# Parametric Vibration Analysis of a Six-Degree-of-Freedom Electro-Hydraulic Stewart Platform

Authors: Xiaoming Yuan; Yue Tang; Weiqi Wang; Lijie Zhang
Journal: Shock and Vibration
 (2021)

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## Abstract

Electro-hydraulic Stewart 6-DOF platform is a 6-DOF parallel mechanism combined with the electro-hydraulic servo control system, which is widely used in the field of construction machinery. In actual working conditions, the flow and pressure pulsation of the hydraulic oil output from the hydraulic leg of the electro-hydraulic Stewart platform are inevitable, so the equivalent stiffness of the platform leg will change, and the stiffness parameters of the transmission system will change, resulting in vibration, which will affect the accuracy of the platform. This paper considering the fluid unit equivalent stiffness cyclical fluctuations and leg, on the basis of the relationship between hydraulic stiffness, constructs the electric hydraulic Stewart platform machine vibration dynamics equation, fluid coupling parameters of vibration parameters using the method of the multiscale approximate analytic formula of the main resonance and combination resonance are derived, and the system parameters vibration time-domain response and frequency response under two different poses are discussed. Results show that the system first to six order natural frequency and the first to the sixth order natural frequency and frequency of hydraulic oil equivalent stiffness of the combination of frequency will have an effect on the parameters of the system vibration. In the main resonance, the dominant frequency is mainly the first to sixth order natural frequency of the system; in the combined resonance, the dominant frequency is the combined frequency. Through the parameter vibration analysis of two different positions of the platform, it is concluded that when the platform is in an asymmetric position, each leg of the system is more involved in vibration. This study can provide the reference for the subsequent dynamic optimization and reliability analysis of the electro-hydraulic Stewart platform.


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## Body

## 1. Introduction

The Stewart 6-DOF platform is a typical parallel mechanism, which connects the base and the moving platform through six kinematic chains and controls the movement of the platform. Because its design and research involve a series of high and new technology fields such as machinery [1], hydraulic [2–4], control [5], computer, signal [6], and sensor [7], it has integrated the knowledge of multiple disciplines such as electromechanical and hydraulic and has been attached great importance by the academic circles [8]. In addition, compared with the series mechanism, the platform has the advantages of stable structure and high precision [9] and is widely applied to machine tools [10], vehicles [11], medical equipment [12], sensors [13], spacecraft [14, 15], telescopes [16], and other fields. Therefore, the Stewart platform also has a strong attraction to the industry. The platform becomes an electro-hydraulic Stewart platform when the hydraulic legs of the Stewart platform are incorporated into the electro-hydraulic servo control system. The electro-hydraulic platform has strong coupling; the platform mass and the equivalent stiffness of the leg will affect the dynamic characteristics of the platform [17]. When the legs are driven by electro-hydraulic servo, the controllability of the platform can be increased. However, compared with the mechanical leg, the equivalent stiffness of the hydraulic leg oil is lower, which leads to the change of the dynamic characteristics of the platform [18–20]. Under the condition that the external excitation frequency approaches to the natural frequency of the platform, vibration and failure will be caused [21, 22]. Therefore, the analysis of platform dynamics is of great significance.In real conditions, the pressure pulsation of the hydraulic oil output by the hydraulic cylinder on the leg of the electro-hydraulic Stewart platform is inevitable, so the equivalent stiffness of the hydraulic oil will change, and the stiffness parameters of the transmission system will change, leading to the generation of vibration [23]. The vibration induced by the time-varying parameters (stiffness, damping, mass, or moment of mass inertia) of the system is called parametric vibration. The time-varying excitation appears in the system differential equation as an inhomogeneous term, which is called the external excitation. When it appears as a coefficient in the equation of motion, it is called a parametric excitation. The main resonance occurs when the natural frequency of the system is close to the excitation frequency of the external excitation. When the natural frequency of the system is close to the sum of the excitation frequency of the external excitation and the pulsation frequency of the equivalent stiffness of the fluid, the combined resonance phenomenon occurs. With the development of science, equipment in various fields have higher requirements for operating accuracy, so it is necessary to reduce the negative impact of vibration on equipment accuracy [24, 25]. Therefore, it is of great significance to study the problem of parametric vibration to reduce system vibration.For parametric vibration problems, scholars mainly use the numerical method and analytical method to solve them, among which the analytical method includes the L-P method [26] (Lindstedt–Poincare method), multiscale method [27], and many other approximate analytical solutions. The multiscale method is an improved approximate analytical method for nonlinear differential equations based on the average method. For the solution of different problems, the multiscale method has a high versatility, so it has been widely used in recent years [28, 29]. He et al. adopted the multiscale method and obtained the nonlinear natural frequency, amplitude-frequency equation, and time history of the beam under the hinged boundary condition by Galerkin approximation discretization [30]. Chen et al. studied the transverse nonlinear vibration of axially accelerated viscoelastic beams with principal parameter resonance and applied the multiscale method to the governing equation to determine the steady-state response [31].This paper considering the hydraulic oil equivalent stiffness cyclical fluctuations and leg, on the basis of relationship between hydraulic stiffness, constructs the electric hydraulic Stewart platform machine vibration dynamics equation, fluid coupling parameters of vibration parameters using the method of the multiscale approximate analytic formula of the primary resonance and combination resonance are derived, and the system parameters vibration time-domain response and frequency response under two different poses are discussed.


## 2. Dynamic Modeling of Electro-Hydraulic Stewart Platform

### 2.1. Position Analysis of Electro-Hydraulic Platform

The geometrical structure diagram of the electro-hydraulic Stewart 6-DOF platform is shown in Figure1. The electro-hydraulic Stewart 6-DOF platform consists of three parts: a top platform, a base platform, and six legs of the same size. Among them, the top platform is a moving platform, and the base platform is a fixed platform. The six supporting legs are connected with the upper and base platforms, respectively, through ball hinges and universal joints. The six legs are composed of hydraulic cylinders and sensors that provide input excitation to the legs to ensure movement of the platform.Figure 1
Schematic diagram of the 6-DOF platform structure.The motional coordinate system A-XYZ is established at the nature heart of the top platform, while the static one B-xyz is established at the nature heart of the base platform. A and B are the nature heart of the top platform and the base one, respectively. The distribution of six balls hinged on the top platform is determined by using ai (i = 1, 2, …, 6), and the six universal joints on the base platform are bi (i = 1, 2, …, 6). By using Ra and E, the rotation matrix and displacement matrix of the motional coordinate system relative to the static one can be expressed. We assumed that m is the mass of the top platform, while Ix, Iy, and Iz are, respectively, the rotation inertia of the top platform about the XYZ axis.By usingPx, Py, and Pz, the movement of the moving coordinate system A-XYZ in three directions relative to the absolute static coordinate system B-xyz can be represented. By using α, β, and γ, the rotation in three directions of the moving coordinate system A-XYZ with respect to the absolute static coordinate system B-xyz can be expressed. The above six variables determine the position of the platform.Therefore, the expression of the rotation matrixRa and displacement matrix E is as follows:(1)Ra=cγcβcγsβsα−sγcαsγsα+cγsβcαsγcβcγcα+sγsβsαsγsβcα−cγcα−sβcβsαcβcα,E=PxPyPzT,where sα=sinα and cα=cosα.The expression of the six leg positions is as follows:(2)li=Ra×Ai+E−ai,where i = 1, 2, …, 6.In this paper, the two poses’ parameter vibration is analyzed. The parameters of the two poses are as follows:Pose 1:(3)x=Px,Py,Pz,α,β,γT=0023000T.Pose 2:(4)x=Px,Py,Pz,α,β,γT=11235510T.The initial position of the platform which is shown in Figure1 is represented by pose 1, and it is symmetrical. Furthermore, the general position of the platform, which has no symmetry, is represented by pose 2, and the position parameters are all changed. Through the analysis of the dynamic characteristics of different postures, whether the change of the platform position has an effect on its parameter vibration is judged.Physical parameters of the six-degree-of-freedom electro-hydraulic platform are shown in Table1:Table 1
Physical parameters of the electro-hydraulic Stewart platform (Unit: Si).
NameSymbolValueTop platform radiusR2Thickness of top platformD0.2Quality of top platformm20Inertia of the top platform about theX axisIx20.07Inertia of the top platform about theY axisIy20.07Inertia of top platform about theZ axisIz40Universal joint at position 1b11/2R3/2R0TUniversal joint at position 2b21/2R3/2R0TUniversal joint at position 3b31/2R−3/2R0TUniversal joint at position 4b41/2R−3/2R0TUniversal joint at position 5b5−R00TUniversal joint at position 6b6−R00TBall hinge position 1a1−1/2R3/2R0TBall hinge position 2a2R00TBall hinge position 3a3R00TBall hinge position 4a4−1/2R−3/2R0TBall hinge position 5a5−1/2R−3/2R0TBall hinge position 6a6−1/2R3/2R0T


### 2.2. Dynamic Equation of Electro-Hydraulic Stewart 6-DOF Platform Transmission System

The dynamic model of the electro-hydraulic platform established in this paper adopts the following assumptions.The dynamic model of the electro-hydraulic Stewart platform adopts the following assumptions:(1)
The components of the Stewart platform are all rigid bodies, that is, the elastic deformation of other components is not considered(2)
In the machine construction of the platform, the supporting legs work as a linear spring along the longitudinal direction of the legs(3)
The leg mass is much smaller than the top platform mass, so the effect caused by the leg mass would become negligible(4)
The friction which is caused by the moving components can be neglected(5)
Ignore the processing error and assembly error of each component


#### 2.2.1. Velocity Analysis and Jacobian Matrix of Electro-Hydraulic Platform

Set the generalized coordinates of the platform to(5)x=Px,Py,Pz,α,β,γT.Thus, the velocityV and angular velocity ω of the moving platform can be expressed as(6)V=P˙=P˙xP˙yP˙z,ω=θ˙=α˙,β˙,γ˙.By assuming thatr is the vector diameter of spherical hinge points relative to the nature heart of the top platform, VS is the speed of ball hinge points, e is the unit direction vector of the leg, and v is the change rate of the length of the leg rod; thus, the expression of them is as follows:(7)ri=RaAi,(8)Vai=V+ω×ri,(9)vi=Vai⋅ei,where i = 1, 2, …, 6.Substituting (8) into (9), we can obtain(10)vi=eiTri×eiTVω.Then, the speed of the six legs is(11)v1v2v3v4v5v6=e1Tr1×e1Te2Tr2×e2Te3Tr3×e3Te4Tr4×e4Te5Tr5×e5Te6Tr6×e6TVω.The shorthand for which is(12)v=JVωT,where [J] is the Jacobian matrix.


#### 2.2.2. Dynamic Equation for Constructing Electro-Hydraulic Platform

The equivalent stiffness of oil in the six-leg hydraulic cylinder is determined byki (i = 1, 2, …, 6), damping is expressed by ci (i = 1, 2, …, 6), and output force of the leg hydraulic cylinder is fai (i = 1, 2, …, 6).The inertial force and its virtual power received by the platform are sorted and calculated, and according to the principle of virtual power, the following can be obtained:(13)δp=fa−kΔl−cl˙⋅δl˙+−mp¨⋅δp˙+−Iaω˙−ω×Iaω⋅δω=0.The Coriolis force and centrifugal force on the platform are ignored. Equation (13) can be simplified to(14)δp=fa−kJx−cJx˙TJδx˙−mp¨Iaω˙Tδx˙=0.In the formula above,x˙=p˙ωT. The components of δx˙ have no effect on each other, so the coefficients are all zero; thus, the dynamic equation can be expressed as(15)Mx¨+Cx˙+Kx=JTfa,where M=mI3×300Ia, K=JTkJ, C=JTcJ, k=diagk1k2k3k4k5k6, c=diagc1c2c3c4c5c6, and fa=fa1fa2fa3fa4fa5fa6T.


