{2} + \frac{1}{2} m_{2} {\dot{y_{2}}}^{2} \\ V = \frac{1}{2} k {(y_{0} - z)}^{2} + \frac{1}{2} k_{2} {(y_{0} - y_{2})}^{2} + \frac{1}{2} k_{1} {(y_{1} - y_{0})}^{2} \\ ℱ = \frac{1}{2} c {(\dot{y_{0}} - \dot{z})}^{2} + \frac{1}{2} c_{2} {(\dot{y_{0}} - \dot{y_{2}})}^{2} + \frac{1}{2} c_{1} {(\dot{y_{1}} - \dot{y_{0}})}^{2} \end{array}

(20)

The dimensionless dynamic equations can be obtained by the Lagrange principle

\begin{array}{l} ε_{0} u_{0}^{″} + β_{0} u_{0} - β_{2} u_{2} - u_{1} + ς_{0} u_{0}^{'} - ς_{2} u_{2}^{'} - ς_{1} u_{1}^{'} = - ε_{0} Z^{″} \\ ε_{0} u_{1}^{″} + (ε_{0} + 1) u_{1} + (ε_{0} + 1) ς_{1} u_{1}^{'} - β_{0} u_{0} + β_{2} u_{2} - ς_{0} u_{0}^{'} + ς_{2} u_{2}^{'} = 0 \\ ε_{0} ε_{2} u_{2}^{″} + (ε_{0} + ε_{2}) β_{2} u_{2} + (ε_{0} + ε_{2}) ς_{2} u_{2}^{'} - ε_{2} β_{0} u_{0} + ε_{2} u_{1} - ε_{2} ς_{0} u_{0}^{'} + ε_{2} ς_{1} u_{1}^{'} = 0 \end{array}

(21)

where

β_{0} = β_{h} + β_{v}

ς_{0} = ς_{h} + ς_{v}

and the other dimensionless parameters are the same as equation (6).

The dynamic equations of the system as shown in Figure 12(b) can be easily obtained from equation (9) as follows

\begin{array}{l} ε_{0} u_{0}^{″} - u_{1} + β_{h} (ρ_{1} u_{0} + ρ_{2} u_{0}^{2} + ρ_{3} u_{0}^{3} + ρ_{4} u_{0}^{4}) + β_{V} u_{0} + ς_{V} u_{0}^{'} + ς_{h} (δ_{0} + δ_{1} u_{0} + δ_{2} u_{0}^{2} + δ_{3} u_{0}^{3} + δ_{4} u_{0}^{4}) u_{0}^{'} - ς_{1} u_{1}^{'} = - ε_{0} Z^{″} \\ ε_{0} u_{1}^{″} + (1 + ε_{0}) u_{1} + (1 + ε_{0}) ς_{1} u_{1}^{'} - β_{h} (ρ_{1} u_{0} + ρ_{2} u_{0}^{2} + ρ_{3} u_{0}^{3} + ρ_{4} u_{0}^{4}) - β_{V} u_{0} - ς_{V} u_{0}^{'} - ς_{h} (δ_{0} + δ_{1} u_{0} + δ_{2} u_{0}^{2} + δ_{3} u_{0}^{3} + δ_{4} u_{0}^{4}) u_{0}^{'} = 0 \end{array}

(22)

The comparison curves of the displacement transmissibility for the three systems are shown in Figure 13. It is clear that the linear system coupled with NES and the X-shaped system only contains one resonant peak $(Ω = 1.1)$ , and their resonant peak value is much higher than the proposed system at this resonant frequency.

Then, based on the parameters in Section 3.1, selecting specific frequency from Figure 13(a) and solving equations (9), (21), and (22) numerically, the vibration amplitude curves of the three systems are displayed in Figure 13(b)–(f). It can be seen from Figure 13(b) and (c) that the system without X-shaped structure has better vibration isolation effect in the frequency band $Ω \in (0.1, 0.3)$ . The proposed system can promote the vibration isolation effect from the frequency $Ω = 0.3$ (Figure 13(f)) and obviously reduce the resonant peak as shown in Figure 13(a), (e), and (f) around the second-order resonant frequency $(Ω = 1.1)$ . The results here demonstrate that the proposed system structure has more excellent dynamic characteristics in more wide frequency bands.

