A Differential Look at the Watt s Governor

Size: px
Start display at page:

Download "A Differential Look at the Watt s Governor"

Transcription

1 Differential Equations Spring /25 A Differential Look at the Watt s Governor by Tim Honn & Seth Stone College of the Redwoods Eureka,CA Math dept. timhonn@cox.net lamentofseth@hotmail.com

2 Introduction Invented by James Watt in the late 1700s, a governor is an automated speed control that ushered in the industrial revolution. 2/25 Mathematical model. Bifurcation. Damping.

3 3/25 The Watt s governor controlling a steam engine.

4 A Simplified Version 4/25 Ball-bearing in a rotating hoop.

5 10 5 ω 0 5 5/ Phase plane for Ω = 1 rad/sec vs. t for Ω = 1 rad/sec t

6 6/25 For Ω > 12 rad/sec the ball moves towards a new equilibrium point.

7 /25 ω vs. t for Ω = 13 rad/sec.

8 Identifying the Forces Identify the forces that always balance. Identify the forces that do not always balance. Sum the forces to derive the equations. 8/25

9 Ω (0, R) 9/25 T R cos (0, 0) F cent cos R sin mg sin Forces opposing the normal force. mg mg cos F cent sin F cent

10 Ω (0, R) 10/25 T (0, 0) F cent cos R cos R sin mg sin mg Tangential forces in the vertical plane. mg cos F cent sin F cent

11 Ω (0, R) 11/25 T (0, 0) F cent cos R cos C R sin mg sin The horizontal path of the ball. mg mg cos F cent sin F cent

12 Finding F cent Recall the kinematic identities, and our values. 12/25 v lin = rv ang a r = v2 lin r F cent = ma r v lin = (R sin )Ω [(R sin )Ω]2 a r = R sin = (R sin )Ω 2 F cent = m(r sin )Ω 2 In our case, Ω is the angular velocity v ang, about the center of C and the radius is R sin. The centrifugal force acting on the ball is the mass times a r. F cent = mω 2 R sin.

13 Ω (0, R) 13/25 T (0, 0) F cent cos R cos R sin F cent mg sin mg cos mg F cent sin ma T = F cent cos mg sin mr = mω 2 R sin cos mg sin mr = mω 2 R sin cos mg sin = Ω 2 sin cos g sin (1) R

14 = Ω 2 sin cos g R sin In order to use this equation we must first transpose it into two first order equations. 14/25 { = ω = = ω ω = = Ω 2 sin cos g R sin An equilibrium angle means that the forces are balanced and the acceleration is zero.

15 Set the right side equal to zero. = 0 Ω 2 sin cos g R sin = 0 15/25 sin (Ω 2 cos g R ) = 0 Therefore, sin = 0 or Ω 2 cos g R = 0. When sin = 0, = 0 or π. To find other equilibrium angles we set the other factor equal to zero.

16 Ω 2 cos g R = 0 cos = g/r Ω 2 (2) 16/25 Cosine is never greater than 1 so we seek Ωs that make the right side less than or equal to 1. g RΩ g R Ω2 0 g R Ω 0 (3)

17 In our case the Ω where bifurcation occurs is, Ω Ω 0. (4) 17/25 Now we find the Ω that produces = π/4. cos = g/r Ω 2 cos π 4 = 9.8/.06 Ω cos π = Ω = Ω. (5)

18 6 4 2 ω 0 18/ For Ω = 15.2 rad/sec. 1 vs. t for Ω = 15.2 rad/sec t

19 19/25

20 Now we have a governor that will maintain the desired angle but oscillates perpetually. How can we improve this performance? = Ω 2 sin cos g mg sin k R m (6) 20/25 The damping term is proportional to the angular velocity (in the vertical plane) and is divided by the mass.

21 6 4 ω / Ω = 15.2 rad/sec with damping term t vs. t for Ω = 15.2 and damping term.

22 / t vs. t for Ω = 15.2, damping term, and m = 50g.

23 / t vs. t for Ω = 15.2, damping term, and m = 5g.

24 / t vs. t for Ω = 15.2, damping term, and m =.25g. As you can see this also would not be a governor of optimum design. When designing a governor one would have to experiment with the parameters and would undoubtedly be somewhere between 5g and 1/4g.

25 Putting It All Together We have a governor design that will maintain the desired Changing R only effects the where the critical Ωs occur but not the oscillatory behavior. Increasing the mass reduces the effects of damping, reducing mass increases the effects of damping. Changing the damping term has an inverse effect as changing the mass. 25/25

Math 240: Spring-Mass Systems

Math 240: Spring-Mass Systems Math 240: Spring-Mass Systems Ryan Blair University of Pennsylvania Wednesday December 5, 2012 Ryan Blair (U Penn) Math 240: Spring-Mass Systems Wednesday December 5, 2012 1 / 13 Outline 1 Today s Goals

More information

THE SINUSOIDAL WAVEFORM

THE SINUSOIDAL WAVEFORM Chapter 11 THE SINUSOIDAL WAVEFORM The sinusoidal waveform or sine wave is the fundamental type of alternating current (ac) and alternating voltage. It is also referred to as a sinusoidal wave or, simply,

More information

Multiple-Angle and Product-to-Sum Formulas

Multiple-Angle and Product-to-Sum Formulas Multiple-Angle and Product-to-Sum Formulas MATH 160, Precalculus J. Robert Buchanan Department of Mathematics Fall 011 Objectives In this lesson we will learn to: use multiple-angle formulas to rewrite

More information

Chapter 6: Periodic Functions

Chapter 6: Periodic Functions Chapter 6: Periodic Functions In the previous chapter, the trigonometric functions were introduced as ratios of sides of a right triangle, and related to points on a circle. We noticed how the x and y

More information

Exercise 1. Consider the following figure. The shaded portion of the circle is called the sector of the circle corresponding to the angle θ.

