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
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
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
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
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
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
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
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
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
/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
Numerical Methods for Solving Differential Equations.
I.
Differential Equation Order
A. First Order Diff Eqn: x=f(t) only
Example:
= 2 2
Solution: x=f(t)
B. 2nd Order Diff Eqn: v=f(x,t)
Example:
II.
=
2
2
=
3
Solutions: = (, ) , = ()
Analytical Solutions
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
ME326 Winter 2014 E. Rossman
Assignment 1
Due Monday, 1/27, at beginning of class
Note: Solve all problems using vector cross products as we did in class.
Monday Lecture Problems:
Beer/Johnston 15.224
Beer/Johnston 15.234
Beer/Johnston 15.236
Wednesday Le
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
. . g 1 B 40 Determine the mass products of inertia Iy ,WI, and In of the steel
GI V. f 3 753 " machine element shown. (The density of steel is 7850 kg/m3 )
EM: L: 1331'"
Dimensions in mm
Fig. 133.40
Solve:
CD W1 Symmfrk
. 550 x; (44: may!
it 3 pl-mc.
Moments and Products of Inertia
I.
Objectives
Review calculation and properties of mass moments of inertia
Introduce products of inertia with calculation and properties
II.
Introduction
We must know how to find moments of inertia and products of inerti
Inertia Tensor
I.
Objectives
Introduce the inertia tensor
Usage and properties of inertia tensor
Example of calculating product of inertia by integration
II.
Inertia Tensor
A. Definition
z
z'
y
y
x
x'
The components of the inertia tensor vary when t
Numerical Methods for Solving Differential Equations.
I.
Objectives
Discuss differential equations used in Dynamics.
Understand how to apply numerical methods for solving differential
equations.
II.
Differential Equations in Dynamics
For a simple one-di
2-Dimensional Kinematics with Non-Rotating Axes
I.
Lesson Objectives:
Review basics of planar kinematics from ME212 (NonRotating Axes).
Provide kinematics example with multiple links.
II.
Fundamental Types of Motion
A. Equations and Analysis based on Mo
3D Angular Momentum
I.
II.
Objectives
Review Concept of Angular Momentum
Introduce 3D Angular Momentum
2D and 3D Kinetic Diagrams
Finding Kinetic Energy in 3D
Review of 2D Angular Momentum
Angular Momentum = Moment of Momentum about the point of
interest