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

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.


Introduction
e 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][3][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. erefore, the Stewart platform also has a strong attraction to the industry. e platform becomes an electro-hydraulic Stewart platform when the hydraulic legs of the Stewart platform are incorporated into the electro-hydraulic servo control system. e 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][19][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]. erefore, 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]. e vibration induced by the time-varying parameters (stiffness, damping, mass, or moment of mass inertia) of the system is called parametric vibration. e 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. e 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]. erefore, 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. e 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, amplitudefrequency 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].
is 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.

Position Analysis of Electro-Hydraulic
Platform. e geometrical structure diagram of the electro-hydraulic Stewart 6-DOF platform is shown in Figure 1. e electrohydraulic 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. e six supporting legs are connected with the upper and base platforms, respectively, through ball hinges and universal joints. e six legs are composed of hydraulic cylinders and sensors that provide input excitation to the legs to ensure movement of the platform. e 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. e distribution of six balls hinged on the top platform is determined by using a i (i � 1, 2, . . ., 6), and the six universal joints on the base platform are b i (i � 1, 2, . . ., 6). By using R a 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 I x , I y , and I z are, respectively, the rotation inertia of the top platform about the XYZ axis.
By using P x , P y , and P z , 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 c, 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. e above six variables determine the position of the platform. erefore, the expression of the rotation matrix R a and displacement matrix E is as follows: where sα � sin α and cα � cos α. e expression of the six leg positions is as follows: where i � 1, 2, . . ., 6. In this paper, the two poses' parameter vibration is analyzed. e parameters of the two poses are as follows: Pose 1: x � P x , P y , P z , α, β, c Pose 2: 2 Shock and Vibration e initial position of the platform which is shown in Figure 1 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.
rough 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 Table 1:

Dynamic Equation of Electro-Hydraulic Stewart 6-DOF Platform Transmission
System. e dynamic model of the electro-hydraulic platform established in this paper adopts the following assumptions. e dynamic model of the electro-hydraulic Stewart platform adopts the following assumptions: (1) e 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) e leg mass is much smaller than the top platform mass, so the effect caused by the leg mass would become negligible (4) e friction which is caused by the moving components can be neglected (5) Ignore the processing error and assembly error of each component

Velocity Analysis and Jacobian Matrix of Electro-Hydraulic Platform.
Set the generalized coordinates of the platform to x � P x , P y , P z , α, β, c T . (5) us, the velocity V and angular velocity ω of the moving platform can be expressed as By assuming that r is the vector diameter of spherical hinge points relative to the nature heart of the top platform, V S 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: where i � 1, 2, . . ., 6. Substituting (8) into (9), we can obtain en, the speed of the six legs is e shorthand for which is where [J] is the Jacobian matrix.

Dynamic Equation for Constructing Electro-Hydraulic
Platform. e equivalent stiffness of oil in the six-leg hydraulic cylinder is determined by k i (i � 1, 2, . . ., 6), damping is expressed by c i (i � 1, 2, . . ., 6), and output force of the leg hydraulic cylinder is f ai (i � 1, 2, . . ., 6). e 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: Shock and Vibration e Coriolis force and centrifugal force on the platform are ignored. Equation (13) can be simplified to In the formula above, _ x � _ p ω T . e components of δ _ x have no effect on each other, so the coefficients are all zero; thus, the dynamic equation can be expressed as

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 where p represents steady-state pressure (Pa), Δp represents pressure pulsation amplitude (Pa), and ω o 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. erefore, 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 where k f represents steady-state fluid stiffness (N/m), k fv represents dynamic fluid stiffness amplitude (N/m), ε represents small parameter, ε � k fv /2k f , ω e 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. erefore, parametric vibration analysis mainly studies the system parametric vibration response when the equivalent stiffness of fluid changes.

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: whereF o represents the pressure pulsation amplitude matrix and F 0 � Δf a1 Δf a2 Δf a3 Δf a4 Δf a5 Δf a6 T . Equation (18) is expressed in the matrix form as

Shock and Vibration
where k represents the steady-state stiffness matrix, k � k f J T J, ΔF represents the excitation force vector, ΔF � (e jω o t + e − jω o t )J T F o , Δk represents the incremental stiffness matrix, and Δk � k f ε(e jω e t + e − jω e t )J T J.

