Question

Imagine that the crankshaft system shown here is that of a piston assembly in a car engine. The crank length AB is R = 40 mm and the position x of the piston C can be shown to vary with the angle $\theta$ as x = R (cos $\theta$ + $\sqrt{2.5^2 - sin^2\theta}$) R B $\theta$ A 2.5R X C $\frac{d\theta}{dt}$ You are trying to find the maximum angular speed $\frac{d\theta}{dt}$ at which your engine can run. Your angular speed is limited by the maximum piston velocity $\frac{dx}{dt}$ possible in the engine, constrained by the material, which is 35 m/s. 1. Work out on paper and upload a) what kind of derivative stencil you need, list of all the possible Taylor series expansions needed, and derivation of a second-order, forward-differenced stencil from those expansions. b) the relationship between piston velocity and angular speed, in a way that can be applied to Python code i. This problem needs to be solved using the principles of numerical differentiation. DO NOT differentiate the function yourself by using calculus. However, answers from manual differentiation can act as a good check on whether your code is doing what it should. ii. $\frac{dx}{dt} = \frac{dx}{d\theta}\frac{d\theta}{dt}$ Piston velocity is displacement x differentiated with respect to time t. Using chain rule of $\frac{dx}{dt} = \frac{dx}{d\theta}\frac{d\theta}{dt}$ gives us the relationship piston velocity and angular speed. Apply stencils derived from Taylor series expansion appropriately, based on the information given.

          Imagine that the crankshaft system shown here is that of a piston assembly in a car engine. The crank
length AB is R = 40 mm and the position x of the piston C can be shown to vary with the angle $\theta$ as
x = R (cos $\theta$ + $\sqrt{2.5^2 - sin^2\theta}$)
R
B
$\theta$
A
2.5R
X
C
$\frac{d\theta}{dt}$
You are trying to find the maximum angular speed $\frac{d\theta}{dt}$ at which your engine can run. Your angular speed
is limited by the maximum piston velocity $\frac{dx}{dt}$ possible in the engine, constrained by the material, which
is 35 m/s.
1. Work out on paper and upload
a) what kind of derivative stencil you need, list of all the possible Taylor series expansions needed,
and derivation of a second-order, forward-differenced stencil from those expansions.
b) the relationship between piston velocity and angular speed, in a way that can be applied to
Python code
i. This problem needs to be solved using the principles of numerical differentiation. DO NOT
differentiate the function yourself by using calculus. However, answers from manual differentiation
can act as a good check on whether your code is doing what it should.
ii.
$\frac{dx}{dt} = \frac{dx}{d\theta}\frac{d\theta}{dt}$
Piston velocity is displacement x differentiated with respect to time t. Using chain rule of $\frac{dx}{dt} = \frac{dx}{d\theta}\frac{d\theta}{dt}$
gives us the relationship piston velocity and angular speed. Apply stencils derived from Taylor series
expansion appropriately, based on the information given.

Imagine that the crankshaft system shown here is that of a piston assembly in a car engine. The crank
length AB is R = 40 mm and the position x of the piston C can be shown to vary with the angle θ as
x = R (cos θ + √(2.5^2 - sin^2θ))
R
B
θ
A
2.5R
X
C
(dθ)/(dt)
You are trying to find the maximum angular speed (dθ)/(dt) at which your engine can run. Your angular speed
is limited by the maximum piston velocity (dx)/(dt) possible in the engine, constrained by the material, which
is 35 m/s.
1. Work out on paper and upload
a) what kind of derivative stencil you need, list of all the possible Taylor series expansions needed,
and derivation of a second-order, forward-differenced stencil from those expansions.
b) the relationship between piston velocity and angular speed, in a way that can be applied to
Python code
i. This problem needs to be solved using the principles of numerical differentiation. DO NOT
differentiate the function yourself by using calculus. However, answers from manual differentiation
can act as a good check on whether your code is doing what it should.
ii.
(dx)/(dt) = (dx)/(dθ)(dθ)/(dt)
Piston velocity is displacement x differentiated with respect to time t. Using chain rule of (dx)/(dt) = (dx)/(dθ)(dθ)/(dt)
gives us the relationship piston velocity and angular speed. Apply stencils derived from Taylor series
expansion appropriately, based on the information given.

Added by Christina F.

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