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Pocket Guide to
SIZING FORMULAE
INTRODUCTION
This article documents the equations of Motion used in CT
SizingSoftware
(CTSS) version 1 to perform motor sizingcalculations for
various
mechanical drive systems.Most formulae are analytical,havingbeen
derived
from first principles,and would benefit fromempirical
confirmation.
COUPLINGInput Parameters:
J = inertiaof coupling
η = efficiency
TorqueQuadrants 1 &3: Motoring
Quadrants 2 &4: Braking
GEARBOX
Input Parameters:
J = inertiaof gearbox
r = reduction ratio
η = efficiency
TorqueQuadrants 1 &3: Motoring
Quadrants 2 &4: Braking
LEAD SCREW
Input Parameters:
m = mass of table +load
θ = incline angle
m b = counter balance mass
µ = coefficient of friction between table and bed
J s = lead screw inertia
p = pitch (turns/m)
F pl = preload force (to be added to thrust load)
F tl = trust load
η = lead screw efficiency
Inertia:
Equivalent inertiaof table and load J L
TorqueQuadrant 1: Motion up incline,motoring,+veω, +ve T
Quadrant 2: Motion up incline,braking,+veω, -ve T
Quadrant 3: Motion down incline,motoring,-veω, -ve T
Quadrant 4: Motion down incline,braking,-veω, +ve T
MISCELLANEOUS INERTIAInput Parameters:
J = inertiaof coupling
η = efficiency
TorqueQuadrants 1 &3: Motoring
Quadrants 2 &4: Braking
BELT &PULLEY
Input Parameters
D 1 = diameter of motor pulley
J 1
= inertiaof motor pulley
D 2 = diameter of driven pulley
J 2 = inertiaof driven pulley
D 3 = inertiaof idler pulley
J 3 = inertiaof idler pulley
m = inertiaof driven pulley
Lb = length of belt
η = efficiency
Inertia
Equivalent inertiaof belt and pulley systemJ bp (as
seen frommotor)
Mass of belt, M = Lb m
TorqueQuadrants 1 &3: Motoring
Quadrants 2 &4: Braking
CYLINDERDRIVE
Input Parameters
J 1 = inertiaof cylinder 1
J 2
= inertiaof cylinder 2
J 3
= inertiaof cylinder 3
J 4 = inertiaof cylinder 4
F 1 = thrust load
d = diameter at which thrust load acts
T f = additional friction loss torque
η = efficiency
Inertia
Total inertia,J T
=J 1
+J 2
+J 3
+J 4
Torque
Quadrant 1: Motoring,+veω, +ve T
Quadrant 2: Braking,+veω, -ve T
Quadrant 3: Motoring,-veω, -ve T
Quadrant 4: Braking,-veω, +ve T
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RACK AND PINION
Input Parameters
D = pitch circle diameter of pinion
J = pinion inertia
m r = mass of rack
F 1 = thrust load
D 3 = inertiaof idler pulley
m = mass of load
Ø = incline angle of load
µ = coefficient of friction between rack &guide
η = efficiency
Inertia
Equivalent inertiaof table and load,J rp
Quadrant 1: Motion up incline,motoring,+veω, +ve T
Quadrant 2: Motion up incline,braking,+veω, -ve T
Quadrant 3: Motion down incline,motoring,-veω, -ve T
Quadrant 4: Motion down incline,brakin,g-veω, +ve T
CONVEYOR
Input Parameters
D 1 = diameter of motor roller
J 1
= inertiaof motor roller
D 2 = diameter of driven roller
J 2
= inertiaof drivn roller
D 3 = diameter of idler roller
J 3
= inertiaof idler roller
Ø = incline angle of load
m b = mass of belt
F 1 = thrust load (acting against belt,e.g.to push
mass off
m = mass of load
µ = coefficient of friction between belt &rollers
η = efficiency
Inertia
Equivalent inertiaof conveyor system,J conv (as seen
fromthe motor)
Torque
Quadrant 1: Motion up incline,motoring,+veω, +ve T
Quadrant 2: Motion up incline,braking,+veω, -ve T
Quadrant 3: Motion down incline,motoring,-veω, -ve T
Quadrant 4: Motion down incline,braking,-veω, +ve T
FEED ROLL
Input Parameters
N 1 = number of drive feed rolls
D 1 = drive feed roll diameter
J 1 = drive feed inertia(each)
N 2 = number of pinch feed rolls
D 2 = pinch feed roll diameter
J 2
= pinch feed inertia (each)
N 1 = number of drive feed rolls
D 1 = drive feed roll diameter
m = mass of load
µ = coefficient of friction between rolls and
materials
η = efficiency
Inertia
Equivalent inertiaof conveyor system,J t(as seen fromthe
motor)
Torque
Quadrant 1: Motoring,+veω, +ve T
Quadrant 2: Braking,+veω, -ve T
Quadrant 3: Motoring,-veω, -ve T
Quadrant 4: Braking,-veω, +ve T