Exercise 1. Consider the following figure. The shaded portion of the circle is called the sector of the circle corresponding to the angle θ. 1 Radian Measures Exercise 1 Consider the following figure. The shaded portion of the circle is called the sector of the circle corresponding to the angle θ. 1. Suppose I know the radian measure of the

More information

Application Note #2442

Application Note #2442 Application Note #2442 Tuning with PL and PID Most closed-loop servo systems are able to achieve satisfactory tuning with the basic Proportional, Integral, and Derivative (PID) tuning parameters. However,

More information

Bakiss Hiyana binti Abu Bakar JKE, POLISAS BHAB

Bakiss Hiyana binti Abu Bakar JKE, POLISAS BHAB 1 Bakiss Hiyana binti Abu Bakar JKE, POLISAS 1. Explain AC circuit concept and their analysis using AC circuit law. 2. Apply the knowledge of AC circuit in solving problem related to AC electrical circuit.

More information

Module 7 : Design of Machine Foundations. Lecture 31 : Basics of soil dynamics [ Section 31.1: Introduction ]

Module 7 : Design of Machine Foundations. Lecture 31 : Basics of soil dynamics [ Section 31.1: Introduction ] Lecture 31 : Basics of soil dynamics [ Section 31.1: Introduction ] Objectives In this section you will learn the following Dynamic loads Degrees of freedom Lecture 31 : Basics of soil dynamics [ Section

More information

Physics 132 Quiz # 23

Physics 132 Quiz # 23 Name (please (please print) print) Physics 132 Quiz # 23 I. I. The The current in in an an ac ac circuit is is represented by by a phasor.the value of of the the current at at some time time t t is is

More information

[ á{tå TÄàt. Chapter Four. Time Domain Analysis of control system

[ á{tå TÄàt. Chapter Four. Time Domain Analysis of control system Chapter Four Time Domain Analysis of control system The time response of a control system consists of two parts: the transient response and the steady-state response. By transient response, we mean that

More information

Phasor. Phasor Diagram of a Sinusoidal Waveform

Phasor. Phasor Diagram of a Sinusoidal Waveform Phasor A phasor is a vector that has an arrow head at one end which signifies partly the maximum value of the vector quantity ( V or I ) and partly the end of the vector that rotates. Generally, vectors

More information

What is a Sine Function Graph? U4 L2 Relate Circle to Sine Activity.pdf

What is a Sine Function Graph? U4 L2 Relate Circle to Sine Activity.pdf Math 3 Unit 6, Trigonometry L04: Amplitude and Period of Sine and Cosine AND Translations of Sine and Cosine Functions WIMD: What I must do: I will find the amplitude and period from a graph of the sine

More information

On the axes of Fig. 4.1, sketch the variation with displacement x of the acceleration a of a particle undergoing simple harmonic motion.

On the axes of Fig. 4.1, sketch the variation with displacement x of the acceleration a of a particle undergoing simple harmonic motion. 1 (a) (i) Define simple harmonic motion. (b)... On the axes of Fig. 4.1, sketch the variation with displacement x of the acceleration a of a particle undergoing simple harmonic motion. Fig. 4.1 A strip

More information

Circuit Analysis-II. Circuit Analysis-II Lecture # 2 Wednesday 28 th Mar, 18

Circuit Analysis-II. Circuit Analysis-II Lecture # 2 Wednesday 28 th Mar, 18 Circuit Analysis-II Angular Measurement Angular Measurement of a Sine Wave ü As we already know that a sinusoidal voltage can be produced by an ac generator. ü As the windings on the rotor of the ac generator

More information

3. Use your unit circle and fill in the exact values of the cosine function for each of the following angles (measured in radians).

