Vibration characteristics of multi-dimensional isolator based on 4-PUU parallel mechanism with joint clearance

For isolating multi-dimensional vibration experienced by vehicle-mounted precise facility, a multi-dimensional vibration isolator based on parallel mechanism with joint clearance is designed and analyzed. The main thrust of this work is establishing kinematic and dynamic equations of the isolator and exploring vibration isolation capability with joint clearance. Firstly, type synthesis of the parallel mechanism with three translations and one rotation is performed. The kinematic and dynamic equations of the isolator with joint clearance are deduced. Subsequently, vibration isolation performance is simulated with different values of joint clearance under harmonic and stochastic excitations in time and frequency domain, respectively. The simulating results demonstrate the proposed isolator with joint clearance inhibit multi-dimensional vibration effectively. The first-order resonance peak is sensitive to the increment in joint clearance. Finally, the proposed multi-dimensional vibration isolator is fabricated. The vibration isolation experiment is conducted. Through experimental results, the isolator reduces multi-dimensional vibration effectively in time and frequency domain. The vibration isolation capability degenerates as the counterweight of moving platform increases.


Introduction
Multi-dimensional vibration appears at least two directions simultaneously and exhibits translational or rotational forms, which exists widely in engineering area and daily life. Taking vehicle-mounted hyperspectral imager, vehicle-mounted radar as examples, due to road roughness and operating process, the vibration in horizontal, longitudinal, vertical, pitch, yaw, and roll directions may generate at the same time [1]. The accuracy and reliability of vehicle-mounted precise instruments are deteriorated because of multidimensional vibration. For isolating multi-dimensional vibration effectively, the isolator exhibited same degrees-of-freedom (Dofs) with multi-dimensional vibration should be introduced. For the traditional multi-dimensional vibration isolator, such as floating raft system [2] and magnetic levitation system [3], which mitigate multi-dimensional vibration effectively. However, the disadvantages, which are dynamic characteristics do not vary with the changing of external excitations, small deformation of isolating elements, complexity of the structures, cannot be solved properly.
In order to overcome the shortcomings of traditional multi-dimensional vibration isolator, parallel mechanism exhibiting high stiffness, dynamic characteristics relating closely to the position, and generating multi-dimensional motions employed minimum actuators has been regarded as a prevalent candidate [4][5][6][7]. Yang et al. [8] established the dynamic equation with derived stiffness of the isolator based on Stewart mechanism. The control force was generated by the branch force and position feedback simultaneously. A novel PI sub-controller was designed, aiming at regulating vibration isolation frequency bandwidth and active viscous damping independently. Gao et al. [9] proposed a novel ambulance stretcher system based on parallel mechanism and magneto-rheological (MR) dampers. The stretcher system isolated multidimensional vibration in horizontal, longitudinal, vertical, and pitch directions effectively. The robust index of linear quadratic regulator (LQR) strategy was established by deducing Lyapunov equation. Tang et al. [10] investigated joint dynamic equation of Stewart vibration isolator by Newton-Euler method. A novel quasi-linear controller was designed by decentralized control theory, which realized mitigation of multi-dimensional vibration in six directions. Wu et al. [11] proposed a X-shape vibration isolator based on Stewart mechanism and deduced kinematic equation by Hamilton theory. The equivalent stiffness and displacement transmissibility were investigated. Zhang et al. [12] investigated six Dofs active-passive composite vibration isolator with hydraulic driving. The fuzzy PID control strategy was employed to inhibit multi-dimensional vibration. Zhao et al. [13] presented a multi-dimensional vibration isolator composed of parallel mechanism with MR damper, which was controlled by sliding mode algorithm in real time and mitigated multi-dimensional vibration in horizontal, longitudinal, and vertical directions significantly. Niu et al. [14] have introduced a novel vibration reduction device based on metamorphic parallel mechanism, which switched Dofs between three and four by varying the arrangement of universal joints. Li et al. [15] proposed a novel five-dimensional vibration isolator, which was controlled by semi-fuzzy optimal control method. The vibration isolation performance was validated by experiment.
Many researches have been reported about multidimensional vibration isolator based on parallel mechanism, which indicate the feasibility of inhibiting multi-dimensional vibration effectively. However, joint clearance cannot be eliminated and neglected in the real manufacturing and assembling process. Many scholars have done many works on exploring kinematics and dynamics of different mechanisms with joint clearance, which can provide a reference for modeling of multi-dimensional isolator with joint clearance.
