gear automation and design revised

Transcription

1 gear automation and design revised Backlash_lookup.5 Backlash_lookup.9 Backlash_lookup Backlash_lookup. Backlash_lookup.9 Backlash_lookup Backlash_lookup. Backlash_lookup.8 6 numerical example... Table page 68 Lynwander, Gear Drive Sysytems for backlash at various diametral values DP diametral_pitch middle of course range... 6 per Shigley Table 9.3 deg N P N G 3 pressure_angle_at_pitch_radius number_of_pinion_teeth number_of_gear_teeth addendum DP.5 dedendum DP Table 9. Sigley for pressure angles,.5 and 5 ************************************ end of input ************************************************************************* C N P N G DP C.5 center_distance R N G N P N G N P R G R DP P R G.5 DP R P BL Backlash_lookup BL. DP T P = T G = CP BL BL = allocate / backlash to each P & G DP R P CP CP.34 N P BL T P T DP P.5 T G T P T G.5 T P.34 R root_p R P dedendum R root_p.875 root_radius_pinion dedendum.5 R root_g R G dedendum R root_g.375 root_radius_gear R add_p R P addendum R add_p. addendum_radius_pinion addendum. R add_g R G addendum R add_g.6 addendum_radius_gear inv tan CTT T P involute_function circular_tooth_thickness_at_pitch_radius inv involute_angle_at_pitch_radius.854 deg /8/6

2 geometry to determine points on involute between root and addendum R, B CTT CTT R R B( R RB A RB A figure. page 3 Lynwander reversed and rotated - values at pitch radius figure. page 3 Lynwander reversed and rotated CTT A = here consider varying from R to a value > design angle = CTT = circular_tooth_thickness = involute_of_ = pressure_angle_design B = A = involute_of_design_pressure_angle R B R = pitch_radius = cos R = R B cos Pinion geometry. R B_P R P cos = circular_range_variable base_radius_pinion R B_P.94 R root_p.875 want to go from base to addendum radius in say points R B_P _add_p acos _add_p 3.3 deg R add_p R root_p R B_P N N = number_of_points_along_involute _add_p i N i ( i ) N involute_angle_at_local_radius inv increment_of_pressure_angle let's put base radius to addendum in :n+ R B_P radius_on_involute R _Pi cos i CTT i R _Pi CTT R P i thickness_at_location /8/6

3 number of teeth N P B_del ( ) angular increment for teeth (offset to angle B) N P thickness B = angle_relative_to_tooth_center = r,l = right,left side of tooth radius_at_location Bl_P i CTT i R _Pi B_del Br_P i B_del CTT i R _Pi i = range_variable_along_involute = tooth_number adding a point at the root radius so we need to add two values of R root and one each of Br and Bl. these are the first points R _P R root_p Bl_P Bl_P Br_P Br_P now i needs to go from to N + N N i N put into vector of R and for polar plot effectiveness N+ points from i up and down. radius up across then down across connecting the dots... R_plot_P R i( N) ( ) _Pi R_plot_P i( N) [ ( ) ] put right data first alternate by and "right" and "left" in sequence R _PN i B_plot_P Br_P B_plot_P Bl_P i( N) ( ) i i( N) [ ( ) ] Ni bug rows( R_plot_P) R_plot_P R_plot_P B_plot_P B_plot_P close curve krhs bug bug Xr_P i Yr_P i convert to X, Y coordinates to allow a closeup of one tooth, cannot restrain in polar plot R _Pi cos Br_P i R _Pi sin Br_P i Xl_P i Yl_P i R _Pi cos Bl_P i R _Pi sin Bl_P i.5 closeup of tooth gear outline root radius base radius /8/6

