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Mathematica Scientific Demonstrations
by Nasser M. Abbasi. Updated August 11, 2014

These are Mathematica demos I wrote over the last 3 or so years. Some of these are also published at Wolfram demonstration web site The Mathematica CDF plugin needs to be installed in your browser to run any of these CDF demos. The plugin is free and easy to install from Wolfram CDF plugin webpage.




1 Wave equation using leap frog

Last changed: July 5, 2014

Solve wave equation        2
utt = c uxx  using leap frog method. boundary conditions are fixed on left and right side. Initial speed is zero. Select initial data and see the wave move.

2 Symmetric top gyro motion

Last changed: July 15, 2014

Adjust the initial wheel spin for symmetric gyro top and watch the effect on motion.

3 Effect on precession due to spinning wheel

Last changed: June 29, 2014

Modify the wheel spin rate, its radius and distance from the support and see the effect on the precession spin rate




4 Spinning wheel

Last changed: June 29, 2014

Adjust the radius, number of rods and angle to see the wheel spin in 3D

5 Rotation stability of 3D cube

Last changed: June 28, 2014

Cube is stable when spinning around either the major or the minor principal axes. This demo illustrates this by solving Euler equations of motion in 3D for zero torque. Select the spin axes, then click on perturbe to see the effect.

6 Rotation stability of cylinder

Last changed: June 28, 2014

Cylinder is a solid object which has one unique moment of inertia, which is around the z-axis. When it is spinning around z-axis, and perturbed it will remain stable. If it was spinning around either the x or the y axes then perturbed, it will become unstable. This demo illustrates this by solving Euler equations of motion in 3D for zero torque.




7 simple spring-mass-damper simulation

Last changed: June 19, 2014

Change the mass, stiffness or damping and see the effect on motion of mass. No stop bar case.

8 simple spring-mass-damper with stopbar

Last changed: June 19, 2014

Change the mass, stiffness or damping and see the effect on motion of mass. When mass hits the bar, it bounces back.

9 three pendulums with two springs

Last changed: 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.




10 Simulation of rigid body dynamics

Last changed: Dec 19, 2013

Final project EMA 542. Two noncollinear rotating bodies

11 Another rigid body dynamics

Last changed: Dec 19, 2013

A small simulation, Solving EMA 542 HW3 problem 1

12 Direct and Shear Strain Deformation in 3D

Last changed: 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)




13 Finite Difference Formulas

Last changed: August 29, 2013

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

14 2D rectangular membrane modal analysis

Last changed: July 31 2013

PDE solved is   Where   is the amplitude of the wave

15 Generalized Single Degree Of Freedom Method

Last changed: 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




16 Vibration analysis of forced single degree freedom system

Last changed: August 29 2013

Solving   vibration, resonance, critical damping, beat beat phenomenon, impulse response, magnification factor

17 Vibration analysis of free unforced single degree freedom system

Last changed: July 24 2013

Solving   vibration system.

18 Velocity and acceleration components for circular motion

Last changed: 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.




19 Adding Gaussian noise to signal using SNR

Last changed: June 21 2014

Specify the SNR and see the Gaussian noise added. The larger the SNR, the less the noise. SNR represents the ratio of the variance of the signal to the noise. For a signal, and cosine is used from zero to 2 PI.

20 elliptical satellite motion demo

Last changed: January 24, 2014

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

21 Double pendulum using heavy spring

Last changed: Sept 10, 2012

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




22 particles collision simulation

Last changed: Sept 25, 2012

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

23 Spring Mass System On a Rotating Table

Last changed: August 8, 2012

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

24 Triple pendulum with damping

Last changed: September 1, 2011

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




25 Chaotic motion of damped driven pendulum

Last changed: September 2, 2011

Bifurcation, Poincare map, Power Spectrum, Phase Portrait

26 Mass on a spring at end of a solid pendulum

Last changed: Nov 10, 2012

Lagrangian method, spring mass, NDSolve

27 Rigid body disk pendulum rotating on moving table

Last changed: June 25, 2011

Lagrangian, principal moments of inertia, angular momentum dL/dt, 3 nonlinear equations of motion, numerical solution with NDSolve