6. Conclusions

This study, an innovative kangaroo-like vibration isolation platform, is presented, analyzed, and evaluated. The HBM technique is applied to analyze approximately on the governing equation of system motion. Based on the results obtained by the HBM, the effect of structural parameters on displacement transmissibility is explored by the numerical method. Besides, the analysis in the time domain and the change of the system energy are supported by numerical calculation. Finally, an Adams simulation model is established, and the simulation data are almost consistent with the theoretical result of the displacement transmissibility, which verifies the correctness of the theoretical analysis. It can be concluded that

the displacement transmissibility is adjustable and designable of structural parameters in a specific frequency band and can provide a useful idea to improve the performance of the proposed design.

The bio-inspired vibration isolation platform can exhibit quasi-periodic motion or strong resonance between the first and second resonant frequencies. It can be inferred that quasi-periodic motion may be a predictor of periodic steady-state response or strong resonance. And the displacement evolution before quasi-periodic motion may be used to distinguish the two phenomena.

The three units independently performed time and frequency response analysis as well as the analysis of system energy change, and some interesting results are found: (i) In the protected object unit, the larger the mass, the greater the frequency domain response, but the system energy is maintained at a relatively low level and (ii) in the NES structure, the change of nonlinear stiffness has a greater impact on the system time domain response and energy change than the change of the mass.

Although the NES parameters of the system have little effect on the displacement transmissibility characteristics, they have a greater influence on the dynamic behavior evolving over time of the system and the system energy dissipation.

Compared with the linear system, the proposed nonlinear geometric structure has more excellent dynamic characteristics in more wide frequency bands.

Finally, the dynamic characteristics of the bio-inspired vibration isolation platform under random excitation might be useful and meaningful, and it can be further studied. Moreover, for the characteristics of the system in the time and frequency domains, some potential design issues can be further research too.

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

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was financially supported by the National Natural Science Foundation of China (Grant No. 51305288,51805347), Natural Science Foundation of Shanxi Province (Grant No. 201901D111245, 201901D111238, 201901D211293, 201801D121171), the Fund for Shanxi “1331 Project” Key Subjects Construction (Grant No.1331KSC), and Shanxi Science and Technology Foundation Platform Construction Projects (Grant No. 201805D121005).

ORCID iD

Chong Zhang

Appendix A

μ_{0} = 0

μ_{1} = η_{2}^{2} tan {(α)}^{2}

μ_{2} = \frac{3 sec {(α)}^{3} η_{2}^{3} tan (α)}{4 L_{2}}

μ_{3} = \frac{(3 - 2 cos (2 α)) sec {(α)}^{6} η_{2}^{4}}{8 L_{2}^{2}}

μ_{4} = - \frac{5 (- 5 + 2 cos (2 α)) sec {(α)}^{7} η_{2}^{5} tan (α)}{64 L_{2}^{3}}

ν_{0} = - \frac{sin {(α)}^{2} η_{2}^{2}}{- 1 + sin {(α)}^{2}}

ν_{1} = \frac{sin (α) η_{2}^{3}}{{(- 1 + sin {(α)}^{2})}^{2} L_{2}}

ν_{2} = - \frac{(1 + 3 sin {(α)}^{2}) η_{2}^{4}}{4 {(- 1 + sin {(α)}^{2})}^{3} L_{2}^{2}}

ν_{3} = \frac{sin (α) (1 + sin {(α)}^{2}) η_{2}^{5}}{2 {(- 1 + sin {(α)}^{2})}^{4} L_{2}^{3}}

ν_{4} = - \frac{(1 + 10 sin {(α)}^{2} + 5 sin {(α)}^{4}) η_{2}^{6}}{16 {(- 1 + sin {(α)}^{2})}^{5} L_{2}^{4}}