3. Use your unit circle and fill in the exact values of the cosine function for each of the following angles (measured in radians). Graphing Sine and Cosine Functions Desmos Activity 1. Use your unit circle and fill in the exact values of the sine function for each of the following angles (measured in radians). sin 0 sin π 2 sin π

More information

3/23/2015. Chapter 11 Oscillations and Waves. Contents of Chapter 11. Contents of Chapter Simple Harmonic Motion Spring Oscillations

3/23/2015. Chapter 11 Oscillations and Waves. Contents of Chapter 11. Contents of Chapter Simple Harmonic Motion Spring Oscillations Lecture PowerPoints Chapter 11 Physics: Principles with Applications, 7 th edition Giancoli Chapter 11 and Waves This work is protected by United States copyright laws and is provided solely for the use

More information

Experiment VI: The LRC Circuit and Resonance

Experiment VI: The LRC Circuit and Resonance Experiment VI: The ircuit and esonance I. eferences Halliday, esnick and Krane, Physics, Vol., 4th Ed., hapters 38,39 Purcell, Electricity and Magnetism, hapter 7,8 II. Equipment Digital Oscilloscope Digital

More information

CHAPTER 6 INTRODUCTION TO SYSTEM IDENTIFICATION

CHAPTER 6 INTRODUCTION TO SYSTEM IDENTIFICATION CHAPTER 6 INTRODUCTION TO SYSTEM IDENTIFICATION Broadly speaking, system identification is the art and science of using measurements obtained from a system to characterize the system. The characterization

More information

MATH 1113 Exam 3 Review. Fall 2017

MATH 1113 Exam 3 Review. Fall 2017 MATH 1113 Exam 3 Review Fall 2017 Topics Covered Section 4.1: Angles and Their Measure Section 4.2: Trigonometric Functions Defined on the Unit Circle Section 4.3: Right Triangle Geometry Section 4.4:

More information

Response spectrum Time history Power Spectral Density, PSD

Response spectrum Time history Power Spectral Density, PSD A description is given of one way to implement an earthquake test where the test severities are specified by time histories. The test is done by using a biaxial computer aided servohydraulic test rig.

More information

Experiment 1 LRC Transients

Experiment 1 LRC Transients Physics 263 Experiment 1 LRC Transients 1 Introduction In this experiment we will study the damped oscillations and other transient waveforms produced in a circuit containing an inductor, a capacitor,

More information

Copyright 2009 Pearson Education, Inc. Slide Section 8.2 and 8.3-1

Copyright 2009 Pearson Education, Inc. Slide Section 8.2 and 8.3-1 8.3-1 Transformation of sine and cosine functions Sections 8.2 and 8.3 Revisit: Page 142; chapter 4 Section 8.2 and 8.3 Graphs of Transformed Sine and Cosine Functions Graph transformations of y = sin

More information

1. If the flux associated with a coil varies at the rate of 1 weber/min,the induced emf is

1. If the flux associated with a coil varies at the rate of 1 weber/min,the induced emf is 1. f the flux associated with a coil varies at the rate of 1 weber/min,the induced emf is 1 1. 1V 2. V 60 3. 60V 4. Zero 2. Lenz s law is the consequence of the law of conservation of 1. Charge 2. Mass

More information

#8A RLC Circuits: Free Oscillations

#8A RLC Circuits: Free Oscillations #8A RL ircuits: Free Oscillations Goals In this lab we investigate the properties of a series RL circuit. Such circuits are interesting, not only for there widespread application in electrical devices,

More information

The Ellipse. PF 1 + PF 2 = constant. Minor Axis. Major Axis. Focus 1 Focus 2. Point 3.4.2

The Ellipse. PF 1 + PF 2 = constant. Minor Axis. Major Axis. Focus 1 Focus 2. Point 3.4.2 Minor Axis The Ellipse An ellipse is the locus of all points in a plane such that the sum of the distances from two given points in the plane, the foci, is constant. Focus 1 Focus 2 Major Axis Point PF

More information

Physics 351 Wednesday, February 7, 2018

Physics 351 Wednesday, February 7, 2018 Physics 351 Wednesday, February 7, 2018 HW3 due Friday. You finished reading ch7 last weekend. You ll read ch8 (Kepler problem) this weekend. HW help: Bill is in DRL 3N6 Wednesdays 4pm 7pm. Grace is in

More information

Electronics and Instrumentation Name ENGR-4220 Fall 1999 Section Modeling the Cantilever Beam Supplemental Info for Project 1.

Electronics and Instrumentation Name ENGR-4220 Fall 1999 Section Modeling the Cantilever Beam Supplemental Info for Project 1. Name ENGR-40 Fall 1999 Section Modeling the Cantilever Beam Supplemental Info for Project 1 The cantilever beam has a simple equation of motion. If we assume that the mass is located at the end of the

More information

The Air Bearing Throughput Edge By Kevin McCarthy, Chief Technology Officer

The Air Bearing Throughput Edge By Kevin McCarthy, Chief Technology Officer 159 Swanson Rd. Boxborough, MA 01719 Phone +1.508.475.3400 dovermotion.com The Air Bearing Throughput Edge By Kevin McCarthy, Chief Technology Officer In addition to the numerous advantages described in

More information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Trigonometry Final Exam Study Guide Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. The graph of a polar equation is given. Select the polar

More information

Resonance in Circuits

Resonance in Circuits Resonance in Circuits Purpose: To map out the analogy between mechanical and electronic resonant systems To discover how relative phase depends on driving frequency To gain experience setting up circuits

More information

Exam III. Solutions. Part A. Multiple choice questions. Check the best answer. Each question carries a value of 4 points.

Exam III. Solutions. Part A. Multiple choice questions. Check the best answer. Each question carries a value of 4 points. Exam III Solutions Part A. Multiple choice questions. Check the best answer. Each question carries a value of 4 points.. In Pascal s demonstration the barrel shown has height h and crosssection area A.

