Problem $4-$ A Simple Application of Simscape

A person wearing a parachute jumps out of an airplane. Assume initially there are no resistive forces

and that the aerodynamic drag of the jumper and parachute can be modeled as viscous damping with

respect to a fixed reference frame. The parachute has a relatively small mass $M_p=10kg$ and large

damping coefficient $B_p=100N*\frac{s}{m}.$ The jumper has a larger mass $M_j=60kg$ and smaller drag

coefficient $B_j=10N*\frac{s}{m}.$ The cords attaching the parachute are called risers and are assumed to be

quite springy. The elastic effect of the risers is represented by the spring constant $K_R=400\frac{N}{m}.$

Draw a model of the jumper and parachute as a simple mechanical system, characterized by

damping, stiffness and mass;

Write a set of equations describing the motion of the jumper and parachute. Choose a reference

so that the positions $x_j(t)$ of the jumper and $x_p(t)$ of the parachute are both zero at $t=0.$ Take

the positive direction of these displacements to be downward;

Build a Simulink model to solve these equations. Assume that at $t=0$ the parachute opens as

the jumper and parachute are falling with initial velocity $v_0=20\frac{m}{s}.$ At that time, the risers are

fully extended, so they are neither slack nor stretched. Choose "ode$15$s$"$ for the solver, simulate

the first $10$ seconds after the parachute opens and plot the velocity and position of the jumper

and parachute against time, as well as the elongation of the risers;

Build a Simscape model that represents the dynamics of the jumper and parachute, using the

same assumptions and settings indicated above. Use the "Ideal Force Source" block to represent

the effect of gravity on the system. Simulate the system and compare the results against the

Simulink model.

Optional $-$ for bonus points: Consider now the $($more realistic$)$ case where the relative wind

exerts an aerodynamic drag force on the parachute. Starting from your Simscape model, substitute

the damping with a force defined as follows:

$F=\frac{1}{2}{C}_x{x}_p^{}{?}^2$

where $C_x=1\mathrm{.}5,=1\mathrm{.}2,$ is the parachute surface area and $x_p^{}$ the parachute velocity in $\frac{m}{s}.$

Assume the parachute has a hemispherical surface area of diameter $D_p.$ Determine $($by trial and

error$)$ an approximate value of the diameter for which the landing speed is close to $7\frac{m}{s}.$