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## Mathematica Scientific Demonstrations

Written by Nasser M. Abbasi. email
updated January 24, 2014

These are some Mathematica CDF apps that I wrote over the last few years. These can be run using the free Mathematica CDF player available from Wolfram CDF plugin webpage

These are updated all the time as needed. The current version of each app is given by the date of the modification posted below the name of the application.

Clicking the image will run a small animation made using an animated gif file. Clicking the run link below the image opens a new web page where the CDF will automatically load and run inside the web page. Many of those pages contain more documentation on the application. Some have separate reports.

The CDF and the notebook are also available using the links below the image to download and run locally.

### 49. three pendulums with two springs

Version: August 16, 2013

This Demonstration simulates the equations of motion of three rigid pendulums A, B, C with a spring attached from the end of A to the end of B and another spring attached from the end of B to the end of C.

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### 48. Simulation of rigid body dynamics

Version: Dec 19, 2013

Final project EMA 542. Two noncollinear rotating bodies.

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### 47. Another rigid body dynamics

Version: Dec 19, 2013

Solving EMA 542 HW3 problem 1

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### 46. Direct and Shear Strain Deformation in 3D

Version: sept 7, 2013

Shows effect of change in $E$ and Poisson ratio on normal stresses and shear stresses. Shows the 3D deformation in an isotropic unit volume (pure shear).

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### 45. Finite Difference Formulas

Version: August 29, 2013

Generated By Interpolating Polynomial. Shows error in approximation based on which formula used.

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### 44. 2D rectangular membrane modal analysis

Version: July 31 2013

PDE solved is $\frac{\partial^2 w}{\partial^2 x}+\frac{\partial^2 w}{\partial^2 y}=\frac{1}{c} \frac{\partial^2 w}{\partial^2 t}$ Where $w(x,y,t)$ is the amplitude of the wave.

more anmations

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### 43. Generalized Single Degree Of Freedom Method

Version: July 28 2013

Generalized Single Degree Of Freedom Method For Wind Tower Structure. Estimating natural frequency for industrial wind tower using the method of generalized single degree of freedom

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### 42 Vibration analysis of forced single degree freedom system

Version: August 29 2013

Solving $m u''+c u'+k u=F \sin(\varpi t)$ vibration, resonance, critical damping, beat beat phenomenon, impulse response, magnification factor

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### 41 Vibration analysis of free unforced single degree freedom system

Version: July 24 2013

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### 40 Showing velocity and acceleration components for circular motion

Version: January 4 2014

Simple demo that shows a particle moving in circular orbit. Vectors show the current velocity and the red colored vectors show the centrifugal and tangential accelerations. The length of the vectors is made to be proportional to the magnitudes.

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### 38 elliptical satellite motion demo

Version: Jan 24, 2014

Change the ellipse eccentricity and see the effect on motion of satellite.

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### 37. Double pendulum using heavy spring

Version: Sept 10, 2012

Three degrees of freedom, Lagrangian method, Rayleigh factor, effective weight of spring

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### 36. particles collision simulation

Version: Sept 25, 2012

collision detection using priori (continuous) detection. Supports up to 8 particles.

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### 35. Spring Mass System On a Rotating Table

Version: August 8, 2012

moment of inertia, rotation dynamics, spring-mass on spinning disk

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### 34. Triple pendulum with damping

Version: September 1, 2011

Triple, double and simple pendulum, damped medium. Lagrangian. phase portrait

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### 33. Chaotic motion of damped driven pendulum

Version: September 2, 2011

Bifurcation, Poincare map, Power Spectrum, Phase Portrait

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### 32. Mass on a spring at end of a solid pendulum

Version: Nov 10, 2012

Lagrangian method, spring mass, NDSolve

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### 31. Rigid body disk pendulum rotating on moving table

Version: June 25, 2011

Lagrangian, principal moments of inertia, angular momentum $\frac{dL}{dt}$, 3 nonlinear equations of motion, numerical solution with NDSolve using direct Dynamics without using Manipulate

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### 30. Rigid body pendulum on a flywheel

Version: June 8, 2011

2 degrees of freedom: $\theta$, the pendulum's swing angle, and $\phi$ the flywheel's rotation angle. two nonlinear equations, Lagrangian energy method

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### 29. Direct dynamics for simulation of pendulum

Version: Nov 10, 2012

simulation of damped and driven pendulum with chaotic motion using direct Dynamics without using Manipulate

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### 28. cylinders with 3 springs

Version: July 26,2013

Simulation of motion of 2 masses connected by 3 springs. Equations derived using the Lagrangian energy method

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### 27. LQR Control of inverted pendulum on moving cart

Version: April 16, 2012

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### 26. PID controller design for second order system

Version: Feb 2, 2012

P, PI, PD, or PID. simulate the plant with and without the controller, Bode, Nyquist, open and closed loop

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### 25. Mohr's Circle For Plane Stress, 2D