## 2.1. Position Analysis of Electro-Hydraulic Platform

The geometrical structure diagram of the electro-hydraulic Stewart 6-DOF platform is shown in Figure1. The electro-hydraulic Stewart 6-DOF platform consists of three parts: a top platform, a base platform, and six legs of the same size. Among them, the top platform is a moving platform, and the base platform is a fixed platform. The six supporting legs are connected with the upper and base platforms, respectively, through ball hinges and universal joints. The six legs are composed of hydraulic cylinders and sensors that provide input excitation to the legs to ensure movement of the platform.Figure 1
Schematic diagram of the 6-DOF platform structure.The motional coordinate system A-XYZ is established at the nature heart of the top platform, while the static one B-xyz is established at the nature heart of the base platform. A and B are the nature heart of the top platform and the base one, respectively. The distribution of six balls hinged on the top platform is determined by using ai (i = 1, 2, …, 6), and the six universal joints on the base platform are bi (i = 1, 2, …, 6). By using Ra and E, the rotation matrix and displacement matrix of the motional coordinate system relative to the static one can be expressed. We assumed that m is the mass of the top platform, while Ix, Iy, and Iz are, respectively, the rotation inertia of the top platform about the XYZ axis.By usingPx, Py, and Pz, the movement of the moving coordinate system A-XYZ in three directions relative to the absolute static coordinate system B-xyz can be represented. By using α, β, and γ, the rotation in three directions of the moving coordinate system A-XYZ with respect to the absolute static coordinate system B-xyz can be expressed. The above six variables determine the position of the platform.Therefore, the expression of the rotation matrixRa and displacement matrix E is as follows:(1)Ra=cγcβcγsβsα−sγcαsγsα+cγsβcαsγcβcγcα+sγsβsαsγsβcα−cγcα−sβcβsαcβcα,E=PxPyPzT,where sα=sinα and cα=cosα.The expression of the six leg positions is as follows:(2)li=Ra×Ai+E−ai,where i = 1, 2, …, 6.In this paper, the two poses’ parameter vibration is analyzed. The parameters of the two poses are as follows:Pose 1:(3)x=Px,Py,Pz,α,β,γT=0023000T.Pose 2:(4)x=Px,Py,Pz,α,β,γT=11235510T.The initial position of the platform which is shown in Figure1 is represented by pose 1, and it is symmetrical. Furthermore, the general position of the platform, which has no symmetry, is represented by pose 2, and the position parameters are all changed. Through the analysis of the dynamic characteristics of different postures, whether the change of the platform position has an effect on its parameter vibration is judged.Physical parameters of the six-degree-of-freedom electro-hydraulic platform are shown in Table1:Table 1
Physical parameters of the electro-hydraulic Stewart platform (Unit: Si).
NameSymbolValueTop platform radiusR2Thickness of top platformD0.2Quality of top platformm20Inertia of the top platform about theX axisIx20.07Inertia of the top platform about theY axisIy20.07Inertia of top platform about theZ axisIz40Universal joint at position 1b11/2R3/2R0TUniversal joint at position 2b21/2R3/2R0TUniversal joint at position 3b31/2R−3/2R0TUniversal joint at position 4b41/2R−3/2R0TUniversal joint at position 5b5−R00TUniversal joint at position 6b6−R00TBall hinge position 1a1−1/2R3/2R0TBall hinge position 2a2R00TBall hinge position 3a3R00TBall hinge position 4a4−1/2R−3/2R0TBall hinge position 5a5−1/2R−3/2R0TBall hinge position 6a6−1/2R3/2R0T


## 2.2. Dynamic Equation of Electro-Hydraulic Stewart 6-DOF Platform Transmission System

The dynamic model of the electro-hydraulic platform established in this paper adopts the following assumptions.The dynamic model of the electro-hydraulic Stewart platform adopts the following assumptions:(1)
The components of the Stewart platform are all rigid bodies, that is, the elastic deformation of other components is not considered(2)
In the machine construction of the platform, the supporting legs work as a linear spring along the longitudinal direction of the legs(3)
The leg mass is much smaller than the top platform mass, so the effect caused by the leg mass would become negligible(4)
The friction which is caused by the moving components can be neglected(5)
Ignore the processing error and assembly error of each component


### 2.2.1. Velocity Analysis and Jacobian Matrix of Electro-Hydraulic Platform

Set the generalized coordinates of the platform to(5)x=Px,Py,Pz,α,β,γT.Thus, the velocityV and angular velocity ω of the moving platform can be expressed as(6)V=P˙=P˙xP˙yP˙z,ω=θ˙=α˙,β˙,γ˙.By assuming thatr is the vector diameter of spherical hinge points relative to the nature heart of the top platform, VS is the speed of ball hinge points, e is the unit direction vector of the leg, and v is the change rate of the length of the leg rod; thus, the expression of them is as follows:(7)ri=RaAi,(8)Vai=V+ω×ri,(9)vi=Vai⋅ei,where i = 1, 2, …, 6.Substituting (8) into (9), we can obtain(10)vi=eiTri×eiTVω.Then, the speed of the six legs is(11)v1v2v3v4v5v6=e1Tr1×e1Te2Tr2×e2Te3Tr3×e3Te4Tr4×e4Te5Tr5×e5Te6Tr6×e6TVω.The shorthand for which is(12)v=JVωT,where [J] is the Jacobian matrix.


### 2.2.2. Dynamic Equation for Constructing Electro-Hydraulic Platform

The equivalent stiffness of oil in the six-leg hydraulic cylinder is determined byki (i = 1, 2, …, 6), damping is expressed by ci (i = 1, 2, …, 6), and output force of the leg hydraulic cylinder is fai (i = 1, 2, …, 6).The inertial force and its virtual power received by the platform are sorted and calculated, and according to the principle of virtual power, the following can be obtained:(13)δp=fa−kΔl−cl˙⋅δl˙+−mp¨⋅δp˙+−Iaω˙−ω×Iaω⋅δω=0.The Coriolis force and centrifugal force on the platform are ignored. Equation (13) can be simplified to(14)δp=fa−kJx−cJx˙TJδx˙−mp¨Iaω˙Tδx˙=0.In the formula above,x˙=p˙ωT. The components of δx˙ have no effect on each other, so the coefficients are all zero; thus, the dynamic equation can be expressed as(15)Mx¨+Cx˙+Kx=JTfa,where M=mI3×300Ia, K=JTkJ, C=JTcJ, k=diagk1k2k3k4k5k6, c=diagc1c2c3c4c5c6, and fa=fa1fa2fa3fa4fa5fa6T.


## 2.2.1. Velocity Analysis and Jacobian Matrix of Electro-Hydraulic Platform

Set the generalized coordinates of the platform to(5)x=Px,Py,Pz,α,β,γT.Thus, the velocityV and angular velocity ω of the moving platform can be expressed as(6)V=P˙=P˙xP˙yP˙z,ω=θ˙=α˙,β˙,γ˙.By assuming thatr is the vector diameter of spherical hinge points relative to the nature heart of the top platform, VS is the speed of ball hinge points, e is the unit direction vector of the leg, and v is the change rate of the length of the leg rod; thus, the expression of them is as follows:(7)ri=RaAi,(8)Vai=V+ω×ri,(9)vi=Vai⋅ei,where i = 1, 2, …, 6.Substituting (8) into (9), we can obtain(10)vi=eiTri×eiTVω.Then, the speed of the six legs is(11)v1v2v3v4v5v6=e1Tr1×e1Te2Tr2×e2Te3Tr3×e3Te4Tr4×e4Te5Tr5×e5Te6Tr6×e6TVω.The shorthand for which is(12)v=JVωT,where [J] is the Jacobian matrix.


## 2.2.2. Dynamic Equation for Constructing Electro-Hydraulic Platform

The equivalent stiffness of oil in the six-leg hydraulic cylinder is determined byki (i = 1, 2, …, 6), damping is expressed by ci (i = 1, 2, …, 6), and output force of the leg hydraulic cylinder is fai (i = 1, 2, …, 6).The inertial force and its virtual power received by the platform are sorted and calculated, and according to the principle of virtual power, the following can be obtained:(13)δp=fa−kΔl−cl˙⋅δl˙+−mp¨⋅δp˙+−Iaω˙−ω×Iaω⋅δω=0.The Coriolis force and centrifugal force on the platform are ignored. Equation (13) can be simplified to(14)δp=fa−kJx−cJx˙TJδx˙−mp¨Iaω˙Tδx˙=0.In the formula above,x˙=p˙ωT. The components of δx˙ have no effect on each other, so the coefficients are all zero; thus, the dynamic equation can be expressed as(15)Mx¨+Cx˙+Kx=JTfa,where M=mI3×300Ia, K=JTkJ, C=JTcJ, k=diagk1k2k3k4k5k6, c=diagc1c2c3c4c5c6, and fa=fa1fa2fa3fa4fa5fa6T.


## 3. Parametric Vibration Equation of Electro-Hydraulic Stewart 6-DOF Platform Transmission System

### 3.1. Time-Varying Equivalent Stiffness of Gas-Liquid Mixture

Under actual working conditions, the pressure fluctuation of the hydraulic oil output by the hydraulic cylinder on the leg of the electro-hydraulic Stewart platform is inevitable; thus, the hydraulic oil equivalent stiffness will change. Assuming that the dynamic pressure changes in accordance with cosine, then the expression of the instantaneous pressure pulsation can be obtained as(16)p=p¯+Δpejωot+e−jωot,where p¯ represents steady-state pressure (Pa), Δp represents pressure pulsation amplitude (Pa), and ωo represents pressure pulsation angular frequency (rad/s).It is known that the hydraulic oil stiffness is a function of the hydraulic oil pressure and the initial gas content under isothermal conditions. Therefore, under a certain initial gas content, the equivalent stiffness fluctuation of hydraulic oil is similar to the pressure fluctuation of hydraulic oil and can be expressed as(17)kf=kf¯+kfvcosωet=kf¯1+εejωet+cc,where kf¯ represents steady-state fluid stiffness (N/m), kfv represents dynamic fluid stiffness amplitude (N/m), ε represents small parameter, ε=kfv/2kf¯, ωe represents angular frequency of fluid stiffness fluctuation (rad/s), and cc represents the complex conjugate.Fluid density is a function of fluid pressure, and the pressure fluctuation is bound to cause the change of fluid mass, but the change of fluid mass is very small under the steady condition. Therefore, parametric vibration analysis mainly studies the system parametric vibration response when the equivalent stiffness of fluid changes.


### 3.2. Construct Parametric Vibration Equations

Based on the lumped parameter method, the fluid pressure is uniformly distributed in the whole control body, and its variation law is consistent. Because of the pressure pulsation, the motion of the platform leg is affected by the dynamic pressure. Assuming that the dynamic pressure is in the law of cosine change and expressed as an exponential function, the system parameter vibration differential equation is shown as follows:(18)δp=ejωot+e−jωotFo−kf¯1+εejωet+ccJx−cJx˙TJδx˙−mp¨Iaω˙Tδx˙=0,whereFo represents the pressure pulsation amplitude matrix and F0=Δfa1Δfa2Δfa3Δfa4Δfa5Δfa6T.Equation (18) is expressed in the matrix form as(19)mx¨+cx˙+kx=ΔF−Δkx,where k represents the steady-state stiffness matrix, k=kf¯JTJ, ΔF represents the excitation force vector, ΔF=ejωot+e−jωotJTFo,Δk represents the incremental stiffness matrix, and Δk=kf¯εejωet+e−jωetJTJ.


## 3.1. Time-Varying Equivalent Stiffness of Gas-Liquid Mixture

Under actual working conditions, the pressure fluctuation of the hydraulic oil output by the hydraulic cylinder on the leg of the electro-hydraulic Stewart platform is inevitable; thus, the hydraulic oil equivalent stiffness will change. Assuming that the dynamic pressure changes in accordance with cosine, then the expression of the instantaneous pressure pulsation can be obtained as(16)p=p¯+Δpejωot+e−jωot,where p¯ represents steady-state pressure (Pa), Δp represents pressure pulsation amplitude (Pa), and ωo represents pressure pulsation angular frequency (rad/s).It is known that the hydraulic oil stiffness is a function of the hydraulic oil pressure and the initial gas content under isothermal conditions. Therefore, under a certain initial gas content, the equivalent stiffness fluctuation of hydraulic oil is similar to the pressure fluctuation of hydraulic oil and can be expressed as(17)kf=kf¯+kfvcosωet=kf¯1+εejωet+cc,where kf¯ represents steady-state fluid stiffness (N/m), kfv represents dynamic fluid stiffness amplitude (N/m), ε represents small parameter, ε=kfv/2kf¯, ωe represents angular frequency of fluid stiffness fluctuation (rad/s), and cc represents the complex conjugate.Fluid density is a function of fluid pressure, and the pressure fluctuation is bound to cause the change of fluid mass, but the change of fluid mass is very small under the steady condition. Therefore, parametric vibration analysis mainly studies the system parametric vibration response when the equivalent stiffness of fluid changes.


