1 Modeling and Experimentation: Compound Pendulum Prof. R.G. Longoria Department of Mechanical Engineering The University of Texas at Austin Fall 2014
2 Overview This lab focuses on developing a mathematical model of the compound pendulum system studied in lab experiments. This lecture reviews the model, then shows how to develop the mathematical model both as a second order ODE and in state space form. A key goal is to learn about how numerical simulations of math models can be used to predict response of a system. A model simulation can be thought of as another way to experiment with a system. Block diagrams are also introduced as a way to describe systems within simulation environments, and LabVIEW examples are provided. The model simulation results from the pendulum system can be compared to measured response data in order to improve the model, especially for determining the dominant type of friction causing the pendulum motion to decay.
3 Lab Objectives 1 Practice system modeling methods, including constitutive modeling for friction effects 2 Become familiar with the National Instruments Control and Simulation module 3 Use a simulation model to compare with measured data and tune system parameters
4 Use modeling to build insight and derive a proper dynamic system model Understand the type of model that is needed. To predict the pendulum response observed in the lab, we need a dynamic system model in the form of ODEs Recognize any special considerations. There is significant friction in the pivot/bearing causing the pendulum oscillation to decay, so the model needs to include torques induced by friction. Make sure the model can be used for the intended purpose. Given a proper model, we need to be able to solve the equations. Since the ODEs cannot be solved in closed-form, we need to put the model in a form suitable for simulation.
5 Modeling Simulation Block Diagrams Experimentation Summary In introductory dynamics courses, you often solved for forces, accelerations, or velocities of bodies at specific instants in time. Consider the example below.
6 Example continued...
7 Modeling Simulation Block Diagrams Experimentation Summary We seek system dynamics models as ordinary differential equations (ODEs) for key variables of interest in order to study how a system behaves over all time. The measured decay of the pendulum oscillations in the lab indicates loss of energy over time due to dissipative effects.
8 Example: the simple pendulum model A simple pendulum model represents dynamics of a point mass at the end of fixed length, massless, rod or string, constrained to move in a plane as shown below. Gravity acts downward. If there are no other forces, the net torque for rotation about the pivot is, T = Tθ = mgl sin θ Now, apply Newton s law, dh dt = ḣ = J θ = T where J = ml 2, so a 2nd order ODE in θ is found, ml 2 θ + mgl sin θ = 0
9 Contrast the simple pendulum with the compound pendulum Simple pendulum: J = ml 2 m is the total mass, l is distance from the pivot to the mass ml 2 θ + mgl sin θ = 0 θ + g l sin θ = 0 Compound pendulum: J 0 = J + ml 2 C m is the total mass, l C is distance from the pivot to the CG J 0 θ + mglc sin θ = 0 θ + mgl C J 0 sin θ = 0 Modeling a compound pendulum as a simple pendulum would incur error that depends on how widely mass is distributed. In some cases, a simple pendulum might be a reasonable approximation.
10 Additional torques in the pendulum model Aside from proper modeling of the inertia effects, the model also needs to consider other torques that cause changes in the angular momentum. Primarily, we are concerned here with torques due to dissipative effects (friction at the pivot, air friction, etc.), or any applied torques or forces on the pendulum body. Both can be included in the Newton s law equation, ḣ = T θ T f T a where T f is the net torques due to friction/losses and T a are applied or actuator torques. It is clear that the pendulum studied in the lab is not lossless since it does not oscillate indefinitely, and there are no other applied forces or actuator torques.
11 Modeling frictional torques Frictional effects can be modeled as frictional torques that act about the pivot. These torques are modeled as functions of angular velocity, T = f(ω). Consider two types: These models represent different types of physical processes. For example, if a bearing has a (viscous) fluid film, induced forces tend to vary linearly with velocity (left). On the other hand, bearings with dry contact result in forces more similar to Coulomb-type friction, being relatively constant with velocity (right).
12 Case 1: Compound pendulum with only linear damping Consider a linear damping torque in the pendulum equation, ḣ = T = T θ T b where where ω = θ. Now, T b = bω = linear damping torque J 0 θ + b θ + mglc sin θ = 0 This equation can be made linear by assuming small motion, so sin θ would be approximately θ and the equation becomes a linear 2nd order ODE with a closed-form solution.
13 Modeling Simulation Block Diagrams Experimentation Summary Case 2: Compound pendulum with linear damping and nonlinear friction Now include a nonlinear friction torque, where ḣ = T = T θ T b T c T c = T o sgn(ω) = Coulombic (nonlinear) frictional torque Linearizing for small motion as done for Case 1 is problematic, because of the sgn (signum) function. The signum function has a very large change in value when its argument is close to zero: sgn(a) = +1 if a > 0 and sgn(a) = 1 if a < 0. Closed form solutions for motion cannot be found as readily as for Case 1. It turns out that this type of friction is more prevalent in many practical systems. Solving this ODE requires numerical integration schemes for simulating the system.
