WENDELSTEIN 7- X Andrea Capriccioli Plasma Vessel vertical supports. Pendulum solution: Equivalent Friction Factor calculation KKS.-Nr.: 1- xxx Max-Planck- Institut für Plasmaphysik Dok.-Kennz.: -x000xx.0 Plasma Vessel Vertical Supports. Pendulum solution: Equivalent Friction Factor calculation 0 A. Capriccioli Rev. Datum erstellt geprüft genehmigt Bemerkungen A. Capriccioli Page 1 of 11
I. Introduction and Description of the Model... 3 II. Results... 5 III. Conclusions... 8 ANNEX 1: PV sliding Stiffness... 9 ANNEX 2: FE Model Validation... 10 ANNEX 3: friction coeff. versus Pendulum length... 11 A. Capriccioli Page 2 of 11
I. INTRODUCTION and DESCRIPTION OF THE MODEL A schematic ANSYS 3-D FE model was created to perform a parametric analysis of the friction coefficient of the Plasma Vessel (PV) vertical supports. The actual design takes into consideration the Pendulum solution for the PV vertical supports. This solution is made up of two ball-and-socket joints connected each other with a long tube. The diameter of the ball-and-socket is 80 mm and the length of the tube varies from ~680 mm (outer PV support AEA) to ~1300-1950 mm (inner PV supports AFF-AEX). The elements used in the model are SOLID45 (3-D Structural Solid) and CONTAC52 (3-D Pointto-Point Contact) for a total of about 15.500 nodes and elements (see Fig.1). In Fig.2 it s possible to see the ball-socket joint and the Figures 3 and 4 show the non linear elements between the components ball and the socket. The gap between the two spherical surfaces (null gap) is set by means of the Real Constant value (GAP). The Fig. 5 shows the constraints on the base of the lower and upper sockets. Fig.1 View of the opened ball-and-socket joint. Fig.2 A. Capriccioli Page 3 of 11
The gap direction is set by means of the real constants NX, NY, and NZ (gap direction vector). The material Property MU defines the coefficient of friction μ (=0.2 in the Ref. case). Fig.3 The nodes on the top of the upper socket are coupled in the vertical direction (Y); they have identical displacements in horizontal (X) direction (1 mm see Fig.6) and null displacement in the other horizontal (Z) direction. Uniform Vertical Forces act on all the nodes. Fig.4 The nodes on the base of the lower socket are fully fixed. Fig.5 A. Capriccioli Page 4 of 11
II. RESULTS The Fig.6 show the upper ball-and-socket joint (the socket is attached to the PV legs) shifted in X (horizontal) direction. The lower and upper socket bases remain horizontal while the internal spherical parts rotate with the fixed friction coefficient. In Fig.7 it s possible to see the vertical (Y) displacements of the lower joint. Fig.6 Fig.7 A. Capriccioli Page 5 of 11
Upper and lower joints. Axial sections (in XY plane): Views of the Vertical (Y) displ. In the present parametric analysis the parameters are the diameter of the spherical surface of contact (ball-socket) and the length of the connection element that joins the two rotation centers. When one of these parameter changes, the reaction force changes too and the ratio between the reaction forces and the total vertical load (applied to the upper nodes) gives a sort of global Equivalent Plane Sliding friction factor (EPSff). A. Capriccioli Page 6 of 11
Three values for the Pendulum Length have been selected (500, 1000 and 2000 mm): per each length there are 4 different diameters (70, 80, 100 and 120 mm). The Table 1 and the Graph 1 show the ratio between the reaction force in X direction and the vertical load: the values represent the EPSlff (the real friction coeff. assigned to the surfaces in contact is ff=0.2). In Graph 5 ANNEX 3 the EPSlff values are reported versus the Pendulum length. Spheres Diam. [mm] Pendulum Length 500 mm Pendulum Length 1000 mm Pendulum Length 2000 mm 70 0.0335 0.0167 0.0083 80 0.0409 0.0195 0.0098 100 0.0464 0.0234 0.0117 120 0.0557 0.0276 0.0139 Table 1 6.0E-02 Equivalent Plane Sliding friction factor 5.0E-02 4.0E-02 3.0E-02 2.0E-02 1.0E-02 0.0E+00 60 70 80 90 100 110 120 130 Graph 1 Sphere Diameter [mm] Pendulum Length L=500 mm Pendulum Length L=1000 mm Pendulum Length L=2000 mm For the actual vertical supports length (with ball-and-socket Diam.=100 mm and ff=0.2), the results are: - AEA port (L~0.680 m): EPSff ~ 0.0343 - AFF port (L~1.300 m) : EPSff ~ 0.018 - AEX port (L~1.950 m): EPSff ~ 0.012 With an average value = 0.0214 A. Capriccioli Page 7 of 11
