Mechanical Engineering

Undercutting of gears, is machining a matched set of dogs/slots that grip into each other - so that they will grip harder the more torque the engine is providing. This will prevent the gears from slipping out again after initially having them engaged. An easy setup, is to fix the gear with its axis center offset in an indexing head - sligthly tipped over(3 to 5 degrees - depending on how much material that can be removed, or how badly the dogs are worn), so that the actual dog to be machined spins precisely around its own vertical axis when table is turned. The finished dog, will have a uniform shape all the way around its contact areafacing the slots/dogs of the interacting gear next to it.

Usually done in a vertical mill, by grinding - or with a dovetail milling bit. Using a dove tail milling bit, will let the machinist fix the gear in a horizontal position, rather than having to tilt it the neccessary amount of degrees. Dogs, slots - or teeth, need to be hardened again, if they are ground beyond the limit of the previous hardened surface of the material.

I'm not sure if you're asking for just the equation for the reactions, or how to calculate the FEM of a cantilever via the Moment Distribution method (aka Hardy Cross method), so I'll anser both.

If you assume clockwise moments are negative...

If your cantilevered span has a length "L", and a uniform load "W" acting downward along the entirelength of the span, then the fixed end moment is (W*L2)/8

If the fixed end is to the left, the equation is positive (the load is acting clockwise, so the reacting FEM is counter-clockwise, therefore positive.) If the fixed end is on the right, the equation becomes -(W*L2)/8

If your uniform load W is acting downward, and is acting along length "d" (which is NOT the entire span), its easiest to convert the uniform load W to a point load "P", where P = W*d

Then your FEM = [P*a*b*(2*L-a)]/(2*L2)

where:

L = entire length of span

a = distance P is acting from the fixed end,

b = distance P is acting from the cantilevered end (also, b = L - a)

OR, if W is centered along the span (therefore Pis also centered), then the equation becomes (3*P*L)/16

When going through the distributive process, the cantilevered end will ALWAYS have a final moment of zero. At the fixed point, the cantilevered side has a distribution factor of zero, while the other side has a D.F. of one. So, if you have length ABCD, where A is a cantelever, and B, C and D are assumed fixed, FEMAB = 0, D.F.BA = 0, D.F.BC = 1, and D.F.CB = D.F.CD = D.F.DC = 0.5

So what ends up happening is that the final FEM for the cantilever ("AB") is zero, and the opposite end of the span ("BA") remains as its initial value. The reason for this is when you sum up the reactions at the point (in this case B) supporting the cantilever and flip the sign, the distribution factors tell you to distribute everything to the side of the support opposite the cantilever ( -(FEMBA + FEMBC) is added to FEMBC). So you can go through the Cross Method as normal, but you don't touch AB or BA, and BC is adjusted only the once that I just mentioned (you don't carry over the 1/2 reactions from CB).

all bodies have same acceleration while coming down from an inclined plane because in such type of case acceleration does not depends upon mass....acceleration can be given as a=gsinθ......θ(theta) is the angle of the inclined plane. g is the acceleration due to gravity or 9.81 m/s2.

*Note: the acceleration due to gravity is actually based on the mass of both bodies and the square of the distance between their centers. While this means that gravity (acceleration) is not equal for all bodies, the mass of the Earth is so much greater than any measurable bodies, that can move down the inclined plane. The mass of the body is negligible in comparison and thus the change in acceleration due to gravity is likewise negligible, and generally not measurable. If a mass the size of the moon were moving down the inclined plane, then that would definitely indicate a greater acceleration.

F = m1m2/r2. Where F = force of gravity, m1 = mass of body 1, m2 = mass of body 2, and r = the distance between the bodies.