More information

(d) If a particle moves at a constant speed, then its velocity and acceleration are perpendicular.

(d) If a particle moves at a constant speed, then its velocity and acceleration are perpendicular. Math 142 -Review Problems II (Sec. 10.2-11.6) Work on concept check on pages 734 and 822. More review problems are on pages 734-735 and 823-825. 2nd In-Class Exam, Wednesday, April 20. 1. True - False

More information

MATH Review Exam II 03/06/11

MATH Review Exam II 03/06/11 MATH 21-259 Review Exam II 03/06/11 1. Find f(t) given that f (t) = sin t i + 3t 2 j and f(0) = i k. 2. Find lim t 0 3(t 2 1) i + cos t j + t t k. 3. Find the points on the curve r(t) at which r(t) and

More information

Precalculus Lesson 9.2 Graphs of Polar Equations Mrs. Snow, Instructor

Precalculus Lesson 9.2 Graphs of Polar Equations Mrs. Snow, Instructor Precalculus Lesson 9.2 Graphs of Polar Equations Mrs. Snow, Instructor As we studied last section points may be described in polar form or rectangular form. Likewise an equation may be written using either

More information

UNIT Explain the radiation from two-wire. Ans: Radiation from Two wire

UNIT Explain the radiation from two-wire. Ans:   Radiation from Two wire UNIT 1 1. Explain the radiation from two-wire. Radiation from Two wire Figure1.1.1 shows a voltage source connected two-wire transmission line which is further connected to an antenna. An electric field

More information

1. Measure angle in degrees and radians 2. Find coterminal angles 3. Determine the arc length of a circle

1. Measure angle in degrees and radians 2. Find coterminal angles 3. Determine the arc length of a circle Pre- Calculus Mathematics 12 5.1 Trigonometric Functions Goal: 1. Measure angle in degrees and radians 2. Find coterminal angles 3. Determine the arc length of a circle Measuring Angles: Angles in Standard

More information

Barrier. (a) State the conditions which must be met for an object to move with simple harmonic motion. (2)

Barrier. (a) State the conditions which must be met for an object to move with simple harmonic motion. (2) 1 In a television game show contestants have to pass under a barrier. The barrier has a vertical height of 0.70m and moves up and down with simple harmonic motion. 3.0m Barrier 0.70m (a) State the conditions

More information

ω d = driving frequency, F m = amplitude of driving force, b = damping constant and ω = natural frequency of undamped, undriven oscillator.

ω d = driving frequency, F m = amplitude of driving force, b = damping constant and ω = natural frequency of undamped, undriven oscillator. Physics 121H Fall 2015 Homework #14 16-November-2015 Due Date : 23-November-2015 Reading : Chapter 15 Note: Problems 7 & 8 are tutorials dealing with damped and driven oscillations, respectively. It may

More information

AC Theory and Electronics

AC Theory and Electronics AC Theory and Electronics An Alternating Current (AC) or Voltage is one whose amplitude is not constant, but varies with time about some mean position (value). Some examples of AC variation are shown below:

More information

Monitoring The Machine Elements In Lathe Using Vibration Signals

Monitoring The Machine Elements In Lathe Using Vibration Signals Monitoring The Machine Elements In Lathe Using Vibration Signals Jagadish. M. S. and H. V. Ravindra Dept. of Mech. Engg. P.E.S.C.E. Mandya 571 401. ABSTRACT: In any manufacturing industry, machine tools

More information

Introduction to Trigonometry. Algebra 2

Introduction to Trigonometry. Algebra 2 Introduction to Trigonometry Algebra 2 Angle Rotation Angle formed by the starting and ending positions of a ray that rotates about its endpoint Use θ to represent the angle measure Greek letter theta

More information

Trigonometric Transformations TEACHER NOTES MATH NSPIRED

Trigonometric Transformations TEACHER NOTES MATH NSPIRED Math Objectives Students will determine the type of function modeled by the height of a capsule on the London Eye observation wheel. Students will translate observational information to use as the parameters

More information

EE 42/100: Lecture 8. 1 st -Order RC Transient Example, Introduction to 2 nd -Order Transients. EE 42/100 Summer 2012, UC Berkeley T.

EE 42/100: Lecture 8. 1 st -Order RC Transient Example, Introduction to 2 nd -Order Transients. EE 42/100 Summer 2012, UC Berkeley T. EE 42/100: Lecture 8 1 st -Order RC Transient Example, Introduction to 2 nd -Order Transients Circuits with non-dc Sources Recall that the solution to our ODEs is Particular solution is constant for DC

More information

The period is the time required for one complete oscillation of the function.