III. CONCLUSIONS The curves shown in Graph 1 can be summarized, in first approximation, with the function [1f]: in the Graph 2 the comparison between the ANSYS and the formula results is reported (only for ff=0.2). It is possible to note that the differences increase when the Pendulum length decreases. Equivalent Plane Sliding friction factor (EPSff) EPSff ff K (d / 80) (1000 / L) [1f] Where: ff is the friction factor of the surfaces in contact K = 0.095 (constant) d is the diameter [mm] of the spheres in the pendulum joints L is the distance [mm] between the two spheres centers ( Pendulum Length) Equivalent Plane Sliding friction factor 6.0E-02 5.0E-02 4.0E-02 3.0E-02 2.0E-02 1.0E-02 0.0E+00 60 70 80 90 100 110 120 130 Graph 2 Sphere Diameter [mm] Pendulum Length L=500 mm Pendulum Length L=1000 mm Pendulum Length L=2000 mm Formula (L=500) Formula (L=1000) Formula (L=2000) In ANNEX 1 is reported the analysis of the PV sliding stiffness: the equivalent friction coefficient used in the calculations was 0.05. In the light of the present analysis, it is necessary decrease the constant components due to the friction. The average of the new equivalent friction factors is 0.0214 and in first approximation the constant component 13.5 kn should be divided by 2.34 (=0.05/0.0214); the other constant 11.5 kn by 1.46 (=0.05/0.0343) while the last 8.3 kn by 3.33 (0.05/0.015). A. Capriccioli Page 8 of 11
Short summary of Plasma Vessel sliding stiffness (PV as Rigid Body ) All the values are referred to a single AEU port. ANNEX 1 X Y PV 72 Young s modulus = 10 3 * E 1 kn UX=1 mm RX=16.4 kn UX=1 mm RX=2.9 kn 2.4 kn/mm + 13.5 kn 1340 mm 8.93 kn/mm 11960 kn/rad UY=1 mm RY=11.7 kn 6638 mm 3.4 kn/mm + 8.3 kn (min) 0.058 kn/mm 0.48 kn/mm (+ 11.5 kn max) AEU Port ~387 kn/mm UY=1 mm RY=3.4 kn Translation along X (CSYS=20) and results in RSYS=1: NO Friction NODE FX [N] FY [N] Ftot [N] UX [mm] UY [mm] UX in csys 20 [mm] UY in csys 20 [mm] 24646-2595.5 1.68E-05 2596-1 -2.05E-04 1 0 24653-1027.5 3162.3 3325-0.30922 0.95099 0.096 0.904 121636 2127.4 1545.7 2630 0.8089 0.58794 0.654 0.346 218493 2035.4-1478.8 2516 0.80914-0.58762 0.655 0.345 315350-1048.8-3228 3394-0.30883-0.95112 0.095 0.905 Ftot [N] 14460 U tot 2.500 2.500 Kr [N/mm]= 2429 UX = 1 mm projection of U tang. in X dir (CSYS=20) U tang U rad Radial Dir Tangential Dir Rotation around a Z and results in RSYS=1: NO Friction 24646 1.96E-05 3353.7 3354 9.96E-07 1 24653 2.75E-05 3354.4 3354 9.63E-07 1 121636 2.80E-05 3357.7 3358 9.97E-07 1 218493 4.10E-13 3351.5 3352 9.70E-07 1 315350 2.92E-05 3358.5 3359 9.78E-07 1 Kt [N/mm]= 3355 1 72 projection of U rad. in X dir (CSYS=20) 1kN
ANNEX 2: FE Model Validation The Fig.8 shows a simplified 2D model of the ball-and-socket joint. In this case the contact is between a cylinder and three points on the socket. In this case it is easy to know the contact forces and to verify the model. With the same parameters (friction coeff., diameter of the ball/cylinder and pendulum length) the results are very close to the 3D Model (and practically identical to the formula [1]). In the Graph 3 the comparison between the 2 Models is reported (only for ff=0.2). 6.0E-02 5.0E-02 4.0E-02 3.0E-02 2.0E-02 1.0E-02 0.0E+00 Graph 3 60 70 80 90 100 110 120 130 Sphere Diameter [mm] Pendulum Length L=500 mm Pendulum Length L=1000 mm Pendulum Length L=2000 mm 2D Model (L=500) 2D Model (L=1000) 2D Model (L=2000) Equivalent Plane Sliding friction factor A. Capriccioli Page 10 of 11
ANNEX 3 EPSlff values versus Pendulum length. Equivalent Plane Sliding friction factor 6.0E-02 5.5E-02 5.0E-02 4.5E-02 4.0E-02 3.5E-02 3.0E-02 2.5E-02 2.0E-02 1.5E-02 1.0E-02 5.0E-03 0.0E+00 500 Graph 5 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 Pendulum Length [mm] Pendulum Sphere d=70 mm Pendulum Sphere d=100 mm Pendulum Sphere d=80 mm Pendulum Sphere d=120 mm A. Capriccioli Page 11 of 11