Flores et al. [16,17] investigated dynamic responses of a slider-crank mechanism with revolute joint clearance by numerical and experimental approaches. The proposed test rig exhibited an adjustable radial clearance in the revolute joint between slider and connecting rod. The maximum value of slider acceleration was selected as the index to evaluate impact severity. Zhang et al. [18] presented a novel methodology to evaluate dynamic response of 3-PRR parallel mechanism with multiple lubricated joints. The lubricated contact forces were calculated by Pinkus-Sternlicht model between journal and bearing. Wang et al. [19] established the dynamic model of 3-CPaRR parallel mechanism considering joint clearance and link flexibility simultaneously. The contact forces of journal and bearing were calculated by Lankarani-Nikravesh and Coulomb friction models, respectively. The results revealed the links with flexibility and joint clearance affecting kinematic response of parallel mechanism significantly. Chen et al. [20] proposed a novel method for establishing dynamic model of 4-UPS-RPU redundant driving spatial parallel mechanism with spherical joint clearance. The influential characteristics of single clearance and multiple clearances on dynamic response of the proposed parallel mechanism were analyzed. Hou et al. [21] established dynamic model of 3-RSR parallel mechanism using Newton-Euler approach. The contact forces were calculated by modified Flores model and Coulomb friction model. The simulation results shown that irregular wear appeared in the spherical joint clearance due to impact and friction forces. Wang et al. [22] presented the dynamic model of 4-SPS/PS parallel mechanism considering flexible link and spherical joint clearance at the same time. The normal and tangential forces were estimated by Lankarani-Nikravesh model and modified Coulomb friction model, respectively. The dynamic response with joint clearance and link flexibility was compared with finite element analysis approach.
The review summarized above is not taken joint clearance in the vibration isolator based on parallel mechanism into account. It is critical to address the influence of joint clearance on vibration reduction capability. Therefore, the main thrust of this work is establishing dynamic model of multi-dimensional vibration isolator based on parallel mechanism with joint clearance and exploring influential characteristics of joint clearance on vibration isolation capability. Additionally, the dynamic model of the proposed isolator with joint clearance is much more realistic, which can make a reference for designing multidimensional vibration isolator based on parallel mechanism. The structure of the work is arranged as follows. Firstly, type synthesis of three translations and one rotation parallel mechanism is conducted by generalized function (G F ) set theory. The kinematics and dynamics of the isolator with joint clearance are deduced. Subsequently, the vibration isolation performance of the proposed isolator with joint clearance is investigated in time and frequency domain, respectively. Thirdly, the proposed isolator with joint clearance is fabricated, and the vibration isolation experiment is conducted. Finally, the conclusions are drawn.
2 Modeling of the isolator based on 4-PUU parallel mechanism with joint clearance

Overview of multi-dimensional vibration isolator
The output characteristics of the proposed isolator are three translations and one rotation (3T1R), which can be synthesized by G F set theory. G F set theory is an effective method to depict the end characteristics of mechanism, which is widely employed in the area of type synthesis. G F set includes six elements of motions, which are three translating characteristics T a ; T b ; T c and three rotating characteristics R a ; R b ; R c , respectively. Subscripts a; b; c indicate axes of translating motions, which are not co-plane, and any two axes are not parallel. Subscripts a; b; c are the rotating axes crossed in a same point. Based on the description of Euler angle and the distance to the frame, the rotating characteristics are described as R a ; R b ; R c , respectively [23,24]. The expected output characteristic is obtained by the constraint of each link. Hence, the end characteristic of parallel mechanism can be described by the intersection of each link end characteristic, which is given by [25,26], where G F is end characteristic of parallel mechanism, and G Fi is end characteristic of ith link, i ¼ 1; 2; 3 Á Á Á ; n.
Since assembling position of active joint is not fixed on different links, number synthesis should be taken into account. The equation of number synthesis is presented as follows [25,26]: where F D is dimension of G F set, N D and n d are numbers of links and active links, respectively, p is number of passive links, and q i is number of actuators in ith active link.
