An underconstrained cable-driven parallel robot (CDPR) suspension system was designed for a virtual flight testing (VFT) model. This mechanism includes two identical upper and lower kinematic chains, each of which comprises a cylindrical pair, rotating pair, and cable parallelogram. The model is pulled via two cables at the top and bottom and fixed by a yaw turntable, which can realize free coupling and decoupling with three rotational degrees of freedom of the model. First, the underconstrained CDPR suspension system of the VFT model was designed according to the mechanics theory, the degrees of freedom were verified, and the support platform was optimized to realize the coincidence between the model’s center of mass and the rotation center of the mechanism during the motion to ensure the stability of the support system. Finally, kinematic and dynamical modeling of the underconstrained CDPR suspension system was conducted; the system stiffness and stability criteria were deduced. Thus, the modeling of an underconstrained, reconfigurable, passively driven CDPR was understood comprehensively. Furthermore, dynamic simulations and experiments were used to verify that the proposed system meets the support requirements of the wind tunnel-based VFT model. This study serves as the foundation for subsequent wind tunnel test research on identifying the aerodynamic parameters of aircraft models, and also provides new avenues for the development of novel support methods for the wind tunnel test model.
Wind tunnel-based virtual flight testing (WTB-VFT) was recently developed to combine traditional wind tunnel testing and flight testing [
Therefore, scholars from several countries have conducted various studies on VFT in wind tunnels. To understand the pitch, roll, and yaw motions of a missile, Gebert et al. [
Lowenberg et al. [
The Russian Central Aerodynamics Research Institute has developed a back-supported VFT support mechanism with three degrees of freedom and performed research on issues such as high angle of attack stall/deviation [
Zhao et al. [
Traditional forms of support devices for WTB-VFT models mainly include an inherent ball joint support, an inherent decoupled support with multiple kinematic joints, and a series of support with multiple kinematic joints outside the model, which require hard support rods or bearing rings to be connected with the model. Large support mounting holes need to be set on the surface of the aircraft model, such that the model can have a large range of motion; the support mounting hole and bearing ring significantly impact the aerodynamic shape of the model. Further, the hard support rod affects the quality of the flow field. These support devices may not fully release the three rotational degrees of freedom of the model nor realize free coupling or decoupling of the three rotational degrees of freedom.
As the hinge points at both ends of the cable function as ball pairs, the cable in this study has little interference with the airflow. Moreover, the developed underconstrained cable-driven parallel robot suspension system of a VFT model (CDPR-VFT), which can release the three rotational degrees of freedom of the aircraft model and maximally restrict the three translational degrees of freedom, has little interference with the flow field, and can realize the free coupling and decoupling motion of the three rotational degrees of freedom of the model. In this study, structural optimization design and stability analysis of the mechanism were performed, and kinematic and dynamic modeling were realized. Finally, corresponding simulation analysis and experiments were performed.
According to the requirements of the WTB-VFT model support, three rotational degrees of freedom of the model should be released and three translational degrees of freedom should be constrained. Therefore, an underconstrained mechanism support model is required. Currently, redundantly constrained supports are used in most cable-driven parallel support wind tunnel testing models [
In this study, according to the mechanism-related theory [
With regard to
To verify the number of degrees of freedom of the above mechanism, the modified G-K degrees of freedom formula [
There were eight local degrees of freedom. The AF, CD, GK, and HI rods all have one local degree of freedom: the local degrees of freedom generated by the combination of the yaw rotation pair and the corresponding branch, and the two local degrees of freedom generated by the rotation of the cross model around its own rod, that is, the rotation of another parallelogram mechanism. Therefore, if a rigid rod is used, the total number of degrees of freedom of this VFT model support mechanism is three; specifically, the cross model has the degrees of freedom of pitch, roll, and yaw, indicating that the mechanism satisfies the support requirements of the VFT model.
