Distributed Parameter Systems Modelling:
We consider several models present in the process and mechanical systems control:
Transport-reaction systems:
1.) Tubular reactor
with boundary conditions , and initial condition
2.) Tubular reactor with recycle
with boundary conditions and initial conditions
3.) Countercurrent heater
4.) Equation of heat conduction (parabolic PDE)
with ,
.
5.) Reaction-diffusion-convection models of dispersion reactor (parabolic PDE)
with and
,
.
Strings & Beams
1.) String Equation (2nd Order Hyperbolic PDE)
Let us consider:
with Derichlet boundary conditions , and
and
. The change of variable leads to
and
, which leads to the following
and also
, from this we obtain
and
, and therefore:
2.) Euler-Bernoulli Beam Equation
Let us consider following version of the beam equation which is a simply supported undamped beam:
where 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
direction at some point
.
3.) Rayleigh Beam Equation
with boundary conditions:
3.) Shear Beam Equation
or
4.) Timoshenko Beam Equation
where the x is translational displacement of the beam and is angular velocity.
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