Appendix B

\begin{array}{l} - r_{1} - \frac{3}{2} a_{2}^{2} r_{2} β_{n} - \frac{3}{2} b_{2}^{2} r_{2} β_{n} - r_{2}^{3} β_{n} + r_{0} β_{V} + r_{0} β_{h} ρ_{1} \\ + \frac{1}{2} a_{0}^{2} β_{h} ρ_{2} + \frac{1}{2} b_{0}^{2} β_{h} ρ_{2} + r_{0}^{2} β_{h} ρ_{2} + \frac{3}{2} a_{0}^{2} r_{0} β_{h} ρ_{3} + \frac{3}{2} b_{0}^{2} r_{0} β_{h} ρ_{3} + r_{0}^{3} β_{h} ρ_{3} \\ + \frac{3}{8} a_{0}^{4} β_{h} ρ_{4} + \frac{3}{4} a_{0}^{2} b_{0}^{2} β_{h} ρ_{4} + \frac{3}{8} b_{0}^{4} β_{h} ρ_{4} + 3 a_{0}^{2} r_{0}^{2} β_{h} ρ_{4} + 3 b_{0}^{2} r_{0}^{2} β_{h} ρ_{4} + r_{0}^{4} β_{h} ρ_{4} = 0 \end{array}

\begin{array}{l} - a_{1} - \frac{3}{4} a_{2}^{3} β_{n} - \frac{3}{4} a_{2}^{3} b_{2}^{2} β_{n} - 3 a_{2} r_{2}^{2} β_{n} + a_{0} β_{V} - Ω^{2} a_{0} ε_{0} + a_{0} β_{h} ρ_{1} + 2 a_{0} r_{0} β_{h} ρ_{2} + \frac{3}{4} a_{2} β_{h} ρ_{3} \\ + \frac{3}{4} a_{0} b_{0}^{2} β_{h} ρ_{3} + 3 a_{0} r_{0}^{2} β_{h} ρ_{3} + 3 a_{0}^{3} r_{0} β_{h} ρ_{4} + 3 a_{0} b_{0}^{2} r_{0} β_{h} ρ_{4} + 4 a_{0} r_{0}^{3} β_{h} ρ_{4} - Ω b_{1} ς_{1} - Ω b_{2} ς_{2} + Ω b_{0} δ_{0} ς_{h} \\ + Ω b_{0} r_{0} δ_{1} ς_{h} + \frac{1}{4} Ω a_{0}^{2} b_{0} δ_{2} ς_{h} + \frac{1}{4} Ω b_{0}^{3} δ_{2} ς_{h} + Ω b_{0} r_{0}^{2} δ_{2} ς_{h} + \frac{3}{4} Ω a_{0}^{2} b_{0} r_{0} δ_{3} ς_{h} + \frac{3}{4} Ω b_{0}^{3} r_{0} δ_{3} ς_{h} + Ω b_{0} r_{0}^{3} δ_{3} ς_{h} \\ + \frac{1}{8} Ω a_{0}^{4} b_{0} δ_{4} ς_{h} + \frac{1}{4} Ω a_{0}^{2} b_{0}^{3} δ_{4} ς_{h} + \frac{1}{8} Ω b_{0}^{5} δ_{4} ς_{h} + \frac{3}{2} Ω a_{0}^{2} b_{0} r_{0}^{2} δ_{4} ς_{h} + \frac{3}{2} Ω b_{0}^{3} r_{0}^{2} δ_{4} ς_{h} + Ω b_{0} r_{0}^{4} δ_{4} ς_{h} + Ω b_{0} ς_{V} = 0 \end{array}