Because of the low interference of the cable with the airflow and the function of the spherical pairs performed by the cable hinge point, as in
In this study, a standard dynamic test model [
Because of the complex aerodynamic shape of the aircraft model, if the cable is directly attached to the fuselage shell, it is difficult to form a complete parallelogram mechanism by combining the support rod, cable, and model, as shown in
Thus, a connection mechanism was designed inside the aircraft model, as shown in
The support rod for rotation around the bearing center was optimized as shown in
To verify the stability of the above CDPR-VFT, motion simulation analysis was performed using Adams software. The dynamic model in
When the longitudinal vertical symmetry plane of the model is simplified to a straight line, that is, the cable is connected to the longitudinal axis, the support rod, two cables, and model jointly form a parallelogram mechanism, as shown in
When the symmetry plane of the model is simplified to a stepped rectangle, that is, the cable is attached to the bearing sleeve rocker, the support rod (which can be a straight rod or a stepped rod), two cables, the model, and two bearing sleeve rockers cannot form a complete parallelogram mechanism, as shown in
The structure was optimized, as depicted in
To verify the feasibility of the above model support mechanism and predict the motion of the model during VFT, kinematic and dynamic modeling were performed for this mechanism.
First, the kinematic relationship between the length of the tow cable and position of the model was determined.
As shown in
In the ground coordinate system
By further deriving
The system dynamics analysis of multiple flexible bodies considering multiple rigid bodies and elastic deformations simultaneously is generally solved by establishing a rigid-flexible coupled dynamic model of the mechanical system (Eulerian–Lagrange differential equations); specifically, based on the first type of Lagrange equation, the Lagrange multiplier method is used to consider the system constraints and solve them simultaneously. To simplify the analysis, according to the Newton–Euler equation, and based on the unconstrained motion equation of the model with six degrees of freedom, the restraining force and restraining moment applied by four cables on the model were added. The frictional dynamic forces of key components in a multibody system should be considered when building dynamic models. For example, Ahmadizadeh et al. [
The aircraft model is a six-degree-of-freedom motion body and its general nonlinear mathematical model can be described by the following differential equation:
According to the momentum theorem, momentum moment theorem, and Gothic theorem, we obtained
Through further calculations, the unconstrained six-degree-of-freedom dynamic equations for the model were determined as [
In
Based on the Euler relation, the unconstrained kinematic equations of the aircraft were established as follows:
The coordinate system shown in
The rigid body was rotated around the
The Euler coordinate system transformation was performed in this study as follows: the ground coordinate system was rotated and transformed in the order of
The cable produces elastic deformation under tension, and the cross model may exhibit small oscillations in movement, which influences the stability of the mechanism. Therefore, elastic damping of the cable was considered in the dynamic modeling, the cable was simplified as a spring model, and its tension equation [
The frictional moment model for the upper support rod rotating bearing [
The resultant force
The acting moment of a single cable on the support rod in the rod system coordinates is given by
The resultant moments
The kinematic and dynamic differential equations for the upper (or lower) support rod rotating around its center of mass are as follows:
In this study, a simulation of the cable-driven VFT model was conducted, and the model shown in
The resultant force
The constrained cable-driven kinematic and dynamic equations of the model were obtained by substituting the resultant force and resultant moment applied to the model into the unconstrained six-degree-of-freedom equations of motion of the model.
From the above equations, the nonlinear functional relationship between the state quantities
The position of the aircraft model is intrinsically related to the cable length and tension; that is, for the underconstrained, reconfigurable (driven point
For further derivation of the relationship between the change rate of cable length obtained using
When
Through the derivation of
In this study, the CDPR-VFT mechanism is an underconstrained parallel support mechanism, and the necessary and sufficient condition for the stability of the system is that the Hessian matrix based on the total stiffness of the system is positive definite [
To verify the stability of the aforementioned underconstrained mechanism, when the support mechanism is locked, the aircraft model attitude angle, cable preload, and other parameters are substituted into
Example 1: pitch angle of aircraft model
Example 2: pitch angle of aircraft model
From
To verify the correctness of the above flight dynamics model and the feasibility of the support mechanism, the Simulink module in MATLAB was used to perform a simulation analysis. To analyze the influence of friction torque on the support mechanism and that of the constraints of the support mechanism on the motion of the model, the stability of the support mechanism was first investigated, and then three different working conditions were used for comparison: the model without friction torque (CDPR-VFT), the motion with friction torque (CDPR-VFT+F friction) under the support constraint, and the unconstrained motion of the model (three degrees of freedom (3DOF) unconstrained). The main parameters of the aircraft model, support rods, and cables are presented in
Aircraft model | Mass (kg) | Inertia (kg·m^{2}) | Length (m) | Span (m) | Chord length (m) | Wing area (m^{2}) |
2.476 | 0.698 | 0.5 | 0.2522 | 0.1064 | ||
Supportrod | Upper rod centroid (m) | Upper rod length (m) | Upper rod inertia (kg·m^{2}) | Lower rod centroid (m) | Lower rod length (m) | Lower rod inertia (kg·m^{2}) |
(0,0,−1.75) | 0.1 | (0,0,−0.6) | 0.3 | |||
Cable | Length (m) | Elastic modulus (N/m^{2}) | Diameter (mm) | Damping ratio | Density (kg/m^{3}) | Preload (N) |
0.375 | 43.9 × 10^{9} | 1.2 | 0.04 | 1440 | 50 |
First, the simulation was performed after initial trimming (trimming angle of 2.1°). Without controlling surface deflection and wind speed, the head-up pitch angle of the model was considered as 1°, and the initial preload force of the cable was 40 N to examine the variation of the model and the pitch support rod.