## 3.2. Construct Parametric Vibration Equations

Based on the lumped parameter method, the fluid pressure is uniformly distributed in the whole control body, and its variation law is consistent. Because of the pressure pulsation, the motion of the platform leg is affected by the dynamic pressure. Assuming that the dynamic pressure is in the law of cosine change and expressed as an exponential function, the system parameter vibration differential equation is shown as follows:(18)δp=ejωot+e−jωotFo−kf¯1+εejωet+ccJx−cJx˙TJδx˙−mp¨Iaω˙Tδx˙=0,whereFo represents the pressure pulsation amplitude matrix and F0=Δfa1Δfa2Δfa3Δfa4Δfa5Δfa6T.Equation (18) is expressed in the matrix form as(19)mx¨+cx˙+kx=ΔF−Δkx,where k represents the steady-state stiffness matrix, k=kf¯JTJ, ΔF represents the excitation force vector, ΔF=ejωot+e−jωotJTFo,Δk represents the incremental stiffness matrix, and Δk=kf¯εejωet+e−jωetJTJ.


## 4. Derivation of Approximate Analytical Formula for Parametric Vibration of Electro-Hydraulic Stewart Platform

Through the relevant knowledge of vibration mechanics, the regular modeψ and spectral matrix Λ of the system of the platform transmission system are solved, and equation (19) is regularized to obtain(20)η¨+CNη˙+Λη=ΔQ−ΔkNη,where η represents the regular generalized displacement vector, η=η1η2η3η4η5η6T, CN represents the canonical damping matrix, CN=CN11⋯CN16⋮⋱⋮CN61⋯CN66, ΔQ represents the regularly excited force vector, ΔQ=ejωot+e−jωotψTJTFo, ΔkN represents the regular stiffness increment matrix, and ΔkN=εejωet+e−jωetΔkN11⋯ΔkN16⋮⋱⋮ΔkN61⋯ΔkN66.


### 4.1. Approximate Analytical Derivation of Main Resonance of Transmission System of Electro-Hydraulic Platform

By using the multiscale method, the quadratic approximate solution, as shown in the following equation, is introduced:(21)ηit=ηi0T0,T1+εηi1T0,T1,where T0=t and T1=εt.For convenience of derivation, the following parameters are introduced into small parameters to obtain(22)cNij=εcNij′,Fo=εFo′.Substitute the above equation into equation (20), and using the same power coefficient equal, the equation of zero power term can be calculated as(23)d2η10dT02+ωn12η10=0,d2η20dT02+ωn22η20=0,d2η30dT02+ωn32η30=0,d2η40dT02+ωn42η40=0,d2η50dT02+ωn52η50=0,d2η60dT02+ωn62η60=0.The first power term is(24)d2η11dT02+ωn12η11=−2∂2η10∂T0∂T1+ejωot2∑j=16∑i=16ψi1JjiFj′−ejωet∑i=16Δk1iηi0−∑i=16CN1i′∂ηi0∂T0+cc,d2η21dT02+ωn22η21=−2∂2η20∂T0∂T1+ejωot2∑j=16∑i=16ψi2JjiFj′−ejωet∑i=16Δk2iηi0−∑i=16CN2i′∂ηi0∂T0+cc,d2η31dT02+ωn32η31=−2∂2η30∂T0∂T1+ejωot2∑j=16∑i=16ψi3JjiFj′−ejωet∑i=16Δk3iηi0−∑i=16CN3i′∂ηi0∂T0+cc,d2η41dT02+ωn42η41=−2∂2η40∂T0∂T1+ejωot2∑j=16∑i=16ψi4JjiFj′−ejωet∑i=16Δk4iηi0−∑i=16CN4i′∂ηi0∂T0+cc,d2η51dT02+ωn52η51=−2∂2η50∂T0∂T1+ejωot2∑j=16∑i=16ψi5JjiFj′−ejωet∑i=16Δk5iηi0−∑i=16CN5i′∂ηi0∂T0+cc,d2η61dT02+ωn62η61=−2∂2η60∂T0∂T1+ejωot2∑j=16∑i=16ψi6JjiFj′−ejωet∑i=16Δk6iηi0−∑i=16CN6i′∂ηi0∂T0+cc,where cc represents the complex conjugate.Let the solution of equation (24) be(25)ηi0=AiejωniT0+cc,where Ai is an unknown function, and the solution can be substituted into equation (24):(26)d2η11dT02+ωn12η11=−2jωn1dA1dT1ejωn1T0+ejωot2∑j=16∑i=16ψi1JjiFj′−∑i=16CN1i′jωniAiejωniT0−∑i=16Δk1iAiejωni+ωeT0+cc,d2η21dT02+ωn22η21=−2jωn2dA1dT1ejωn2T0+ejωot2∑j=16∑i=16ψi2JjiFj′−∑i=16CN2i′jωniAiejωniT0−∑i=16Δk2iAiejωni+ωeT0+cc,d2η31dT02+ωn32η31=−2jωn3dA1dT1ejωn3T0+ejωot2∑j=16∑i=16ψi3JjiFj′−∑i=16CN3i′jωniAiejωniT0−∑i=16Δk3iAiejωni+ωeT0+cc,d2η41dT02+ωn42η41=−2jωn4dA1dT1ejωn4T0+ejωot2∑j=16∑i=16ψi4JjiFj′−∑i=16CN4i′jωniAiejωniT0−∑i=16Δk4iAiejωni+ωeT0+cc,d2η51dT02+ωn52η51=−2jωn5dA1dT1ejωn5T0+ejωot2∑j=16∑i=16ψi5JjiFj′−∑i=16CN5i′jωniAiejωniT0−∑i=16Δk5iAiejωni+ωeT0+cc,d2η61dT02+ωn62η61=−2jωn6dA1dT1ejωn6T0+ejωot2∑j=16∑i=16ψi6JjiFj′−∑i=16CN6i′jωniAiejωniT0−∑i=16Δk6iAiejωni+ωeT0+cc.Whenωo approaches ωn1, the harmonic parameter σ is introduced, and the frequency relationship of each angle is shown in the following formula:(27)ωo=ωn1+εσ.Substitute equation (27) into equation (26) and eliminate the perpetual term:(28)−jωn12dA1dT1+cN11′A1+∑j=16∑i=16ψi1JjiFj′ejσT12=0,−jωn22dA2dT1+cN22′A2=0,−jωn32dA3dT1+cN33′A3=0,−jωn42dA4dT1+cN44′A4=0,−jωn52dA5dT1+cN55′A5=0,−jωn62dA6dT1+cN66′A6=0.The solution of equation (28) is obtained by using the constant conversion method:(29)A1=C1e−CN11′T1/2+∑j=16∑i=16ψi1JjiFJ′j2ωn12Jσ+CN11′ejσT1,A2=C2e−CN22′T1/2,A3=C3e−CN33′T1/2,A4=C4e−CN44′T1/2,A5=C5e−CN55′T1/2,A6=C6e−CN66′T1/2,where C1, …, C6 are constant.By using the relationship between trigonometric functions and complex numbers and combining with Euler’s formula, the first equation in equation (29) can be reduced to(30)A1=C1e−CN11′/2T1−∑j=16∑i=16ψi1JjiFJ′2ωn1CN11′2+4σ2ejθ+σT1,where θ=arctanCN11′/2σ.With the increase of time, equation (29) will gradually approach zero. Therefore, by substituting equation (29) into equation (25), the steady-state zeroth degree approximate analytical solution of the electro-hydraulic platform system can be obtained as(31)η10=−∑j=16∑i=16ψi1JjiFJ′ωn1CN11′2+4σ2cosθ+ωn1+εσt,η20=0,η30=0,η40=0,η50=0,η60=0.Substituting equations (29) and (30) into equation (26), the first-order approximate analytical solution of the steady-state electro-hydraulic system can be obtained as(32)η11=−2ΔkN11A1cosωn1−ωetωe2ωn1−ωe−cosωn1−ωetωe2ωn1+ωe,η21=∑j=16∑i=16ψi2JjiFJ′cosω0tωn22−ω02−2ΔkN21A1cosωn1+ωetωn22−ωn1+ωe2+cosωn1−ωetωn22−ωn1−ωe2−CN21′ωn1A1sinωn1tωn22−ωn12,η31=∑j=16∑i=16ψi3JjiFJ′cosω0tωn32−ω02−2ΔkN31A1cosωn1+ωetωn32−ωn1+ωe2+cosωn1−ωetωn32−ωn1−ωe2−CN31′ωn1A1sinωn1tωn32−ωn12,η41=∑j=16∑i=16ψi4JjiFJ′cosω0tωn42−ω02−2ΔkN41A1cosωn1+ωetωn42−ωn1+ωe2+cosωn1−ωetωn42−ωn1−ωe2−CN41′ωn1A1sinωn1tωn42−ωn12,η51=∑j=16∑i=16ψi5JjiFJ′cosω0tωn52−ω02−2ΔkN51A1cosωn1+ωetωn52−ωn1+ωe2+cosωn1−ωetωn52−ωn1−ωe2−CN51′ωn1A1sinωn1tωn52−ωn12,η61=∑j=16∑i=16ψi6JjiFJ′cosω0tωn62−ω02−2ΔkN61A1cosωn1+ωetωn62−ωn1+ωe2+cosωn1−ωetωn62−ωn1−ωe2−CN61′ωn1A1sinωn1tωn62−ωn12.Then, the steady-state response of the main resonance of the electro-hydraulic platform system in rectangular coordinates is(33)x=ψη0+εη1.In the same way, the main resonance responses can be obtained, while the external excitation frequency approaches to the second-order to the sixth-order natural frequency of the current electro-hydraulic platform system.