The period is the time required for one complete oscillation of the function. Trigonometric Curves with Sines & Cosines + Envelopes Terminology: AMPLITUDE the maximum height of the curve For any periodic function, the amplitude is defined as M m /2 where M is the maximum value and

More information

Vibrations on a String and Resonance

Vibrations on a String and Resonance Vibrations on a String and Resonance Umer Hassan and Muhammad Sabieh Anwar LUMS School of Science and Engineering September 7, 2010 How does our radio tune into different channels? Can a music maestro

More information

5-5 Multiple-Angle and Product-to-Sum Identities

5-5 Multiple-Angle and Product-to-Sum Identities Find the values of sin 2, cos 2, and tan 2 for the given value and interval. 1. cos =, (270, 360 ) Since on the interval (270, 360 ), one point on the terminal side of θ has x-coordinate 3 and a distance

More information

Name Date Class. Identify whether each function is periodic. If the function is periodic, give the period

Name Date Class. Identify whether each function is periodic. If the function is periodic, give the period Name Date Class 14-1 Practice A Graphs of Sine and Cosine Identify whether each function is periodic. If the function is periodic, give the period. 1.. Use f(x) = sinx or g(x) = cosx as a guide. Identify

More information

Graphing Sine and Cosine

Graphing Sine and Cosine The problem with average monthly temperatures on the preview worksheet is an example of a periodic function. Periodic functions are defined on p.254 Periodic functions repeat themselves each period. The

More information

Characterizing the Frequency Response of a Damped, Forced Two-Mass Mechanical Oscillator

Characterizing the Frequency Response of a Damped, Forced Two-Mass Mechanical Oscillator Characterizing the Frequency Response of a Damped, Forced Two-Mass Mechanical Oscillator Shanel Wu Harvey Mudd College 3 November 013 Abstract A two-mass oscillator was constructed using two carts, springs,

More information

StandingWaves_P2 [41 marks]

StandingWaves_P2 [41 marks] StandingWaves_P2 [41 marks] A loudspeaker emits sound towards the open end of a pipe. The other end is closed. A standing wave is formed in the pipe. The diagram represents the displacement of molecules

More information

CHAPTER WAVE MOTION

CHAPTER WAVE MOTION Solutions--Ch. 12 (Wave Motion) CHAPTER 12 -- WAVE MOTION 12.1) The relationship between a wave's frequency ν, its wavelength λ, and its wave velocity v is v = λν. For sound in air, the wave velocity is

More information

4) Drive Mechanisms. Techno_Isel H830 Catalog

4) Drive Mechanisms. Techno_Isel H830 Catalog 4) Drive Mechanisms This section will introduce most of the more common types of drive mechanisms found in linear motion machinery. Ideally, a drive system should not support any loads, with all the loads

More information

EFFECTS OF PHASE AND AMPLITUDE ERRORS ON QAM SYSTEMS WITH ERROR- CONTROL CODING AND SOFT DECISION DECODING

EFFECTS OF PHASE AND AMPLITUDE ERRORS ON QAM SYSTEMS WITH ERROR- CONTROL CODING AND SOFT DECISION DECODING Clemson University TigerPrints All Theses Theses 8-2009 EFFECTS OF PHASE AND AMPLITUDE ERRORS ON QAM SYSTEMS WITH ERROR- CONTROL CODING AND SOFT DECISION DECODING Jason Ellis Clemson University, jellis@clemson.edu

More information

DEPARTMENT OF ELECTRICAL AND ELECTRONIC ENGINEERING BANGLADESH UNIVERSITY OF ENGINEERING & TECHNOLOGY EEE 402 : CONTROL SYSTEMS SESSIONAL

DEPARTMENT OF ELECTRICAL AND ELECTRONIC ENGINEERING BANGLADESH UNIVERSITY OF ENGINEERING & TECHNOLOGY EEE 402 : CONTROL SYSTEMS SESSIONAL DEPARTMENT OF ELECTRICAL AND ELECTRONIC ENGINEERING BANGLADESH UNIVERSITY OF ENGINEERING & TECHNOLOGY EEE 402 : CONTROL SYSTEMS SESSIONAL Experiment No. 1(a) : Modeling of physical systems and study of

More information

Algebra and Trig. I. The graph of

Algebra and Trig. I. The graph of Algebra and Trig. I 4.5 Graphs of Sine and Cosine Functions The graph of The graph of. The trigonometric functions can be graphed in a rectangular coordinate system by plotting points whose coordinates

More information

15. the power factor of an a.c circuit is.5 what will be the phase difference between voltage and current in this

15. the power factor of an a.c circuit is.5 what will be the phase difference between voltage and current in this 1 1. In a series LCR circuit the voltage across inductor, a capacitor and a resistor are 30 V, 30 V and 60 V respectively. What is the phase difference between applied voltage and current in the circuit?