As shown in
The above results demonstrate that owing to the elastic damping of the cable, slight oscillations occur during the movement of the mechanism, but they quickly stabilize. If the cable is tightened by a moving pair in the screw assembly of the mechanism, the preload force of the cable can be increased to reduce this slight oscillation.
Owing to space limitations, only the typical longitudinal manipulation response of the CDPR-VFT was investigated in this study.
The typical longitudinal manipulation input signal was selected, that is, when the wind speed was stabilized at 20 m/s, the elevator step of 1° was input and the step point was at 10 s. The manipulation response of the elevator is depicted in
To analyze the influence of the frictional moment of the crossbar bearing on the mechanism of the support system, two cases of the support mechanism without and with frictional moment were selected for comparison.
The simulation results in
To analyze the response characteristics of the entire support mechanism to the model, a simulation of unconstrained rotation with three degrees of freedom was introduced for comparison data to verify the correctness of the mechanism.
The variation in the center of mass displacement of the model under the longitudinal manipulation input is shown in
The change in the preload of the different cables under the above longitudinal manipulation input is shown in
Because the tension change caused by the elastic deformation of the cable in this study was relatively small compared with the aerodynamic force and control torque, the effect of increasing the cable preload on the longitudinal control response was very small. Owing to the elasticity of the cable, the mechanism inevitably had a small translation, but this allowed for the translation requirements of the supporting mechanism of the VFT model because WTB-VFT focuses more on releasing the three rotational degrees of freedom of the model.
According to the dynamic similarity criterion [
After the model was trimmed, an elevator dual square-wave maneuver was performed to excite the short-period modal motion of the model. The initial pitch angle of model trim was −4.5°, and the elevator moved from 0° to the negative direction of the control surface (i.e., the positive direction of Δ
In this study, CDPR-VFT was designed. Through mechanism design, dynamic modeling, simulation analysis, and experimental verification, the following conclusions were obtained:
The CDPR-VFT mechanism was designed according to the mechanism theory. Through the degree of freedom verification, it is proved that the mechanism can realize the free coupling and decoupling of the model with three rotational degrees of freedom and meet the requirements of the WTB-VFT model support;
Structural optimization and stability analysis of the CDPR-VFT were performed, proving that the optimized mechanical structure has good stability;
A mathematical model of the kinematics and dynamics of the cable-driven underconstrained, reconfigurable, passively driven mechanism was developed for the CDPR-VFT, which considers the cable elasticity and friction moment, on which the stiffness and stability criteria of the system are derived, and an example is provided herein. Finally, simulation and experimental methods were used to analyze the response of the model to typical longitudinal maneuvers in VFT. The results show that the CDPR-VFT model support mechanism has little effect on the motion of the vehicle model, which verifies the feasibility and validity of the mathematical model.
The CDPR-VFT developed in this study can provide conditions for conducting integrated research on aerodynamics/motion/control of aircraft models, exploring the mechanism of aerodynamics/motion coupling of aircraft models, identifying aerodynamic parameters of models, and providing a reference for designing cable-driven WTB-VFT models.