\begin{array}{l} - b_{1} - \frac{3}{4} a_{2}^{2} b_{2} β_{n} - \frac{3}{4} b_{2}^{3} β_{n} - 3 b_{2} r_{2}^{2} β_{n} + b_{0} β_{V} - F Ω^{2} ε_{0} - Ω^{2} b_{0} ε_{0} + b_{0} β_{h} ρ_{1} + 2 b_{0} r_{0} β_{h} ρ_{2} \\ + \frac{3}{4} a_{0}^{2} b_{0} β_{h} ρ_{3} + \frac{3}{4} b_{0}^{3} β_{h} ρ_{3} + 3 b_{0} r_{0}^{2} β_{h} ρ_{3} + 3 a_{0}^{2} b_{0} r_{0} β_{h} ρ_{4} + 3 b_{0}^{3} r_{0} β_{h} ρ_{4} + 4 b_{0} r_{0}^{3} β_{h} ρ_{4} + Ω a_{1} ς_{1} + Ω a_{2} ς_{2} - Ω a_{0} δ_{0} ς_{h} \\ - Ω a_{0} r_{0} δ_{1} ς_{h} - \frac{1}{4} Ω a_{0}^{3} δ_{2} ς_{h} - \frac{1}{4} Ω a_{0} b_{0}^{2} δ_{2} ς_{h} - Ω a_{0} r_{0}^{2} δ_{2} ς_{h} - \frac{3}{4} Ω a_{0}^{3} r_{0} δ_{3} ς_{h} - \frac{3}{4} Ω a_{0} b_{0}^{2} r_{0} δ_{3} ς_{h} - Ω b_{0} r_{0}^{3} δ_{3} ς_{h} \\ - \frac{1}{8} Ω a_{0}^{5} δ_{4} ς_{h} - \frac{1}{4} Ω a_{0}^{3} b_{0}^{2} δ_{4} ς_{h} - \frac{1}{8} Ω a_{0} b_{0}^{4} δ_{4} ς_{h} - \frac{3}{2} Ω a_{0}^{3} r_{0}^{2} δ_{4} ς_{h} - \frac{3}{2} Ω a_{0} b_{0}^{2} r_{0}^{2} δ_{4} ς_{h} - Ω a_{0} r_{0}^{4} δ_{4} ς_{h} - Ω a_{0} ς_{V} = 0 \end{array}

\begin{array}{l} r_{1} + \frac{3}{2} a_{2}^{2} r_{2} β_{n} + \frac{3}{2} b_{2}^{2} r_{2} β_{n} + r_{2}^{3} β_{n} - r_{0} β_{V} + r_{1} ε_{0} - r_{0} β_{h} ρ_{1} \\ - \frac{1}{2} a_{0}^{2} β_{h} ρ_{2} - \frac{1}{2} b_{0}^{2} β_{h} ρ_{2} - r_{0}^{2} β_{h} ρ_{2} - \frac{3}{2} a_{0}^{2} r_{0} β_{h} ρ_{3} - \frac{3}{2} b_{0}^{2} r_{0} β_{h} ρ_{3} - r_{0}^{3} β_{h} ρ_{3} \\ - \frac{3}{8} a_{0}^{4} β_{h} ρ_{4} - \frac{3}{4} a_{0}^{2} b_{0}^{2} β_{h} ρ_{4} - \frac{3}{8} b_{0}^{4} β_{h} ρ_{4} - 3 a_{0}^{2} r_{0}^{2} β_{h} ρ_{4} - 3 b_{0}^{2} r_{0}^{2} β_{h} ρ_{4} - r_{0}^{4} β_{h} ρ_{4} = 0 \end{array}

