Posts Tagged ‘gear tooth’

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.


Spur Gears In Motion

Wednesday, February 12th, 2014

      Last time we learned about forces generated when spur gear teeth mesh and move along a specific line of action.   Today we’ll see them in movement.

      Looking at the illustration below it might appear that there are three teeth in contact, but this isn’t the case.   As the gears rotate, only two teeth make contact at any given time, although the third tooth comes very close.   The actual point of contact between the teeth is represented by a black dot on the illustration below.   This is where two opposing forces, F1 and F2, meet.

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      Now let’s animate the illustration to see how the line of action remains constant the entire time the gear teeth are in motion.   By constant I mean that this imaginary line’s position and angle does not change relative to the gears throughout the course of their movement.

mechanical design engineer

      In the animation, the point of contact moves along the line of action as the gears turn.   Each tooth’s involute profile ensures that smooth contact is maintained along the faces and flanks of the gear teeth.   The involute profile’s unique shape facilitates opposing teeth remaining in constant contact along the line of action for the duration of their movement together.

      If the gear tooth profile wasn’t involute in its shape, say for example it was square or triangular, the forces acting upon the meshed teeth during movement would vary in direction and intensity as a result of uneven contact between the teeth.   For example, consider the square shaped tooth profile in the gear train below.

Gear expert

      As the gears rotate, the pointed tip of one tooth strikes the flat face of another.   As they  continue to turn, the two flat faces of the teeth slap together, then the pointed tip of one tooth will strike the flat face of the other tooth, and so forth.   The result is movement that is jerky and destructive.   There would be excessive vibration and wear and tear on the teeth, resulting in rapid gear tooth erosion and decreased efficiency overall.

      Next time we’ll introduce the gear ratio, a formula which allows us to alter the rotational speed of the driven gear in relation to that of the driving gear, something which comes in handy when designing things that require this differential.


Overcoming Inertia

Monday, February 3rd, 2014

      Inertia.   It’s the force that keeps us in bed after the alarm has rung.   It seems to have a life of its own, and today we’ll see how it comes into play in keeping other stationary objects at rest.

      Last time we identified a specific point of contact between spur gear teeth in a gear train and introduced the opposing forces, F1 and F 2, generated there.   Today we’ll see what these forces represent, identifying one of them as inertia.

gear engineering expert witness

      So where do these forces come from?   They’re forces generated by different means that converge at the same point of contact, the point at which gear teeth mesh.   They follow a very specific geometric path to meet there, an imaginary straight line referred to as the line of action.

      F1 is always generated by a source of mechanical energy.   In our locomotive example introduced earlier in this blog series that source is an electric traction motor, upon which a driving gear is mounted.   When the motor is energized, a driving force F1 is generated, which causes gear teeth on the driving gear to push against gear teeth of the driven gear.

      Force F2 is not as straightforward to understand, because it’s not generated by a motor.   Instead, it’s the resisting force that the weight of a stationary object poses against its being moved from an at-rest position, known as inertia.   The heavier the object, the more inertia it presents with.   Trains, of course, are extremely heavy, and to get them to move a great deal of inertia must be overcome.   Inertia is also a factor in attempting to stop objects already in motion.

      To get a stationary locomotive to move, mechanical energy must be transmitted from the driving gear that’s attached to its traction motor, then on to the driven gear attached to its axle.    At their point of contact, the driving force of the motor, F1, is met by the resisting force of inertia, F2. In order for the train to move, F1 must be greater than F2.   If F1 is less than or equal to F2, then the train won’t leave the station.

      Next week we’ll animate our static image and watch the interplay between gear teeth, taking note of the line of action during their movement.


Gear Terminology

Sunday, January 5th, 2014

      Last time we reviewed the ancient origins of gears and saw how they’ve been around a lot longer than most people realize.    Now let’s familiarize ourselves with the terminology of modern gears by taking a look at the most basic and commonly used gear construction, the spur gear.

      A spur gear is shown below, so named due to its resemblance to spurs commonly found attached to horse riding boots.

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Spur Gear

      At their most basic gears are wheels containing many projections which resemble teeth.    These teeth are equally spaced around the wheel’s circumference and are designed to mesh, or fit together, with the teeth of other like gears.

      Looking more closely at the teeth of a modern spur gear, we see they have a rather complex and peculiar curved shape, along with their own terminology.

Gear Tooth Terminology

      There’s a pitch circle that intersects each gear tooth between the root of the tooth, or bottom land, and the tip of the tooth, or top land.    Above the pitch circle each tooth side bears a face.    Below the pitch circle and under each face is a flank.

      Spur gear teeth don’t necessarily have to have this shape.    All that’s required is that the teeth fit together in such a way so as to permit fluid interaction between them as they rotate.    As a matter of fact, some primitive gears consisted of wooden wheels with teeth made of wooden pegs.    These pegs were inserted into evenly spaced holes which were drilled around the circumference of the wheel.    The wooden pegs of each wheel would mesh with one another, and when one gear wheel was caused to rotate, its pegs would press against the pegs of the other gear, making it rotate along with it.

      So if simple pegs worked well enough, then why are modern gear teeth so specifically shaped?     We’ll see why next time when we join gears together to form a gear train.