14 Basic idea behind ODE system simulation When we talk about system simulation in DSC, this typically refers to solving an ODE initial value problem. This means that we have a set of ODEs that model our system of interest 1. Consider the simplest type of dynamic system, a first order ODE, dx dt = f(x, u, t) which we ll call a state equation for the state variable, x. The right hand side is a function of x itself, an input to the system, u, and time, t. The modeling process provides f(x, u, t). Numerical integration requires an initial value at initial time, t 0, x(t 0 ) = x 0. Then, the solution at the next step, t = t 0 + t, is to be approximated. 1 Some systems may involve both differential and algebraic equations (DAEs), which won t be considered in this course.
15 Euler ODE Solver You may recall that numerically solving ODEs for initial value problems is simply marching forward in time. The solution is found at discrete time steps, given the initial value at t 0. Given x 0, the value at initial time, an estimate of the value at t 1 is, ˆx 1 = x 0 + x 0 The job of the solver is to estimate x 0. The simplest algorithm is the Euler solver, which is basically a Taylor series approximation. The Euler solve estimates x 0 by, x 0 = f(x 0, u 0, t 0 ) t using an approximation from the ODE x/ t f(x, u, t), and given all initial values. There is always error in the estimate of the true value, x 1.
16 Selection of ODE Solvers The Euler method is the simplest ODE solver, and it usually uses a fixed time step. To get good solutions (more accurate, stable), usually you need to make the time step very small. The Euler method is a 1st order Runge-Kutta (RK) method. You may have learned about Runge-Kutta methods in a computational methods course. The most commonly used RK algorithm is 4th order, which uses four evaluations of the ODEs to estimate the next value of x, as opposed to the single evaluation made by the Euler routine. The RK4 algorithm allows you to take larger time steps than Euler, is more stable, but as for Euler the error must be managed by the user. More sophisticated algorithms use variable time step to control numerical errors. Many commercial and open-source software packages have built-in fixed-step and variable step algorithms that can be used for ODE simulation. This course will provide practice in using the algorithms available in LabVIEW.
17 Modeling Simulation Block Diagrams Experimentation Summary ODE solvers require that the equations be put into 1st order form The Euler solver was described using the simple 1st order ODE in the variable x. This is the form required by ODE solvers. Consider the single x variable generalized as a vector x formed by n state variables of a system. ODE algorithms are designed to accept descriptions of the system ODEs as a vector formed by the model equations. In general, the 1st order vector of state equations is, ẋ = f(x, u, t) where u is now a vector of r inputs. Writing the equations in this form can either be done by converting the nth order ODE to n 1st order ODEs, or the n 1st order ODEs can be derived directly. The latter method is the way equations are directly derived when the bond graph method is used.
18 Modeling Simulation Block Diagrams Experimentation Summary Example: Convert the simple pendulum model from 2nd order to 1st order form Recall the simple pendulum model ml 2 θ + mgl sin θ = 0 To convert this 2nd order ODE to 1st order, first define n = 2 state variables as x 1 = θ and x 2 = θ. We now want to write two 1st order ODEs in terms of these new state variables. The first one is found by taking the derivative of the first variable, ẋ 1 = θ, which is recognized as x 2. Therefore, the first equation is, ẋ 1 = x 2 Now, take derivative of second variable, ẋ 2 = θ. To find this equation, we must use the original 2nd order ODE, which has θ as well as θ. Solve for θ, θ = (g/l) sin θ = (g/l) sin x 1 This gives us the second state equation, ẋ 2 = (g/l) sin x 1 These two equations are the state space equations for the simple pendulum.
19 State space form Here are the simple pendulum state space equations in matrix form: [ ] [ ] ẋ1 x ẋ = = f(x, u, t) = 2 g l sin(x 1) ẋ 2 The rightmost term is a vector of the n nonlinear, state equations. These are in the form required by ODE solvers. The state equations are generally coupled; i.e., each equation can be a function of any and all system states. Each equation quantifies how each state changes over time (the slope at each time step). In a numerical algorithm, each equation is used to estimate the value of the state at each time step, as illustrated for the simple Euler routine.
20 Using solvers with script files Commercial software packages such as Matlab and LabVIEW have solvers that will numerically integrate ODEs. The ODEs are usually formatted in a script function file. For example, for the simple pendulum, a script file may take the form: function file function f = PendulumEqs(t,x) global g l f1 = x(2); f2 = -g*sin(x(1))/l; f = [f1;f2]; Here, the values returned by this function are sent to the solver (e.g., ode45). The use of script files is very effective. In this course, we will be learning a different approach that uses block diagrams to graphically describe the ODEs. Using this approach allows use of LabVIEW for simulation.
21 Block diagram representations of systems Block diagrams are used to graphically represent signal flow and functional relationships between signals. A directed line (with arrowhead) indicates a signal, which can represent a system variable or a control input. Nodes, shown as blocks, represent input-output relationships between signal variables.. Basic functions are gains, summers, integrators and differentiators, and as such block diagrams are effective in representing differential equation models. Modern software programs use block diagrams as a way to communicate system representations to computer-aided analysis tools.
22 Block diagram programming environments have become very popular as part of computer-aided engineering packages Why? They allow us to describe systems using a graphical form, which can be useful for communicating how different components interact. Block diagrams are functional and not just schematics (as we will see). There is a rich history not only in describing control systems but also in how (analog) computational algorithms were originally designed.