Since the Dofs of proposed parallel mechanism is four, the parameters in Eq. (2) are, respectively, given by: Considering favorable stability, the parallel mechanism exhibited four links with symmetrical structure is selected. The end characteristics of the parallel mechanism above-mentioned are described by G F intersection rule [26]: where T Ã means the translational motion, and R Ã presents rotational axis, respectively. According to Eq. (4), a part of parallel mechanism with four symmetrical links are listed in Table 1. Table 1 describes configuration of 3T1R parallel mechanism with symmetrical structure. In Table 1, P is prismatic pair, R 1 and R 2 are revolute joints with different axes, and U is universal joint. Based on spatial axis migrating rule, the revolute joint in Table 1 can be arranged discretionarily. Thus, the derivative mechanisms are not presented. The prismatic pairs are essential to assemble springs and dampers in the proposed 3T1R parallel mechanism. Thus, the mechanism configuration is selected as 4-PUR 1 R 2 . Because universal joint is composed of two revolute joints with different axes, for describing conveniently, the configuration of proposed parallel mechanism is defined as 4-PUU.
The main structure of the proposed isolator is 4-PUU parallel mechanism composed of moving platform, fixed platform and four links L i i ¼ 1; 2; 3; 4 ð Þ , respectively. The moving and fixed platforms' shapes are both rectangles with the lengths 2a Â 2b and 2c Â 2d, respectively. From fixed platform to moving platform, the joints are composed of prismatic pair, universal joint, and universal joint, respectively. Two axes in one universal joint exhibit 45°offset value. The viscous dampers and springs are arranged on four prismatic pairs. The schematic diagram of the proposed isolator is shown in Fig. 1a. For demonstrating connecting details and configuration of universal joint clearly, part I and part II in Fig. 1a have been amplified and presented in Fig. 1b. As above-mentioned, the universal joint can be regarded as the combination of two revolute joints with different axes. In part I of Fig. 1b, the axes of revolute joints C 11 and C 12 , B 11 and B 12 exhibit 45°angle. The axes of C 11 and B 11 are in vertical direction. In part II of Fig. 1b, the axes of revolute joints C 21 and C 22 , B 21 and B 22 exhibit 45°angle. Any two revolute joints above-mentioned exist joint clearance. Because of symmetry of the proposed parallel mechanism, the configurations of another two links are same with part I and part II. The moving and fixed Cartesian coordinates (P À x p y p z p and O À xyz) are established on moving and fixed platforms, respectively. The Dofs of the proposed isolator is four, which can be validated by G-K formula with correction coefficient [27]. Since universal joint is regarded as the combination of two revolute joints with axes offset, so in the following parts, the kinematic and dynamic models of the isolator mainly take revolute joint clearance into account.

Kinematics of the isolator with joint clearance
Joint clearance cannot be neglected in real condition because of manufacturing error and assembling requirement. For the proposed isolator based on 4-PUU parallel mechanism, the universal joint clearance model can be illustrated as the combination of two revolute joints with clearance. Generally, the centers of journal and bearing are coincided in initial position for simplicity. The joint clearance in initial position can be calculated as c ¼ R J À R B j j , where R J and R B are the radius vectors of journal and bearing, respectively. The clearance limits the journal orbit staying inside the bearing's boundary [28].
The journal generates relative motion freely inside the bearing until the contact between two bodies occur. The schematic diagram of the stochastic motion is presented exaggeratively in Fig. 2. Because the lengths of journal and bearing in axial direction are similar, the inclination contact is negligible. Thus, the local and total Cartesian coordinates are defined as o À xy and O À XY in the radial direction of journal and bearing, respectively. Q j and Q i are the centers of journal and bearing, respectively.
In the total coordinate, the eccentric value of the centers between journal and bearing is expressed as follows: where r Q j and r Q i are position vectors in total coordinate. The penetration depth between journal and bearing is defined as d¼ e j j À c, which is employed to judge the states of journal and bearing. d\0 represents free flight mode, d ¼ 0 represents critical state of contact or separate, and d [ 0 represents contact state.
For the vibration isolator, in a certain position x p ; y p ; z p Â Ã T of the moving platform, B Pi ; B Oi and C Pi ; C Oi denote the coordinates of points B i ; C i in moving and fixed Cartesian frames, respectively, i ¼ 1; 2; 3; 4. The coordinate C i is expressed as follows: where . T x is transfer matrix around x-axis, which is given by where h is the angle around x-axis. The original distance between B i and C i without joint clearance is deduced by geometric relation and expressed as follows: The inverse kinematic equation without joint clearance can be calculated by Eq. (8), which is given by: where D i is expressed as follows: For the proposed vibration isolator, points C i connect to the moving platform through universal (revolute) joint. According to Fig. 2, the position vectors of contact state in points C i can be illustrated as follows: where n ¼ e= e j j is unit vector of eccentric value and represents the normal direction of collision. Operating derivation on Eq. (11), the velocity vectors in contact state are expressed as follows: Thus, the velocity vectors in normal and tangential directions are described as follows: where For another connecting points B i , the positions vectors (r B i ; r B j ) and velocity vectors ( _ r B i ; _ r B j ) can be presented as Eq. (11)-(13) with similar expressions. Thus, the actual distance between B i and C i is calculated as follows: Due to the output characteristics of parallel mech- inverse Jacobian matrix of the isolator with joint clearance is given by: whereD 1 ¼ã 2 À d 2 À d 4 À x p À Á 2 ,D 3 ¼ã 2 À Àd 2 ð þd 4 À x p Þ 2 .