More information

5.1 Graphing Sine and Cosine Functions.notebook. Chapter 5: Trigonometric Functions and Graphs

5.1 Graphing Sine and Cosine Functions.notebook. Chapter 5: Trigonometric Functions and Graphs Chapter 5: Trigonometric Functions and Graphs 1 Chapter 5 5.1 Graphing Sine and Cosine Functions Pages 222 237 Complete the following table using your calculator. Round answers to the nearest tenth. 2

More information

Math 1205 Trigonometry Review

Math 1205 Trigonometry Review Math 105 Trigonometry Review We begin with the unit circle. The definition of a unit circle is: x + y =1 where the center is (0, 0) and the radius is 1. An angle of 1 radian is an angle at the center of

More information

Simple Path Planning Algorithm for Two-Wheeled Differentially Driven (2WDD) Soccer Robots

Simple Path Planning Algorithm for Two-Wheeled Differentially Driven (2WDD) Soccer Robots Simple Path Planning Algorithm for Two-Wheeled Differentially Driven (2WDD) Soccer Robots Gregor Novak 1 and Martin Seyr 2 1 Vienna University of Technology, Vienna, Austria novak@bluetechnix.at 2 Institute

More information

Mock final exam Math fall 2007

Mock final exam Math fall 2007 Mock final exam Math - fall 7 Fernando Guevara Vasquez December 5 7. Consider the curve r(t) = ti + tj + 5 t t k, t. (a) Show that the curve lies on a sphere centered at the origin. (b) Where does the

More information

Trigonometric Equations

Trigonometric Equations Chapter Three Trigonometric Equations Solving Simple Trigonometric Equations Algebraically Solving Complicated Trigonometric Equations Algebraically Graphs of Sine and Cosine Functions Solving Trigonometric

More information

1 Graphs of Sine and Cosine

1 Graphs of Sine and Cosine 1 Graphs of Sine and Cosine Exercise 1 Sketch a graph of y = cos(t). Label the multiples of π 2 and π 4 on your plot, as well as the amplitude and the period of the function. (Feel free to sketch the unit

More information

EE 3TP4: Signals and Systems Lab 5: Control of a Servomechanism

EE 3TP4: Signals and Systems Lab 5: Control of a Servomechanism EE 3TP4: Signals and Systems Lab 5: Control of a Servomechanism Tim Davidson Ext. 27352 davidson@mcmaster.ca Objective To identify the plant model of a servomechanism, and explore the trade-off between

More information

Nomograms for Synthesizing Crank Rocker Mechanism with a Desired Optimum Range of Transmission Angle

Nomograms for Synthesizing Crank Rocker Mechanism with a Desired Optimum Range of Transmission Angle International Journal of Mining, Metallurgy & Mechanical Engineering (IJMMME Volume 3, Issue 3 (015 ISSN 30 4060 (Online Nomograms for Synthesizing Crank Rocker Mechanism with a Desired Optimum Range of

More information

CONTROLLING THE OSCILLATIONS OF A SWINGING BELL BY USING THE DRIVING INDUCTION MOTOR AS A SENSOR

CONTROLLING THE OSCILLATIONS OF A SWINGING BELL BY USING THE DRIVING INDUCTION MOTOR AS A SENSOR Proceedings, XVII IMEKO World Congress, June 7,, Dubrovnik, Croatia Proceedings, XVII IMEKO World Congress, June 7,, Dubrovnik, Croatia XVII IMEKO World Congress Metrology in the rd Millennium June 7,,

More information

Basic Analog Circuits

Basic Analog Circuits Basic Analog Circuits Overview This tutorial is part of the National Instruments Measurement Fundamentals series. Each tutorial in this series, will teach you a specific topic of common measurement applications,

More information

Chapter Moving Charges and Magnetism

Chapter Moving Charges and Magnetism 100 Chapter Moving Charges and Magnetism 1. The power factor of an AC circuit having resistance (R) and inductance (L) connected in series and an angular velocity ω is [2013] 2. [2002] zero RvB vbl/r vbl

More information

Poles and Zeros of H(s), Analog Computers and Active Filters

Poles and Zeros of H(s), Analog Computers and Active Filters Poles and Zeros of H(s), Analog Computers and Active Filters Physics116A, Draft10/28/09 D. Pellett LRC Filter Poles and Zeros Pole structure same for all three functions (two poles) HR has two poles and

More information

How to Graph Trigonometric Functions

How to Graph Trigonometric Functions How to Graph Trigonometric Functions This handout includes instructions for graphing processes of basic, amplitude shifts, horizontal shifts, and vertical shifts of trigonometric functions. The Unit Circle

More information

Plasma Confinement by Pressure of Rotating Magnetic Field in Toroidal Device

Plasma Confinement by Pressure of Rotating Magnetic Field in Toroidal Device 1 ICC/P5-41 Plasma Confinement by Pressure of Rotating Magnetic Field in Toroidal Device V. Svidzinski 1 1 FAR-TECH, Inc., San Diego, USA Corresponding Author: svidzinski@far-tech.com Abstract: Plasma

More information

Basic Electronics Learning by doing Prof. T.S. Natarajan Department of Physics Indian Institute of Technology, Madras

Basic Electronics Learning by doing Prof. T.S. Natarajan Department of Physics Indian Institute of Technology, Madras Basic Electronics Learning by doing Prof. T.S. Natarajan Department of Physics Indian Institute of Technology, Madras Lecture 26 Mathematical operations Hello everybody! In our series of lectures on basic

More information

Intermediate and Advanced Labs PHY3802L/PHY4822L

Intermediate and Advanced Labs PHY3802L/PHY4822L Intermediate and Advanced Labs PHY3802L/PHY4822L Torsional Oscillator and Torque Magnetometry Lab manual and related literature The torsional oscillator and torque magnetometry 1. Purpose Study the torsional

