Problem $4(12+6$ points$)-$ Impedance of systems under force

feedback

$(4$ points$)$ Consider the two$-$mass system shown below. Write the equations of motion in

symbolic form, and calculate the output impedance $($transfer function between applied force and

velocity of the distal mass $m_2).$ You should use a symbolic solver like Matlab's to solve the system

of two equations $(F=$ ma for each mass$),$ with the unknowns $v_1($velocity of mass $1)$ and $v_2($velocity

of mass $2).$

$(4$ points$)$ Consider now the same system under force feedback, i$.$e$.$ when an input $u=GF$ is

applied to the proximal mass. Write the equations of motion in symbolic form, and calculate the

output impedance.

$(2$ points$)$ Assuming $M_1=1kg,M_2=0\mathrm{.}1kg,K=10\frac{N}{m},C=0\mathrm{.}2N\frac{s}{m},G=8,$ calculate the two

transfer functions numerically, and identify poles, zeroes, and the Bode plot of both transfer

functions.

$(2$ points$)$ What is the effect of applying force feedback on the output impedance of the system?

$(2$ points BONUS FOR BMEG$441,$ required for BMEG$641)$ Consider now G$=20.$ What

happens to this new system?

$(4$ points BONUS FOR BMEG$441,$ required for BMEG$641)$ What is the range of gain values

that allows the system to be passive? Why is passivity a desired property of human$-$interacting

systems?

$($Hint: calculate the real part of the impedance for the symbolic system, and impose that the

numerator needs to be greater than zero. The denominator is always $>0,$ but complex symbolic

math is needed to show that is the case$)$