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Thermal expansion:
The coefficient of thermal expansion of a rod is equal to the ((change in length) / (length)) for a change in temperature in a given unit (Centigrade or Fahrenheit, for example)
The coefficient of thermal expansion for aluminum is about 22ppm (parts per million) / degree C or 12 ppm / degree F
A design question: What is the allowable temperature rise for a 5″ long aluminum pushrod if the Tappet clearance is 0.010″ ?
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Principle of Virtual Work:
WORK IN = WORK OUT; Work = Force x Distance
A 1/4-20 jack screw is used to lift a weight; it is turned CCW with 1 ft-lb of torque. What is the vertical force?
1 ft-lb torque could be produced by turning the screw with a 6″ wrench with 2 lb Force applied to the end of the wrench – one rotation would be 12 pi inch travel of the end of the wrench – so the WORK IN would be 2 lb times 12pi inches or 75.4 inch-lb; the screw rotated one revolution CCW would move 1/20th of an inch upward, so the WORK out would be the LIFTING FORCE x 0.05″; since the WORK IN = WORK OUT, the LIFTING FORCE would be 75.4 inch-lb / 0.05 inch = 1508 lb.
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Coulomb Friction – also know as Stick-Slip Friction:
Horizontal Force = (Coefficient of Friction) x (Normal Force)
The coefficient of friction for rubber on asphalt is about 1. The motorcycle weighs 200 pounds. The rider weighs 200 pounds. The distance between the rear axle and the ground is 1 foot. How much torque on the rear wheel will cause the rear wheel to slip? What is the maximum force accelerating the motorcycle forward?
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Gears, and Chain-and-Sprocket – decreases speed, increases torque
Top speed of crank = 6000 RPM = 100 Revolutions per second
Rear wheel radius (axle to ground) = 1 foot
Wheel circumference = 6.28 feet
Top motorcycle speed in top gear is (100/6.84)*6.28 = 91.8 feet per second or about 63 mph
Coefficient of friction for rubber tires on asphalt is about 1 – normal load is about 200 lbs
So, the torque on the rear wheel where the tire starts to slip is about 200 ft-lbs.
In the lowest gear the overall gear ratio (engine crank to rear wheel) is 20.3.
So, the torque on the crank that produces 200 ft-lbs is 200/20.3 = 9.85 ft-lbs.
Where does the 6.84 come from in the table above? It is (48/19)*(46/17) = 6.836.
That is, 6.836 is the turns ratio of the crank reduced by primary chain linking Engine to the Clutch times the turns ratio of Gearbox reduced by the drive chain linking the Gearbox to the Rear Wheel.
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Stick-Slip Friction:
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Gravitational Lensing (curvature of spacetime)
Einstein Cross – Multiple images of same Quasar by gravitational lensing due to galaxy in line of sight.
Binary Black Hole video simulation showing gravitational lensing
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Balanced Equation for Octane (Gasoline) Combustion:
One mole of C8H18 weighs 114 grams (8 x 12 + 18 x 1).
According to the balanced octane equation, to react one mole of C8H18 we need 12.5 moles of O2. 12.5 moles of O2 weighs 400 grams (12.5 x 32).
Since air is 21% O2, there will be 3.76 N2 along for the ride, which means 47 moles (3.76 * 12.5) of N2. 47 moles of N2 weighs 1316 grams (47 x 28).
The combined weight of O2 and N2 needed to react one mole of C8H18 is 1716 grams (400 + 1316). The air to fuel ratio (AFR) by weight is then 15 (1716 / 114).
REMEMBER THIS NUMBER.
That means, 15 grams of air are needed to react every 1 gram of gasoline.
(The number used for AFR in most industry calculations is 14.7.)
Our engine displacement is 200cc or 1/5th of a liter.
How much does 200cc of air weigh?
One mole of air weighs 28.8 grams ((.21 x 32) + (.79 x 28)), and occupies 22.4 liters at STP.
Therefore, 1/5th liter of air, a single gulp in our engine, weighs 0.257 grams (0.2 x 28.8 / 22.4).
This means that the fuel needed to exactly react with this gulp weighs 0.0171 grams (0.257 / 15).
So, the maximum energy release per explosion in our 200cc engine is 759 Joules (0.0171 x 44,400).
(Recall that octane has an energy content of 44,400 Joules / gram — see above)
Less fuel (lean mixture) will give less energy per explosion and result in air in the exhaust. More fuel (rich mixture) will not completely react and cause unburned hydrocarbons in exhaust.
The top speed of the engine is 6000 RPM (revolutions per minute) or 100 RPS (revolutions per second).
There is one explosion for every two crankshaft revolutions – recall that we have a 4-stroke engine.
Thus, at top engine speed, there are 50 explosions per second.
The power is the energy per time.
This results in 37,950 Watts (759 Joules x 50 per second).
Note that a Watt is a Joule / second, and one horsepower (Hp) is 746 Watt.
Thus, the engine Hp at 100% efficiency is 50.9 Hp (37,950 / 746).
The manufacturer spec is about 10 Hp.
The engine efficiency is 20% (10 / 50.9).
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Stress (pounds per square inch) versus Strain (fractional change in length over length)
Tensile Strength of mild steel is about 60,000 pounds per square inch – that is, 60K psi or 60 ksi
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Forces