### 4.2. Approximate Analytical Derivation of Combined Resonance of Electro-Hydraulic Platform Transmission System

The combined resonance response of the electro-hydraulic platform transmission system is derived by the multiscale method, which can be expressed as(34)ηiT0,T1=ηi0T0,T1+εηi1T0,T1,where T0=t and T1=εt.A small parameter is introduced into the damping:(35)cNij=εcNij′.Substituting equations (34) and (35) into equation (20), the zeroth power term is sorted out and the equation is shown as follows:(36)∂2η10∂T02+ωn12η10=ejωoT02∑j=16∑i=16ψi1JjiFj+cc,∂2η20∂T02+ωn22η20=ejωoT02∑j=16∑i=16ψi2JjiFj+cc,∂2η30∂T02+ωn32η30=ejωoT02∑j=16∑i=16ψi3JjiFj+cc,∂2η40∂T02+ωn42η40=ejωoT02∑j=16∑i=16ψi4JjiFj+cc,∂2η50∂T02+ωn52η50=ejωoT02∑j=16∑i=16ψi5JjiFj+cc,∂2η60∂T02+ωn62η60=ejωoT02∑j=16∑i=16ψi6JjiFj+cc.The equation for sorting out the first power term is shown as follows:(37)∂2η11∂T02+ωn12η11=−2∂2η10∂T0∂T1−ejωet∑i=16Δk1iηi0−∑i=16CN1i′∂ηi0∂T0+cc,∂2η21∂T02+ωn22η21=−2∂2η20∂T0∂T1−ejωet∑i=16Δk2iηi0−∑i=16CN2i′∂ηi0∂T0+cc,∂2η31∂T02+ωn32η31=−2∂2η30∂T0∂T1−ejωet∑i=16Δk3iηi0−∑i=16CN3i′∂ηi0∂T0+cc,∂2η41∂T02+ωn42η41=−2∂2η40∂T0∂T1−ejωet∑i=16Δk4iηi0−∑i=16CN4i′∂ηi0∂T0+cc,∂2η51∂T02+ωn52η51=−2∂2η50∂T0∂T1−ejωet∑i=16Δk5iηi0−∑i=16CN5i′∂ηi0∂T0+cc,∂2η61∂T02+ωn62η61=−2∂2η60∂T0∂T1−ejωet∑i=16Δk6iηi0−∑i=16CN6i′∂ηi0∂T0+cc,where ccrepresents the complex conjugate.Let the analytic solution of equation (37) be as follows:(38)ηi0=BiejωniT0+Diejω0T0+cc,where Di=∑j=16∑i=16ψi1JjiFj/2ωni2−ωo2.Substitute equation (38) into equation (37) to obtain the equation as shown below:(39)d2η11dT02+ωn12η11=−2jωn1dB1dT1ejωn1T0−∑i=16CN1i′jωniBiejωniT0+jω0Diejω0T0−∑i=16Δk1iBiejωni+ωeT0+Diejω0+ωeT0+cc,d2η21dT02+ωn22η21=−2jωn2dB2dT1ejωn2T0−∑i=16CN2i′jωniBiejωniT0+jω0Diejω0T0−∑i=16Δk2iBiejωni+ωeT0+Diejω0+ωeT0+cc,d2η31dT02+ωn32η31=−2jωn3dB3dT1ejωn3T0−∑i=16CN3i′jωniBiejωniT0+jω0Diejω0T0−∑i=16Δk3iBiejωni+ωeT0+Diejω0+ωeT0+cc,d2η41dT02+ωn42η41=−2jωn4dB4dT1ejωn4T0−∑i=16CN4i′jωniBiejωniT0+jω0Diejω0T0−∑i=16Δk4iBiejωni+ωeT0+Diejω0+ωeT0+cc,d2η51dT02+ωn52η51=−2jωn5dB5dT1ejωn5T0−∑i=16CN5i′jωniBiejωniT0+jω0Diejω0T0−∑i=16Δk5iBiejωni+ωeT0+Diejω0+ωeT0+cc,d2η61dT02+ωn62η61=−2jωn6dB6dT1ejωn6T0−∑i=16CN6i′jωniBiejωniT0+jω0Diejω0T0−∑i=16Δk6iBiejωni+ωeT0+Diejω0+ωeT0+cc.By analyzing the above equation, we can see that whenωo≈ωni, the main resonance occurs in the system. In addition, when ω0≈ωni±ωe, the system will also have combinatorial resonance. Under the condition that the external excitation frequency approaches to the sum of the first-order natural frequency and the equivalent stiffness pulsation frequency of hydraulic oil, the harmonic parameters are introduced as follows:(40)ω0=ωn1+ωe+εσ.Substitute equation (40) into equation (39) and eliminate the perpetual term:(41)2jωn12dB1dT1+CN11′B1+∑i=16Δk1iDiejσT1=0,2jωn22dB2dT1+CN22′B2=0,2jωn32dB3dT1+CN33′B3=0,2jωn42dB4dT1+CN44′B4=0,2jωn52dB5dT1+CN55′B5=0,2jωn62dB6dT1+CN66′B6=0.The general solution of equation (41) is(42)B1=E1e−CN11′/2T1+∑i=16ΔkN1iDiωn14σ2+CN11′2ejθ+σT1,B2=E2e−CN22′/2T1,B3=E3e−CN33′/2T1,B4=E4e−CN44′/2T1,B5=E5e−CN55′/2T1,B6=E6e−CN66′/2T1,where E1, …, E6 are constant and θ=arctanCN11′/2σ.Similar to the principal vibration analysis, by substituting equation (42) into equation (38), the zeroth degree approximate analytical solution of the combined resonance of the 6-DOF platform system can be obtained as follows:(43)η10=2×∑i=16ΔkN1iDiωn14σ2+CN11′2cosθ+ωn1+εσt+2D1cosω0t,η20=2D2cosω0t,η30=2D3cosω0t,η40=2D4cosω0t,η50=2D5cosω0t,η60=2D6cosω0t.Substituting equation (42) into equation (39), the steady-state first-order approximate analytical solution of the combined resonance of the transmission system can be obtained as follows:(44)η11=−2ηa1+ηb1−ηc1,η21=−2ηa2+ηb2−ηc2−ωn1CN21′B1sinωn1tωn22−ωn12,η31=−2ηa3+ηb3−ηc3−ωn1CN31′B1sinωn1tωn32−ωn12,η21=−2ηa4+ηb4−ηc4−ωn1CN41′B1sinωn1tωn42−ωn12,η21=−2ηa5+ηb5−ηc5−ωn1CN51′B1sinωn1tωn52−ωn12,η21=−2ηa6+ηb6−ηc6−ωn1CN61′B1sinωn1tωn62−ωn12,where(45)ηa1=ΔkN11B1cosωn1−ωftωf2ωn1−ωf−cosωn1+ωftωf2ωn1+ωf,ηa2=ΔkN21B1cosωn1+ωftωn22−ωn1+ωf2−cosωn1−ωftωn22−ωn1−ωf2,ηa3=ΔkN31B1cosωn1+ωftωn32−ωn1+ωf2−cosωn1−ωftωn32−ωn1−ωf2,ηa4=ΔkN41B1cosωn1+ωftωn42−ωn1+ωf2−cosωn1−ωftωn42−ωn1−ωf2,ηa5=ΔkN51B1cosωn1+ωftωn52−ωn1+ωf2−cosωn1−ωftωn52−ωn1−ωf2,ηa6=ΔkN61B1cosωn1+ωftωn62−ωn1+ωf2−cosωn1−ωftωn62−ωn1−ωf2,ηb1=∑i=16ΔkN1iDicosωn1−ωftωn12−ωn1+ωf2,ηb2=∑i=16ΔkN2iDicosωn1+ωftωn22−ωn1+ωf2+cosωn1−ωftωn22−ωn1−ωf2,ηb3=∑i=16ΔkN3iDicosωn1+ωftωn32−ωn1+ωf2+cosωn1−ωftωn32−ωn1−ωf2,ηb4=∑i=16ΔkN4iDicosωn1+ωftωn42−ωn1+ωf2+cosωn1−ωftωn42−ωn1−ωf2,ηb5=∑i=16ΔkN5iDicosωn1+ωftωn52−ωn1+ωf2+cosωn1−ωftωn52−ωn1−ωf2,ηb6=∑i=16ΔkN6iDicosωn1+ωftωn62−ωn1+ωf2+cosωn1−ωftωn62−ωn1−ωf2,ηc1=ω0∑i=16CN1i′Disinω0tωn12−ω02,ηc2=ω0∑i=16CN2i′Disinω0tωn22−ω02,ηc3=ω0∑i=16CN3i′Disinω0tωn32−ω02,ηc4=ω0∑i=16CN4i′Disinω0tωn42−ω02,ηc5=ω0∑i=16CN5i′Disinω0tωn52−ω02,ηc6=ω0∑i=16CN6i′Disinω0tωn62−ω02.Then, the combined resonant steady-state response of the 6-DOF platform system in rectangular coordinates is(46)x=ψη0+εη1.Similarly, the combined resonance response can be obtained by referring to the above solution process, while the external excitation frequency is close to the sum of the second order to the sixth-order natural frequency of the 6-DOF platform system and the equivalent stiffness fluctuation frequency of the hydraulic oil unit.


## 4.1. Approximate Analytical Derivation of Main Resonance of Transmission System of Electro-Hydraulic Platform

By using the multiscale method, the quadratic approximate solution, as shown in the following equation, is introduced:(21)ηit=ηi0T0,T1+εηi1T0,T1,where T0=t and T1=εt.For convenience of derivation, the following parameters are introduced into small parameters to obtain(22)cNij=εcNij′,Fo=εFo′.Substitute the above equation into equation (20), and using the same power coefficient equal, the equation of zero power term can be calculated as(23)d2η10dT02+ωn12η10=0,d2η20dT02+ωn22η20=0,d2η30dT02+ωn32η30=0,d2η40dT02+ωn42η40=0,d2η50dT02+ωn52η50=0,d2η60dT02+ωn62η60=0.The first power term is(24)d2η11dT02+ωn12η11=−2∂2η10∂T0∂T1+ejωot2∑j=16∑i=16ψi1JjiFj′−ejωet∑i=16Δk1iηi0−∑i=16CN1i′∂ηi0∂T0+cc,d2η21dT02+ωn22η21=−2∂2η20∂T0∂T1+ejωot2∑j=16∑i=16ψi2JjiFj′−ejωet∑i=16Δk2iηi0−∑i=16CN2i′∂ηi0∂T0+cc,d2η31dT02+ωn32η31=−2∂2η30∂T0∂T1+ejωot2∑j=16∑i=16ψi3JjiFj′−ejωet∑i=16Δk3iηi0−∑i=16CN3i′∂ηi0∂T0+cc,d2η41dT02+ωn42η41=−2∂2η40∂T0∂T1+ejωot2∑j=16∑i=16ψi4JjiFj′−ejωet∑i=16Δk4iηi0−∑i=16CN4i′∂ηi0∂T0+cc,d2η51dT02+ωn52η51=−2∂2η50∂T0∂T1+ejωot2∑j=16∑i=16ψi5JjiFj′−ejωet∑i=16Δk5iηi0−∑i=16CN5i′∂ηi0∂T0+cc,d2η61dT02+ωn62η61=−2∂2η60∂T0∂T1+ejωot2∑j=16∑i=16ψi6JjiFj′−ejωet∑i=16Δk6iηi0−∑i=16CN6i′∂ηi0∂T0+cc,where cc represents the complex conjugate.Let the solution of equation (24) be(25)ηi0=AiejωniT0+cc,where Ai is an unknown function, and the solution can be substituted into equation (24):(26)d2η11dT02+ωn12η11=−2jωn1dA1dT1ejωn1T0+ejωot2∑j=16∑i=16ψi1JjiFj′−∑i=16CN1i′jωniAiejωniT0−∑i=16Δk1iAiejωni+ωeT0+cc,d2η21dT02+ωn22η21=−2jωn2dA1dT1ejωn2T0+ejωot2∑j=16∑i=16ψi2JjiFj′−∑i=16CN2i′jωniAiejωniT0−∑i=16Δk2iAiejωni+ωeT0+cc,d2η31dT02+ωn32η31=−2jωn3dA1dT1ejωn3T0+ejωot2∑j=16∑i=16ψi3JjiFj′−∑i=16CN3i′jωniAiejωniT0−∑i=16Δk3iAiejωni+ωeT0+cc,d2η41dT02+ωn42η41=−2jωn4dA1dT1ejωn4T0+ejωot2∑j=16∑i=16ψi4JjiFj′−∑i=16CN4i′jωniAiejωniT0−∑i=16Δk4iAiejωni+ωeT0+cc,d2η51dT02+ωn52η51=−2jωn5dA1dT1ejωn5T0+ejωot2∑j=16∑i=16ψi5JjiFj′−∑i=16CN5i′jωniAiejωniT0−∑i=16Δk5iAiejωni+ωeT0+cc,d2η61dT02+ωn62η61=−2jωn6dA1dT1ejωn6T0+ejωot2∑j=16∑i=16ψi6JjiFj′−∑i=16CN6i′jωniAiejωniT0−∑i=16Δk6iAiejωni+ωeT0+cc.Whenωo approaches ωn1, the harmonic parameter σ is introduced, and the frequency relationship of each angle is shown in the following formula:(27)ωo=ωn1+εσ.Substitute equation (27) into equation (26) and eliminate the perpetual term:(28)−jωn12dA1dT1+cN11′A1+∑j=16∑i=16ψi1JjiFj′ejσT12=0,−jωn22dA2dT1+cN22′A2=0,−jωn32dA3dT1+cN33′A3=0,−jωn42dA4dT1+cN44′A4=0,−jωn52dA5dT1+cN55′A5=0,−jωn62dA6dT1+cN66′A6=0.The solution of equation (28) is obtained by using the constant conversion method:(29)A1=C1e−CN11′T1/2+∑j=16∑i=16ψi1JjiFJ′j2ωn12Jσ+CN11′ejσT1,A2=C2e−CN22′T1/2,A3=C3e−CN33′T1/2,A4=C4e−CN44′T1/2,A5=C5e−CN55′T1/2,A6=C6e−CN66′T1/2,where C1, …, C6 are constant.By using the relationship between trigonometric functions and complex numbers and combining with Euler’s formula, the first equation in equation (29) can be reduced to(30)A1=C1e−CN11′/2T1−∑j=16∑i=16ψi1JjiFJ′2ωn1CN11′2+4σ2ejθ+σT1,where θ=arctanCN11′/2σ.With the increase of time, equation (29) will gradually approach zero. Therefore, by substituting equation (29) into equation (25), the steady-state zeroth degree approximate analytical solution of the electro-hydraulic platform system can be obtained as(31)η10=−∑j=16∑i=16ψi1JjiFJ′ωn1CN11′2+4σ2cosθ+ωn1+εσt,η20=0,η30=0,η40=0,η50=0,η60=0.Substituting equations (29) and (30) into equation (26), the first-order approximate analytical solution of the steady-state electro-hydraulic system can be obtained as(32)η11=−2ΔkN11A1cosωn1−ωetωe2ωn1−ωe−cosωn1−ωetωe2ωn1+ωe,η21=∑j=16∑i=16ψi2JjiFJ′cosω0tωn22−ω02−2ΔkN21A1cosωn1+ωetωn22−ωn1+ωe2+cosωn1−ωetωn22−ωn1−ωe2−CN21′ωn1A1sinωn1tωn22−ωn12,η31=∑j=16∑i=16ψi3JjiFJ′cosω0tωn32−ω02−2ΔkN31A1cosωn1+ωetωn32−ωn1+ωe2+cosωn1−ωetωn32−ωn1−ωe2−CN31′ωn1A1sinωn1tωn32−ωn12,η41=∑j=16∑i=16ψi4JjiFJ′cosω0tωn42−ω02−2ΔkN41A1cosωn1+ωetωn42−ωn1+ωe2+cosωn1−ωetωn42−ωn1−ωe2−CN41′ωn1A1sinωn1tωn42−ωn12,η51=∑j=16∑i=16ψi5JjiFJ′cosω0tωn52−ω02−2ΔkN51A1cosωn1+ωetωn52−ωn1+ωe2+cosωn1−ωetωn52−ωn1−ωe2−CN51′ωn1A1sinωn1tωn52−ωn12,η61=∑j=16∑i=16ψi6JjiFJ′cosω0tωn62−ω02−2ΔkN61A1cosωn1+ωetωn62−ωn1+ωe2+cosωn1−ωetωn62−ωn1−ωe2−CN61′ωn1A1sinωn1tωn62−ωn12.Then, the steady-state response of the main resonance of the electro-hydraulic platform system in rectangular coordinates is(33)x=ψη0+εη1.In the same way, the main resonance responses can be obtained, while the external excitation frequency approaches to the second-order to the sixth-order natural frequency of the current electro-hydraulic platform system.