\begin{array}{l} a_{1} + \frac{3}{4} a_{2}^{3} β_{n} + \frac{3}{4} a_{2} b_{2}^{2} β_{n} + 3 a_{2} r_{2}^{2} β_{n} - a_{0} β_{V} + a_{1} ε_{0} - Ω^{2} a_{1} ε_{0} - a_{0} β_{h} ρ_{1} - 2 a_{0} r_{0} β_{h} ρ_{2} - \frac{3}{4} a_{0}^{3} β_{h} ρ_{3} - \frac{3}{4} a_{0} b_{0}^{2} β_{h} ρ_{3} - 3 a_{0} r_{0}^{2} β_{h} ρ_{3} \\ - 3 a_{0}^{3} r_{0} β_{h} ρ_{4} - 3 a_{0} b_{0}^{2} r_{0} β_{h} ρ_{4} - 4 a_{0} r_{0}^{3} β_{h} ρ_{4} + Ω b_{1} ς_{1} + Ω b_{1} ε_{0} ς_{1} + Ω b_{2} ς_{2} - Ω b_{0} δ_{0} ς_{h} - Ω b_{0} r_{0} δ_{1} ς_{h} - \frac{1}{4} Ω a_{0}^{2} b_{0} δ_{2} ς_{h} - \frac{1}{4} Ω b_{0}^{3} δ_{2} ς_{h} \\ - Ω b_{0} r_{0}^{2} δ_{2} ς_{h} - \frac{3}{4} Ω a_{0}^{2} b_{0} r_{0} δ_{3} ς_{h} - \frac{3}{4} Ω b_{0}^{3} r_{0} δ_{3} ς_{h} - Ω b_{0} r_{0}^{3} δ_{3} ς_{h} - \frac{1}{8} Ω a_{0}^{4} b_{0} δ_{4} ς_{h} - \frac{1}{4} Ω a_{0}^{2} b_{0}^{3} δ_{4} ς_{h} - \frac{1}{8} Ω a_{0}^{5} δ_{4} ς_{h} \\ - \frac{3}{2} Ω a_{0}^{2} b_{0} r_{0}^{2} δ_{4} ς_{h} - \frac{3}{2} Ω b_{0}^{3} r_{o}^{2} δ_{4} ς_{h} - Ω b_{0} r_{0}^{4} δ_{4} ς_{h} - Ω b_{0} ς_{V} = 0 \end{array}

\begin{array}{l} b_{1} + \frac{3}{4} a_{2}^{2} b_{2} β_{n} + \frac{3}{4} b_{2}^{3} β_{n} + 3 b_{2} r_{2}^{2} β_{n} - b_{0} β_{V} + b_{1} ε_{0} - Ω^{2} b_{1} ε_{0} - b_{0} β_{h} ρ_{1} - 2 b_{0} r_{0} β_{h} ρ_{2} - \frac{3}{4} a_{0}^{2} b_{0} β_{h} ρ_{3} - \frac{3}{4} b_{0}^{3} β_{h} ρ_{3} - 3 b_{0} r_{0}^{2} β_{h} ρ_{3} \\ - 3 a_{0}^{2} b_{0} r_{0} β_{h} ρ_{4} - 3 b_{0}^{3} r_{0} β_{h} ρ_{4} - 4 b_{0} r_{0}^{3} β_{h} ρ_{4} - Ω a_{1} ς_{1} - Ω a_{1} ε_{0} ς_{1} - Ω a_{2} ς_{2} + Ω a_{0} δ_{0} ς_{h} + Ω a_{0} r_{0} δ_{1} ς_{h} + \frac{1}{4} Ω a_{0}^{3} δ_{2} ς_{h} + \frac{1}{4} Ω a_{0} b_{0}^{2} δ_{2} ς_{h} \\ + Ω a_{0} r_{0}^{2} δ_{2} ς_{h} + \frac{3}{4} Ω a_{0}^{3} r_{0} δ_{3} ς_{h} + \frac{3}{4} Ω a_{0} b_{0}^{2} r_{0} δ_{3} ς_{h} + Ω a_{0} r_{0}^{3} δ_{3} ς_{h} + \frac{1}{8} Ω a_{0}^{5} δ_{4} ς_{h} + \frac{1}{4} Ω a_{0}^{3} b_{0}^{2} δ_{4} ς_{h} + \frac{1}{8} Ω a_{0} b_{0}^{4} δ_{4} ς_{h} \\ + \frac{3}{2} Ω a_{0}^{3} r_{0}^{2} δ_{4} ς_{h} + \frac{3}{2} Ω a_{0} b_{0}^{2} r_{o}^{2} δ_{4} ς_{h} + Ω a_{0} r_{0}^{4} δ_{4} ς_{h} - Ω a_{0} ς_{V} = 0 \end{array}

