Research on the vibration characteristics of the mechanical systems is a necessary step for the stable and reliable operation of high-end equipment. The 6 degrees of freedom parallel mechanism is proposed for the supporting mechanism of the antenna. First, the dynamic equations of the moving platform and branch of the mechanism are established. The closed-form dynamics of the mechanism are derived based on the Newton–Euler method. In addition, the vibration equation of the parallel antenna is established based on the vibration theory, and the relationship between the natural frequency, displacement response, and vibration frequency is obtained. Afterward, the pitch and roll poses of a 1.8 m aperture antenna are developed based on a MATLAB software simulation. The actuation forces of parallel antenna under no-load and load conditions are simulated. Finally, the natural frequencies and vibration modes of the initial position and roll (20

Compared with the series mechanism, the parallel mechanism contains multiple legs and a closed chain structure. With the characteristics of a compact structure, excellent bearing performance, low motion inertia, and small error accumulation (Huang et al., 2013), the parallel mechanism is widely used in motion simulators, shaking tables, space-docking mechanisms, satellite antennas, and other fields (Furqan et al., 2017). The Stewart mechanism is the most classic of the parallel mechanisms and contains a fixed platform, a
moving platform, and six actuation branches connecting the fixed and moving
platforms. The 6 degrees of freedom (DOF) parallel mechanism was first applied as a flight simulator (Stewart, 1965). A wave compensation platform (Zhan et al., 2020), based on a 3-SPR (where S is the spherical joint, P is the prismatic joint, and R is the revolute joint) parallel platform, is designed for marine ships with a dynamic positioning system. A 3-CPU (where C is the cylindrical joint, P is the prismatic joint, and U is the universal joint; Corinaldi et al., 2017) parallel pointing mechanism for a singularity-free path planning problem is studied. Based on the modular design concept, a new truss antenna mechanism is proposed (Guo et al., 2020), and the expansion rate performance is compared and analyzed. The rigid–flexible coupled dynamic modeling and dynamic characterization of the 4SRS

Dynamic models can improve the response velocity and accuracy of parallel mechanisms. In the design of a dynamic control scheme, it is often necessary to obtain the display form of the dynamic model. The kinematics and dynamics of the 6-SPS (where S is the spherical joint and P is the prismatic joint) parallel mechanism were modeled based on the Kane method (Wu et al., 2012). The closed-form dynamic model of the Stewart mechanism is obtained using the Kane method (Asadi et al., 2018). The dynamics and motion characteristics of a 4 DOF hybrid antenna mechanism are analyzed (Zhang et al., 2021). The dynamics of a 6 DOF micro-vibration simulator are modeled by the Kane method, and dynamic joint simulations are performed (Yang et al., 2016). A general method for the improved dynamic modeling of the Stewart platform based on the virtual working principle is proposed (Kalani et al., 2016). The dynamics model of a parallel mechanism with flexible hinges is developed based on the principle of virtual work (Jiao et al., 2019). The PU-2UPS 3 DOF parallel antenna is proposed, and the dynamics of the mechanism is analyzed (Zhang et al., 2021). The inverse dynamics model of the series–parallel dynamics simulator, based on the principle of virtual work and the kinematic model, is investigated (Hu et al., 2016). By applying a combination of the Lagrangian formulation and the Newton–Euler method, explicit compact, closed-form dynamic equations are presented (Guo et al., 2006). The improved Stewart parallel manipulator dynamic equations are generated using the Newton–Euler method, considering the rotational DOF of the pod around the axial direction (Pedrammehr et al., 2012). A systematic approach is proposed to solve the inverse dynamics of the Stewart–Gough manipulator utilizing the virtual work principle (Tsal et al., 2000). The kinetic equations of the 6–6 Stewart mechanism were established using the principle of virtual work, and the steps of establishing the kinetic model were summarized and verified using the Lagrangian method (Staicu, 2011). The representation of the screw velocity and acceleration in the object coordinate system and the spatial coordinate system is given, the Hessian matrix of the parallel mechanism is derived by combining the principle of virtual work, and the dynamics model is established (Zhao, 2015). The parameter optimization of the 6 DOF parallel vibration isolated platform is investigated (Cheng, 2019). A dynamic and vibration analysis of seismically isolated platforms based on displacement and force characteristics is carried out (Chen, 2021). A novel method for analyzing the natural frequencies of parallel robots is proposed, which focuses on the lowest natural frequencies and expresses the corresponding natural modes in the Cartesian reference system of the end-effector (Hoevenaars, 2019). The dynamic performance of the Stewart mechanism is evaluated by natural frequencies and local dynamic anisotropy indices (Bang, 2014).