## 4.2. Approximate Analytical Derivation of Combined Resonance of Electro-Hydraulic Platform Transmission System

The combined resonance response of the electro-hydraulic platform transmission system is derived by the multiscale method, which can be expressed as(34)ηiT0,T1=ηi0T0,T1+εηi1T0,T1,where T0=t and T1=εt.A small parameter is introduced into the damping:(35)cNij=εcNij′.Substituting equations (34) and (35) into equation (20), the zeroth power term is sorted out and the equation is shown as follows:(36)∂2η10∂T02+ωn12η10=ejωoT02∑j=16∑i=16ψi1JjiFj+cc,∂2η20∂T02+ωn22η20=ejωoT02∑j=16∑i=16ψi2JjiFj+cc,∂2η30∂T02+ωn32η30=ejωoT02∑j=16∑i=16ψi3JjiFj+cc,∂2η40∂T02+ωn42η40=ejωoT02∑j=16∑i=16ψi4JjiFj+cc,∂2η50∂T02+ωn52η50=ejωoT02∑j=16∑i=16ψi5JjiFj+cc,∂2η60∂T02+ωn62η60=ejωoT02∑j=16∑i=16ψi6JjiFj+cc.The equation for sorting out the first power term is shown as follows:(37)∂2η11∂T02+ωn12η11=−2∂2η10∂T0∂T1−ejωet∑i=16Δk1iηi0−∑i=16CN1i′∂ηi0∂T0+cc,∂2η21∂T02+ωn22η21=−2∂2η20∂T0∂T1−ejωet∑i=16Δk2iηi0−∑i=16CN2i′∂ηi0∂T0+cc,∂2η31∂T02+ωn32η31=−2∂2η30∂T0∂T1−ejωet∑i=16Δk3iηi0−∑i=16CN3i′∂ηi0∂T0+cc,∂2η41∂T02+ωn42η41=−2∂2η40∂T0∂T1−ejωet∑i=16Δk4iηi0−∑i=16CN4i′∂ηi0∂T0+cc,∂2η51∂T02+ωn52η51=−2∂2η50∂T0∂T1−ejωet∑i=16Δk5iηi0−∑i=16CN5i′∂ηi0∂T0+cc,∂2η61∂T02+ωn62η61=−2∂2η60∂T0∂T1−ejωet∑i=16Δk6iηi0−∑i=16CN6i′∂ηi0∂T0+cc,where ccrepresents the complex conjugate.Let the analytic solution of equation (37) be as follows:(38)ηi0=BiejωniT0+Diejω0T0+cc,where Di=∑j=16∑i=16ψi1JjiFj/2ωni2−ωo2.Substitute equation (38) into equation (37) to obtain the equation as shown below:(39)d2η11dT02+ωn12η11=−2jωn1dB1dT1ejωn1T0−∑i=16CN1i′jωniBiejωniT0+jω0Diejω0T0−∑i=16Δk1iBiejωni+ωeT0+Diejω0+ωeT0+cc,d2η21dT02+ωn22η21=−2jωn2dB2dT1ejωn2T0−∑i=16CN2i′jωniBiejωniT0+jω0Diejω0T0−∑i=16Δk2iBiejωni+ωeT0+Diejω0+ωeT0+cc,d2η31dT02+ωn32η31=−2jωn3dB3dT1ejωn3T0−∑i=16CN3i′jωniBiejωniT0+jω0Diejω0T0−∑i=16Δk3iBiejωni+ωeT0+Diejω0+ωeT0+cc,d2η41dT02+ωn42η41=−2jωn4dB4dT1ejωn4T0−∑i=16CN4i′jωniBiejωniT0+jω0Diejω0T0−∑i=16Δk4iBiejωni+ωeT0+Diejω0+ωeT0+cc,d2η51dT02+ωn52η51=−2jωn5dB5dT1ejωn5T0−∑i=16CN5i′jωniBiejωniT0+jω0Diejω0T0−∑i=16Δk5iBiejωni+ωeT0+Diejω0+ωeT0+cc,d2η61dT02+ωn62η61=−2jωn6dB6dT1ejωn6T0−∑i=16CN6i′jωniBiejωniT0+jω0Diejω0T0−∑i=16Δk6iBiejωni+ωeT0+Diejω0+ωeT0+cc.By analyzing the above equation, we can see that whenωo≈ωni, the main resonance occurs in the system. In addition, when ω0≈ωni±ωe, the system will also have combinatorial resonance. Under the condition that the external excitation frequency approaches to the sum of the first-order natural frequency and the equivalent stiffness pulsation frequency of hydraulic oil, the harmonic parameters are introduced as follows:(40)ω0=ωn1+ωe+εσ.Substitute equation (40) into equation (39) and eliminate the perpetual term:(41)2jωn12dB1dT1+CN11′B1+∑i=16Δk1iDiejσT1=0,2jωn22dB2dT1+CN22′B2=0,2jωn32dB3dT1+CN33′B3=0,2jωn42dB4dT1+CN44′B4=0,2jωn52dB5dT1+CN55′B5=0,2jωn62dB6dT1+CN66′B6=0.The general solution of equation (41) is(42)B1=E1e−CN11′/2T1+∑i=16ΔkN1iDiωn14σ2+CN11′2ejθ+σT1,B2=E2e−CN22′/2T1,B3=E3e−CN33′/2T1,B4=E4e−CN44′/2T1,B5=E5e−CN55′/2T1,B6=E6e−CN66′/2T1,where E1, …, E6 are constant and θ=arctanCN11′/2σ.Similar to the principal vibration analysis, by substituting equation (42) into equation (38), the zeroth degree approximate analytical solution of the combined resonance of the 6-DOF platform system can be obtained as follows:(43)η10=2×∑i=16ΔkN1iDiωn14σ2+CN11′2cosθ+ωn1+εσt+2D1cosω0t,η20=2D2cosω0t,η30=2D3cosω0t,η40=2D4cosω0t,η50=2D5cosω0t,η60=2D6cosω0t.Substituting equation (42) into equation (39), the steady-state first-order approximate analytical solution of the combined resonance of the transmission system can be obtained as follows:(44)η11=−2ηa1+ηb1−ηc1,η21=−2ηa2+ηb2−ηc2−ωn1CN21′B1sinωn1tωn22−ωn12,η31=−2ηa3+ηb3−ηc3−ωn1CN31′B1sinωn1tωn32−ωn12,η21=−2ηa4+ηb4−ηc4−ωn1CN41′B1sinωn1tωn42−ωn12,η21=−2ηa5+ηb5−ηc5−ωn1CN51′B1sinωn1tωn52−ωn12,η21=−2ηa6+ηb6−ηc6−ωn1CN61′B1sinωn1tωn62−ωn12,where(45)ηa1=ΔkN11B1cosωn1−ωftωf2ωn1−ωf−cosωn1+ωftωf2ωn1+ωf,ηa2=ΔkN21B1cosωn1+ωftωn22−ωn1+ωf2−cosωn1−ωftωn22−ωn1−ωf2,ηa3=ΔkN31B1cosωn1+ωftωn32−ωn1+ωf2−cosωn1−ωftωn32−ωn1−ωf2,ηa4=ΔkN41B1cosωn1+ωftωn42−ωn1+ωf2−cosωn1−ωftωn42−ωn1−ωf2,ηa5=ΔkN51B1cosωn1+ωftωn52−ωn1+ωf2−cosωn1−ωftωn52−ωn1−ωf2,ηa6=ΔkN61B1cosωn1+ωftωn62−ωn1+ωf2−cosωn1−ωftωn62−ωn1−ωf2,ηb1=∑i=16ΔkN1iDicosωn1−ωftωn12−ωn1+ωf2,ηb2=∑i=16ΔkN2iDicosωn1+ωftωn22−ωn1+ωf2+cosωn1−ωftωn22−ωn1−ωf2,ηb3=∑i=16ΔkN3iDicosωn1+ωftωn32−ωn1+ωf2+cosωn1−ωftωn32−ωn1−ωf2,ηb4=∑i=16ΔkN4iDicosωn1+ωftωn42−ωn1+ωf2+cosωn1−ωftωn42−ωn1−ωf2,ηb5=∑i=16ΔkN5iDicosωn1+ωftωn52−ωn1+ωf2+cosωn1−ωftωn52−ωn1−ωf2,ηb6=∑i=16ΔkN6iDicosωn1+ωftωn62−ωn1+ωf2+cosωn1−ωftωn62−ωn1−ωf2,ηc1=ω0∑i=16CN1i′Disinω0tωn12−ω02,ηc2=ω0∑i=16CN2i′Disinω0tωn22−ω02,ηc3=ω0∑i=16CN3i′Disinω0tωn32−ω02,ηc4=ω0∑i=16CN4i′Disinω0tωn42−ω02,ηc5=ω0∑i=16CN5i′Disinω0tωn52−ω02,ηc6=ω0∑i=16CN6i′Disinω0tωn62−ω02.Then, the combined resonant steady-state response of the 6-DOF platform system in rectangular coordinates is(46)x=ψη0+εη1.Similarly, the combined resonance response can be obtained by referring to the above solution process, while the external excitation frequency is close to the sum of the second order to the sixth-order natural frequency of the 6-DOF platform system and the equivalent stiffness fluctuation frequency of the hydraulic oil unit.


