Research on multidimensional loading device of material mechanical test

A multidimensional loading device for the material mechanical test based on Stewart Platform was proposed and manufactured in this paper. A determination method of mechanism parameters toward specific engineering requirements was analysed via inverse kinematics and contact interference boundary condition of mechanism components. A set of optimized mechanism parameters was achieved and then the pose space was obtained. Meanwhile, five repeated quasi-static standard tensile tests were performed on the multidimensional loading device and the standard tensile testing machine, respectively. The average yield strength and ultimate strength of the specimen achieved from the different testers were both merely 1.13%. Results indicated that material mechanical multidimensional loading could be conducted by the parallel mechanism.


Introduction
To guarantee engineering structure or machinery working in order, components should possess enough capacity to bear the loads on them [1].Mechanics of material mainly focuses on deformation, force, and failure analysis.It provides related theory for optimizing component design, such as strength, stiffness, and stability [2].Majority of formulas in mechanics of material were derived from mechanical tests and were established with many hypotheses and simplifications.Therefore, errors were inevitably generated and may deeply affect the validity and accuracy of the theoretical calculation.Yield strength derived by Von Miss were 15% greater than that obtained by Tresca in the pure torsional experiment [3].Furthermore, many failure analyses, in practical application, could not be conducted by theoretical calculation [4].Consequently, to pursue more accurate mechanical properties and the effective utilization of materials, a considerable amount of different material mechanical experiments were done.In addition to standard single axis tests, two-step uniaxial tests [5], plane biaxial tests [6], combined tension/torsion strain path change tests [7][8] were carried out.However, multidimensional loading experiments that are more in line with the practical engineering application were rarely performed.The main reason that these experiments were not conducted was the lack of multidimensional loading device (MLD) [9].
In this work, a MLD for material mechanical test based on Stewart Platform was proposed and manufactured.A brief architecture description and the inverse kinematics of the MLD were presented at the beginning and then the detailed method of designing and optimizing mechanism parameters were given.Finally, five repeated standard quasi-static tensile experiments were conducted on the MLD and Zwick Z100, respectively.Results indicated that it was feasible to perform material mechanical experiments on the MLD.

Architecture description of the MLD
The prototype of the MLD, shown in fig. 1, was composed of a fixed base, a mobile platform, and six identical motor-driven kinematic chains.Each limb, possessing a ball screw prismatic joint, connected the base with a Hook hinge and a spherical joint to the mobile platform.eq.( 1) described the coordinates of the spherical joint center B i (i=1~6).(2)

Inverse kinematics analysis
To make the description of the moving Cartesian frame's pose, which was expressed in the fixed Cartesian frame, consistent with the rotational motion along axes of the fixed Cartesian frame, the RPY rotational matrix was selected, see eq.( 3).
Where, c=cos, s=sin.φ1, φ2 and φ3 were rotational angles of the moving Cartesian frame corresponding to the X, Y and Z axes, respectively.φ φ φ φ φ φ φ φ φ φ φ φ (3) The inverse kinematics of this MLD was the solution of the limbs' length (|P i B i |) based on the mobile platform' pose.Take any limb, in fig.2(a), as a research object.According to the vector algebra, eq.( 4) could be expressed as below. ( Substituting eq.( 4) into eq.(5), the vector equation of the inverse kinematics was given bellow.

Determination of the MLD structure parameters
The workspace of parallel mechanism including the position and orientation spaces was used to determine the structure parameters.Considering position and orientation simultaneously could result in a considerable calculation, and make it impossible to intuitively describe the pose space.A two-step method of the pose solution was introduced to design the MLD parameters.Firstly, selected a set of initial structure parameters based on the engineering performance indexes, and determined the interference conditions of the parallel mechanism.Secondly, using the inverse kinematics to obtain the position workspace based on the previous results; unless the initial parameters could meet the requirements of the engineering indexes, the program went down to the next step or went back.Thirdly, selected a subspace from the position workspace to verify whether or not the orientation workspace could meet the engineering indexes; the final structure parameters was obtained only if the orientation conditions was met.Detailed process of the two-step method was described in fig. 3.

