Chapter 2 The Laplace Transform
Objective: Mathematical preparation for the study of next chapters
Contents:
2.1 Introduction
2.2 Complex Numbers and Harmonic Motion
2.3 Laplace Transformation
2.4 Inverse Laplace Transformation
2.5 Solving Linear Differen
3.2 Mechanical Elements
In system dynamics, mechanical systems can be modeled as a
combination of following three elements:
Mass (inertial elements)
It represents the inertia of mechanical elements
Spring
It represents the flexibility of mechanical ele
Chapter 2 The Laplace Transform
Objective: Mathematical preparation for the study of next chapters
Contents:
2.1 Introduction
2.2 Complex Numbers and Harmonic Motion
2.3 Laplace Transformation
2.4 Inverse Laplace Transformation
2.5 Solving Linear Differen
2.2 (b) Harmonic Motion
What is Harmonic Motion?
Consider a point Z, which moves along the circular orbit R with constant speed
Im (y)
z
R
t+
Re (x)
R radius of orbit or magnitude of Z (m or in.)
circular frequency (rad/s)
initial phase angle (rad)
ZG2
9.4 Vibration Isolation (Base Excitation)
1 Force Excitation and Motion Excitation
2 Transmissibility
3 Transmissibility for Force Excitation
4 Transmissibility for Motion Excitation
5 Design of Vibration Isolation
ZG2006
1
Two Kinds of Vibration Isolatio
Chapter 3 Mechanical Systems
Objective: Basic elements and modeling of Mechanical systems
Contents:
3.1 Unit System (SI and EE) and Newtons Laws of Mechanics
3.2 Basic Elements (Mass, Spring, Damper)
3.3 Modeling of Mechanical Systems
3.4 Application of E
9. 3 Vibration in Rotating Mechanical Systems
1
Centripetal force and centrifugal force
2
Imbalance
3
EOM
4
Amplitude-frequency and phase-frequency plots
5
Frequency and speed of rotation
6
Natural frequency and critical speed
ZG2006
1
Centripetal Force &
4.2 Block Diagram (also CH10.2)
Block Diagram can be regarded as a graphical form of Transfer Function
G ( s)
Y (s)
X ( s)
X(s)
Y ( s ) X ( s ) G ( s )
TF(s)
Y(s)
Key Points
1
in s domain
2
output Y(s)=X(s) * TF(s)
ZG2006
1
Basic Elements in Block Diagra
4.3 PFE with MATLAB
F ( s)
N ( s)
D( s)
Both N(s) and D(s) are polynomials in s
N ( s ) b0 s m b1s m 1 bm 1s bm
D( s ) a0 s n a1s n 1 an 1s an
In MATLAB, polynomials are represented by arrays (row vectors) of numbers as follows, which
contains the coeffi
2.5 Solve ODE by Laplace Transform
Example
x 2 x 10 x 0
x(0) 0
x (0) 1
solution
Laplace:
L x 2 x 10 x L 0
s 2 X s sx(0) x(0) 2 sX s x(0) 10 X s 0
1
s 2 2s 10
1
1
3
Xs
( s 1) 2 32 3 ( s 1) 2 32
Xs
PFE:
1
Inverse Lapalce:x (t ) L
X s 1 e t sin(3t )
3
Nee
General Chemistry 1035 Homework 5 Due on October 14 in class Make sure you have bubbled the correct student number. Failure to do so will result in a zero 1. Manganese has the oxidation number of +5 in (A) (B) [MnF6]3 Mn2O7 (C) (D) [MnO4]2 [Mn(CN)6]
2. An
General Chemistry 1035 Homework 4 Due on October 7 in class Make sure you have bubbled the correct student number. Failure to do so from this homework onwards will result in a zero for that homework 1. If 0.50 mol of Na3PO4 is mixed with 0.30 mol of BaCl2
General Chemistry 1035 Homework 3 Due on Sept 23 in class 1. The chemical equation 2CO2 2CO + O2 informs us that (A) when 2 mol of CO2 decompose, less than 2 mol of CO will be formed and the quantity of O2 formed will be less than 1 mol. (B) only under sp
Chapter 10: Dynamics of Rotational Motion
1. Torque
A force applied at a right angle to a lever will generate a torque. The distance from the pivot to the point of force application will be linearly proportional to the torque produced.