## 5. Parametric Vibration Response Analysis of an Electro-Hydraulic Stewart Platform

### 5.1. Vibration Response Analysis of System Parameters in Pose 1

Position and pose 1 parameter:(47)x=Px,Py,Pz,α,β,γT=0023000T.When the platform is in the position of pose 1, the vibration response of the system parameter is analyzed.Parameter vibration-related calculation parameters of the 6-DOF platform transmission system are shown in Table2. Substituting the platform mass, leg stiffness, and parameters shown in Table 1 into equation (20), the parametric vibration equation of the 6-DOF platform transmission system is obtained.Table 2
Calculation parameters related to vibration of 6-DOF platform transmission system parameters.
NameValueSteady-state value of the first-order natural frequencyω1 (rad·s−1)128.4811Steady-state value of the second-order natural frequencyω2 (rad·s−1)128.4811Steady-state value of the third-order natural frequencyω3 (rad·s−1)259.8076Steady-state value of the fourth-order natural frequencyω4 (rad·s−1)367.4235Steady-state value of the fifth-order natural frequencyω5 (rad·s−1)524.4956Steady-state value of the sixth-order natural frequencyω6 (rad·s−1)524.4956Fluctuation frequency of equivalent stiffness of fluid elementωe (rad·s−1)40.7150Amplitude of external excitationF0 (N)[20 20 20 20 20 20]TStructural dampingC/N·(m/s)[20 20 20 20 20 20]T


#### 5.1.1. Main Resonance Response Analysis

The time-domain and frequency-domain responses of the main resonance when the external excitation frequency approaches to the stable value of the first-order to the sixth-order natural angular frequency of the electro-hydraulic platform transmission system are shown in Figures2–7. As can be seen from Figures 2(a)–7(a), while the external excitation frequency approaches to the steady-state value of the natural frequency of the first to sixth order of the transmission system, respectively, the mode corresponding to the natural frequency of the system will have a relatively intense main resonance phenomenon, and its maximum displacement is 0.00002712 mm. As can be seen from Figures 2(b)–7(b), when the main resonance is generated in the system, there is a spike in the frequency-domain image, which corresponds to the external excitation frequency close to the natural frequency. This indicates that it plays a leading role in the main resonance of the system. In this case, the dominant frequency is the sum of the natural frequencies of each order and the fluctuation frequency of the equivalent stiffness of hydraulic oil, and other natural frequencies also reach their maximum amplitudes at the dominant frequency.Figure 2
Main vibration response of the electro-hydraulic Stewart platform transmission system atωo≈ω1. (a) ωo≈ω1 time-domain response. (b) ωo≈ω1 frequency-domain response.
(a)(b)Figure 3
Main vibration response of the electro-hydraulic Stewart platform transmission system atωo≈ω2. (a) ωo≈ω2 time-domain response. (b) ωo≈ω2 frequency-domain response.
(a)(b)Figure 4
Main vibration response of the electro-hydraulic Stewart platform transmission system atωo≈ω3. (a) ωo≈ω3 time-domain response. (b) ωo≈ω3 frequency-domain response.
(a)(b)Figure 5
Main vibration response of the electro-hydraulic Stewart platform transmission system atωo≈ω4. (a) ωo≈ω4 time-domain response. (b) ωo≈ω4 frequency-domain response.
(a)(b)Figure 6
Main vibration response of the electro-hydraulic Stewart platform transmission system atωo≈ω5. (a) ωo≈ω5 time-domain response. (b) ωo≈ω5 frequency-domain response.
(a)(b)Figure 7
Main vibration response of the electro-hydraulic Stewart platform transmission system atωo≈ω6. (a) ωo≈ω6 time-domain response. (b) ωo≈ω6 frequency-domain response.
(a)(b)


#### 5.1.2. Combined Resonance Response

While the external excitation frequency approaches to the combined frequency of the stable value of the first- to sixth-order natural angular frequency of the electro-hydraulic platform transmission system and the fluctuation angular frequency of the equivalent stiffness of hydraulic oil, the combined resonance time-domain response and frequency-domain response are shown in Figures8–13(a). As can be seen from Figures 8(a)–13(a), the combined resonance phenomenon occurs in the system, but its amplitude is larger than that of the system when the main resonance occurs, and the maximum displacement is 0.09427 mm. According to Figures 8(b)–13(b), when the system produces combined resonance, there are three spikes in the frequency-domain image. Among them, the combined frequency ωo ≈ ωi + ωe, which is the stable value of the natural angular frequency of each order of the system and the fluctuation angular frequency of the equivalent stiffness of hydraulic oil, has the highest peak, indicating that it plays a leading role in the combined resonance of the system, while the other frequencies play a regulating role. There is a lower peak on the left and right side of the peak corresponding to the combined frequencies, with the left corresponding frequency ωo ≈ ωi and the right corresponding frequency ωo ≈ ωi + 2ωe (i = 1, 2, …, 6).Figure 8
Combined vibration response of the electro-hydraulic Stewart platform transmission system atωo ≈ ω1 + ωe. (a) ωo ≈ ω1 + ωe time-domain response. (b) ωo ≈ ω1 + ωe frequency-domain response.
(a)(b)Figure 9
Combined vibration response of the electro-hydraulic Stewart platform transmission system atωo ≈ ω2 + ωe. (a) ωo ≈ ω2 + ωe time-domain response. (b) ωo ≈ ω2 + ωe frequency-domain response.
(a)(b)Figure 10
Combined vibration response of the electro-hydraulic Stewart platform transmission system atωo ≈ ω3 + ωe. (a) ωo ≈ ω3 + ωe time-domain response. (b) ωo ≈ ω3 + ωe frequency-domain response.
(a)(b)Figure 11
Combined vibration response of the electro-hydraulic Stewart platform transmission system atωo ≈ ω4 + ωe. (a) ωo ≈ ω4 + ωe time-domain response. (b) ωo ≈ ω4 + ωe frequency-domain response.
(a)(b)Figure 12
Combined vibration response of the electro-hydraulic Stewart platform transmission system atωo ≈ ω5 + ωe. (a) ωo ≈ ω5 + ωe time-domain response. (b) ωo ≈ ω5 + ωe frequency-domain response.
(a)(b)Figure 13
Combined vibration response of the electro-hydraulic Stewart platform transmission system atωo ≈ ω6 + ωe. (a) ωo ≈ ω6 + ωe time-domain response. (b) ωo ≈ ω6 + ωe frequency-domain response.
(a)(b)


### 5.2. Vibration Response Analysis of System Parameters in Pose 2

Position and pose 2 parameter:(48)x=Px,Py,Pz,α,β,γT=11235510T.When the platform is in the position of pose 2, the parameter vibration response of the system is analyzed.Parameter vibration-related calculation parameters of the 6-DOF platform transmission system are shown in Table3. Substituting the platform mass, leg stiffness, and parameters shown in Table 1 into equation (20), the parametric vibration equation of the 6-DOF platform transmission system is obtained.Table 3
Calculation parameters related to vibration of 6-DOF platform transmission system parameters.
NameValueSteady-state value of the first-order natural angular frequencyω1 (rad·s−1)111.1024Steady-state value of the second-order natural angular frequencyω2 (rad·s−1)124.4199Steady-state value of the third-order natural angular frequencyω3 (rad·s−1)235.3889Steady-state value of the fourth-order natural angular frequencyω4 (rad·s−1)375.2415Steady-state value of the fifth-order natural angular frequencyω5 (rad·s−1)505.6999Steady-state value of the sixth-order natural angular frequencyω6 (rad·s−1)518.4187Fluctuation angular frequency of equivalent stiffness of fluid elementωe (rad·s−1)40.7150Amplitude of external excitationF0 (N)[20 20 20 20 20 20]TStructural dampingC/N·(m/s)[20 20 20 20 20 20]T


#### 5.2.1. Main Resonance Response Analysis

The time-domain and frequency-domain responses of the main resonance, while the external excitation frequency is close to the stable value of the first-order to the sixth-order natural angular frequency of the electro-hydraulic platform transmission system can be seen from Figures14–19. With the change of position and pose, the natural frequency of the system changes and the main resonance response also changes. As can be seen from Figures 14(a)–19(a), while the external excitation frequency is close to the steady-state value of the natural frequency of the first to sixth order of the transmission system, the system will experience a relatively intense primary resonance with a maximum amplitude of 0.00002923 mm. Compared with pose 1, the modal amplitudes of the system are all increased. This indicates that, in the asymmetric pose, the connection between each leg is closer, the coupling is increased, and the degree of participation in vibration is increased. As can be seen from Figures 14(b)–19(b), when the main resonance is generated in the system, there is a spike in the frequency-domain image, and the corresponding frequency of the peak corresponds to the approximate analytical formula of the main resonance of the system. Its peak corresponds to the external excitation frequency close to the natural frequency. This indicates that it plays a leading role in the main resonance of the system. In this case, the dominant frequency is the sum of the natural frequencies of each order and the fluctuation frequency of the equivalent stiffness of hydraulic oil, and other natural frequencies also reach their maximum amplitudes at the dominant frequency.Figure 14
Main vibration response of the electro-hydraulic Stewart platform transmission system atωo≈ω1. (a) ωo≈ω1 time-domain response. (b) ωo≈ω1 frequency-domain response.
(a)(b)Figure 15
Main vibration response of the electro-hydraulic Stewart platform transmission system atωo≈ω2. (a) ωo≈ω2 time-domain response. (b) ωo≈ω2 frequency-domain response.
(a)(b)Figure 16
Main vibration response of the electro-hydraulic Stewart platform transmission system atωo≈ω3. (a) ωo≈ω3 time-domain response. (b) ωo≈ω3 frequency-domain response.
(a)(b)Figure 17
Main vibration response of the electro-hydraulic Stewart platform transmission system atωo≈ω4. (a) ωo≈ω4 time-domain response. (b) ωo≈ω4 frequency-domain response.
(a)(b)Figure 18
Main vibration response of the electro-hydraulic Stewart platform transmission system atωo≈ω5. (a) ωo≈ω5 time-domain response. (b) ωo≈ω5 frequency-domain response.
(a)(b)Figure 19
Main vibration response of the electro-hydraulic Stewart platform transmission system atωo≈ω6. (a) ωo≈ω6 time-domain response. (b) ωo≈ω6 frequency-domain response.
(a)(b)


#### 5.2.2. Combined Resonance Response

As can be seen from Figures20(a)–25(a), when the external excitation frequency is close to the combination frequency of the stable value of the first to sixth order natural angular frequency of the electro-hydraulic platform transmission system and the fluctuation angular frequency of the equivalent stiffness of hydraulic oil, the combined resonance phenomenon occurs in the system and the amplitude is relatively severe. Compared with pose 1, the modal amplitudes of the system are all increased. This indicates that, in the asymmetric pose, the connection between each leg is closer, the coupling is increased, and the degree of participation in vibration is increased. This indicates that, in the asymmetric posture, the connection between each leg is more close, and the degree of participation in vibration is increased. As can be seen from Figures 20(b)–25(b), when the system produces combined resonance, there are three spikes in the frequency-domain image, and the corresponding frequencies of each spike correspond to the approximate analytical formula of the system’s main resonance one by one. Among them, the peak corresponding to the combined frequency ωo ≈ ωi + ωe of the stable value of the natural angular frequency of each order of the system and the fluctuation angular frequency of the equivalent stiffness of hydraulic oil is the highest. It shows that it plays a leading role in the combined resonance of the system and the other frequencies play a regulating role. There is a lower peak on the left and right side of the peak corresponding to the combined frequencies, with the left corresponding frequency ωo ≈ ωi and the right corresponding frequency ωo ≈ ωi + 2ωe (i = 1, 2, …, 6).Figure 20
Combined vibration response of the electro-hydraulic Stewart platform transmission system atωo ≈ ω1 + ωe. (a) ωo ≈ ω1 + ωe time-domain response. (b) ωo ≈ ω1 + ωe frequency-domain response.
(a)(b)Figure 21
Combined vibration response of the electro-hydraulic Stewart platform transmission system atωo ≈ ω2 + ωe. (a) ωo ≈ ω2 + ωe time-domain response. (b) ωo ≈ ω2 + ωe frequency-domain response.
(a)(b)Figure 22
Combined vibration response of the electro-hydraulic Stewart platform transmission system atωo ≈ ω3 + ωe. (a) ωo ≈ ω3 + ωe time-domain response. (b) ωo ≈ ω3 + ωe frequency-domain response.
(a)(b)Figure 23
Combined vibration response of the electro-hydraulic Stewart platform transmission system atωo ≈ ω4 + ωe. (a) ωo ≈ ω4 + ωe time-domain response. (b) ωo ≈ ω4 + ωe frequency-domain response.
(a)(b)Figure 24
Combined vibration response of the electro-hydraulic Stewart platform transmission system atωo ≈ ω5 + ωe. (a) ωo ≈ ω5 + ωe time-domain response. (b) ωo ≈ ω5 + ωe frequency-domain response.
(a)(b)Figure 25
Combined vibration response of the electro-hydraulic Stewart platform transmission system atωo ≈ ω6 + ωe. (a) ωo ≈ ω6 + ωe time-domain response. (b) ωo ≈ ω6 + ωe frequency-domain response.
(a)(b)