Contact interference boundary conditions
Contact interference conditions of the mentioned MLD were composed of three kinds of interference conditions: the angle range of joints, limbs' length range, and contact interference conditions of components.
According to fig. 4, the spherical joint interference condition could be expressed by eq.( 7).Note here that vector n denoted the unit vector along the z axis of the mobile Cartesian frame.
Similar to the spherical joint, the interference conditions of the Hook hinge were given by eq.( 8).
Where N pi represented the unit vector along the connecting line of two adjacent Hook hinges' rotary center, see fig. 1.
Each length range of limbs could be calculated from eq.( 9), as bellow.
According to the structure of the MLD, the contact interferences only could occur at the end of the motors or at positions where limbs were close to the upper grip.Since the principle of these two kinds of interferences was the same, only the interferences at the end of the motors were analysed in this paper.To simplify the interference analysis, assuming that the motor was a cylinder whose radius was r.Therefore, the contact interference analysis between two motors was transferred to the problem of the contact of two cylinders.Let dmin be the minimum distance between two cylinders, d be the length of the common normal line of these two cylinders' centreline, shown in fig. 5.The radius of the circumscribed circle of the motor, denoted by r 1 , could be obtained by eq.(10).Let s be the side length of the motor and r'=r 1 .If the thick line segment a i b i and a i+1 b i+1 , shown in fig.5, did not contact with each other, adjacent motors would not occur contact interference.Let m be the vector along a i b i .It could be expressed by the parametric equation, see eq.( 12).Meanwhile, the feet of perpendicular c i and c i+1 could be obtained by solving eq.( 13).

Structure parameters and workplace of MLD
Based on the inverse kinematics and the interference boundary condition, structure parameters and the position workspace were obtained.Optimized parameters were summarized in table 1; different views of the position workspace were plotted in fig.7, respectively.Different from the serial mechanism, the position and orientation workspaces of the parallel mechanism possessed strong coupled effects with each other.Consequently, the pose workspace usually be expressed by specific engineering

Pose workspace parameters
-8～8 performance indexes.As for this MLD, the pose workspace could be described that the orientation space should be as large as possible under the premise of meeting the requirements of the position space, see table 2.

Material mechanical experiments
Since the mobile platform of this MLD had six degrees of freedom, it could act more general loads on the material specimen, such as F x , F y , F z , M x , M y , and M z .As for this mechanism, as long as the MLD possessed enough stiffness, strength, motion space, and measurement accuracy, it could be used as a material testing machine.
Translations and rotations along the three axes of the fixed Cartesian frame were the same in mechanism theory.Therefore, only tensile tests were performed to verify the loading capacity in this paper.In this work, specimen was manufactured using AISI 1045 steel widely used in practical application.All samples were referenced      Mechanical properties derived from the two testers were compared and summarized in table 3.
Errors of the elastic segment in fig.11 and fig.12 were caused by that extensometer was not installed on the MLD while Zwick Z100 possessed an extensometer.Meanwhile, table 3 manifested that the maximum errors of σ s and σ b were merely both 1.13%.Therefore, results indicated that it was feasible to conduct material mechanical experiments on the MLD.

Summary
Aiming at the defects of the traditional material testers, on which the more general loads could not be applied, a new material mechanical MLD based on the parallel mechanism was proposed.Meanwhile, pure tensile experiments of AISI 1045 steel were performed on the MLD and on the standard tensile tester Zwick Z100, respectively.Main conclusions could be drawn as follows: (1) Detailed contact interference boundary conditions were analysed which could be applied to the design of the parallel mechanism; (2) The method of designing and optimizing structure parameters of the parallel mechanism was proposed based on the inverse kinematics and the contact interference boundary condition; (3) Errors of the average yield and ultimate strengths obtained from the two devices were merely both 1.13%.
Results indicated that the MLD could meet the demands of material mechanical testing machine.It was feasible to perform material mechanical experiments via parallel mechanism.

Figure 3 .
Figure 3. Two-step method of designing the MLD parameters.
It could be divided into two main situations according to the relative position of the two thick lines: (1) If d＞ 2r', motors would not contact each other; (2) if d≤2r', contact interference could be subdivided into four cases, shown in fig. 5. Detailed analysis of the cases, in fig.6, were summarized:

Figure 6 .
Figure 6.Flow chart of the analysis of contact interference.

(
Where a i , b i , a i+1 and b i+1 were the vectors described in the fix Cartesian frame.
Unless k i ∈ [0, 1], point m i lied between the points a i and b i ; unless k i+1 ∈ [0,1], point m i+1 lied between the points a i+1 and b i+1 .Let b i M i+1 be the perpendicular line of a i+1 b i+1 .m i+1 could be derived from eq. (

Figure 7 .
Figure 7. Different views of the position space.

Table 1 .
Structure parameters of the MLD.

Table 2 .
Pose workspace of the MLD.

Table 3 .
Comparison of mechanical properties derived from MLD and Zwick Z100.