28 Rigid body pendulum on a flywheel

Last changed: 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

29 Direct dynamics for simulation of pendulum

Last changed: Nov 10, 2012

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

30 cylinders with 3 springs

Last changed: July 26,2013

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




31 LQR Control of inverted pendulum on moving cart

Last changed: April 16, 2012

Simulation of LQR control

32 PID controller design for second order system

Last changed: Feb 2, 2012

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

33 Mohr's Circle For Plane Stress, 2D

Last changed: Nov 10, 2013

principle stresses and Mohr's Circle




34 Single span Euler Bernoulli beam

Last changed: Oct 21 2009

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

35 finite element Ritz method for axially loaded beam

Last changed: 6/2/2012

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

36 Solving 2D Poisson PDE on non-uniform rectangle grid

Last changed: May 30, 2014

2D Poisson PDE on non-uniform rectangle grid.   . Neummann, Dirichlet B.C. direct, Jacobi, Gauss-Seidel, Gauss-Seidel red/black, SOR, Chebyshev, steepest descent, conjugate gradient, GMRES, BiCGSTAB, sparse matrices




37 solving Helmholtz differential equation in 2D using finite difference

Last changed: Feb 2, 2012

solving Helmholtz differential equation in 2D. Dirichlet, Sommerfeld B.C.   in 2D, Dirichlet, Sommerfeld B.C.

38 solving Helmholtz differential equation in 1D using finite difference

Last changed: March 6, 2012

solving   in 1D, Dirichlet, Sommerfeld B.C.

39 Solving diffusion-advection-reaction (heat) in 1D

Last changed: Feb 20, 2012

solving   in 1D. Neummann, Dirichlet B.C.




40 Finite difference solution of the diffusion-convection in 1D

Last changed: Feb 10, 2012

solving   in 1D, Dirichlet, Neummann B.C.

41 Minimal example to solve Poisson 2D using Jacobi method

Last changed: March 6, 2012

Illustrate Jacobi method

42 Finite difference for solving Poisson PDE on unit squared

Last changed: Nov 14, 2010

small example, numerical solution of Poisson PDE on unit square. Dirichlet B.C.    on unit square, Dirichlet B.C.




43 Design a digital filter by geometric means

Last changed: 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   to

44 From continuous-to-discrete time Fourier transform by sampling

Last changed: April 7 2010

Relation between continuous time Fourier transform (CTFT) of a continuous time signal   and discrete time Fourier transform (DTFT) of the discrete signal   sampling

45 Fourier series coefficients of a rectangular pulse signal

Last changed: April 12 2009

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




46 Phase plane plot of the Van der Pol differential equation

Last changed: April 12 2009

phase plot   vs.   , Van der Pol nonlinear differential equation for different initial conditions

47 Analog-to-discrete system conversion using impulse invariance

Last changed: May 3, 2010

Applied to second order system, sampling, poles, zeros.    ,

48 ImageData using rows and columns

Last changed: August 7, 2013

Change image size using matrix notation




49 Power efficiency of amplitude modulation

Last changed: August 31 2009

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

50 Rectangular pulse and its Fourier transform

Last changed: December 27 2009

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

51 sinc interpolation to reconstruct a signal from its sample

Last changed: Feb 18 2010

sinc interpolation




52 Power content of frequency modulation and phase modulation

Last changed: September 6 2009

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

53 IIR digital low-pass Filter Design by Butterworth method

Last changed: Sept 25, 2010

passband end frequency, attenuation, transformation of   to   , bilinear or impulse invariance

54 Computed tomography using Radon Transform (incremental angles)

Last changed: July 5, 2011

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




55 Computed tomography using Radon Transform

Last changed: July 4, 2011

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

56 Illustrating the use of discrete distributions in Version 8

Last changed: September 1, 2008

Illustrating the use of discrete distributions in Version 8.

57 Illustrating the use of discrete distributions in Version 7

Last changed: December 28, 2013

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