\begin{array}{l} \frac{3}{2} a_{2}^{2} r_{2} β_{n} ε_{0} + \frac{3}{2} b_{2}^{2} r_{2} β_{n} ε_{0} + r_{2}^{3} β_{n} ε_{0} + r_{1} ε_{2} + \frac{3}{2} a_{2}^{2} r_{2} β_{n} ε_{2} + \frac{3}{2} b_{2}^{2} r_{2} β_{n} ε_{2} + r_{2}^{3} β_{n} ε_{2} - r_{0} β_{V} ε_{2} - r_{0} β_{h} ε_{2} ρ_{1} - \frac{1}{2} a_{0}^{2} β_{h} ε_{2} ρ_{2} - \frac{1}{2} b_{0}^{2} β_{h} ε_{2} ρ_{2} \\ - r_{0}^{2} β_{h} ε_{2} ρ_{2} - \frac{3}{2} a_{0}^{2} r_{0} β_{h} ε_{2} ρ_{3} - \frac{3}{2} b_{0}^{2} r_{0} β_{h} ε_{2} ρ_{3} - r_{0}^{3} β_{h} ε_{2} ρ_{3} - \frac{3}{8} a_{0}^{4} β_{h} ε_{2} ρ_{4} - \frac{3}{4} a_{0}^{2} b_{0}^{2} β_{h} ε_{2} ρ_{4} - \frac{3}{8} b_{0}^{4} β_{h} ε_{2} ρ_{4} - 3 a_{0}^{2} r_{0}^{2} β_{h} ε_{2} ρ_{4} - 3 b_{0}^{2} r_{0}^{2} β_{h} ε_{2} ρ_{4} \\ - r_{0}^{4} β_{h} ε_{2} ρ_{4} = 0 \end{array}

\begin{array}{l} \frac{3}{4} a_{2}^{3} β_{n} ε_{0} + \frac{3}{4} a_{2} b_{2}^{2} β_{n} ε_{0} + 3 a_{2} r_{2}^{2} β_{n} ε_{0} + a_{1} ε_{2} + \frac{3}{4} a_{2}^{3} β_{n} ε_{2} + \frac{3}{4} a_{2} b_{2}^{2} β_{n} ε_{2} + 3 a_{2} r_{2}^{2} β_{n} ε_{2} - a_{0} β_{V} ε_{2} - Ω^{2} a_{2} ε_{0} ε_{2} - a_{0} β_{h} ε_{2} ρ_{1} \\ - 2 a_{0} r_{0}^{} β_{h} ε_{2} ρ_{2} - \frac{3}{4} a_{0}^{3} β_{h} ε_{2} ρ_{3} - \frac{3}{4} a_{0} b_{0}^{2} β_{h} ε_{2} ρ_{3} - 3 a_{0} r_{0}^{2} β_{h} ε_{2} ρ_{3} - 3 a_{0}^{3} r_{o} β_{h} ε_{2} ρ_{4} - 3 a_{0} b_{0}^{2} r_{0} β_{h} ε_{2} ρ_{4} - 4 a_{0} r_{0}^{3} β_{h} ε_{2} ρ_{4} + Ω b_{1} ε_{2} ς_{2} + Ω b_{2} ε_{0} ς_{2} \\ + Ω b_{2} ε_{2} ς_{2} - Ω b_{0} δ_{0} ε_{2} ς_{h} - Ω b_{0} r_{0} δ_{1} ε_{2} ς_{h} - \frac{1}{4} Ω a_{0}^{2} b_{0} δ_{2} ε_{2} ς_{h} - \frac{1}{4} Ω b_{0}^{3} δ_{2} ε_{2} ς_{h} - Ω b_{0} r_{0}^{2} δ_{2} ε_{2} ς_{h} - \frac{3}{4} Ω a_{0}^{2} b_{0} r_{0} δ_{3} ε_{2} ς_{h} - \frac{3}{4} Ω b_{0}^{3} r_{0} δ_{3} ε_{2} ς_{h} \\ - Ω b_{0} r_{0}^{3} δ_{3} ε_{2} ς_{h} - \frac{1}{8} Ω a_{0}^{4} b_{0} δ_{4} ε_{2} ς_{h} - - \frac{1}{4} Ω a_{0}^{2} b_{0}^{3} δ_{4} ε_{2} ς_{h} - \frac{1}{8} Ω b_{0}^{5} δ_{4} ε_{2} ς_{h} - \frac{3}{2} Ω a_{0}^{2} b_{0} r_{0}^{2} δ_{4} ε_{2} ς_{h} - \frac{3}{2} Ω b_{0}^{3} r_{0}^{2} δ_{4} ε_{2} ς_{h} - Ω b_{0} r_{0}^{4} δ_{4} ε_{2} ς_{h} - Ω b_{0} ε_{2} ς_{V} = 0 \end{array}

