MAE 242Dynamics – Section I

Dr. Kostas Sierros

Important information

• E-mail: Kostas.Sierros@mail.wvu.edu

• Room: G-19 ESB

• Phone: 293-3111 ext 2310

• HELP: Tuesday & Thursday 9:30-11:00 AM and by appointment

• MAE 242 – Section 1: Every Tuesday & Thursday 8:00-9:15 AM

• All lectures will take place at 113 MRB-E

Syllabus

Date Material covered Homework Exam weight21-Aug-07 Statics revision and 12.1-12.2    

23-Aug-07 12.3 TBA in class  28-Aug-07 12.4-12.6    

30-Aug-07 12.7-12.8 TBA in class  04-Sep-07 12.9-12.10    

06-Sep-07 13.1-13.3 TBA in class  11-Sep-07 13.4-13.6    

13-Sep-07 14.1-14.3 TBA in class  18-Sep-07 14.4-14.6    20-Sep-07 Review    25-Sep-07 Midterm 1   15%

27-Sep-07 15.1-15.3 TBA in class  02-Oct-07 15.4    

04-Oct-07 15.5-15.7 TBA in class  09-Oct-07 16.1-16.3    

11-Oct-07 16.4-16.5 TBA in class  16-Oct-07 16.6-16.7    

18-Oct-07 17.1 TBA in class  23-Oct-07 17.2-17.3    25-Oct-07 Review    30-Oct-07 Midterm 2   15%

01-Nov-07 17.4 TBA in class  06-Nov-07 17.5    

08-Nov-07 18.1-18.2 TBA in class  13-Nov-07 18.3-18.4    

15-Nov-07 18.5 TBA in class  27-Nov-07 19.1-19.2    

29-Nov-07 19.3-19.4 TBA in class  04-Dec-07 General concepts on vibrations (Ch 22)    06-Dec-07 Review    12-Dec-07 FINAL EXAM   25%

Assessment & important datesAssessment

Midterm 1: 15%, Midterm 2: 15%, Structural project and Report: 15%, Quiz: 10%, Computer assignment and report: 10%

Homework: 10%, Final: 25%

Important dates

Oct 26th is the last day to drop class

Nov 17-25th Thanksgiving

Dec 7th last day of class

Dec 16th Winter break

Text books

Engineering Mechanics: Dynamics

C. Hibbeler, 11th Edition, Prentice Hall, 2006

…and probably some more…

Problem 2

Problem 3

Kinematics of a particle: Objectives

• Concepts such as position, displacement, velocity and acceleration are introduced

• Study the motion of particles along a straight line. Graphical representation

• Investigation of a particle motion along a curved path. Use of different coordinate systems• Analysis of dependent motion of two particles

• Principles of relative motion of two particles. Use of translating axis

Lecture 2

• Kinematics of a particle (Chapter 12)

- 12.3

Material covered

• Kinematics of a particle

-Rectilinear kinematics: Erratic motion

-Next lecture; General curvilinear motion, rectangular components and motion of a projectile

Today’s Objectives

Students should be able to:

1. Determine position, velocity, and acceleration of a particle using graphs (12.3)

Erratic (discontinuous) motion

Graphing provides a good way to handle complex motions that would be difficult to describe with formulas. Graphs also provide a visual description of motion and reinforce the calculus concepts of differentiation and integration as used in dynamics

The approach builds on the facts that slope and differentiation are linked and that integration can be thought of as finding the area under a curve

s-t graph construct v-t

Plots of position vs. time can be used to find velocity vs. time curves. Finding the slope of the line tangent to the motion curve at any point is the velocity at that point (or v = ds/dt)

Therefore, the v-t graph can be constructed by finding the slope at various points along the s-t graph

v-t graph construct a-t

Plots of velocity vs. time can be used to find acceleration vs. time curves. Finding the slope of the line tangent to the velocity curve at any point is the acceleration at that point (or a = dv/dt)

Therefore, the a-t graph can be constructed by finding the slope at various points along the v-t graph

Also, the distance moved (displacement) of the particle is the area under the v-t graph during time t

Given the a-t curve, the change in velocity (v) during a time period is the area under the a-t curve.

So we can construct a v-t graph from an a-t graph if we know the initial velocity of the particle

a-t graph construct v-t

v-t graph construct s-t

We begin with initial position S0 and add algebraically increments Δs determined from the

v-t graph

Equations described by v-t graphs may be integrated in order to yield equations that describe segments of the s-t graph

graphing

handle complex motions

visual description of motion

differentiation and integration

slope and area under curve

Please remember the link!!!

Explanation of Example 12.7 (A)

Explanation of Example 12.7 (B)

A couple of cases that are a bit more …COMPLEX… and therefore need more attention!!!

A couple of cases more…

a-s graph construct v-s

A more complex case is presented by the a-s graph. The area under the acceleration versus position curve represents the change in velocity

(recall a ds = v dv )

This equation can be solved for v1, allowing you to solve for the velocity at a point. By doing this repeatedly, you can create a plot of velocity versus distance

v-s graph construct a-s

Another complex case is presented by the v-s graph. By reading the velocity v at a point on the curve and multiplying it by the slope of the curve (dv/ds) at this same point, we can obtain the acceleration at that point.

a = v (dv/ds)

Thus, we can obtain a plot of a vs. s from the v-s curve

Please think about it

If a particle in rectilinear motion has zero speed at some instant in time, is the acceleration necessarily zero at the same instant ?

Groups think about this problem please

Given: The v-t graph shown

Find: The a-t graph, average speed, and distance traveled for the 30 s interval

Find slopes of the curves and draw the a-t graph. Find the area under the curve--that is the distance traveled. Finally, calculate average speed (using basic definitions!)

Hint

Solution to the problem (A)

1

8

a(ft/s²)

10 30t(s)

For 0 ≤ t ≤ 10 a = dv/dt = 0.8 t ft/s²

For 10 ≤ t ≤ 30 a = dv/dt = 1 ft/s²

Solution to the problem (B)

s0-10 = v dt = (1/3) (0.4)(10)3 = 400/3 ft

s10-30 = v dt = (0.5)(30)2 + 30(30) – 0.5(10)2 – 30(10)

= 1000 ft

s0-30 = 1000 + 400/3 = 1133.3 ft

vavg(0-30) = total distance / time

= 1133.3/30

= 37.78 ft/s

Try at home please (I)

Try at home please (II)

Try at home please (III)