PROBLEM 18.25
Three slender rods, each of mass m and length 2a, are welded
together to form the assembly shown. The assembly is hit at A in a
vertical downward direction. Denoting the corresponding impulse
by F At, determine immediately after the impact (
ME326 Spring 2015
E. Rossman
Assignment 8
This homework will not be collected. All answers will be posted on Polylearn.
Problem 1: Beer/Johnston Problem 18.25
Problem 2: Beer/Johnston Problem 18.33
Problem 3:
The uniform thin disk of radius R= 6-in and we
Transformation Matrices for Vectors
y
Y
X
x
P
Y
y
x
z
X
Z
Z (System B)
z (System I)
Figure 1. Position vector of point P with two frames, Inertial System I and Rotating System B
Position vector of P in Inertial System (System I)
= + +
Position vector of
Mechanisms 4-bar linkages
Figure 1. Four-bar linkage
Link Names
1.
2.
3.
4.
Input link
Output link
Coupler link
Fixed link
Grashofs Law
If the sum of the shortest and longest link of a planar quadrilateral linkage
is less than or equal to the sum of the r
Fixed Point Rotation Examples
Example 1
The platform centered at O rotates about the z axis at a constant rate of 1 rad/s.
The disk with point A rotates at a rate of 3 rad/s. Find the angular velocity and
acceleration of the disk with point A, and the vel
3 Dimensional Motion Rotation about a fixed point.
Angular Velocities
Figure 1. Rotation about 2 axes (Meriam)
For the drum,
=
Velocity of a Point, Fixed Point Rotation
/
=
Space Cone and Body Cone
Figure 2. Instantaneous Center of Rotation creates s
Velocities and Accelerations
Look at the components of velocity of the slider on a turntable:
Let =.
r/ +
= +
For each of the velocity components, what are the
possible changes that can cause acceleration of P?
+
+
+ 2
= +
Coriolis Accelera
Rotating Axes in 2D
I.
Class Demonstration of the movement of a point at the end of a
telescoping rod:
Demo 1: Point A acts as a pivot point. Point B is NOT allowed to
telescope, i.e., point B is rigidly attached via the rod to point A.
y
A
x
B
Questions
Introduction to Mechanisms
A mechanism consists of links which are connected by joints. The purpose of a
mechanism is to transfer motion and/or force.
A Joint Pair may be categorized by the number of Degrees of
Freedom (DOF) that it allows. For two-dimens
Rigid Body Impulse and Momentum
Derive Impulse/Momentum Relation from Newtons 2nd Law:
Principle of Linear Impulse and Momentum:
1 + = 2
1 + = 2
Final Momentum-Show on
MVD
Initial Momentum-Show
on Kinetic Diagram/MVD
Linear ImpulseGet from FBD
Angular Imp
Newtons 2nd Law with Rigid Bodies
1. Make FBD and Kinetic/MAD diagrams. Axes may depend on type of
motion. See 3 below.
FBD
o Shows applied and reactive forces.
o Shows applied and reactive moments.
MAD (Mass Acceleration Diagram or Kinetic Diagram)
o M
Equations of Motion for 3D Rigid Bodies
Introduction: Recall 2D Equations of Motion:
=
=
If taking moments with respect to G, this becomes:
=
A more general way to write this is:
=
):
Finding the Derivative of Angular Momentum at G (
Figure 1: 3D
Gyroscopes
Steady Precession Equation of Motion
= ( + )
Steady Precession: Spin, Moment, and Precession follow the Right Hand Rule.
When =90,
=
=
Example
The top consists of a thin disk that has a weight of 8 lb and a radius of 0.3 ft. The
rod has n
Angular Momentum and Kinetic Energy
Kinetic Energy
2D Kinetic Energy
1 2
1
= 2 +
2
2
3D Kinetic Energy
1
1
= 2 +
2
2
3D Kinetic Energy for fixed point rotation (Special Case)
1
=
2
3D Kinetics
Work/Energy Relation
1 + 12 = 2
Impulse Momentum Prin
3D Angular Momentum
Review of 2D Angular Momentum
Angular Momentum = Moment of Momentum
Use an MVD to find the Moment of Momentum with respect to a point.
