Archive for May, 2014

Equating Torques and Pitch Circle Radii Within a Gear Train

Thursday, May 29th, 2014

      Last time we developed torque equations for the driving and driven gears within a simple gear train.  They are,

T1 = D1 × F

T2 = D2 × F

where, T1 and T2 are the driving and driven gear torques, D1 and D2 are the driving and driven gear pitch radii, and F is the resultant Force vector, the common factor between the two equations.

gear train exoert

      Now we’ll combine these two equations relative to F to arrive at a single equation which equates the torques and pitch circle radii of the driving and driven gears in the gear train.   This type of computation is commonly used to design gear trains to ensure they perform at a given level.

      As a first step we’ll use algebra to rearrange terms and place the two equations equal to F.    First we’ll do it for the driving gear, dividing both sides of the equation by the pitch circle radius, D1.

T1 ÷  D1 = D1 ÷ D1 × F

T1 ÷ D1= 1 × F

F = T1 ÷ D1

      In a similar fashion, we’ll do it for the driven gear by dividing both sides of the equation by the pitch circle radius, D2.

T2 = D2 × F   →   F = T2 ÷ D2

      Since F is the common term between the two equations, we can set them up as equal to each other,

F = T1 ÷ D1 = T2 ÷ D2

which means that,

T1 ÷ D1 = T2 ÷ D2

      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 T2 with respect to T1 and the ratio of D1 to D2, thus providing a mechanical advantage to the electric motor the gear train is attached to.


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 F, known as the resultant Force vector.   This simplification allows us to achieve common ground between F and the two Distance vectors of our driving and driven gears, represented as D1 and D2.   We can then use this commonality to develop individual torque equations for both gears in the train.

gear train exoert

      In this illustration we clearly see that the Force vector, F, is at a 90º angle to the two Distance vectors, D1 and D2.   Let’s see why this angular relationship between them is crucial to the development of torque calculations.

      First a review of the basic torque formula, presented in a previous blog,

Torque = Distance × Force × sin(ϴ)

      By inserting D1, F, and ϴ = 90º into this formula we arrive at the torque calculation, T1 , for the driving gear in our gear train:

T1 = D1 × F × sin(90º)

      From a previous blog in this series we know that sin(90º) = 1, so it becomes,

T1 = D1 × F

      By inserting D2, F, and ϴ = 90º into the torque formula, we arrive at the torque calculation, T2 , for the driven gear:

T2 = D2 × F × sin(90º)

T2 = D2 × F × 1

T2 = D2 × F

      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.

gears and torque

      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 F1 to the driven gear Force vector F2, then use that linking equation to develop a separate torque formula for each of the gears in the train.

      We learned in the previous blog in this series that F1 and F2 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 ϴ with respect to their Distance vectors D1 and D2.   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:

F = [F1 × sin(ϴ)] – [F2 × sin(ϴ)]

where F is called a resultant Force vector, so named because it represents the force that results when the dead, or inert, weight that’s present in the resisting force F2 cancels out some of the positive force of F1.

      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.

mechanical engineering expert witness

      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, D1, and the driven gear Distance vector, D2.   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 F1 and F2 are aligned.

      As we learned last time, the Force vector, F1, results from the torque that’s created at the pivot point located at the center of the driving gear.   This driving gear is mounted on a shaft that’s attached to an electric motor, the ultimate powering source behind the torque.  F1 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 F2, a resisting force, which extends along the same line of action, but in a direction opposite to that of F1.   These two Force vectors butt heads, pushing back against one another.

      F2 is essentially a negative force manifested by the dead weight of the mechanical load of the machinery components resting upon the shaft of the driving gear.   Its unmoving inertia resists being put into motion.   In order for the gears in the gear train to turn, F1 must be greater than F2, in other words, it must be great enough to overcome the resistance presented by F2.

      With the two Force vectors pushing against each other along the line of action, the angle ϴ between vectors F2 and D2, is the same as the angle ϴ between F1 and D1.   Next time we’ll use the angular relationship between these four vectors to develop torque calculations for both gears in the gear train.