Dynamics of the isolator with joint clearance
Generally, the rotating speed of journal and bearing in the isolator based on parallel mechanism is low, thus the temperature rise and wear between journal and bearing are tiny. In another aspect, for avoiding influence of fluid lubrication on vibration isolation capability, thus the collision between journal and bearing wall is dry contact. The contact forces are generated and transmitted to the mechanical system [29]. The contact force F c can be illustrated by penetration depth, which is given by [30], where F N and F T are normal and tangential forces, respectively. The normal and tangential forces can be described by Lankarani-Nikravesh (L-N) model and modified Coulomb friction force model. For the revolute joint with clearance, the energy dissipation is generally tiny [31]. The clearance between journal and bearing is larger compared to contacting area, and the load is smaller compared to other contacting models [32]. Thus, L-N model is diffusely employed in investigating contact phenomenon of revolute joint with clearance. The L-N model is given by [33], where d is penetration depth, _ d is relative penetration velocity, _ d À ð Þ is initial impact velocity, e r is restitution coefficient, and the exponent n equals to 1.5 for most metal contact. K c is contact stiffness depended on materials and geometric parameters of the contact bodies, which is given by, The coefficients h B ; h J in Eq. (18) are described as follows: where v k and E k are Poisson's ratio and Young's modulus, respectively. Aiming at improving stability of dynamics integration, the modified Coulomb friction force model is given by [34,35]: where c f is friction coefficient, and c d is correction coefficient, which is given by where v 0 and v 1 are the boundaries of tangential velocity, respectively. The boundaries of tangential velocity v 0 ; v 1 ð Þ are employed as the switching condition. However, the method to determine accurate values of v 0 ; v 1 ð Þin the friction process is not proposed yet properly [36]. Therefore, the values of v 0 ; v 1 ð Þ adopted in simulation are refereed from reference [35].
Because of coupling characteristics of parallel mechanism, the solution of isolator is hard to be obtained. Therefore, the simplified conditions, which are moving platform oscillating near equilibrium position and neglecting mass of links, should be taken into account. The dynamic equation of multi-dimensional vibration isolator with joint clearance can be expressed by Lagrange equation with damping energy, which is given by where coordinate, and C is generalized force vector including contact force. M ¼ diag m p ; m p ; m p ; I x À Á is inertia matrix of moving platform, and m p and I x are mass of moving platform and moment inertia around x axis, respectively. K ¼ J ÀT diag k ð ÞJ À1 and C ¼ J ÀT diag c e ð ÞJ À1 are stiffness and damping matrices, respectively. k and c e are stiffness of spring and damping coefficient, respectively. J ÀT denotes force Jacobian matrix. The dynamic equation of the proposed isolator with joint clearance is synthesized as follows: 3 Simulation of the isolator with joint clearance The main parameters and values of the proposed multi-dimensional isolator are shown in Table 2. The parameters in Table 2 are same as the parameters in experiment in Sect. 4.

Time response of the isolator with joint clearance
Defining Z ¼ X; _ X Â Ã T as state variable of the multidimensional vibration isolator, the dynamic equation in Eq. (23) is rewritten as state-space formulation, which is given by where the system matrix, contact force transfer matrix, and input matrix are, respectively, given by For exploring influential characteristics of vibration isolation capability with joint clearance changing, meanwhile, avoiding influence of unknown frequency on vibration isolation performance, the harmonic acceleration with certain frequency is adopted, which is described as € x 1 ¼ 5 sin 4pt ð Þþ3 cos 1:5pt ð Þ. The time response of the isolator with different values of joint clearance in all directions can be solved by Runge-Kutta method, which is shown in Fig. 3. Figure 3 depicts the output velocity in time domain of moving platform under harmonic excitation in all directions. For showing clearly in the picture, the curve of excitation is not presented. The changing trend with or without joint clearance is generally consistent. As the value of joint clearance increasing, the output velocity of moving platform increases simultaneously. The phenomenon indicates that the vibration isolation performance deteriorates significantly due to the existence of joint clearance. The contact forces are applied to the dynamic equation of the isolator by Jacobian force matrix, which can counteract a part of damping force, that is, the main reason of vibration isolation performance degeneration. In the maximum value of velocity, the fluctuation is observed clearly. The reason can be explained that the maximum of kinetic energy results in larger normal and tangential contact forces. Because of the random motion between journal and bearing, the contact position and contact force value are different.
