## Syllabus and Lecture Notes

**Course Goals: on completing EN0040, students will:**

- Be able to idealize a simple mechanical system or component as a collection of particles or rigid bodies, and to use Newtonian mechanics, with the aid of analytical or computational methods, to analyze forces and motion in the idealized system. Relates to ABET outcomes (a), (e), (k)
- Be familiar with the characteristics of vibrations in linear systems; and have the ability to analyze the free, damped, and forced vibrations of a 1 degree of freedom system. Relates to ABET outcome (a)
- Be able to design and conduct simple experiments to measure the dynamical properties of a mechanical system or components. Relates to ABET outcome (b)
- Be able to apply Newtonian mechanics to design a mechanical system to meet specified constraints, including: to function effectively in teams of 3-5 students; to communicate design specifications through clear and effective oral and written reports, to perform appropriate design calculations and optimization where appropriate, and to successfully manufacture and test a completed design Relates to ABET outcome (a), (b), (c), (g)

**Class lecture notes**, 2015

L2 Jan 29 Dynamics of Particles- position-velocity-accel relations; straight line motion; integrating accels with MATLAB, formulas for position, velocity acceleration for circular motion at constant speed. Matlab script to integrate accels from a data file Sample data file (accels measured by the notepad)L3 Feb 3 Dynamics of Particles- Example circular motion problem; circular motion at arbitrary speed; motion along a curved path with normal-tangential vectors; formulas for polar coordinates; Newton's laws for a particle/rigid body with no rotation; Example of calculating forces using Newton's laws.L4 Feb 5 Dynamics of Particles- Example of F=ma and Mc=0; example circular motion with n-t coords; calculating position of a particle with time dependent forces; trajectory equations; example using trajectory equations.L5 Feb 10 Solving equations of motion for particles with Mupad and Matlab(Centrifugal pump and spring-mass system examples) mupad matlab for impeller matlab for spring-mass systemL6 Feb 10 Solving equations of motion for particles with Mupad and Matlab(Squirrel problem, MATLAB 'event' function; building vibratin problem). Squirrel matlab, building matlab.L7 Conservation laws for particles:Definitions of power; work done by a force; kinetic energy; power/work-kinetic energy relation for a single particle; calculating forces from a measured atomic binding energy; car crash exampleL8 Conservative forces; energy conservation equation for conservative systems;Definition of a conservative force; Potential Energy; Examples of using force-potential energy relations; conservative systems; energy conservation equation for conservative systems; bungee jumper exampleL9 Impulse-Momentum relations for systems of particlesDefinitions of impulse and momentum; impulse-momentum relations for a single particle; force-plate measurement example; impulse-momentum for a system of particles; application to collisions; restitution coefficient; collisions with straight-line motion.L10 3D frictionless collsisions, angular momentumCalculating satellite velocities, baby walker problem.L11: Vibration of undamped 1DOF systems:General features of a vibration signal; Simple Harmonic Motion (SHM); Harmonic Oscillator solution; using tables of solutions to solve EOM; qualitative discussion of natural frequencies and mode shapes for systems with several degrees of freedomL12: Vibration of undamped 1DOF systems:Counting DOF and vibration modes; combining springs; relationship between static deflection and natural frequency; using energy conservation for calcuating EOM, EOM for a nonlinear system (pendulum)L13 Free vibration of damped 1DOF systems: Solving EOM for nonlinear systems (continued); The Dashpot; Vibration of a spring-mass-dashpot system; damping factor; underdamped, overdamped and critically damped system.L14 Forced vibrations:Using damped vibration respnose to measure natural frequencey and damping coefficient; response of a spring-mass system subjected to a harmonic external force; magnification factor; resonance.L15 Forced Vibrations:Base and rotor excited systems, qualitative discussion of forced systems with several degrees of freedom (and natural frequencies/mode shapes)L16 Rigid body kinematicsAngular velocity and acceleration; formulas for velocity and acceleration of a point in a rigid body rotating about a stationary axis in 2D/3D, gear kinematics.L17 Rigid Body KinematicsL18 Rigid Body Dynamics(rotational equation of motion for rigid bodies, mass moments of inertia)L19 Rigid body dynamics/energy methods for rigid bodiesLadder problem; pendulum problem; rotation about a fixed axis; kinetic energy of a rigid bodyL20 Rigid Body Dynamics(Energy methods)L21 Rigid Body Dynamics(Energy and angular momentum)L22 Solar CarProjectADDITIONAL RIGID BODY SLIDES:

SUMMARY SLIDES FOR MIDTERM (ppt) (pdf)

SUMMARY SLIDES FOR FINAL (ppt) (pdf)

**Detailed notes (electronic text)**

1. Brief introduction to the objectives and methods of dynamics

2. Review of forces and Moments (pdf version) (reading assignment - not covered in lectures2.1 Forces

2.2 Moments

2.3 Force couples, pure moments and Torques

2.4 Constraint and reaction forces and moments

2.5 Friction forces

3. Analyzing motion of systems of particles(pdf version)3.1 Equations of motion for a particle

3.2 Calculating forces required to cause prescribed motion of particles

3.3 Solving equations of motion for systems of particles with MATLAB

4. Conservation Laws for Particles(pdf version)4.1 Work, power, potential energy and kinetic energy relations

4.2 Linear impulse-momentum relations

4.3 Angular impulse-momentum relations

5. Vibrations(pdf version)5.1 Features of vibrations and overview of issues in controlling vibrations

5.2 Free vibration of conservative single degree of freedom systems

5.3 Free vibration of damped single degree of freedom systems

5.4 Forced vibration of single degree of freedom systems

5.5 Introduction to vibration of multi-degree of freedom systems (advanced topic - not covered in lecture)

Summary of Solutions to EOM for vibration problems (pdf)

Java forced vibration simulator

(The Java Applets were written way back in the day when love was free, and the internet was innocent and trusting, and the word 'hacker' did not yet exist. Getting them to run on browsers with modern security features is a chore. You need to

(i) Download the latest Java (go to java.com);

(ii) Go to your computer security settings (Control Panel);

(iii) select Programs from the menu on the left and then click the Java icon;

(iv) select the Security tab from the top of the menu; click the 'Edit Site List' button;

(v) Click Add

(vi) Enter the web address of the Notes page of the course website into the box provided

http://www.brown.edu/Departments/Engineering/Courses/En4/Notes/notes.html

(vi) Move the Security bar to 'Medium'

(vi) Click OK.

(vii) Ignore the dire warning (taking this course might mess with your brain but we wont do anything to your computer)

You should then be able to run the applets. If any of you CS150 wizards have the interest, time, expertise and infrastructure to package our old Java codes into signed JAR files we would love to hear from you!

6. Analyzing motion of systems of rigid bodies6.1 Introduction to Rigid Body Motion

6.2 Describing (two dimensional) rigid body motion

6.3 Equations of motion for a rigid body moving in a plane

6.4 Solving equations of motion for a rigid body

6.5 Energy and Momentum for rigid bodies

6.6 Application to power transmission