## 5.1. Vibration Response Analysis of System Parameters in Pose 1

Position and pose 1 parameter:(47)x=Px,Py,Pz,α,β,γT=0023000T.When the platform is in the position of pose 1, the vibration response of the system parameter is analyzed.Parameter vibration-related calculation parameters of the 6-DOF platform transmission system are shown in Table2. Substituting the platform mass, leg stiffness, and parameters shown in Table 1 into equation (20), the parametric vibration equation of the 6-DOF platform transmission system is obtained.Table 2
Calculation parameters related to vibration of 6-DOF platform transmission system parameters.
NameValueSteady-state value of the first-order natural frequencyω1 (rad·s−1)128.4811Steady-state value of the second-order natural frequencyω2 (rad·s−1)128.4811Steady-state value of the third-order natural frequencyω3 (rad·s−1)259.8076Steady-state value of the fourth-order natural frequencyω4 (rad·s−1)367.4235Steady-state value of the fifth-order natural frequencyω5 (rad·s−1)524.4956Steady-state value of the sixth-order natural frequencyω6 (rad·s−1)524.4956Fluctuation frequency of equivalent stiffness of fluid elementωe (rad·s−1)40.7150Amplitude of external excitationF0 (N)[20 20 20 20 20 20]TStructural dampingC/N·(m/s)[20 20 20 20 20 20]T


### 5.1.1. Main Resonance Response Analysis

The time-domain and frequency-domain responses of the main resonance when the external excitation frequency approaches to the stable value of the first-order to the sixth-order natural angular frequency of the electro-hydraulic platform transmission system are shown in Figures2–7. As can be seen from Figures 2(a)–7(a), while the external excitation frequency approaches to the steady-state value of the natural frequency of the first to sixth order of the transmission system, respectively, the mode corresponding to the natural frequency of the system will have a relatively intense main resonance phenomenon, and its maximum displacement is 0.00002712 mm. As can be seen from Figures 2(b)–7(b), when the main resonance is generated in the system, there is a spike in the frequency-domain image, which corresponds to the external excitation frequency close to the natural frequency. This indicates that it plays a leading role in the main resonance of the system. In this case, the dominant frequency is the sum of the natural frequencies of each order and the fluctuation frequency of the equivalent stiffness of hydraulic oil, and other natural frequencies also reach their maximum amplitudes at the dominant frequency.Figure 2
Main vibration response of the electro-hydraulic Stewart platform transmission system atωo≈ω1. (a) ωo≈ω1 time-domain response. (b) ωo≈ω1 frequency-domain response.
(a)(b)Figure 3
Main vibration response of the electro-hydraulic Stewart platform transmission system atωo≈ω2. (a) ωo≈ω2 time-domain response. (b) ωo≈ω2 frequency-domain response.
(a)(b)Figure 4
Main vibration response of the electro-hydraulic Stewart platform transmission system atωo≈ω3. (a) ωo≈ω3 time-domain response. (b) ωo≈ω3 frequency-domain response.
(a)(b)Figure 5
Main vibration response of the electro-hydraulic Stewart platform transmission system atωo≈ω4. (a) ωo≈ω4 time-domain response. (b) ωo≈ω4 frequency-domain response.
(a)(b)Figure 6
Main vibration response of the electro-hydraulic Stewart platform transmission system atωo≈ω5. (a) ωo≈ω5 time-domain response. (b) ωo≈ω5 frequency-domain response.
(a)(b)Figure 7
Main vibration response of the electro-hydraulic Stewart platform transmission system atωo≈ω6. (a) ωo≈ω6 time-domain response. (b) ωo≈ω6 frequency-domain response.
(a)(b)


### 5.1.2. Combined Resonance Response

While the external excitation frequency approaches to the combined frequency of the stable value of the first- to sixth-order natural angular frequency of the electro-hydraulic platform transmission system and the fluctuation angular frequency of the equivalent stiffness of hydraulic oil, the combined resonance time-domain response and frequency-domain response are shown in Figures8–13(a). As can be seen from Figures 8(a)–13(a), the combined resonance phenomenon occurs in the system, but its amplitude is larger than that of the system when the main resonance occurs, and the maximum displacement is 0.09427 mm. According to Figures 8(b)–13(b), when the system produces combined resonance, there are three spikes in the frequency-domain image. Among them, the combined frequency ωo ≈ ωi + ωe, which is the stable value of the natural angular frequency of each order of the system and the fluctuation angular frequency of the equivalent stiffness of hydraulic oil, has the highest peak, indicating that it plays a leading role in the combined resonance of the system, while the other frequencies play a regulating role. There is a lower peak on the left and right side of the peak corresponding to the combined frequencies, with the left corresponding frequency ωo ≈ ωi and the right corresponding frequency ωo ≈ ωi + 2ωe (i = 1, 2, …, 6).Figure 8
Combined vibration response of the electro-hydraulic Stewart platform transmission system atωo ≈ ω1 + ωe. (a) ωo ≈ ω1 + ωe time-domain response. (b) ωo ≈ ω1 + ωe frequency-domain response.
(a)(b)Figure 9
Combined vibration response of the electro-hydraulic Stewart platform transmission system atωo ≈ ω2 + ωe. (a) ωo ≈ ω2 + ωe time-domain response. (b) ωo ≈ ω2 + ωe frequency-domain response.
(a)(b)Figure 10
Combined vibration response of the electro-hydraulic Stewart platform transmission system atωo ≈ ω3 + ωe. (a) ωo ≈ ω3 + ωe time-domain response. (b) ωo ≈ ω3 + ωe frequency-domain response.
(a)(b)Figure 11
Combined vibration response of the electro-hydraulic Stewart platform transmission system atωo ≈ ω4 + ωe. (a) ωo ≈ ω4 + ωe time-domain response. (b) ωo ≈ ω4 + ωe frequency-domain response.
(a)(b)Figure 12
Combined vibration response of the electro-hydraulic Stewart platform transmission system atωo ≈ ω5 + ωe. (a) ωo ≈ ω5 + ωe time-domain response. (b) ωo ≈ ω5 + ωe frequency-domain response.
(a)(b)Figure 13
Combined vibration response of the electro-hydraulic Stewart platform transmission system atωo ≈ ω6 + ωe. (a) ωo ≈ ω6 + ωe time-domain response. (b) ωo ≈ ω6 + ωe frequency-domain response.
(a)(b)


## 5.1.1. Main Resonance Response Analysis

The time-domain and frequency-domain responses of the main resonance when the external excitation frequency approaches to the stable value of the first-order to the sixth-order natural angular frequency of the electro-hydraulic platform transmission system are shown in Figures2–7. As can be seen from Figures 2(a)–7(a), while the external excitation frequency approaches to the steady-state value of the natural frequency of the first to sixth order of the transmission system, respectively, the mode corresponding to the natural frequency of the system will have a relatively intense main resonance phenomenon, and its maximum displacement is 0.00002712 mm. As can be seen from Figures 2(b)–7(b), when the main resonance is generated in the system, there is a spike in the frequency-domain image, which corresponds to the external excitation frequency close to the natural frequency. This indicates that it plays a leading role in the main resonance of the system. In this case, the dominant frequency is the sum of the natural frequencies of each order and the fluctuation frequency of the equivalent stiffness of hydraulic oil, and other natural frequencies also reach their maximum amplitudes at the dominant frequency.Figure 2
Main vibration response of the electro-hydraulic Stewart platform transmission system atωo≈ω1. (a) ωo≈ω1 time-domain response. (b) ωo≈ω1 frequency-domain response.
(a)(b)Figure 3
Main vibration response of the electro-hydraulic Stewart platform transmission system atωo≈ω2. (a) ωo≈ω2 time-domain response. (b) ωo≈ω2 frequency-domain response.
(a)(b)Figure 4
Main vibration response of the electro-hydraulic Stewart platform transmission system atωo≈ω3. (a) ωo≈ω3 time-domain response. (b) ωo≈ω3 frequency-domain response.
(a)(b)Figure 5
Main vibration response of the electro-hydraulic Stewart platform transmission system atωo≈ω4. (a) ωo≈ω4 time-domain response. (b) ωo≈ω4 frequency-domain response.
(a)(b)Figure 6
Main vibration response of the electro-hydraulic Stewart platform transmission system atωo≈ω5. (a) ωo≈ω5 time-domain response. (b) ωo≈ω5 frequency-domain response.
(a)(b)Figure 7
Main vibration response of the electro-hydraulic Stewart platform transmission system atωo≈ω6. (a) ωo≈ω6 time-domain response. (b) ωo≈ω6 frequency-domain response.
(a)(b)


## 5.1.2. Combined Resonance Response

While the external excitation frequency approaches to the combined frequency of the stable value of the first- to sixth-order natural angular frequency of the electro-hydraulic platform transmission system and the fluctuation angular frequency of the equivalent stiffness of hydraulic oil, the combined resonance time-domain response and frequency-domain response are shown in Figures8–13(a). As can be seen from Figures 8(a)–13(a), the combined resonance phenomenon occurs in the system, but its amplitude is larger than that of the system when the main resonance occurs, and the maximum displacement is 0.09427 mm. According to Figures 8(b)–13(b), when the system produces combined resonance, there are three spikes in the frequency-domain image. Among them, the combined frequency ωo ≈ ωi + ωe, which is the stable value of the natural angular frequency of each order of the system and the fluctuation angular frequency of the equivalent stiffness of hydraulic oil, has the highest peak, indicating that it plays a leading role in the combined resonance of the system, while the other frequencies play a regulating role. There is a lower peak on the left and right side of the peak corresponding to the combined frequencies, with the left corresponding frequency ωo ≈ ωi and the right corresponding frequency ωo ≈ ωi + 2ωe (i = 1, 2, …, 6).Figure 8
Combined vibration response of the electro-hydraulic Stewart platform transmission system atωo ≈ ω1 + ωe. (a) ωo ≈ ω1 + ωe time-domain response. (b) ωo ≈ ω1 + ωe frequency-domain response.
(a)(b)Figure 9
Combined vibration response of the electro-hydraulic Stewart platform transmission system atωo ≈ ω2 + ωe. (a) ωo ≈ ω2 + ωe time-domain response. (b) ωo ≈ ω2 + ωe frequency-domain response.
(a)(b)Figure 10
Combined vibration response of the electro-hydraulic Stewart platform transmission system atωo ≈ ω3 + ωe. (a) ωo ≈ ω3 + ωe time-domain response. (b) ωo ≈ ω3 + ωe frequency-domain response.
(a)(b)Figure 11
Combined vibration response of the electro-hydraulic Stewart platform transmission system atωo ≈ ω4 + ωe. (a) ωo ≈ ω4 + ωe time-domain response. (b) ωo ≈ ω4 + ωe frequency-domain response.
(a)(b)Figure 12
Combined vibration response of the electro-hydraulic Stewart platform transmission system atωo ≈ ω5 + ωe. (a) ωo ≈ ω5 + ωe time-domain response. (b) ωo ≈ ω5 + ωe frequency-domain response.
(a)(b)Figure 13
Combined vibration response of the electro-hydraulic Stewart platform transmission system atωo ≈ ω6 + ωe. (a) ωo ≈ ω6 + ωe time-domain response. (b) ωo ≈ ω6 + ωe frequency-domain response.
(a)(b)


## 5.2. Vibration Response Analysis of System Parameters in Pose 2

Position and pose 2 parameter:(48)x=Px,Py,Pz,α,β,γT=11235510T.When the platform is in the position of pose 2, the parameter vibration response of the system is analyzed.Parameter vibration-related calculation parameters of the 6-DOF platform transmission system are shown in Table3. Substituting the platform mass, leg stiffness, and parameters shown in Table 1 into equation (20), the parametric vibration equation of the 6-DOF platform transmission system is obtained.Table 3
Calculation parameters related to vibration of 6-DOF platform transmission system parameters.
NameValueSteady-state value of the first-order natural angular frequencyω1 (rad·s−1)111.1024Steady-state value of the second-order natural angular frequencyω2 (rad·s−1)124.4199Steady-state value of the third-order natural angular frequencyω3 (rad·s−1)235.3889Steady-state value of the fourth-order natural angular frequencyω4 (rad·s−1)375.2415Steady-state value of the fifth-order natural angular frequencyω5 (rad·s−1)505.6999Steady-state value of the sixth-order natural angular frequencyω6 (rad·s−1)518.4187Fluctuation angular frequency of equivalent stiffness of fluid elementωe (rad·s−1)40.7150Amplitude of external excitationF0 (N)[20 20 20 20 20 20]TStructural dampingC/N·(m/s)[20 20 20 20 20 20]T