In order to evaluate influence degree of joint clearance on vibration isolation capability, the velocity root-mean-square (RMS) value is calculated and employed as the index. Table 3 illustrates output velocity RMS values of moving platform. The velocity excitation RMS value has been calculated by integrating the acceleration, which is given by 0.534 ms À1 or rads À1 À Á . Compared to input excitation, the output velocity of moving platform has been reduced significantly, which implies the effectiveness of the proposed multi-dimensional isolator with joint clearance. The maximum changing rates have been calculated in x; y; z directions and around x-axes, which are 80, 31.37, 48.15, and 53.33%, respectively. The changing rates indicate the sensitivity of isolating performance to joint clearance. According to the results, the vibration isolation performance is most sensitive in x direction, and most insensitive in y direction to joint clearance.

Frequency response of the isolator with joint clearance
Assumed the proposed isolator fixed on the vehicle, the fixed platform experienced stochastic excitation due to road roughness. Additionally, stochastic excitation illustrates dynamic characteristics of the isolator clearly because of exhibiting sufficient broadband frequency information. The road random excitation is given by [37] _ where x 2 t ð Þ is road roughness amplitude, G x is road roughness coefficient, v e is vehicle velocity, n 0 and n 1 are the lowest cut-off frequency and reference frequency, respectively, and w is white Gauss noise with zero mean value. The parameters and values in Eq. (26) are shown in Table 4 Defining T ¼ 0 4Â4 ; I 4Â4 ½ and conducting Fourier transformation on Eq. (24), frequency response of the isolator is obtained, which is given by If joint clearance is not taken into account, i.e., joint contact force is neglected, Eq. (27) is transformed to ideal passive control system, which is expressed as follows: where x 2 x ð Þ is Fourier transformation of road random excitation. Figure 4 depicts the output displacement of moving platform with joint clearance in frequency domain. For illustrating the influence of joint clearance on vibration isolation capability in frequency domain clearly, the curve without joint clearance and damping coefficient is exhibited simultaneously. Combined with the parameters in Table 2, the fourth-order natural frequencies are calculated as f 1 = 3.82 Hz, f 2 = 9.53 Hz, f 3 = 16.98 Hz, and f 4 = 62.12 Hz. The abscissas of resonance peaks c ¼ 0; c e ¼ 0 ð Þin Fig. 4 are equal to the natural frequencies exactly. This phenomenon implies the correctness of analysis. Passive control curve has been added for illustrating the effectiveness of the isolator in frequency domain. It is shown that all resonance peaks have been inhibited in passive control mode. As the value of joint clearance increasing, the inhibiting effect of first-order resonance peak degenerates, and the first-order resonance peak shifts to lower frequency range, except in z-direction. The phenomenon indicates the first-order resonance peak is sensitive to joint clearance. Other resonance peaks do not change obviously. The reason can be clarified by the varying of Jacobian matrix. The existence of joint clearance can be regarded as the increment of equivalent mass or decrement in equivalent stiffness, respectively. The proposed isolator is passive control system because of without energy input to the isolation system, and low-frequency isolation performance is limited. Due to the existence of joint clearance, additional contact force is transmitted to the isolation system. The parameters of spring and viscous dampers cannot be regulated properly; furthermore, the output damping force is not optimal in real time. Thus, the resonance peaks are not controlled significantly.

Vibration reduction experiment of the isolator
The proposed isolator based on 4-PUU parallel mechanism is manufactured and assembled. The journal of universal joint is pin with diameter 4 mm. For assembling the pin in the bearing, the diameter of the bearing is 4 þ0:5 þ0:1 mm. Thus, the isolator with joint clearance is obtained. This work is a preliminary discussion of multi-dimensional vibration isolator with joint clearance. Thus, the effectiveness of vibration isolation capability with joint clearance is concerned first and foremost. Therefore, the variable joint clearance is not designed. The schematic diagrams of four universal joints are presented in Fig. 5. Figure 5 describes arrangement of the proposed universal joints. For the rotational axes of universal joint U 1 , one is along z-axis, another is in x À y plane with 45°offset value from z-axis. For the rotational axes of universal joint U 2 , one axis is along x-axis, another is in x À y plane with 45°offset value from zaxis. For the rotational axes of universal joint U 3 , one axis is along y-axis, another is in x À y plane with 45°o ffset value from z-axis. For the universal joint U 4 , one axis is along z-axis, another is in x À y plane with 45°o ffset value from z-axis. Four groups of springs and viscous dampers are regarded as prismatic pairs. The main parameters and values of the prototype are same as Table 2.
