For an artery fixed at both ends that has a uniformly distributed mass $m,$ length $l,$ a constant bending rigidity $EI,$ and is subjected to axial pulsatile pressure $p(t)=p_{}+p_{a}cos(t),$ in which $p_g$ is the mean pressure, $p_a$ is the pulse pressure. The time dependent displacement $y(t)$ of the artery is given by

$\frac{d^{2}y(t)}{d{t}^2}+({}_n-{}_{n}cost)y(t)=0$

${}_n=\frac{{}_{n0}^2}{{}^2}(1-\frac{l^2}{{}^2}\frac{p_0{A}_{\text{l}\text{u}\text{m}}-N}{EI}),{}_n=\frac{{}_{n0}^2}{{}^2}\frac{l^2}{{}^2}\frac{p_a{A}_{\text{l}\text{u}\text{m}}}{EI}$

${}_{n0}=\frac{{}^2}{l^2}\sqrt[\phantom{2}]{\frac{EI}{{}_w{A}_w+{}_f{A}_{\text{l}\text{u}\text{m}}}},A_{\text{l}\text{u}\text{m}}=r_i^2$

where, $r_i$ is the lumen radius of the artery, $N$ is the axial tension, ${}_n$ is called the $n$th natural angular frequency of the artery fixed at both ends. For the artery at any given pressure and excitation frequency, $y(t)$ can be determined by using Eq$.(1).$

Project description

A dynamic system is stable if the displacements under small perturbation are bounded; it is unstable if the displacements of the system are amplified with time. The critical pressure is defined as the minimum pressure at which the displacement is amplified with time. In this project, determine the critical mean pressure $p_0$ by solving Eq$.(1)$ numerically that will cause the artery to lose stability, for $p_a=30mmHg.$ The simulation step can be started as below:

To do so$,$ first assign a small value to $p^0,$ solve the displacement $y(t)$ using Eq$.1.$ The $y(t)$ is a function of time $t.$ If $y(t)$ is not amplified with time, increase the $p_{}$ by $2mmHg(266\mathrm{.}6Pa),$ solve the $y(t)$ again.

Continue the procedure until reach a value of $p_{}$ such that the displacement $y(t)$ is amplified with time $($loss of stability$).$ This is the critical mean pressure $p_0$ at given pulse pressure $p_a.$

Check the displacement $y(t)$ for time $t$ from $0$ to $2000$ time steps $($may plot the displacement vs$.$ time$).$ May assume the system loses stability if the displacement $>0\mathrm{.}05.$

Submit your Matlab programs and the simulation results using following parameter values. Also show result $($of critical p$0)$ by plotting the displacement $y$ vs$.$ time $t$ where $y$ is amplified with time.

Parameter values are given as below $($all units are SI units$)$

axial force $N=162\mathrm{.}5{10}^{-3}N,$

artery lumen radius $r_i=2\mathrm{.}7546{10}^{-3}m$; outer radius $r_v=3\mathrm{.}28298{10}^{-3}m$;

artery lumen area $A_{\text{l}\text{u}\text{m}}=r_i^2$; cross sectional area $A_w=(ro{l}^2-r_i^2)$

artery original length $L=22{10}^{-3}m,$ stretch ratio $=1\mathrm{.}5,$ and the length $I={}^{**}L$;

bending rigidity $EI=1\mathrm{.}378{10}^{-6}$;

density of the arterial wall ${}_w=1\mathrm{.}05{10}^3,$ blood density ${}_f=1\mathrm{.}06{10}^{3}k\frac{g}{m^3}$

$$ is calculated by $=2f,$ where $f$ is the frequency of the pulsatile pressure, for the simulation, use $f=3\mathrm{.}5Hz.$

$p_a=30mmHg,$ need to convert to Pascal.

Initial conditions are $y(0)=0\mathrm{.}01,dy\frac{0}{d}t=0$

Following is an example of the plot of displacement vs$.$ time $($with different parameter values$),$ where the system is currently stable since the displacements are bounded between $-0\mathrm{.}01$ and $0\mathrm{.}01.$