Angular Momentum in 3D
Figure 1. 3-dimensional MVD. Ref: Beer/Johnson 10th Ed
3D Angular Momentum ab
Inertia Tensor.
z
z'
y
y
x'
x
z
Diagonalized Inertia Tensor
0
0
0
0
0
0
Z
Y
G
y
x
X
Finding the Principal Axes of Inertia
For a given set of axes, xyz, for which the moments and products of
inertia are known, there are three roots of I in the following
Rigid Body Work and Energy
Work /Energy Relation derived from Newtons 2nd Law
Work (U)
Work done by a force:
12 =
12 = cos
Work done by a couple:
12 =
Kinetic Energy (T)
1 2
1
= 2 +
2
2
Principle of Work and Energy
1 + 12 = 2
Conservation of Energy (
Moments and Products of Inertia
Review of Mass Moments of Inertia
z
I xx = rx dm = ( y 2 + z 2 )dm
2
I yy = ry dm = ( x 2 + z 2 )dm
2
rz
I zz = rz dm = ( x 2 + y 2 )dm
2
dm
Parallel Axis Theorem:
ry
I = I + md 2
rx
y
x
Products of Inertia
z
x = distance o
Gyroscopes
Demonstration: Bike Wheel
Figure 1. Bike Wheel held by handles
Figure 2. Bike wheel supported by rope
Discussion:
When bike wheel is spinning and supported as in Figure 1, what is the
direction of the angular momentum?
When bike wheel is spin
ME326
Homework Assignment 3
Hand in Problems 1-6, shown below. Work the additional problems shown below Problem 1-6, but do
not hand them in.
Problem 1
Refer to the following Beer/Johnston problems shown below.
For each of the problems, show calculations
/Volumes/204/MHDQ078/work%0/indd%0
Moments of Inertia of
Common Geometric Shapes
Rectangle
Ix 121 bh3
Iy 121 b3h
Ix 13bh3
Iy 13b3h
JC 121 bh1b2 h2 2
Mass Moments of Inertia of
Common Geometric Shapes
y
Slender rod
y'
G
Iy Iz 121 mL2
h
z
x'
C
L
x
x
Thin re
ME326 Winter 2013 E. Rossman
Assignment 4
Due Monday, 2/3 at beginning of class
Problem 1: Find the mobility (number of degrees of freedom) for each of the devices below:
jl'o-v~:_f~
j .\V\l S
b. Link 1 attached to "ground. Links 1, 4, and 6 are solid
uABXCg/A - ~z
" 4 A
wMKx (Income; +10$thoJ> :
L.)
~2rcxMJ + 23?
A A 4
23'. MJ rual : ~32:
ME326 Winter 2014 Name KE Y
Exam 1 Class Time (Circle) 8 AM 9 AM
Instructions: Show all work as necessary to demonstrate your understanding, starting with.the
approp
13/
J
1/8,]
PROBLEM 18.3
Two unifoml rods AB and CE, each of weight 3 lb and
length '2 fl, are welded to each other at their midpoints.
Knowing that this assembly has an angular velocity of
constant magnitude (0 = 12 rad/s, determine the magnitude
and dir
Exam 2 Winter 2014 Name
Instructions: Show all work neatly in a single column, starting with the appropriate governing
equation. Clearly show all necessary diagrams.
Problem 1:
The rod assembly shown is supported at G by a ball-and-socketjoint. The rod ha
PROBLEM 18.68
The S-kg shaft shown has :1 uniform cross section. Knowing
that the shaft rotates at. the constant rate a) :12 rad/s, determine
. . |
the dynamlc macnons at A and B '
w: .3 r2;
andl Dyno/WL Rwdtmj a+ A i [Z l
goiu"
IV =ij =0 (swwwj on y
$0
ME 329 Fall 2015 Exam #1
Name K5 1 Section
Problem 1: (50 points) The gure shows a pair of shaft-mounted straight cut spur
gears having a diametral pitch of 5 teeth/in with a 2 face width, an 18-tooth 20
pinion driving a 45-tooth gear. The power input
ME326 Winter 2014 E. Rossman
Assignment 1
Due Tuesday, 1/21, at beginning of class
Solve all problems using the vector cross products. Review the formatting requirements discussed in the
syllabus.
Monday Lecture Problems:
Beer/Johnston 15.158
Beer/Johnsto