A positive vibration isolation strategy is conducted. The vibration excitation is supplied by eccentric motor, which exhibits voltage governor with the range 0 V; 250 V ½ . Two three-axis accelerometers are arranged on moving and fixed platform, respectively. The three-axis accelerometers connect to dynamic signal acquisition. The vibration signals are analyzed and stored in the computer. The diagram of principle and test rig of positive vibration isolation experiment is shown in Fig. 6.
Regulating input voltage and coefficient of damper are 125 V and 2 Nsm À1 , respectively. The sampling frequency is 5120 Hz. After first sampling, regulating input voltage as 140 and 155 V, other parameters remain the same, repeating the experimental process and recording the data. Figure 7 describes the vibration isolation capability of the isolator with joint clearance in time domain. The three-axis accelerometers 1 and 2 record the accelerations of rigid foundation and eccentric motor, respectively. It is shown significantly that the proposed vibration isolator with joint clearance can  Due to the limitation of experimental devices, angular accelerometer is not available, then the rotating characteristic around x-axis is not presented.
Generally, the moving platform connects to the isolated object, and the mass of isolated object is different. For describing the vibration isolation performance with joint clearance and different mass, the experiment is conducted by adding counterweight 5, 10, and 15 kg on moving platform in different voltage, respectively. Figures 8, 9, and 10 depict the vibration isolation performance with different counterweights in frequency domain. As the voltage increases, the rotational speed of eccentric motor raises at the same time. Due to stepless speed regulation mode of eccentric motor, the relation between voltage and rotational speed is hard to be obtained. From Figs. 8, 9, and 10,   decrement in resonance peaks is limited because of passive control characteristics. The phenomenon of frequency multiplication appears because of the nonlinear characteristics of excitations and the isolator. In addition, due to the disturbance and the power frequency of current is 50 Hz, the frequency multiplication of 50 Hz exists. Compared to the amplitude value of excitation, the frequency multiplication of 50 Hz can be neglected. In order to investigate the vibration isolation capability with different counterweights, the RMS values of acceleration in different directions are calculated and compared. Table 5 describes the RMS values of acceleration in xÀ; yÀ and z-directions with different counterweights. As the counterweights of moving platform increase, the RMS value of output acceleration increases simultaneously, which indicates the vibration isolation capability getting worse. The experimental result can verify the theoretical research, which has been reported by Gao et al. [9].

Conclusion
The main goal of this work is establishing dynamic model of multi-dimensional vibration isolator based on 4-PUU parallel mechanism with joint clearance and addressing influential characteristics of joint clearance on vibration isolation capability. The kinematic and dynamic equations are deduced by geometric relation and Lagrange approach, respectively. The contact forces are expressed by Lankarani-Nikravesh model and modified Coulomb friction force model, respectively, which are applied to isolating system by force Jacobian matrix.
The vibration isolation performance with different values of joint clearance is addressed in time and frequency domain, respectively. The proposed isolator mitigates multi-dimensional vibration significantly with joint clearance. The isolation capability with joint clearance is most sensitive in x-direction, and most stable in y-direction. Because of the maximum value of kinetic energy, the contact phenomenon is much more significantly. Hence, fluctuation generates near the maximum value of velocity obviously. As the value of joint clearance increases, the first-order resonance peak shifts to lower frequency range. For the proposed isolator, the existence of joint clearance can be regarded as the increment in equivalent mass or decrement in equivalent stiffness, respectively. The additional contact forces decrease a part of damping force, which is a main reason of vibration isolation performance degeneration.
The proposed isolator is manufactured and tested. The vibration isolation performance is effective in time and frequency domain, respectively. As the counterweights of moving platform increase, the vibration isolation capability becomes worse.
Funding This research is supported by Shandong Provincial Natural Science Foundation, China (ZR2021QE203) and National Natural Science Foundation of China (52075294).
Data availability The data that support the findings of this study are available from the corresponding author upon reasonable request.