Last time we developed torque equations for the driving and driven gears within a simple gear train. They are,
× F
Fwhere, T are the driving and driven gear torques, _{2}D and _{1}D are the driving and driven gear pitch radii, and _{2}F is the resultant Force vector, the common factor between the two equations. Now we’ll combine these two equations relative to As a first step we’ll use algebra to rearrange terms and place the two equations equal to
In a similar fashion, we’ll do it for the driven gear by dividing both sides of the equation by the pitch circle radius,
Since
which means that,
Next time we’ll see how to use this equation to manipulate our gear train so that it acts as a torque converter by increasing T and the ratio of _{1}D to _{1}D, thus providing a mechanical advantage to the electric motor the gear train is attached to._{2}_______________________________________ |

## Archive for May, 2014

### Equating Torques and Pitch Circle Radii Within a Gear Train

Thursday, May 29th, 2014### Gear Train Torque Equations

Thursday, May 22nd, 2014
In our last blog we mathematically linked the driving and driven gear Force vectors to arrive at a single common vector D. We can then use this commonality to develop individual torque equations for both gears in the train._{2} In this illustration we clearly see that the Force vector, D. Let’s see why this angular relationship between them is crucial to the development of torque calculations._{2}First a review of the basic torque formula, presented in a previous blog,
By inserting F, and ϴ = 90º into this formula we arrive at the torque calculation, T, for the driving gear in our gear train:_{1 }
From a previous blog in this series we know that
By inserting F, and ϴ = 90º into the torque formula, we arrive at the torque calculation, T, for the driven gear:_{2 }
Next week we’ll combine these two equations relative to F, the common link between them, and obtain a single equation equating the torques and pitch circle radii of the driving and driven gears in the gear train. _______________________________________ |

### The Mathematical Link Between Gears in a Gear Train

Wednesday, May 14th, 2014
Last time we analyzed the angular relationship between the Force and Distance vectors in this simple gear train. Today we’ll discover a commonality between the two gears in this train which will later enable us to develop individual torque calculations for them. From the illustration it’s clear that the driving gear is mechanically linked to the driven gear by their teeth. Because they’re linked, force, and hence torque, is transmitted by way of the driving gear to the driven gear. Knowing this we can develop a mathematical equation to link the driving gear Force vector F, then use that linking equation to develop a separate torque formula for each of the gears in the train._{2} We learned in the previous blog in this series that F travel in opposite directions to each other along the same line of action. As such, both of these Force vectors are situated in the same way so that they are each at an angle value _{2}ϴ with respect to their Distance vectors D and _{1}D This fact allows us to build an equation with like terms, and that in turn allows us to use trigonometry to link the two force vectors into a single equation:_{2. }
where Fcancels out some of the positive force of _{2 }F._{1}Next week we’ll simplify our gear train illustration and delve into more math in order to develop separate torque computations for each gear in the train. _______________________________________ |

### Distance and Force Vectors of a Simple Gear Train

Monday, May 5th, 2014
Last time we examined how torque and force are created upon the driving gear within a simple gear train. Today we’ll see how they affect the driven gear. Looking at the gear train illustration above, we see that each gear has both distance and force vectors. We’ll call the driving gear Distance vector, D. Each of these Distance vectors extend from pivot points located at the centers of their respective gear shafts. From there they extend in opposite directions until they meet at the line of action, the imaginary line which represents the geometric path along which Force vectors _{2}F and _{1}F are aligned._{2} As we learned last time, the Force vector, F follows a path along the line of action until it meets with the driven gear teeth, where it then exerts its pushing force upon them. It’s met by Force vector _{1}F, a resisting force, which extends along the same line of action, but in a direction opposite to that of _{2}F. These two Force vectors butt heads, pushing back against one another._{1} F must be greater than _{1}F, in other words, it must be great enough to overcome the resistance presented by _{2}F._{2} With the two Force vectors pushing against each other along the line of action, the angle D, is the same as the angle _{2}ϴ between F. Next time we’ll use the angular relationship between these four vectors to develop torque calculations for both gears in the gear train._{1} and D_{1}_______________________________________ |