More information

Inverted Pendulum Swing Up Controller

Inverted Pendulum Swing Up Controller Dublin Institute of Technology ARROW@DIT Conference Papers School of Mechanical and Design Engineering 2011-09-29 Inverted Pendulum Swing Up Controller David Kennedy Dublin Institute of Technology, david.kennedy@dit.ie

More information

Chapter 6: Periodic Functions

Chapter 6: Periodic Functions Chapter 6: Periodic Functions In the previous chapter, the trigonometric functions were introduced as ratios of sides of a triangle, and related to points on a circle. We noticed how the x and y values

More information

Mekanisme Robot - 3 SKS (Robot Mechanism)

Mekanisme Robot - 3 SKS (Robot Mechanism) Mekanisme Robot - 3 SKS (Robot Mechanism) Latifah Nurahmi, PhD!! latifah.nurahmi@gmail.com!! C.250 First Term - 2016/2017 Velocity Rate of change of position and orientation with respect to time Linear

More information

Electromagnetic Oscillations and Currents. March 23, 2014 Chapter 30 1

Electromagnetic Oscillations and Currents. March 23, 2014 Chapter 30 1 Electromagnetic Oscillations and Currents March 23, 2014 Chapter 30 1 Driven LC Circuit! The voltage V can be thought of as the projection of the vertical axis of the phasor V m representing the time-varying

More information

Chapter 1. Trigonometry Week 6 pp

Chapter 1. Trigonometry Week 6 pp Fall, Triginometry 5-, Week -7 Chapter. Trigonometry Week pp.-8 What is the TRIGONOMETRY o TrigonometryAngle+ Three sides + triangle + circle. Trigonometry: Measurement of Triangles (derived form Greek

More information

PHYS 1444 Section 003 Lecture #19

PHYS 1444 Section 003 Lecture #19 PHYS 1444 Section 003 Lecture #19 Monday, Nov. 14, 2005 Electric Generators DC Generator Eddy Currents Transformer Mutual Inductance Today s homework is homework #10, due noon, next Tuesday!! 1 Announcements

More information

MAE106 Laboratory Exercises Lab # 5 - PD Control of DC motor position

MAE106 Laboratory Exercises Lab # 5 - PD Control of DC motor position MAE106 Laboratory Exercises Lab # 5 - PD Control of DC motor position University of California, Irvine Department of Mechanical and Aerospace Engineering Goals Understand how to implement and tune a PD

More information

CHAPTER 11 TEST REVIEW -- MARKSCHEME

CHAPTER 11 TEST REVIEW -- MARKSCHEME AP PHYSICS Name: Period: Date: 50 Multiple Choice 45 Single Response 5 Multi-Response Free Response 3 Short Free Response 2 Long Free Response MULTIPLE CHOICE DEVIL PHYSICS BADDEST CLASS ON CAMPUS AP EXAM

More information

Chapter 33. Alternating Current Circuits

Chapter 33. Alternating Current Circuits Chapter 33 Alternating Current Circuits Alternating Current Circuits Electrical appliances in the house use alternating current (AC) circuits. If an AC source applies an alternating voltage to a series

More information

Lab 9 AC FILTERS AND RESONANCE

Lab 9 AC FILTERS AND RESONANCE 09-1 Name Date Partners ab 9 A FITES AND ESONANE OBJETIES OEIEW To understand the design of capacitive and inductive filters To understand resonance in circuits driven by A signals In a previous lab, you

More information

Model Correlation of Dynamic Non-linear Bearing Behavior in a Generator

Model Correlation of Dynamic Non-linear Bearing Behavior in a Generator Model Correlation of Dynamic Non-linear Bearing Behavior in a Generator Dean Ford, Greg Holbrook, Steve Shields and Kevin Whitacre Delphi Automotive Systems, Energy & Chassis Systems Abstract Efforts to

More information

ROOT CAUSE FAILURE ANALYSIS

ROOT CAUSE FAILURE ANALYSIS ROOT CAUSE FAILURE ANALYSIS PLANT ENGINEERING MAINTENANCE SERIES Vibration Fundamentals R. Keith Mobley Root Cause Failure Analysis R. Keith Mobley Maintenance Fundamentals R. Keith Mobley ROOT CAUSE FAILURE

More information

13.2 Define General Angles and Use Radian Measure. standard position:

13.2 Define General Angles and Use Radian Measure. standard position: 3.2 Define General Angles and Use Radian Measure standard position: Examples: Draw an angle with the given measure in standard position..) 240 o 2.) 500 o 3.) -50 o Apr 7 9:55 AM coterminal angles: Examples:

More information

The Discussion of this exercise covers the following points: Angular position control block diagram and fundamentals. Power amplifier 0.

