The distance from TDC (top dead centre) for the pin is given by x when the crank is
at angle . Hence :
i) Give an equation for x in terms of the other variables (r, L and ).
ii) If the crank rod is rotating at a constant value of ω calculate the speed of the
piston
i.e. calculate
dt
dx
in terms of ω, r, and L
iii) Hence calculate the x value when the speed of the piston is at a maximum.
iv) If L is 1 metre, r is 0.25 metres and the crank is rotating at 5,000 rpm
calculate the maximum velocity of the piston in m/s.
i) To find an equation for x in terms of the other variables (r, L, and θ), we can use the concept of trigonometry and the geometry of the crank mechanism. Let's assume that the piston pin is connected to the crankshaft at a distance r from its center. The length of the connecting rod is L.
From the given information, we can see that the distance from the top dead center (TDC) for the pin is given by x when the crank is at angle θ.
Now, if we draw a right-angled triangle with the crank, connecting rod, and x as the sides, we can use trigonometric functions to relate the sides of the triangle. In particular, the sine function will be useful.
Looking at the triangle, the opposite side is x, the adjacent side is L, and the hypotenuse is r. Therefore, the equation for x in terms of L, r, and θ is:
sin(θ) = x / L
ii) To calculate the speed of the piston, we need to find the derivative of x with respect to time (t). We can use the chain rule to differentiate the equation from part (i).
Differentiating both sides of the equation from part (i) with respect to time, we get:
d(sin(θ)) / dt = d(x / L) / dt
Applying the chain rule, we have:
cos(θ) * dθ / dt = (dx / dt) / L
Simplifying, we get:
dx / dt = L * cos(θ) * dθ / dt
This equation gives us the speed of the piston (dx/dt) in terms of ω, r, θ, and L.
iii) To find the x value when the speed of the piston is at a maximum, we need to find the maximum value of dx/dt. To do this, we can take the derivative of dx/dt with respect to θ and set it equal to zero.
Differentiating the equation from part (ii) with respect to θ, we get:
d²x / dt² = -L * sin(θ) * (dθ / dt)²
Setting d²x/dt² equal to zero, we have:
-L * sin(θ) * (dθ / dt)² = 0
Since L and sin(θ) are always positive, we can conclude that (dθ / dt) must be zero for dx/dt to reach its maximum value. Therefore, the x value when the speed of the piston is at a maximum occurs at θ = 0 or θ = π radians (180 degrees).
iv) To calculate the maximum velocity of the piston in m/s, we need to know the value of ω (angular velocity) in radians per second. Given that the crank is rotating at 5,000 rpm (revolutions per minute), we can convert it to radians per second.
ω = (5000 rpm) * (2π radians/rev) * (1 min/60 s)
Simplifying, we find:
ω = 500π radians/s
Now we can substitute the values of r, L, ω, and θ (θ = 0) into the equation from part (ii) to calculate the maximum velocity (dx/dt) of the piston:
dx / dt = L * cos(θ) * (dθ / dt)
= L * cos(0) * ω
Substituting L = 1 m and ω = 500π radians/s, we get:
dx / dt = (1 m) * cos(0) * (500π radians/s)
= 500π m/s
Therefore, the maximum velocity of the piston is 500π m/s.