Problem $1)(30$ pts $)$ Determine the types of the systems represented by the following equations. Determine whether the given systems are

Lumped or distributed parameter

Linear or non$-$linear

Time$-$invariant or time$-$variant

Free or forced

Continuous$-$time or discrete$-$time

Deterministic or stochastic

Indicate the terms causing non$-$linearity, time dependency, forcing etc. with full reasoning. Assume that the system parameters and initial conditions are known exactly.

$1.$a$)$ Equation of motion of a mass$-$spring$-$damper system attached on a Skotch$-$Yoke mechanism as shown in Figure $1.$ You can consider Force F as the input and angular position $$ as the output of the system.

$I_o{}^{}+m_3{L}^2(co{s}^2({}^{})-sincos{}^{}{?}^2)$

$+(m_4+m_3)L^2(cossin{}^{}{?}^2+si{n}^2({}^{}))$

$+c{L}^{2}co{s}^2{}^{}+k{L}^{2}sincos=M-$FLsin$$

$1.$b$)$ Equation of motion for the rocket while the engine is burning fuel, i$.$e$.$ the rocket equation.

$v(t)=v_i+$uln$(\frac{m_i}{m(t)})$

where

$v$ and $v_i$ are the current and initial velocities of the rocket respectively, $m$ and $m_i$ are the current and initial masses of the fuel respectively, $u$ is the velocity of the ejected gas, i$.$e$.$ the fuel.

$1.$e$)$ The transfer function between voltage input and speed output of DC motor model is

$G(s)=\frac{K}{(Js+b)(Ls+R)+K^2}$

where

J and b are the inertia and the damping coefficient of the shaft of the motor,

L and R are the inductance and resistance of the electrical circuit, K is the gain of the system.

$1.$d$)$ The equation that describes the exponentially weighted running average filter. Consider the system output as y $,$ and input as x $,$ and n as the number of sampled instances.

$1.$e$)$ The Laplace's Equation described in cylindrical coordinates $(r,,z.$

$\frac{de{l}^{2}u}{del{r}^2}+\frac{1}{r}\frac{\text{d}\text{e}\text{l}\text{u}}{\text{d}\text{e}\text{l}\text{r}}+\frac{1}{r^2}\frac{de{l}^{2}u}{\text{d}\text{e}\text{l}{}^2}+\frac{de{l}^{2}u}{del{z}^2}=0$