## 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 notes from lecture will be posted here** (note that the lectures are also recorded on lecture-capture - to access them go to the Canvas site for the course, and click the 'EchoCenter' button on the left - so you can find the explanations that went with the formulas in class if you need them)

- L2: Motion of particles - position,velocity acceleration relations, math review, straight line motion, integrating experimental measurements of acceleration using MATLAB matlab file from class, data file from class (the lecture capture failed for this class - sorry!)
- L3: Motion of particles - circular motion with normal/tangential coords; polar coords; Newton's laws, tablecloth example
- L4: Motion of particles - examples of F=ma and M_c=0; example of F=ma and n-t coordinates; calculating the path of a particle using Newton's laws; trajectory formulas
- L5 Motion of particles - calculating particle motion using F=ma and solving the differential equation with Mupad and Matlab. Centrifugal pump example and suspension example. Impeller matlab Spring-mass matlab
- L6: Motion of particles - squirrel trajectories; event functions in MATLAB; building in an earthquake event_template, squirrel_matlab, building_matlab
- L7: Conservation laws for particles: Definitions of power, work and kinetic energy; work-power-KE relations for particles; examples.
- L8: Conservative forces; Potential Energy of a conservative force; conservative systems; energy equations for conservative systems. Examples.
- L9: Linear momentum: Definitions of linear impulse and momentum; force-plate jump example; momentum equation for systems of particles; collisions - definition of restitution coefficient; general formulas for 1-D collision
- L10: 3D frictionless collisions; Angular Momentum; satellite example, baby walker example.
- L11: Free vibration of conservative systems I: Features of vibrations; Simple Harmonic Motion; Harmonic Oscillator; meaning of Natural Frequencies and Mode Shapes
- L12: Free vibration of conservative systems II: How to count degrees of freedom and vibration modes; combining springs; using energy to obtain EOM; Finding natural frequencies for nonlinear systems (pendulum)
- L13: Damped Free Vibrations minus-K vibration isolator; dashpots; canonical damped vibration problem (damped spring-mass system); definitions of damping factor and damped natural frequency; thrust measurement test stand example.
- L14: Damped Free Vibrations II, Forced Vibrations I: Measuring damping and natural frequency from a free vibration signal; solution to spring-mass system subjected to harmonic external force; 'steady state' and 'transient' response of a forced system; magnification factor; behavior of forced system for low and high frequency, resonance.
- L15: Forced Vibrations II: Base and rotor excited systems; resonance and anti-resonance in systems with two DOF, the tuned mass damper, exciting vibration modes in musical instruments....
- L16 Rigid bodies 1: kinematics of a rigid body rotating about a point; formulas for acceleration and velocity on a rotating rigid body; gears
- L17 Rigid bodies 2: kinematics of combined rotation and translation; robot arm kinematics; head impacts
- L18 Rigid bodies 3: Rolling wheels, equations of motion for rigid bodies, calculating mass moments of inertia
- L19 Rigid bodies 4: equations of rotational motion for rigid bodies, ladder, rolling/slipping disk and rigid body pendulum examples.
- L20 Rigid bodies 5: Equations of motion for objects rotating about a fixed point; pendulum example revisited; cylinder races; rotational KE.
- L21 Rigid bodies 6: 2 bar/collar example; linear and angular momentum for rigid bodies.
- L22 Rigid bodies 7: energy methods examples, hydrofoil example
- L23 Solar Car

**Summary slides for Midterm Review**

**Summary slides for Final** (slides in ppt format - needed to see the animations)

**Worked problems from Section**

- S4: Motion of Particles - Straight-line motion with variable acceleration; example using n-t coordinates; example using polar coordinates; example of F=ma and Mc=0 to calculate reaction forces
- S6: Solving particle motion problems with MATLAB: quadrupole filter; event function; forced pendulum. Quadrupole filter MATLAB; forced pendulum MATLAB
- S9: Using Energy Conservation: spider-web problem; electric motorcycle problem
- S11 Free vibration of undamped systems: Solving EOM using solution sheet; buoy problem; frictional oscillator
- S12 Free vibration of undamped systems: Derivation of solution to standard vibration EOM; counting DOF; examples of energy approach to EOM and nonlinear systems
- S13 Damped free vibrations Simple examples of using damped vibration formulas; measuring impulses from explosions using damped vibration response
- S14: Rigid bodies 1: Rigid body kinematics formulas - application to gears.
- S15: Rigid bodies 2: Using rigid body kinematics formulas to find velocities and accelerations in mechanisms
- S16: Rigid bodies 3: Procedure for solving rigid body dynamics problems, examples, center of percussion of a bat
- S17: Rigid bodies 4: Energy methods - calculating angular/linear velocitities using energy methods; deriving EOM with energy methods

**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