4 Gear geometry reset reset to avoid extra values in gear R B_G R G cos base_radius_gear R B_G.4 N 4 R root_g.375 want to go from base to addendum radius in say points retained separate number N R B_G _add_g acos _add_g 8.4 deg R add_g N N = number_of_points_along_involute R B_G R root_g R B_G _ded_g acos _ded_g.86i deg R root_g _root_g if R root_g R B_G _ded_g i N i _root_g if root is > base, start involute at root not base. to allow the pposite, insert extra point as in pinion. _add_g _root_g N ( i ) increment_of_pressure_angle inv involute_angle_at_local_radius R B_G radius_on_involute R _G cos CTT CTT i R _Gi i R thickness_at_location G N G B_del ( ) angular increment for teeth (offset to angle B) N G thickness B = angle_relative_to_tooth_center = r,l = right,left side of tooth radius_at_location CTT i = range_variable_along_involute i CTT i Bl_G B_del Br_G B_del i R i _Gi R = tooth_number _Gi adding a point at the root radius, R root is max(r B,R root ) and one each of Br and Bl. these are the first points. in either case, added point is R root_g. R _G R root_g put into vector of R and for polar plot effectiveness Bl_G Bl_G Br_G Br_G now i needs to go from to N + N N i N N points from i up and down. radius up across then down across connecting the dots... R_plot_G R i( N) ( ) _Gi R_plot_G i( N) [ ( ) ] put right data first, alternate by and "right" and "left" in sequence B_plot_G Br_G B_plot_G i( N) ( ) i i( N) [ ( ) ] bug R _GN i rows( R_plot_G) R_plot_G R_plot_G B_plot_G B_plot_G bug bug Bl_G Ni close curve convert to X, Y coordinates to allow a closeup of one tooth, cannot restrain in polar plot Xr_G i Yr_G i R _Gi cos Br_G i R _Gi sin Br_G i Xl_G i Yl_G i krhs R _Gi cos Bl_G i R _Gi sin Bl_G i 4 /8/6

5 9.5 6 closeup of tooth R_plot_G B_plot_G geometry to shift gear to appropriate center 3 relationships to shift a circle R, from center at, to C, atan(r*sin(/(r*cos(c)) R, R*cos(C R*sin( Rcos C Rsin R R C = angle_circle_center_rotated atan R C Rsin Rcos C shift gear a distance C, no rotation of center but rotate gear (B P ) by / circular pitch angle to mesh Gear i rows( R_plot_G) add rotation dependent on FRAME start at / -CP go to CP/?? B_shift B_shift 6 deg N G B_rot B_shift B_shift FRAME B_rot 6 deg BL B_ad_P 4R P and finally... remove BL for meshing, applying half thedistance on each of pinion and gear pinion is rotating CCW so adust BL/4*R P and gear is CW so add BL/4*RG (CCW) to gear B_shift B_rot B_ad_G = total_rotation_of_gear B_ad_G applied before translation BL 4R G 5 /8/6

6 R_plot_G_ R i R_plot_G B_plot_G B_shift B_rot B_ad_G C i i B_plot_G_ i R_plot_G B_plot_G B_shift B_rot B_ad_G C i i now add tangent to the mix... pinion is rotating ccw therefore tangent is at: R B_P gear is rotating cw therefore tangent before shift is at: R B_G and we need to translate it R tan_g R R B_G C tan_g R B_G C so tangent plot is t_plot R B_P R tan_g tan_g t_plot R_plot_G_ R_plot_P t_plot B_plot_G_ B_plot_PB_rotRB_ad_P t_plot 6 /8/6

7 X_G reset Y_G reset shift to X,Y so can get closeup of mesh in animation rows( R_plot_P) X_G R_plot_G_ cos B_plot_G_ X_P R_plot_P cos B_plot_P B_rotR B_ad_P i i i Y_G R_plot_G_ sin B_plot_G_ Y_P R_plot_P sin B_plot_P B_rotR B_ad_P i i i X_tan_G R tan_g cos tan_g X_tan_P R B_P cos Y_tan_G R tan_g sin tan_g Y_tan_P R B_P sin t_xy_pl Y_tan_G Y_tan_P X_tan_G X_tan_P.5 Y_G Y_P t_xy_pl X_GX_P t_xy_pl R.5 N P N G 3 B_rotR 9 deg 7 /8/6

8 .5. Y_G Y_P t_xy_pl X_GX_P t_xy_pl These last two figures are animated in gear mesh video revised. In animating, the variable FRAME is incremented from :, the calculations highlighted above are carried out and plotted, the plots updated and a video screen captured. 8 /8/6