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. _______________________________________ |

## Posts Tagged ‘vector’

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

Wednesday, May 14th, 2014### 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}_______________________________________ |

### Torque Formula Symplified

Wednesday, April 2nd, 2014
Last time we introduced the mathematical formula for
We learned that the factors Vectors have both a magnitude, that is, a size or extent, and a direction, and they are typically represented in physics and engineering problems by straight arrows. In our illustration the vector for distance is represented by an orange arrow, while the vector for force is represented by a red arrow. The orange distance vector has a magnitude of 6 inches, while the red force vector has a magnitude of 10 pounds, which is being supplied by the user’s arm muscle manipulating the nut. That muscle force follows a path from the arm to the pivot point located at the center of the nut, a distance of 6 inches. Vector arrows point in a specific direction, a direction which is indicative of the way in which the vectors’ magnitudes — in our case inches of distance vs. pounds of force — are oriented with respect to one another. In our illustration the orange distance vector points away from the pivot point. This is according to engineering and physics convention, which dictates that, when a force vector is acting upon an object to produce a torque, the distance vector always points from the object’s pivot point to the line of force associated with the force vector. The angle, Next we must determine the trigonometric value for For our angle of 90 degrees we find that,
Thus the formula for torque in our example, because the
Next time we’ll insert numerical values into the equation and see how easily torque can be manipulated. _______________________________________ |

### Vectors, Sin(ϴ), and the Torque Formula

Wednesday, March 26th, 2014
Last time we introduced a physics concept known as and vectors,sin(ϴ).
In this formula, ϴ, or thay-tah, is a Greek symbol used to represent the angle present between the Force and Distance vectors as they interact to create torque. The value of sin(ϴ) depends upon the number of degrees in the angle ϴ. Sin(ϴ) can be found by measuring the angle ϴ, entering its value into a scientific calculator, and pressing the Sin button.We’ll dive into the math behind the vectors next time, when we return to our wrench and nut example and apply vector force quantities.
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