### 5.2.1. Main Resonance Response Analysis

The time-domain and frequency-domain responses of the main resonance, while the external excitation frequency is close to the stable value of the first-order to the sixth-order natural angular frequency of the electro-hydraulic platform transmission system can be seen from Figures14–19. With the change of position and pose, the natural frequency of the system changes and the main resonance response also changes. As can be seen from Figures 14(a)–19(a), while the external excitation frequency is close to the steady-state value of the natural frequency of the first to sixth order of the transmission system, the system will experience a relatively intense primary resonance with a maximum amplitude of 0.00002923 mm. Compared with pose 1, the modal amplitudes of the system are all increased. This indicates that, in the asymmetric pose, the connection between each leg is closer, the coupling is increased, and the degree of participation in vibration is increased. As can be seen from Figures 14(b)–19(b), when the main resonance is generated in the system, there is a spike in the frequency-domain image, and the corresponding frequency of the peak corresponds to the approximate analytical formula of the main resonance of the system. Its peak corresponds to the external excitation frequency close to the natural frequency. This indicates that it plays a leading role in the main resonance of the system. In this case, the dominant frequency is the sum of the natural frequencies of each order and the fluctuation frequency of the equivalent stiffness of hydraulic oil, and other natural frequencies also reach their maximum amplitudes at the dominant frequency.Figure 14
Main vibration response of the electro-hydraulic Stewart platform transmission system atωo≈ω1. (a) ωo≈ω1 time-domain response. (b) ωo≈ω1 frequency-domain response.
(a)(b)Figure 15
Main vibration response of the electro-hydraulic Stewart platform transmission system atωo≈ω2. (a) ωo≈ω2 time-domain response. (b) ωo≈ω2 frequency-domain response.
(a)(b)Figure 16
Main vibration response of the electro-hydraulic Stewart platform transmission system atωo≈ω3. (a) ωo≈ω3 time-domain response. (b) ωo≈ω3 frequency-domain response.
(a)(b)Figure 17
Main vibration response of the electro-hydraulic Stewart platform transmission system atωo≈ω4. (a) ωo≈ω4 time-domain response. (b) ωo≈ω4 frequency-domain response.
(a)(b)Figure 18
Main vibration response of the electro-hydraulic Stewart platform transmission system atωo≈ω5. (a) ωo≈ω5 time-domain response. (b) ωo≈ω5 frequency-domain response.
(a)(b)Figure 19
Main vibration response of the electro-hydraulic Stewart platform transmission system atωo≈ω6. (a) ωo≈ω6 time-domain response. (b) ωo≈ω6 frequency-domain response.
(a)(b)


### 5.2.2. Combined Resonance Response

As can be seen from Figures20(a)–25(a), when the external excitation frequency is close to the combination frequency of the stable value of the first to sixth order natural angular frequency of the electro-hydraulic platform transmission system and the fluctuation angular frequency of the equivalent stiffness of hydraulic oil, the combined resonance phenomenon occurs in the system and the amplitude is relatively severe. Compared with pose 1, the modal amplitudes of the system are all increased. This indicates that, in the asymmetric pose, the connection between each leg is closer, the coupling is increased, and the degree of participation in vibration is increased. This indicates that, in the asymmetric posture, the connection between each leg is more close, and the degree of participation in vibration is increased. As can be seen from Figures 20(b)–25(b), when the system produces combined resonance, there are three spikes in the frequency-domain image, and the corresponding frequencies of each spike correspond to the approximate analytical formula of the system’s main resonance one by one. Among them, the peak corresponding to the combined frequency ωo ≈ ωi + ωe of the stable value of the natural angular frequency of each order of the system and the fluctuation angular frequency of the equivalent stiffness of hydraulic oil is the highest. It shows that it plays a leading role in the combined resonance of the system and the other frequencies play a regulating role. There is a lower peak on the left and right side of the peak corresponding to the combined frequencies, with the left corresponding frequency ωo ≈ ωi and the right corresponding frequency ωo ≈ ωi + 2ωe (i = 1, 2, …, 6).Figure 20
Combined vibration response of the electro-hydraulic Stewart platform transmission system atωo ≈ ω1 + ωe. (a) ωo ≈ ω1 + ωe time-domain response. (b) ωo ≈ ω1 + ωe frequency-domain response.
(a)(b)Figure 21
Combined vibration response of the electro-hydraulic Stewart platform transmission system atωo ≈ ω2 + ωe. (a) ωo ≈ ω2 + ωe time-domain response. (b) ωo ≈ ω2 + ωe frequency-domain response.
(a)(b)Figure 22
Combined vibration response of the electro-hydraulic Stewart platform transmission system atωo ≈ ω3 + ωe. (a) ωo ≈ ω3 + ωe time-domain response. (b) ωo ≈ ω3 + ωe frequency-domain response.
(a)(b)Figure 23
Combined vibration response of the electro-hydraulic Stewart platform transmission system atωo ≈ ω4 + ωe. (a) ωo ≈ ω4 + ωe time-domain response. (b) ωo ≈ ω4 + ωe frequency-domain response.
(a)(b)Figure 24
Combined vibration response of the electro-hydraulic Stewart platform transmission system atωo ≈ ω5 + ωe. (a) ωo ≈ ω5 + ωe time-domain response. (b) ωo ≈ ω5 + ωe frequency-domain response.
(a)(b)Figure 25
Combined vibration response of the electro-hydraulic Stewart platform transmission system atωo ≈ ω6 + ωe. (a) ωo ≈ ω6 + ωe time-domain response. (b) ωo ≈ ω6 + ωe frequency-domain response.
(a)(b)


## 5.2.1. Main Resonance Response Analysis

The time-domain and frequency-domain responses of the main resonance, while the external excitation frequency is close to the stable value of the first-order to the sixth-order natural angular frequency of the electro-hydraulic platform transmission system can be seen from Figures14–19. With the change of position and pose, the natural frequency of the system changes and the main resonance response also changes. As can be seen from Figures 14(a)–19(a), while the external excitation frequency is close to the steady-state value of the natural frequency of the first to sixth order of the transmission system, the system will experience a relatively intense primary resonance with a maximum amplitude of 0.00002923 mm. Compared with pose 1, the modal amplitudes of the system are all increased. This indicates that, in the asymmetric pose, the connection between each leg is closer, the coupling is increased, and the degree of participation in vibration is increased. As can be seen from Figures 14(b)–19(b), when the main resonance is generated in the system, there is a spike in the frequency-domain image, and the corresponding frequency of the peak corresponds to the approximate analytical formula of the main resonance of the system. Its peak corresponds to the external excitation frequency close to the natural frequency. This indicates that it plays a leading role in the main resonance of the system. In this case, the dominant frequency is the sum of the natural frequencies of each order and the fluctuation frequency of the equivalent stiffness of hydraulic oil, and other natural frequencies also reach their maximum amplitudes at the dominant frequency.Figure 14
Main vibration response of the electro-hydraulic Stewart platform transmission system atωo≈ω1. (a) ωo≈ω1 time-domain response. (b) ωo≈ω1 frequency-domain response.
(a)(b)Figure 15
Main vibration response of the electro-hydraulic Stewart platform transmission system atωo≈ω2. (a) ωo≈ω2 time-domain response. (b) ωo≈ω2 frequency-domain response.
(a)(b)Figure 16
Main vibration response of the electro-hydraulic Stewart platform transmission system atωo≈ω3. (a) ωo≈ω3 time-domain response. (b) ωo≈ω3 frequency-domain response.
(a)(b)Figure 17
Main vibration response of the electro-hydraulic Stewart platform transmission system atωo≈ω4. (a) ωo≈ω4 time-domain response. (b) ωo≈ω4 frequency-domain response.
(a)(b)Figure 18
Main vibration response of the electro-hydraulic Stewart platform transmission system atωo≈ω5. (a) ωo≈ω5 time-domain response. (b) ωo≈ω5 frequency-domain response.
(a)(b)Figure 19
Main vibration response of the electro-hydraulic Stewart platform transmission system atωo≈ω6. (a) ωo≈ω6 time-domain response. (b) ωo≈ω6 frequency-domain response.
(a)(b)


## 5.2.2. Combined Resonance Response

As can be seen from Figures20(a)–25(a), when the external excitation frequency is close to the combination frequency of the stable value of the first to sixth order natural angular frequency of the electro-hydraulic platform transmission system and the fluctuation angular frequency of the equivalent stiffness of hydraulic oil, the combined resonance phenomenon occurs in the system and the amplitude is relatively severe. Compared with pose 1, the modal amplitudes of the system are all increased. This indicates that, in the asymmetric pose, the connection between each leg is closer, the coupling is increased, and the degree of participation in vibration is increased. This indicates that, in the asymmetric posture, the connection between each leg is more close, and the degree of participation in vibration is increased. As can be seen from Figures 20(b)–25(b), when the system produces combined resonance, there are three spikes in the frequency-domain image, and the corresponding frequencies of each spike correspond to the approximate analytical formula of the system’s main resonance one by one. Among them, the peak corresponding to the combined frequency ωo ≈ ωi + ωe of the stable value of the natural angular frequency of each order of the system and the fluctuation angular frequency of the equivalent stiffness of hydraulic oil is the highest. It shows that it plays a leading role in the combined resonance of the system and the other frequencies play a regulating role. There is a lower peak on the left and right side of the peak corresponding to the combined frequencies, with the left corresponding frequency ωo ≈ ωi and the right corresponding frequency ωo ≈ ωi + 2ωe (i = 1, 2, …, 6).Figure 20
Combined vibration response of the electro-hydraulic Stewart platform transmission system atωo ≈ ω1 + ωe. (a) ωo ≈ ω1 + ωe time-domain response. (b) ωo ≈ ω1 + ωe frequency-domain response.
(a)(b)Figure 21
Combined vibration response of the electro-hydraulic Stewart platform transmission system atωo ≈ ω2 + ωe. (a) ωo ≈ ω2 + ωe time-domain response. (b) ωo ≈ ω2 + ωe frequency-domain response.
(a)(b)Figure 22
Combined vibration response of the electro-hydraulic Stewart platform transmission system atωo ≈ ω3 + ωe. (a) ωo ≈ ω3 + ωe time-domain response. (b) ωo ≈ ω3 + ωe frequency-domain response.
(a)(b)Figure 23
Combined vibration response of the electro-hydraulic Stewart platform transmission system atωo ≈ ω4 + ωe. (a) ωo ≈ ω4 + ωe time-domain response. (b) ωo ≈ ω4 + ωe frequency-domain response.
(a)(b)Figure 24
Combined vibration response of the electro-hydraulic Stewart platform transmission system atωo ≈ ω5 + ωe. (a) ωo ≈ ω5 + ωe time-domain response. (b) ωo ≈ ω5 + ωe frequency-domain response.
(a)(b)Figure 25
Combined vibration response of the electro-hydraulic Stewart platform transmission system atωo ≈ ω6 + ωe. (a) ωo ≈ ω6 + ωe time-domain response. (b) ωo ≈ ω6 + ωe frequency-domain response.
(a)(b)


## 6. Conclusion

In this paper, the parametric vibration dynamics equation of the Stewart 6-DOF platform is established. Based on the multiscale method, the theoretical formulas of the main resonance and the combined resonance response of the system are derived, and the parametric vibration response law of the system is analyzed. Its conclusions are as follows:(1)
The response characteristics of the main resonance response of the electro-hydraulic Stewart 6-DOF platform transmission system are mainly determined by the modal characteristics of the system; the combined resonance response of the transmission system is more complex. The natural frequencies of each order are the dominant frequencies, and the combined frequencies play a regulating role in the response.(2)
Through the platform of two different pose parameter vibration analysis, the platform position in the asymmetrical position compared with the symmetric position, the time-domain response of the system, and the frequency response analysis is more close to the actual situation; when the external excitation frequency close to the system natural frequency system of six order modal vibration amplitude is higher, each leg of the system participate in the higher degree of vibration with enhanced coupling.The results provide a theoretical basis for the resonance phenomenon of the electro-hydraulic Stewart platform. It provides theoretical support for subsequent dynamic optimization and reliability analysis. Through analysis, it is suggested that the platform should be installed with a vibration absorber to reduce the excitation frequency only at a specific frequency to avoid resonance.



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*Source: 9994786-2021-08-12.xml*