15.164 At the instant shown the hang-t]; at? the boom AB is beingdecmased
at the. constant rate of 0.2 m/s and the beam is being lowered at the
Want rate of 0.08 rad/s.- Determine (a) the velocity'of point B,
(b) the acceleration of point B.
8 - “,otf
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
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
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
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
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
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
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 (
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
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
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 Fall 2015
E. Rossman
Assignment 8
Hand in Problems 1-6, below.
Problem 1:
In Example 2 from Mondays lecture, we solved the problem below using the angular
+
=
impulse/momentum equation with respect to point O. Now, solve it again with using
1
,
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
2-Dimensional Kinematics with Non-Rotating Axes
Different types of motion are analyzed as follows:
1. Translation
=
=
2. Rotation about a fixed point/axis (Velocity of point A in a circular
path about point O):
=
= /
(
)
= +
= / + ( / )
3. G
ME326 Fall 2015
E. Rossman
Homework Assignment 1 due Monday 9/28/15
This homework assignment may be turned in either in class on Monday, 9/28/15, or it may be scanned
as a single file and submitted to the Polylearn link.
Notes on Polylearn submission:
1.
ME326 Fall 2015
E. Rossman
Assignment 2 Due Monday, 10/5, at beginning of class (or up to 8 PM by PolyLearn)
Hand in Problems 1-6 as detailed below. Solve all kinematics problems using the vector cross
products. Review the formatting requirements discusse
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
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
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 Moments and Axes of Inertia
For a given set of axes, xyz, where the moments and products of
inertia are known (Ixx, Iyy, Izz, Ixy, Iyz, Ixz), there
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
/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
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
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
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