Mathematica Scientiﬁc Demonstrations
by Nasser M. Abbasi. Updated April 1, 2015

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 Equation of motion simulation of RRR robot arm in 3D

Last changed: April 1, 2015

Equations of motion of 3 degrees of freedom robot arm (3 revolute joints) is derived and simulated. You can adjust the torque at each joint, and add damping and initial angles.

The derivation is PDF and notebook

### 2 Kharitonov triangle for interval polynomial

Last changed: Nov 28, 2014

Simulation of motion of Kharitonov triangle for using two intervals polynomial as illustrations

### 3 Simple example of 3D rotation

Last changed: April 8, 2015

Cube is rotated in 3D

Last changed: Sept 22, 2014

Select the Radon transform method and the inverse Radon transform ﬁlter and change the cut oﬀ frequency and see the eﬀect on the image

### 5 Wave equation using leap frog

Last changed: July 5, 2014

Solve wave equation 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.

### 6 Symmetric top gyro motion

Last changed: July 15, 2014

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

### 7 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

### 8 Spinning wheel

Last changed: June 29, 2014

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

### 9 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.

### 10 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.

### 11 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.

### 12 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.

### 13 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.

### 14 Simulation of rigid body dynamics

Last changed: Dec 19, 2013

Final project EMA 542. Two noncollinear rotating bodies

### 15 Another rigid body dynamics

Last changed: Dec 19, 2013

A small simulation, Solving EMA 542 HW3 problem 1

### 16 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)

### 17 Finite Diﬀerence Formulas

Last changed: August 29, 2013

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

### 18 2D rectangular membrane modal analysis

Last changed: July 31 2013

PDE solved is Where is the amplitude of the wave

### 19 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

### 20 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

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

Last changed: July 24 2013

Solving vibration system.

### 22 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.

### 23 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.

### 24 elliptical satellite motion demo

Last changed: January 24, 2014

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

### 25 Double pendulum using heavy spring

Last changed: January 18, 2015

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

### 26 particles collision simulation

Last changed: Sept 25, 2012

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

### 27 Spring Mass System On a Rotating Table

Last changed: August 8, 2012

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

### 28 Triple pendulum with damping

Last changed: September 1, 2011

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

### 29 Chaotic motion of damped driven pendulum

Last changed: September 2, 2011

Bifurcation, Poincare map, Power Spectrum, Phase Portrait

### 30 Mass on a spring at end of a solid pendulum

Last changed: Nov 10, 2012

Lagrangian method, spring mass, NDSolve

### 31 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

### 32 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

### 33 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

### 34 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

### 35 LQR Control of inverted pendulum on moving cart

Last changed: April 16, 2012

Simulation of LQR control

### 36 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

### 37 Mohr's Circle For Plane Stress, 2D

Last changed: Nov 10, 2013

principle stresses and Mohr's Circle

### 38 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

### 39 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

### 40 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

### 41 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.

### 42 solving Helmholtz diﬀerential equation in 1D using ﬁnite diﬀerence

Last changed: March 6, 2012

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

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

Last changed: Feb 20, 2012

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

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

Last changed: Feb 10, 2012

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

### 45 Minimal example to solve Poisson 2D using Jacobi method

Last changed: March 6, 2012

Illustrate Jacobi method

### 46 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.

### 47 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

### 48 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

### 49 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

### 50 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

### 51 Analog-to-discrete system conversion using impulse invariance

Last changed: May 3, 2010

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

### 52 ImageData using rows and columns

Last changed: August 7, 2013

Change image size using matrix notation

### 53 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)

### 54 Rectangular pulse and its Fourier transform

Last changed: December 27 2009

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

### 55 sinc interpolation to reconstruct a signal from its sample

Last changed: Feb 18 2010

sinc interpolation

### 56 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.

### 57 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

### 58 Computed tomography using Radon Transform (incremental angles)

Last changed: July 5, 2011

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

### 59 Computed tomography using Radon Transform

Last changed: July 4, 2011

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

### 60 Illustrating the use of discrete distributions in Version 8

Last changed: Oct 18, 2014

Illustrating the use of discrete distributions in Version 8.

### 61 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