Program Idler Gear Center Distance (Intersection of Circles) Introduction

Transcription

1 Program Idler Gear Center Distance (Intersection of Circles) Introduction Finding the two possible idler gear center locations is a recurring problem when designing gear trains. The problem is really one of finding the intersections of two circles. The radii of the circles are the center distances between driver and idler and between driven and idler. The usual solution from analytic geometry involves the simultaneous solution of the quadratic equations for the two circles. (x-xc1) 2 +(y-yc1) 2 = c1 2 (x-xc2) 2 +(y-yc2) 2 = c2 2 Where: xc1 = x coordinate to center #1 yc1 = y coordinate to center #1 xc2 = x coordinate to center #2 yc2 = y coordinate to center #2 c1 = center #1 to intersections c2 = center #2 to intersections x = x coordinate to intersections y = y coordinate to intersections Methods for finding the two intersections (two sets of values for x and y) from these equations requires a considerable amount of algebraic manipulation. An alternative solution uses trigonometry and is quite straight forward. Since TK Solver has the ability to add and subtract with number pairs and convert between polar and rectangular coordinates it further simplifies the procedure. In the following steps the usual equations are given first and then the TK equation. (The usual equations can also be used in TK if desired.) Step 1) ix = xc2-xc1 horizontal dist between centers #1 & #2 iy = yc2-yc1 vertical dist between centers #1 & #2 TK Step 1) (ix,iy) = (xc2,yc2)-(xc1,yc1) Step 2) c = sqrt(ix2+iy2) center distance, centers #1 to #2 Step 3) B = arctan(iy/ix) angle: x-axis to CL between #1 & #2 (adjust angle B for proper quadrant with respect to co-ordinate system centered on center #1) TK Step 1&2) (ix,iy) = ptor(c,b)

2 UTS Integrated Gear Software Step 4) D = arccos((c 2 +c1 2 -c2 2 )/(2*c*c1)) TK Step 4) 2*c*c1*cos(D) = c 2 +c1 2 -c2 2 angle: #1, #2 CL to idler CL's about center #1 Step 5) ix1 = c1*cos(b-d) center #1 to 1st intersection, horizontal iy1 = c1*sin(b-d) center #1 to 1st intersection, vertical ix2 = c1*cos(b+d) center #1 to 2nd intersection, horizontal iy2 = c1*sin(b+d) center #1 to 2nd intersection, vertical TK Step 5) (ix1,iy1) = ptor(c1,b-d) (ix2,iy2) = ptor(c1,b+d) Step 6) x1 = xc1+ix1 x coordinate to idler center #1 y1 = yc1+iy1 y coordinate to idler center #1 x2 = xc2+ix2 x coordinate to idler center #2 y2 = yc2+iy2 x coordinate to idler center #2 TK Step 6) (x1,y1) = (xc1,yc1)+(ix1,iy1) (x2,y2) = (xc1,yc1)+(ix2,iy2) These steps will find the two possible idler locations with the centers c1 and c2 located in any quadrant as long as the proper angle is found in step 3 (TK step 3 finds the quadrant automatically). Example If you are using UTS TK Model for the first time you may wish to run the following example. In this example all centers are in the first quadrant. In the wizard data entry form, enter the coordinates of the main gear centers #1 and #2 along with the required center distances to the idlers, as shown in Figure 1. The solved model is shown in Report 1. 2

3 Idler Gear Center Distance (Intersection of Circles) Fig. 1 Report 1 None 3

4 UTS Integrated Gear Software Coordinates to Center #1 x co-ordinate y co-ordinate Center #1 to Idler Center Center #1 Quadrant Coordinates to Center #2 x co-ordinate y co-ordinate Center #2 to Idler Center Center #2 Quadrant CENTER DISTANCE: Ctr #1 to Ctr #2 CENTER DISTANCE: Ctr #1 to Ctr #2 IDLER CENTER LOCATIONS: Location #1 x coordinate 1 y coordinate 1 Idler #1 Quadrant IDLER CENTER LOCATIONS: Location #2 x co-ordinate 2 y co-ordinate 2 Idler #2 Quadrant AUXILIARY DATA x distance: Ctr #1 to Ctr #2 y distance: Ctr #1 to Ctr # in in in Three in in in One in in in One in in Two in in 4

5 Idler Gear Center Distance (Intersection of Circles) Angle: x-axis to CL, Ctr #1 & Ctr #2 Angle: CL to idler CL's about Ctr #1 x distance: Ctr #1 to idler loc #1 y distance: Ctr #1 to idler loc #1 x distance: Ctr #1 to idler loc #2 y distance: Ctr #1 to idler loc #2 We have a complete set of data for both idler locations. Note that there are no caution messages in the error message area at the top of the report. (If you wish to see the error conditions which are checked they are in procedure function "msg".) A plot of the centers is available. See Figure 2 for a plot of this example. Fig deg deg in in in in 5