\begin{array}{l} \frac{3}{4} a_{2}^{2} b_{2} β_{n} ε_{0} + \frac{3}{4} b_{2}^{3} β_{n} ε_{0} + 3 b_{2} r_{2}^{2} β_{n} ε_{0} + b_{1} ε_{2} + \frac{3}{4} a_{2}^{2} b_{2} β_{n} ε_{2} + \frac{3}{4} b_{2}^{3} β_{n} ε_{2} + 3 b_{2} r_{2}^{2} β_{n} ε_{2} - b_{0} β_{V} ε_{2} - Ω^{2} b_{2} ε_{0} ε_{2} - b_{0} β_{h} ε_{2} ρ_{1} \\ - 2 b_{0} r_{0} β_{h} ε_{2} ρ_{2} - \frac{3}{4} a_{0}^{2} b_{0} β_{h} ε_{2} ρ_{3} - \frac{3}{4} b_{0}^{3} β_{h} ε_{2} ρ_{3} - 3 b_{0} r_{0}^{2} β_{h} ε_{2} ρ_{3} - 3 a_{0}^{2} b_{0} r_{0} β_{h} ε_{2} ρ_{4} - 3 b_{0}^{3} r_{0} β_{h} ε_{2} ρ_{4} - 4 b_{0} r_{0}^{3} β_{h} ε_{2} ρ_{4} - Ω a_{1} ε_{2} ς_{1} - Ω a_{2} ε_{0} ς_{2} \\ - Ω a_{2} ε_{2} ς_{2} + Ω a_{0} δ_{0} ε_{2} ς_{h} + Ω a_{0} r_{0} δ_{1} ε_{2} ς_{h} + \frac{1}{4} Ω a_{0}^{3} δ_{2} ε_{2} ς_{h} + \frac{1}{4} Ω a_{0} b_{0}^{2} δ_{2} ε_{2} ς_{h} + Ω a_{0} r_{0}^{2} δ_{2} ε_{2} ς_{h} + \frac{3}{4} Ω a_{0}^{3} b_{0} r_{0} δ_{3} ε_{2} ς_{h} + \frac{3}{4} Ω a_{0} b_{0}^{2} r_{0} δ_{3} ε_{2} ς_{h} \\ + Ω a_{0} r_{0}^{3} δ_{3} ε_{2} ς_{h} + \frac{1}{8} Ω a_{0} b_{0}^{5} b_{0} δ_{4} ε_{2} ς_{h} + \frac{1}{4} Ω a_{0}^{3} r_{0}^{2} δ_{4} ε_{2} ς_{h} + \frac{1}{8} Ω a_{0} b_{0}^{4} δ_{4} ε_{2} ς_{h} + \frac{3}{2} Ω a_{0}^{3} b_{0} r_{0}^{2} δ_{4} ε_{2} ς_{h} + \frac{3}{2} Ω a_{0} b_{0}^{4} r_{0}^{2} δ_{4} ε_{2} ς_{h} + Ω a_{0} r_{0}^{4} δ_{4} ε_{2} ς_{h} + Ω a_{0} ε_{2} ς_{V} = 0 \end{array}

Appendix C

l_{0} = l_{1} = 0

l_{2} = \frac{1}{4} η_{2}^{2} tan {(α)}^{2}

l_{3} = \frac{1}{8} sec {(α)}^{3} η_{2}^{3} tan (α)

l_{4} = \frac{1}{64} sec {(α)}^{6} (1 + 4 sin {(α)}^{2}) η_{2}^{4}

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