Due to the complex kinematic chain structure of parallel mechanisms, its dynamic model often presents the characteristics of many coupling parameters and strong nonlinearity. Currently, the research on parallel antennas mainly focuses on configuration design and dynamic modeling. There are few reports on the vibration characteristics and harmonic response analysis combined with the specific working conditions of parallel antenna. The working environment of the antenna system is often inadequate and vulnerable to external interference. The research on vibration characteristics is a vital link for the stable and reliable operation of parallel antennas. Considering the practical working requirements of a 6 DOF antenna system, this paper establishes the closed dynamic model of the mechanism, simulates the motion and attitude of the parallel antenna, and carries out the modal and harmonic response analysis of the parallel antenna.

Considering the pitching and azimuth motion requirements of an antenna, the
self-weight of the antenna reflector, and the influence of wind load, a 6 DOF
parallel mechanism is adopted to support the position antenna. The structural diagram of the parallel antenna support mechanism is shown in Fig. 1a. The force relationship of each component of the parallel antenna under the condition of ignoring the mass and inertia of Hooke hinge is shown in Fig. 1b.

Force diagram of the parallel antenna.

Assuming that the centroid of the moving platform is located at the center
point

According to the differential equations for the linear and angular velocities of the rigid body,

Each branch of the 6 DOF antenna mechanism consists of two parts, i.e., the
actuation link

Structure diagram of the actuation branch.

The force relationship of the actuation link

Force diagram of the actuation link.

According to the force relationship of the actuation link

Force diagram of the actuation link.

According to the force relationship of the actuation link

According to the differential formula of the linear velocity and angular
velocity of the rigid body,

To obtain the closed-form dynamic model of the parallel antenna, it is necessary to integrate the dynamic models of the moving platform and each branch, complete the transformation of the intermediate generalized coordinates and the main generalized coordinates through coordinate transformation, and consider the force relationship between the moving platform and each branch.

Calculate the relationship based on the parallel mechanism velocity
Jacobian as follows:

Combined with the relevant knowledge of vibration theory, the dynamic model
of the parallel antenna described in Eq. (16) can be sorted into the form of a motion differential equation, as follows:

Since the damping does not have an effect on the natural frequency and vibration mode of the system, the undamped free vibration equation of the system is sorted out as follows:

The basic dynamic equation of the harmonic response analysis of the parallel antenna is expressed as follows:

The formula (Eq. 20) can be obtained in the parallel antenna without damping the simple harmonic excitation of the structure amplitude and frequency response relationship.

When considering the actual working requirements of a 1.8 m aperture antenna system, it is necessary to realize the

Three motion forms of the parallel antenna.

The simulation results show that the parallel antenna can achieve the motion
in a roll (

According to the requirements of antenna motion range and bearing performance, the virtual prototype modeling and assembly are completed. The prototype model of the 1.8 m aperture parallel antenna mechanism is shown in Fig. 6.

Prototype model of the 1.8 m parallel antenna.

Combined with the actual working conditions of the antenna on the swing index and swing period requirements, the parallel antenna dynamics simulation is carried out based on the analysis of the parallel antenna horizontal roll and pitch motion state. Based on Adams software, the following dynamics simulation is carried out for two kinds of motion states of parallel antenna. The structure and mass parameters of the parallel antenna are shown in Table 1, based on SolidWorks software measurements.

Mass and inertia parameters of parallel antenna.

Further considering the load carrying capacity of the actual operating link
of the parallel antenna, the parallel antenna no load and 10 t load dynamics
simulations are carried out, respectively. The simulation obtained the
actuation force of each branch of the roll (20

Dynamic simulation curves of roll (20

Dynamic simulation curves of pitch (20

According to Figs. 7 and 8, the parallel antenna, in accordance with the roll (20

Maximum actuation forces of three kinds of swing mode actuation.

