Distributed Parameter Systems (DPS) Lab DPS Lab

Distributed Parameter Systems Modelling:

We consider several models present in the process and mechanical systems control:

Transport-reaction systems:

1.) Tubular reactor

$<br /> \frac{\partial x}{\partial t}=-v(\zeta)\frac{\partial x}{\partial \zeta}+\psi(\zeta) x$

with boundary conditions $x(0,t)=u(t)$, and initial condition $x(\zeta,0)=x_0(\zeta)$

2.) Tubular reactor with recycle

$\frac{\partial x}{\partial t}=-v(\zeta)\frac{\partial x}{\partial \zeta}+\psi(\zeta)x$

with boundary conditions $x(0,t)=Rx(1,t)+(1-R)u(t)$  and initial conditions  $x(\zeta,0)=x_0(\zeta)$

3.) Countercurrent heater

$\frac{\partial x_1}{\partial t} =-\frac{\partial x_1}{\partial \zeta}+k(x_2-x_1),x_1(t,0)=u_1(t)$

$\frac{\partial x_2}{\partial t} =\frac{\partial x_2}{\partial \zeta}-k(x_2-x_1),x_2(t,1)=u_2(t)$

$\dot{x}(t)={\mathcal{A}}x(t)$

${\mathcal{B}}x(t)=u(t)$

4.) Equation of heat conduction (parabolic PDE)

$<br /> \frac{\partial x}{\partial t}=\frac{\partial^2 x}{\partial \zeta^2}$

with $x(0,t)=0=x(1,t)$, $x(\zeta,0)=x_0(\zeta)$.

5.) Reaction-diffusion-convection models of dispersion reactor (parabolic PDE)

$\frac{\partial x}{\partial t}=\frac{\partial^2 x}{\partial \zeta^2}-v\frac{\partial x}{\partial \zeta}+\psi x$

with $\frac{\partial x}{\partial \zeta}(0,t)=Pe(x(0,t)-u(t))$ and $\frac{\partial x}{\partial \zeta}(1,t)=0$,  $x(\zeta,0)=x_0(\zeta)$.

Strings & Beams

1.) String Equation (2nd Order Hyperbolic PDE)

Let us consider:
$\frac{\partial ^2 x}{\partial t^2}=c^2\frac{\partial ^2 x}{\partial \zeta ^2}$
with Derichlet boundary conditions $x(0,t)=0=x(1,t)$, and  $x(\zeta,0)=\phi(\zeta)$ and $\dot{x}(\zeta,0)=\psi(\zeta)$. The change of variable leads to $\frac{\partial x}{\partial \zeta}=x_1,$ and $\frac{\partial x}{\partial t}=x_2$, which leads to the following $\frac{\partial }{\partial t}\frac{\partial x}{\partial \zeta}=\frac{\partial x_1}{\partial t}$ and also $\frac{\partial }{\partial \zeta}\frac{\partial x}{\partial t}=\frac{\partial x_2}{\partial \zeta}$, from this we obtain $\frac{\partial x_1}{\partial t}=\frac{\partial x_2}{\partial \zeta}$ and $\frac{\partial }{\partial t}[x_2]=c^2\frac{\partial x_1}{\partial \zeta}$, and therefore:

$\dot{x}(t)={\mathcal{A}}x(t)$

2.) Euler-Bernoulli Beam Equation 
Let us consider following version of the beam equation which is a simply supported undamped beam:
$\mu \frac{\partial^2 x}{\partial t^2}+EI\frac{\partial^4 x}{\partial \zeta^4}=q$

where $\mu$ in the mass per unit length, E is the elastic modulus and I is the second moment of area of the beam's cross-section.The x represents deflection of the beam in the $z$ direction at some point $\zeta$. 

3.) Rayleigh Beam Equation

$\frac{\partial ^2 x}{\partial t^2}-\alpha \frac{\partial^4 x}{\partial\zeta^2 \partial t^2}-a \frac{\partial^3 x}{\partial\zeta^2 \partial t}+\frac{\partial^4 x}{\partial\zeta^4}=0$

with boundary conditions:

$x(0,t)=0=x(1,t)$
$\frac{\partial^2 x}{\partial \zeta^2}(0,t)=u(t)$
$\frac{\partial^2 x}{\partial \zeta^2}(1,t)=0$

3.) Shear Beam Equation

$\frac{\partial ^2 x}{\partial t^2}-\alpha \frac{\partial^4 x}{\partial\zeta^2 \partial t^2}+\frac{\partial^4 x}{\partial\zeta^4}=0$

or

$\alpha\frac{\partial ^2 x}{\partial t^2}=\frac{\partial^2 x}{\partial \zeta^2}-\frac{\partial \psi}{\partial \zeta}$

$0=\alpha \frac{\partial^2 \psi}{\partial \zeta^2}-\psi+\frac{\partial x}{\partial \zeta}$

4.) Timoshenko Beam Equation

$\rho A\frac{\partial ^2 x}{\partial t^2}-q(\zeta,t)=\frac{\partial }{\partial \zeta}[\kappa AG (\frac{\partial x}{\partial \zeta}-\phi)]$
$\rho I \frac{\partial^2 \phi}{\partial t^2}=\frac{\partial }{\partial \zeta} (EI\frac{\partial \phi}{\partial \zeta})+\kappa AG (\frac{\partial x}{\partial \zeta}-\phi)$

where the x is translational displacement of the beam and $\phi$ is angular velocity.

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