Index to Partial Differential Equation Animation Scripts written in MATLAB at the University of Wyoming

Question and comments should be directed to Raymond G. Jacquot at quot@uwyo.edu.

• Electrical Transmission Lines
• Beam Vibration
• Heat Conduction
• Beach Nourishment
• String Vibration
• Groundwater Drawdown
• Wave Propagation in Elastic Bars
• Static Beam Bending Problems
• Fluid Dynamics
• General Mathematics
• References

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Electrical Transmission Lines:

• tls.m Displays solution to lossless, sinusoidally driven transmission line, has GUI. Requires MATLAB 7.0 or newer.
• tls.png Contains a drawing used by tls.m.
• tls.fig Contains the graphics for the GIU used by tls.m.
• transmline2.m Displays solution to lossless, sinusoidally driven transmission line the same as tls.m. Does not have a GUI and user must change parameters in the script. Specific source and line parameters are specified in the script and the input is the load impedance Z_L.
• lossytransmline.m Displays solution to a lossy, sinusoidally driven transmission line. Specific source and line parameters are specified in the script. Input is the load impedance Z_L.
• transmwave3.m Displays the solution to lossless line driven by a d.c. source. Specific source and line parameters are specified in the script and the load resistance R_L is an input.
• transmlinepulse.m Displays the solution to lossless line driven by a rectangular pulse. Source and line parameters are specified in the script and the pulse width and the load resistance R_L are inputs.
• TmlSqWv.m Displays the line voltage for a lossless transmission line driven by a square wave source. All waveforms are represented by Fourier series truncated after the 41^st term.

Beam Vibration:

• beamvibration.m Displays the solution to a free vibration of a cantilever beam from an initial displacement. Uses generalized Fourier series in the orthogonal beam functions. The initial deflection shape is y(x,0)=y₀[0.667(x/L)²+0.333(x/L)³].
• cantvib3.m Solves the same problem as beamvibration.m except the beam is discretized spatially using 16 nodes and finite differences in space.
• clampedclampedbeam.m Displays the free vibration solution to a clamped-clamped beam starting with an initial condition y(x,0)=2(x/L)²-(x/L)³-4(x/L)⁴+3(x/L)⁵. Solution is in generalized Fourier series in the orthogonal beam functions.
• cantbeamimpulse.m Displays the response of a cantilever beam driven by an impulse function of intensity I₀ at a location x=a. Input quantity is the impulse location a/L. Solution is in generalized Fourier series in the orthogonal beam functions.
• forcedbeamvibration.m Displays the vibration of an undamped cantilever beam driven by a uniform distributed force f₀ which is constant in time and suddenly applied at t=0. Solution is in generalized Fourier series in the orthogonal beam functions.
• dampedbeams.m Displays the motions of damped (or undamped) simply supported, cantilever, clamped-pinned and clamped-clamped beams to point or uniformly distributed loads or initial conditions. The temporal form of the load can be an impulse, step or sinusoid. The damping is assumed to be proportional to local velocity (viscous). Generalized Fourier series solution.
• BR.m Displays the motion of a damped beam as in dampedbeams.m except it is done with a GUI.
• BR.fig Contains the graphihcs for the GUI used by BR.m
• cantbeamanimation.m Displays the steady-state motion of a cantilever beam excited by a sinusoidal displacement of amplitude Y₀ at the fixed end.
• canttipforceanimation.m Displays the steady-state sinusoidal vibration of a cantilever beam forced at the free end with a sinusoidal force of amplitude F₀.
• movingload2.m Displays the motion of a simply supported beam with a moving load P starting from the left end with a user controlled velocity. The input variable is the ratio of the transit time to the first natural period of the beam.
• ssbeamdispex.m Displays the steady-state motion of a simply supported beam driven by a sinusoidal displacement of amplitude Y₀ at the left end.

Heat Conduction:

• conduction.m Displays solution to the diffusion equation for T=T₀ at left boundary and T=0 at right boundary, zero initial temperature, Fourier series solution.
• conduction2.m Displays solution to the diffusion equation for T=T₀ at left boundary and T=T₀ at right boundary, zero initial temperature, Fourier series solution.
• conduction3.m Displays solution to the diffusion equation for T=T₀ at left boundary and T=0 at right boundary, zero initial temperature. This is a finite difference solution in space.
• conduction4.m Displays solution to the diffusion equation with convective boundaries for T=T₀ at left boundary and T=0 at right boundary, zero initial temperature. This is a finite difference solution in space.
• conduction5.m Displays solution to the diffusion equation for T=0 at left boundary and T=0 at right boundary, T₀ initial temperature. Fourier series solution.
• conduction6.m Displays temperature distribution in a slab with both faces insulated and initially the left half at T₀ and the right half at zero temperature. Fourier series solution.
• infiniteslab.m Displays the temperature in an infinite slab with initial temperature T₀ between -L and L at t=0 and zero elsewhere. Fourier series solution.
• convboundaries.m Displays temperatures in a finite slab with convective heat transfer coefficients h₁ and h₂ on the left and right boundaries respectively. The film coefficients are assumed to be the same on both the left and right. Fourier series solution.
• conductioncyl.m Displays radial temperatures in an infinite cylinder with zero initial temperature and temperature at r = R suddenly elevated to T₀ at t = 0. Generalized Fourier series solution in Bessel functions.
• heatedcyl.m Displays radial temperature distribution in an infinite cylinder of radius R that is heated by a uniform volumetric generation of heat q as in ohmic heating of an electrical conductor. The initial temperature is zero and the boundary temperature is zero. Generalized Fourier series solution in Bessel functions.
• semiinfiniteslabstep.m Displays temperatures in semiinfinite medium (halfspace) when the temperature at x = 0 suddenly changes from 0 to T₀ at t = 0 with the halfspace initially at zero temperature. Fourier transform solution.
• semiinfiniteslab.m Displays steady-state sinusoidal temperatures in a semi-infinite slab when the surface at x=0 temperature varies sinusoidally. Phasor Solution.
• conductionsphere.m Displays the thermal response of a homogeneous sphere driven from zero initial temperature T₀ by a sudden temperature change T_H at the surface r=R. Fourier series solution.
• conductionspheresinusoid.m Displays the steady-state thermal response of a sphere to a sinusoidal temperature variation of amplitude T₀ at the surface r=R. The input is dimensionless sinusoid frequency ωR²/κ where κ is the diffusivity of the material of the sphere. Phasor solution.
• conductionspherecooling.m Displays the thermal response of a homogeneous sphere driven from initial temperature T₀ by a sudden temperature change to zero at the surface at r=R. Fourier series solution.
• conductionspheresource.m Displays the temperature in a sphere of radius R with a point source of heat Q₀ and the origin and a temperature of zero at r=R and zero initial temperature everywhere. Fourier series solution.
• insulatedslab.m Displays the temperature in a slab of thickness L and insulated on the x = L face with a suddenly imposed temperature T_H on the x = 0 boundary. Initial temperature is T₀. Fourier series solution.
• conductionslabsinusoid.m Displays the steady-state thermal response of a slab to a sinusoidal temperature variation of amplitude T₀ at the surface x = 0. Tha slab is insulated at x=L. The input is dimensionless sinusoid frequency ωL²/κ where κ is the diffusivity of the material of the sphere. Phasor solution.

Beach Nourishment:

• beachnourishment.m Displays the solution to the diffusion equation for an infinite domain with an initial rectangular beach projection planform. Fourier transform solution.

String Vibration:

• stringanimation.m Displays the d’Alembert solution to the plucked string problem. Input is the nondimensional location of the pluck, a/L.
• stringvibration.m Displays the Fourier series solution to the plucked string problem. Input is the nondimensional location of the pluck, a/L
• displacementexcitedstring.m Displays standing waves in a taut string fixed at the right end and with sinusoidal motion of amplitude Y₀ at the left end. Input is the ratio of the frequency of excitation to the first natural frequency of the string. Phasor solution.
• forcedstring.m Displays the motion of a taut string forced at x=a by a sinusoidal force with amplitude P₀. Inputs are the ratio of the forcing frequency to the first natural frequency of the string and the nondimensional location of the force a/L. Phasor solution.
• stringspring.m Displays the motion of a taut string, fixed at x = 0 and terminated at x = L with a transverse spring. Input is the stiffness parameter related to the stiffness of the spring. Generalized Fourier series solution.

Groundwater Drawdown:

• gndw1.m Solves two layer aquifer problem and stores the solution in a data file framedata.mat to be played back by gndw2.m (Jacob model).
• framedata.mat Contains data file generated by the gndw1.m script to be played back in gndw2.m.
• gndw2.m Displays data generated in gndw1.m which is loaded as an array saved in framedata.mat.
• gndw3.m Solves and displays the solution to the one layer aquifer (Theis model).

Wave Propagation in Elastic Bars

• elasticbar.m Displays the solution to an elastic bar fixed on the left end and free on the right end with zero initial velocity and an initial linear displacement field (constant strain) at t = 0. Fourier series solution.
• elasticbar2.m Displays the solution to an elastic bar fixed on the left end with a suddenly applied constant force F on the right end and no initial velocity or displacement. Fourier series solution.
• elasticbar4.m Displays the solution to an elastic bar fixed at the left end, free at the right end, driven by a compressive impulse of intensity I₀ at the free end. The bar has no initial deformation or velocity.
• torsionbar.m Displays torsional wave propagation in a round bar with an initial twist proportional to the distance from the fixed end and no initial velocity. Fourier series solution.
• torsionbar2.m Displays torsional wave propagation in a round bar fixed at the left end with a suddenly applied constant torque T to the right end and no initial angular twist or velocity. Fourier series solution.

Static Beam Bending Problems

• ssbeamoneload.m Displays the static shear, bending moment and deflection for a simply-supported beam as a load P traverses from left to right.
• ssbeamtwoloads.m Displays the static shear, bending moment and deflection as two loads of value P traverse a simply-supported beam. The loads are 30% of the span length apart in spacing.
• cantileverbeamoneload.m Displays the shear, bending moment and deflection as a load P traverses a cantilever beam from the fixed end to the free end.

Fluid Dynamics

• Blasius.m Displays the flowfield for the viscous, incompressible flow over a flat plate by first solving the Blasius equation. The solutions for the velocity field come from the solution to the Blasius equation and the derivatives thereof.

General Mathematics

• centrallimit.m Displays the probability density functions (from a histogram) for the sum of n random uniformly distributed variables where n varies from zero to 20. At each step the sum is scaled so as to have zero mean and unity variance.
• GibbsPhenom.m Animates the evolution of the Gibbs phenomenon for a Fourier series representation of a square wave by continusously summing the terms and displaying the resulting waveform for 61 terms.
• criticalspeed.m Calculates the critical speed of a shaft of length L with two simple supports located at arbitrary locations. The shaft can have step changes in cross-section and can have discrete masses at arbitrary locations. Uses Rayleigh's method. Displays the shaft and the fundamental mode shape.