By looking at the results of the simulation analysis of the parallel antenna's roll and pitch dynamics, it becomes clear that, when the parallel antenna is in roll motion, the maximum actuation force is about 7000.5 N at no load, and the maximum actuation force is about 30517.1 N at 10 t load. The results of the dynamics simulation analysis provide a reference for the selection of the parallel antenna actuator.

Vibration analysis seeks to suppress the vibration response of a system by analyzing the frequency response signal and modal analysis. The vibration characteristics of the parallel antenna are studied using the vibration module in Adams software. According to the structural characteristics and operating requirements of the parallel antenna, the vibration characteristics of the parallel antenna are simulated in a typical operating attitude with the help of the Adams vibration analysis module.

In the initial position of the parallel antenna, three sinusoidal excitation
forces with a zero-phase angle along the coordinate axis are set at the centroid of the moving platform. The output channel is set at the centroid
of the moving platform, and the tension spring dampers with the stiffness coefficients

Natural frequencies of rolling position.

According to the simulation analysis results of vibration characteristics of
parallel antenna in Table 3, the natural frequencies of each order of the
parallel antenna are related to the branch stiffness. When the branch stiffness increases, the natural frequencies of each order increase. With
the parallel antenna roll angle increase, the second-order natural frequency
becomes larger, and the rest of the order of the natural frequency is reduced. When the parallel antenna is in the roll (20

The harmonic response analysis can learn the deformation of the parallel antenna under a specific simple harmonic load. The following harmonic response analysis is performed for parallel antennas under different branch actuation stiffness and different positional conditions, respectively. We apply stress with an amplitude of 500 N in the

Displacement responses of the initial position.

Displacement responses of the roll (20

According to Figs. 9 and 10, the maximum displacement response and
corresponding frequency of the parallel antenna in the initial position and
roll (20

Harmonic response analysis results.

It can be seen from Table 4 that, when the initial position branch stiffness
is

The control strategy based on the dynamic model is helpful to improve the
dynamic performance of the parallel antenna. The inverse dynamics control
framework of parallel antenna based on workspace is shown in Fig. 11.
According to the requirements of the pitch and azimuth motion of the
parallel antenna, we set the expected motion track

Inverse dynamics control block diagram of the parallel antenna.

The control system simulation of parallel antenna mechanism is carried out based on MATLAB. The simulation time is set as 2 s, and the simulation step is set as 0.1 s. The trajectory tracking error in the parallel antenna for the completing pitch, azimuth, and yaw is shown in Fig. 12.

Trajectory tracking errors in the parallel antenna.

As shown in Fig. 12, the tracking error range of the parallel antenna, based on inverse dynamics control in the workspace, is

The closed-form dynamics equation of the 1.8 m aperture 6 DOF parallel antenna is established. Based on the vibration theory, the vibration equation of the parallel antenna system is derived, and the relationship between the mechanism's natural frequency, displacement response, and resonance frequency is obtained. The simulation verifies the pitch and roll motion attitude of the parallel antenna. And, considering the actual operation carrying capacity of parallel antenna, a parallel antenna no load and 10 t load dynamics simulation is carried out. The simulation, based on the Adams vibration analysis module, obtains the natural frequency and vibration pattern of the initial position and roll (20

The code of the paper is not publicly accessible. According to the requirements of the research project, relevant codes cannot be disclosed temporarily and are available upon reasonable request from the corresponding author.

No data sets were used in this article.

GZ and XX conceptualized the project, developed the methodology, prepared the original draft, and developed the software. JH curated the data, reviewed and edited the paper, developed the software, visualized the project, and validated the results. JG supervised the project and was responsible for the project administration. All authors have read and agreed to the published version of the paper.

The contact author has declared that none of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This research has been supported by the Natural Science Foundation of Jiangsu Province of China (grant no. BK20220649) and Key R&D Program of Jiangsu Province (grant no. BE2022062).

This paper was edited by Daniel Condurache and reviewed by three anonymous referees.