Version: Nov 10, 2013

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### 24. Single span Euler Bernoulli beam

Version: Oct 21 2009

deflection curve of beam, bending moment, shear diagrams, point load, distributed load, external moment, deflection ratio

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### 23. finite element Ritz method for axially loaded beam

Version: 6/2/2012

deflection curve of beam, bending moment, shear diagrams, point load, distributed load, external moment, deflection ratio

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### 22. Solving 2D Poisson PDE on non-uniform rectangle grid

Version: May 20, 2013

$-\nabla^2 u=f(x,y)$, Neummann, Dirichlet B.C. direct, Jacobi, Gauss-Seidel, Gauss-Seidel red/black, SOR, Chebyshev, steepest descent, conjugate gradient, GMRES, BiCGSTAB, sparse matrices

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### 21. solving Helmholtz differential equation in 2D using finite difference

Version: Feb 2, 2012

solving $-\nabla^2 u-k^2 u=f(x,y)$ in 2D, Dirichlet, Sommerfeld B.C.

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### 20. solving Helmholtz differential equation in 1D using finite difference

Version: March 6, 2012

solving $-U_{xx} -k^2 U=f(x)$ in 1D, Dirichlet, Sommerfeld B.C.

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### 19. Solving diffusion-advection-reaction (heat) in 1D

Version: Feb 20, 2012

solving $c\,u_{xx}=d\,u_t+a\,u+f(x,t)$ in 1D. Neummann, Dirichlet B.C.

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### 18. Finite difference solution of the diffusion-convection in 1D

Version: Feb 10, 2012

solving $c\, u_{xx}=d\, u_t+a\, u_x$ in 1D, Dirichlet, Neummann B.C.

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### 17. Minimal example to solve Poisson 2D using Jacobi method

Version: March 6, 2012

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### 16. Finite difference for solving Poisson PDE on unit square

Version: Nov 14, 2010

small example, numerical solution of $\Delta^2u=-e^{-(x-0.25)^2-(y-0.6)^2}$ on unit square, Dirichlet B.C.

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### 15. Phase plane plot of the Van der Pol differential equation

Version: April 12 2009

phase plot ($x'(t)$ vs. $x(t)$), Van der Pol nonlinear differential equation for different initial conditions

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### 14. Design a digital filter by geometric means

Version: April 12 2009

select poles and zero locations with mouse, magnitude spectrum, BodePlot, Stability Margins, Nyquist, Nichols, root locus, S-plane map,region of convergence, convert H(z) to H(s)

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### 13. Fourier series coefficients of a rectangular pulse signal

Version: April 12 2009

plot magnitude and phase of the Fourier coefficients for a rectangular pulse train signal, sinc function

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### 12. From continuous-to-discrete time Fourier transform by sampling

Version: April 7 2010

Relation between continuous time Fourier transform (CTFT) of a continuous time signal $x_a(t)$ and discrete time Fourier transform (DTFT) of the discrete signal $x(n)$, sampling

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### 11. Analog-to-discrete system conversion using impulse invariance

Version: May 3, 2010

Applied to second order system, sampling, poles, zeros, $H(z)= \sum\limits_{i=1}^N \frac{T\, A_i}{1-z^{-1} \exp(p_i\, T)}$, $H(s)= \sum\limits_{i=1}^N \frac{A_i}{s-p_i}$

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### 10. ImageData using rows and columns

Version: August 7, 2013

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### 9. Power efficiency of amplitude modulation

Version: August 31 2009

Illustrates the effect of changing the modulation factor on the power efficiency of ordinary amplitude modulation (AM)

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### 8. Rectangular pulse and its Fourier transform

Version: December 27 2009

Manipulate a rectangular pulse signal and observe how its Fourier transform changes

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### 7. sinc interpolation to reconstruct a signal from its sample

Version: Feb 18 2010

Manipulate a rectangular pulse signal and observe how its Fourier transform changes

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### 6. Power content of frequency modulation and phase modulation

Version: September 6 2009

BesselJ, modulation index, carrier frequency, carrier amplitude, deviation constant, bandwidth of the modulated carrier, power ratio.

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### 5. IIR digital low-pass Filter Design by Butterworth method

Version: Sept 25, 2010

passband end frequency, attenuation, transformation of $H(s)$ to $H(z)$, bilinear or impulse invariance

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### 4. Computed tomography using Radon Transform (incremental angles)

Version: July 5, 2011

Simulation of Computed Tomography (CT), Radon transform. Incremental angle version

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### 3. Computed tomography using Radon Transform

Version: July 4, 2011

Simulation of Computed Tomography (CT), Radon transform. Incremental angle version

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### 2. Illustrating the use of discrete distributions in Version 8

Version: September 1, 2008

Simulation of Computed Tomography (CT), Radon transform. Incremental angle version

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### 1. Illustrating the use of discrete distributions in Version 7

Version: December 28, 2013

The probability mass function and the cumulative distribution function are displayed for all Mathematica 7 discrete distributions

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