Question

Consider the slider-crank mechanism shown in the following figure. The slider-crank mechanism is made of a crank (of length r ), a connecting rod (of length \( l \) ) and a slider that is constrained so that it only moves in the x -direction. A motor is attached to the crank, and the crank rotates with a constant angular velocity \( (\omega=70 \mathrm{rad} / \mathrm{s}) \). The distance from the motor to the connection between the connecting rod and the slider is given by \( x \). We have also defined the terms \( a \) and \( b \), which are the \( \mathrm{x}- \) axis projections of the crank length and the connecting rod length, respectively. The angle that the crank makes with the horizontal is given by \( \theta \). The mechanism must satisfy Groshof's criterion \( l \geq 2.5 r \) to ensure \( 360^{\circ} \) rotation of the crank. Additional constraints on the mechanism are given by \( 0.5 \leq r \leq 10,2.5 \leq l \leq 25 \), and \( 10 \leq x \leq 20 \). When the system is at \( \theta=40^{\circ} \) and the system rotates with a constant angular velocity of \( \omega=70 \mathrm{rad} / \mathrm{s} \) : a) Derive an expression for the slider's velocity hint 1: derive an expression for \( a \) and \( b \) in terms of the design variables \( r \) and \( l \), and the constant \( \theta \) hint 2: use geometry to relate position ( \( x \) ) and the expressions that you have derived for \( a \) and \( b \) hint 3: think about how velocity (v) can be determined from position \( (x) \) with respect to time b) Maximize the speed of the slider. Speed, here, can be represented by the magnitude of the velocity (not worrying about sign/direction). Your goal is to find the lengths of the crank and connecting rod that correspond to this maximum speed. Formulate an optimization problem in standard form (with normalized constraints) c) Plot, using MATLAB, the two-variable design space. Plot only the contours associated with the following objective function values (where negative numbers represent \( -1 * \mathrm{~F} \) ) Also, identify the constraints on this figure (you can draw the constraints by hand) d) Identify the constrained minimum on the figure. Demonstrate that optimality conditions hold. At the minimum you identified, what are the Lagrange multipliers? \[ \begin{array}{l} \operatorname{contour}\left(\mathrm{x}_{1}, \mathrm{x}_{2}, \mathrm{~F}, \mathrm{v}\right) \end{array} \]

          Consider the slider-crank mechanism shown in the following figure. The slider-crank mechanism is made of a crank (of length r ), a connecting rod (of length \( l \) ) and a slider that is constrained so that it only moves in the x -direction. A motor is attached to the crank, and the crank rotates with a constant angular velocity \( (\omega=70 \mathrm{rad} / \mathrm{s}) \). The distance from the motor to the connection between the connecting rod and the slider is given by \( x \). We have also defined the terms \( a \) and \( b \), which are the \( \mathrm{x}- \) axis projections of the crank length and the connecting rod length, respectively. The angle that the crank makes with the horizontal is given by \( \theta \).

The mechanism must satisfy Groshof's criterion \( l \geq 2.5 r \) to ensure \( 360^{\circ} \) rotation of the crank. Additional constraints on the mechanism are given by \( 0.5 \leq r \leq 10,2.5 \leq l \leq 25 \), and \( 10 \leq x \leq 20 \).

When the system is at \( \theta=40^{\circ} \) and the system rotates with a constant angular velocity of \( \omega=70 \mathrm{rad} / \mathrm{s} \) :
a) Derive an expression for the slider's velocity
hint 1: derive an expression for \( a \) and \( b \) in terms of the design variables \( r \) and \( l \), and the constant \( \theta \)
hint 2: use geometry to relate position ( \( x \) ) and the expressions that you have derived for \( a \) and \( b \)
hint 3: think about how velocity (v) can be determined from position \( (x) \) with respect to time
b) Maximize the speed of the slider. Speed, here, can be represented by the magnitude of the velocity (not worrying about sign/direction). Your goal is to find the lengths of the crank and connecting rod that correspond to this maximum speed. Formulate an optimization problem in standard form (with normalized constraints)
c) Plot, using MATLAB, the two-variable design space. Plot only the contours associated with the following objective function values (where negative numbers represent \( -1 * \mathrm{~F} \) )

Also, identify the constraints on this figure (you can draw the constraints by hand)
d) Identify the constrained minimum on the figure. Demonstrate that optimality conditions hold. At the minimum you identified, what are the Lagrange multipliers?
\[
\begin{array}{l}
\operatorname{contour}\left(\mathrm{x}_{1}, \mathrm{x}_{2}, \mathrm{~F}, \mathrm{v}\right)
\end{array}
\]

Consider the slider-crank mechanism shown in the following figure. The slider-crank mechanism is made of a crank (of length r ), a connecting rod (of length l ) and a slider that is constrained so that it only moves in the x -direction. A motor is attached to the crank, and the crank rotates with a constant angular velocity (ω=70 rad / s). The distance from the motor to the connection between the connecting rod and the slider is given by x. We have also defined the terms a and b, which are the x- axis projections of the crank length and the connecting rod length, respectively. The angle that the crank makes with the horizontal is given by θ.

The mechanism must satisfy Groshof's criterion l ≥ 2.5 r to ensure 360^∘ rotation of the crank. Additional constraints on the mechanism are given by 0.5 ≤ r ≤ 10,2.5 ≤ l ≤ 25, and 10 ≤ x ≤ 20.

When the system is at θ=40^∘ and the system rotates with a constant angular velocity of ω=70 rad / s :
a) Derive an expression for the slider's velocity
hint 1: derive an expression for a and b in terms of the design variables r and l, and the constant θ
hint 2: use geometry to relate position ( x ) and the expressions that you have derived for a and b
hint 3: think about how velocity (v) can be determined from position (x) with respect to time
b) Maximize the speed of the slider. Speed, here, can be represented by the magnitude of the velocity (not worrying about sign/direction). Your goal is to find the lengths of the crank and connecting rod that correspond to this maximum speed. Formulate an optimization problem in standard form (with normalized constraints)
c) Plot, using MATLAB, the two-variable design space. Plot only the contours associated with the following objective function values (where negative numbers represent -1 * F )

Also, identify the constraints on this figure (you can draw the constraints by hand)
d) Identify the constrained minimum on the figure. Demonstrate that optimality conditions hold. At the minimum you identified, what are the Lagrange multipliers?

contour(x1, x2, F, v)

Added by Lisa A.

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