The Discussion of this exercise covers the following points: Angular position control block diagram and fundamentals. Power amplifier 0. Exercise 6 Motor Shaft Angular Position Control EXERCISE OBJECTIVE When you have completed this exercise, you will be able to associate the pulses generated by a position sensing incremental encoder with

More information

MATH Week 10. Ferenc Balogh Winter. Concordia University

MATH Week 10. Ferenc Balogh Winter. Concordia University MATH 20 - Week 0 Ferenc Balogh Concordia University 2008 Winter Based on the textbook J. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics for Calculus, 5th Edition, Thomson All figures and videos

More information

Magnitude & Intensity

Magnitude & Intensity Magnitude & Intensity Lecture 7 Seismometer, Magnitude & Intensity Vibrations: Simple Harmonic Motion Simplest vibrating system: 2 u( x) 2 + ω u( x) = 0 2 t x Displacement u ω is the angular frequency,

More information

Optimal Control System Design

Optimal Control System Design Chapter 6 Optimal Control System Design 6.1 INTRODUCTION The active AFO consists of sensor unit, control system and an actuator. While designing the control system for an AFO, a trade-off between the transient

More information

A study of Vibration Analysis for Gearbox Casing Using Finite Element Analysis

A study of Vibration Analysis for Gearbox Casing Using Finite Element Analysis A study of Vibration Analysis for Gearbox Casing Using Finite Element Analysis M. Sofian D. Hazry K. Saifullah M. Tasyrif K.Salleh I.Ishak Autonomous System and Machine Vision Laboratory, School of Mechatronic,

More information

Midterm 1. Total. Name of Student on Your Left: Name of Student on Your Right: EE 20N: Structure and Interpretation of Signals and Systems

Midterm 1. Total. Name of Student on Your Left: Name of Student on Your Right: EE 20N: Structure and Interpretation of Signals and Systems EE 20N: Structure and Interpretation of Signals and Systems Midterm 1 12:40-2:00, February 19 Notes: There are five questions on this midterm. Answer each question part in the space below it, using the

More information

Radiometry I: Illumination. cs348b Matt Pharr

Radiometry I: Illumination. cs348b Matt Pharr Radiometry I: Illumination cs348b Matt Pharr Administrivia Extra copies of lrt book Bug fix for assignment 1 polynomial.h file Onward To The Physical Description of Light Four key quantities Power Radiant

More information

VIBRATION ANALYSIS OF DRILLING OPERATION

VIBRATION ANALYSIS OF DRILLING OPERATION VIBRATION ANALYSIS OF DRILLING OPERATION Amit S. Wani 1, Gayatri S. Sagavkar 2, Vaibhav K. Bhate 3 Department of Mechanical Engineering, Fr.Conceiceo Rodrigues Institute of Technology, Vashi, Navi Mumbai,

More information

Experiment 7: Frequency Modulation and Phase Locked Loops

Experiment 7: Frequency Modulation and Phase Locked Loops Experiment 7: Frequency Modulation and Phase Locked Loops Frequency Modulation Background Normally, we consider a voltage wave form with a fixed frequency of the form v(t) = V sin( ct + ), (1) where c

More information

13-1 Practice. Trigonometric Identities. Find the exact value of each expression if 0 < θ < 90. 1, find sin θ. 1. If cos θ = 1, find cot θ.

13-1 Practice. Trigonometric Identities. Find the exact value of each expression if 0 < θ < 90. 1, find sin θ. 1. If cos θ = 1, find cot θ. 1-1 Practice Trigonometric Identities Find the exact value of each expression if 0 < θ < 90. 1. If cos θ = 5 1, find sin θ.. If cot θ = 1, find sin θ.. If tan θ = 4, find sec θ. 4. If tan θ =, find cot

More information

Torque on a Current Loop: Motors. and Meters

Torque on a Current Loop: Motors. and Meters OpenStax-CNX module: m61560 1 Torque on a Current Loop: Motors * and Meters OpenStax Physics with Courseware Based on Torque on a Current Loop: Motors and Meters by OpenStax This work is produced by OpenStax-CNX

More information

Magnetic Field of the Earth

Magnetic Field of the Earth Magnetic Field of the Earth Name Section Theory The earth has a magnetic field with which compass needles and bar magnets will align themselves. This field can be approximated by assuming there is a large

More information

Automatic Control Systems 2017 Spring Semester

Automatic Control Systems 2017 Spring Semester Automatic Control Systems 2017 Spring Semester Assignment Set 1 Dr. Kalyana C. Veluvolu Deadline: 11-APR - 16:00 hours @ IT1-815 1) Find the transfer function / for the following system using block diagram

More information

AUTOMATIC VOLTAGE REGULATOR AND AUTOMATIC LOAD FREQUENCY CONTROL IN TWO-AREA POWER SYSTEM

AUTOMATIC VOLTAGE REGULATOR AND AUTOMATIC LOAD FREQUENCY CONTROL IN TWO-AREA POWER SYSTEM AUTOMATIC VOLTAGE REGULATOR AND AUTOMATIC LOAD FREQUENCY CONTROL IN TWO-AREA POWER SYSTEM ABSTRACT [1] Nitesh Thapa, [2] Nilu Murmu, [3] Aditya Narayan, [4] Birju Besra Dept. of Electrical and Electronics

More information

The Mathematics of the Stewart Platform

The Mathematics of the Stewart Platform The Mathematics of the Stewart Platform The Stewart Platform consists of 2 rigid frames connected by 6 variable length legs. The Base is considered to be the reference frame work, with orthogonal axes

More information