Derivation of Approximate Analytical Formula for Parametric Vibration of Electro-Hydraulic Stewart Platform
rough the relevant knowledge of vibration mechanics, the regular mode ψ and spectral matrix Λ of the system of the platform transmission system are solved, and equation (19) is regularized to obtain where η represents the regular generalized displacement

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: where T 0 � t and T 1 � εt.
For convenience of derivation, the following parameters are introduced into small parameters to obtain Substitute the above equation into equation (20), and using the same power coefficient equal, the equation of zero power term can be calculated as e first power term is Shock and Vibration 5 where cc represents the complex conjugate. Let the solution of equation (24) be where A i is an unknown function, and the solution can be substituted into equation (24): Δk 1i A i e j ω ni +ω e ( )T0 + cc, Δk 2i A i e j ω ni +ω e ( )T0 + cc, Δk 3i A i e j ω ni +ω e ( )T0 + cc, Δk 4i A i e j ω ni +ω e ( )T0 + cc, Δk 5i A i e j ω ni +ω e ( )T0 + cc, Δk 6i A i e j ω ni +ω e ( )T0 + cc.
When ω o approaches ω n1 , the harmonic parameter σ is introduced, and the frequency relationship of each angle is shown in the following formula: Substitute equation (27) into equation (26) and eliminate the perpetual term: e solution of equation (28) is obtained by using the constant conversion method: where C 1 , . . ., C 6 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 where θ � arctan(C N11 ′ /2σ).
With the increase of time, equation (29) will gradually approach zero. erefore, 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 Shock and Vibration 7 Substituting equations (29) and (30) into equation (26), the first-order approximate analytical solution of the steadystate electro-hydraulic system can be obtained as en, the steady-state response of the main resonance of the electro-hydraulic platform system in rectangular coordinates is 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.

Approximate Analytical Derivation of Combined Resonance of Electro-Hydraulic Platform Transmission System.
e combined resonance response of the electro-hydraulic platform transmission system is derived by the multiscale method, which can be expressed as where T 0 � t and T 1 � εt. A small parameter is introduced into the damping: Substituting equations (34) and (35) into equation (20), the zeroth power term is sorted out and the equation is shown as follows: Shock and Vibration e equation for sorting out the first power term is shown as follows: where ccrepresents the complex conjugate.
By analyzing the above equation, we can see that when ω o ≈ ω ni , the main resonance occurs in the system. In addition, when ω 0 ≈ ω ni ± ω e , 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: Substitute equation (40) into equation (39) and eliminate the perpetual term: e general solution of equation (41) is where E 1 , . . ., E 6 are constant and θ � arctan(C N11 ′ /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: Substituting equation (42) into equation (39), the steadystate first-order approximate analytical solution of the combined resonance of the transmission system can be obtained as follows: where Shock and Vibration Shock and Vibration en, the combined resonant steady-state response of the 6-DOF platform system in rectangular coordinates is 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.

Vibration Response Analysis of System Parameters in Pose 1.
Position and pose 1 parameter: 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 Table 2. 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.

Main Resonance Response Analysis.
e 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 Figures 2-7. 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 frequencydomain image, which corresponds to the external excitation frequency close to the natural frequency. is 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.

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 Figures 8-13(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 ω o ≈ ω i + ω e , 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. ere is a lower peak on the left and right side of the peak corresponding to the combined frequencies, with the left corresponding frequency ω o ≈ ω i and the right corresponding frequency ω o ≈ ω i + 2ω e (i � 1, 2, . . ., 6).

Vibration Response Analysis of System Parameters in Pose 2. Position and pose 2 parameter:
x � P x , P y , P z , α, β, c T � 1 1 2 � 3 √ 5 5 10 T . (48) 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 Table 3. 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.

Main Resonance Response Analysis.
e time-domain and frequency-domain responses of the main resonance, while the external excitation frequency is close to the Shock and Vibration 13 Table 2: Calculation parameters related to vibration of 6-DOF platform transmission system parameters.
Name Value Steady-state value of the first-order natural frequency ω 1 (rad·s − 1 ) 128.4811 Steady-state value of the second-order natural frequency ω 2 (rad·s − 1 ) 128.4811 Steady-state value of the third-order natural frequency ω 3 (rad·s − 1 ) 259.8076 Steady-state value of the fourth-order natural frequency ω 4 (rad·s − 1 ) 367.4235 Steady-state value of the fifth-order natural frequency ω 5 (rad·s − 1 ) 524.4956 Steady-state value of the sixth-order natural frequency ω 6 (rad·s  200 300 400 500 600 External excitation frequency ω o /(rad/s) stable value of the first-order to the sixth-order natural angular frequency of the electro-hydraulic platform transmission system can be seen from Figures 14-19. 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.
is 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   200 300 400 500 600 External excitation frequency ω o /(rad/s) 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.
is 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. Figures 20(a)-25(a), when the external excitation frequency is close to the combination frequency of the stable value of 200 300 400 500 600 External excitation frequency ω o /(rad/s)  200 300 400 500 600 External excitation frequency ω o /(rad/s) 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. is 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. is 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   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. ere is a lower peak on the left and right side of the peak corresponding to the combined frequencies, with the left corresponding frequency ω o ≈ ω i and the right corresponding frequency ω o ≈ ω i + 2ω e (i � 1, 2, . . ., 6).       Table 3: Calculation parameters related to vibration of 6-DOF platform transmission system parameters.

24
Shock and Vibration

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) e 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. e natural frequencies of each order are the dominant frequencies, and the combined frequencies play a regulating role in the response. 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. e 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. rough 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.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.