Q$1)$ Posture Analysis:

a$.$ Numerical solution of position variables: Write a computer program to solve the

unknowns using the Newton$-$Raphson technique.

$$ Your written work must legibly show all the intermediate steps, such as

the cartesian components of the equations, Jacobian, etc.

$$ Run your program to calculate the position variables against the input

angle for all the links $($Rotation of links $2,3,$ and $4$; and X$-$coordinate of

link $5).$

$$ Present your numerical results for each link both in tabular format and as

graphs. Discretize the range of input angle with $10\backslash$deg and show your results

for each input angle.

b$.$ Expand your vector loop solution to capture the motion of points B$,$ P $($coupler

point$),$ and C$.$

i$.$ Determine the stroke of the piston $($Link $5)$

ii$.$ Determine the total distance travelled by C for a single rotation of the

input link.

iii. Plot the locus of the coupler point P $($XY plot$)$ for a complete rotation of

the input link.

iv$.$ Calculate the input posture when the coupler point P has the maximum X

and Y displacements and the minimum X and Y displacements.

Q$2)$ Velocity and Acceleration Analysis.

a$.$ Kinematic coefficients

i$.$ Differentiate your vector loop equations to determine the first$-$order and

second$-$order kinematic coefficients of the mechanism. Show your work.

ii$.$ Calculate these coefficients for each input angle using a computer code.

Present your numerical results for each link both in tabular format and as

graphs. Discretize the range of input angle with $10\backslash$deg and show your results

for each input angle.

iii. Show the instant centers on a scaled drawing of the mechanism for two

input angles. Determine the first$-$order kinematic coefficients using

instant centers for those two input angles. Verify the accuracy of your

computer code by comparing the answer to this question to the values

calculated in part ii$.$

b$.$ Plot the angular velocities and accelerations of links $3$ and $4$ and the linear velocity

and accelerations of point C and Link $5$ against the input posture. Present your

numerical results for each link both in tabular format and as graphs. Discretize the

range of input angle with $10\backslash$deg and show your results for each input angle.

c$.$ Velocity of point P

i$.$ Determine the first$-$order and second$-$order kinematic coefficient of point

P$.$ Present your numerical results both in tabular format and as graphs.

Discretize the range of input angle with $10\backslash$deg and show your results for each

input angle.

ii$.$ Plot the velocity and acceleration of point P against the input angle.

Present your numerical results both in tabular format and as graphs.

Discretize the range of input angle with $10\backslash$deg and show your results for each

input angle.

Q$3)$ Static Force Analysis of the Mechanism

a$.$ Draw free body diagrams of links $2,3,4,$ and $5.$ Identify the type of members for

these links $($two force, two force with a moment, three force, three force with a

moment, four force, etc.$).$

b$.$ Perform a manual graphical static force analysis for one posture of the input link.

Ensure that the analysis is performed on an accurately scaled drawing of the

mechanism. For the initial iteration neglect the effects of gravity. Then for the final

iteration include the effects of gravity. Comment on the significance of gravity.

I WILL GIVE YOU GOOD REVIEW IF SOLVED CORRECTLY.

$\backslash$table$[[$Parameter$,$Value$],[O_{2}J,0\mathrm{.}110$ m$],[O_{2}H,0\mathrm{.}140$ m$],[O_{2}A,0\mathrm{.}100$ m$],[AC,0\mathrm{.}200$ m$],[AB,0\mathrm{.}250$ m$],[AP,0\mathrm{.}200$ m$],[BP,0\mathrm{.}150$ m$]]$

The following external forces act on the mechanism:

A constant force vec$(F{)}_P=750$hat$()N$ at point $P$

A torque vec$(T{)}_2$ on Link $2$ from the crankshaft

Spring force between point $C$ and $O_S.$ Spring has a stiffness coefficient of $2000\frac{N}{m}$ with a free length of $17\mathrm{.}5$ cm

Viscous damping force due to a damper between $B$ and $O_C.$ Line connecting pin $B$ and $O_C$ is parallel to X$-$axis. Damping coefficient is $15N\frac{s}{m}$

Gravity acts in the $-$ve Y$-$direction

Coefficient of friction between point C and ground is $=0\mathrm{.}25.$ The remaining friction in the mechanism can be neglected

The inertial properties of the links are tabulated below:

$\backslash$table$[[$Link$,$Mass $($kg$),$Mass moment of inertia $($Nms ${?}^2),$Center of Mass$],[2,m_2,5,I_{G2},0\mathrm{.}01,$Pivot $O_2$