Force causes a chan
Chapter 9: Rotation of Rigid Bodies
1. Angular Velocity and Acceleration
Radian
360 1rad = = 57.3 2
s = r
s = r
Angular Velocity : (omega) not w
av , z =
t d z = lim = t 0 t dt
Direction of angular velocity : Right-hand rule
Angular Acceleration :
z av
Chapter 8: Momentum, Impulse, and Collisions
Goals for Chapter 8
To determine the momentum of a particle To add time and study the relationship of impulse and momentum To see when momentum is conserved and examine the implications of conservation To u
PHYS2305: Practice Test 3
For the best result of the test, you need to understand completely the example problems in the lecture and the homework problems. This practice test serves as a complementary tool. Please note that the actual test will not necess
Review 2
Using Newton's 1st Law : Particles in Equilibrium
Particle in equilibrium
r F =0
Fx = 0 Fy = 0
Using Newton's 2nd Law: Dynamics of Particles
Dynamics : the relationship of motion to the forces that cause it
r r F = ma
Fx = max Fy = ma y
: New
Chapter 7: Potential Energy and Energy Conservation
1. Gravitational Potential Energy
Gravity does _ work on mass. Gravity does _ work on mass. potential energy _ potential energy _
Work done by gravity
Wgravity
r r ^ ( = F = (- mgj ) y2 - y1 ) j ) s = mg
Example 5.24) A passenger on a carnival Ferris wheel moves in a vertical circle of radius R with constant speed. The seat remains upright during the motion. Find expressions for the force the seat exerts on the passenger at the top and the bottom.
Chapte
Chapter 5 : Applying Newton's Law
1. Using Newton's 1st Law : Particles in Equilibrium
Until Chapter 9, we treat an object as a point particle: Rotational motion is not taken into account. Particle in equilibrium
r F =0
Fx = 0 Fy = 0
One dimensional equil
Problem 3.21) In a carnival booth, you win a stuffed giraffe if you
toss a quarter into a small dish. The dish is on a shelf above the point where the quarter leaves your hand and is a horizontal distance of 2.1 m from this point. If you toss the coin wi
Graphs of the motion with constant acceleration
Example) The figure shows a plot of vx(t) for a car traveling in a
straight line. (a) What is aav,x between t = 6 s and t = 11 s? (b) What is vav,x for the same time interval? (c) What is vav,x for the inter
Graphs of the motion with constant acceleration
Example) The figure shows a plot of vx(t) for a car traveling in a
straight line. (a) What is aav,x between t = 6 s and t = 11 s? (b) What is vav,x for the same time interval? (c) What is vav,x for the inter
Chapter 2 : Motion along a Straight Line
1. Displacement and Average Velocity
Position : the location of an object Displacement : the direction and distance of the shortest path between an initial and final position: x f - xi Velocity : speed and directio
Chapter 2 : Motion along a Straight Line
1. Displacement and Average Velocity
Position : the location of an object Displacement : the direction and distance of the shortest path between an initial and final position: x f - xi Velocity : speed and directio
Chapter 1 Units, Physical Quantities, and Vectors
3. Standards and Units
SI (Systme International (french) unit (or MKS unit) Time in s (second) Length in m (meter) Mass in kg An equation must be dimensionally consistent (be sure you're "adding apples to
Chapter 1 Units, Physical Quantities, and Vectors
3. Standards and Units
SI (Systme International (french) unit (or MKS unit) Time in s (second) Length in m (meter) Mass in kg An equation must be dimensionally consistent (be sure you're "adding apples to
Chapter 14: Fluid Mechanics
1. Density
Fluid: A substance that can flow and conform to the shape of a container. Liquids and gases are fluids.
M Density : mass per unit volume, unit: kg/m = V
3
For water at 4C, =1000 kg/m3=1 g/cm3 Originally, g/cm3 was