23 Block diagram algebra Summing point Let x(t) input and y(t) output. For general nonlinear systems, the output is simply, y = g(x). Branching point If the system is linear, y = G x G (gain) may be a constant or a system transfer function, G(s). The s represents the Laplace operator.
24 Block diagram calculus Other blocks especially useful for representing physical system models include the integrator: These are common blocks, but they are sometimes shown in software products using the s Laplace operator. For example, the integrator is: There is also a derivative and the derivative block is Important: these are just symbols - the blocks perform numerical integration/differentiation (not Laplace transforms).
25 Modeling Simulation Block Diagrams Experimentation Summary How do you build a block diagram of system state equations? For a given complete set of state equations (as you will learn to derive in ME 344), the following steps are taken: 1 Identify an integrator for each first order equation, so each integrator generates a state 2 Form the terms of each equation using system parameters, gain blocks, and functional blocks 3 Use summing blocks form the state derivatives (i.e., add terms as needed to form the right hand side of each 1st order equation) 4 Specify initial conditions, x(0), and system inputs, u NOTE: There are some systems that result in equations where the ODEs do not take this desirable form. You may learn about algebraic loops and derivative causality in your DSC course.
26 Block diagram of nth order state space system The system equations generate the dx/dt for each state, and the n integrators produce the states, which are then passed back into the system equations. Note each integrator requires an initial condition, x(0).
27 Example: block diagram for linear 1st order ODE Re-write the 1st order ODE in the form: dx dt = 1 [ x + u(t)] τ Then we can use basic block elements to describe the algebra and the calculus in the equations, as shown below. Remember that to find x the integrator needs to have an initial condition, x(0).
28 Two popular software products that use block diagrams are LabVIEW and Matlab/Simulink. These programs feature: combined structured and block diagram programming a capacity to communicate with physical hardware efficient means for designing a human-user interface modeling and simulation tools for physical and engineering systems We have adopted LabVIEW for use in this course.
29 LabVIEW Control and Simulation Module
30 LabVIEW Control and Simulation Module Build your block diagrams within the C&S Loop: Basic block diagram VIs are found under Signal Arithmetic: and Continuous Linear Systems:
31 Modeling Simulation Block Diagrams Experimentation Summary The integrator block is a key element for integrating your state equations Remember that the input to the integrator block will be the derivative of your state: By double-clicking the integrator VI block, you can access settings: and the output is the state. The settings allow you to configure how you will set key integration parameters and whether you will set them in the dialog box or wire to a terminal.
32 Modeling Simulation Block Diagrams Experimentation Summary Example: LV simulation diagram for standard 1st order system
33 Example: LV simulation diagram for the simple pendulum In this example VI for the simple pendulum, a formula node used as an alternate way to code the state equations rather than with block diagrams.
34 Experimentation The experimentation is now to be conducted using a numerical model. Here are some suggestions for using your simulation model: determine if your estimates of the pendulum moment of inertia allows predictions to compare well with measured response data; show how the simulation can be tuned to improve the model show that the simulation gives the same type of response characteristics, especially with proper frictional torque models; weigh the effect of linear versus nonlinear type friction calculate stored energy, and improve prediction on how energy decreases with each cycle
35 Summary Build experience formulating system models from physical laws Use a known physical problem for learning about constitutive models, especially linear vs. nonlinear types More opportunity to experiment with LabVIEW VIs for simulation, learning about Control and Simulation Module Use LabVIEW for analyzing data from experiments, relating to modeling results
Modeling and Experimentation: Mass-Spring-Damper System Dynamics Prof. R.G. Longoria Department of Mechanical Engineering The University of Texas at Austin July 20, 2014 Overview 1 This lab is meant to
Assignments VIII and IX, PHYS 301 (Classical Mechanics) Spring 2014 Due 3/21/14 at start of class Homeworks VIII and IX both center on Lagrangian mechanics and involve many of the same skills. Therefore,
NONLINEAR MECHANICAL SYSTEMS (MECHANISMS) The analogy between dynamic behavior in different energy domains can be useful. Closer inspection reveals that the analogy is not complete. One key distinction
Dynamics and control of mechanical systems Date Day 1 (01/08) Day 2 (03/08) Day 3 (05/08) Day 4 (07/08) Day 5 (09/08) Day 6 (11/08) Content Review of the basics of mechanics. Kinematics of rigid bodies
Simple and Physical Pendulums Challenge Problem Solutions Problem 1 Solutions: For this problem, the answers to parts a) through d) will rely on an analysis of the pendulum motion. There are two conventional
Torsion Spring Oscillator with Dry Friction Manual Eugene Butikov Annotation. The manual includes a description of the simulated physical system and a summary of the relevant theoretical material for students
PH 120 Project # 2: Pendulum and chaos Due: Friday, January 16, 2004 In PH109, you studied a simple pendulum, which is an effectively massless rod of length l that is fixed at one end with a small mass
The Simple Double Pendulum Austin Graf December 13, 2013 Abstract The double pendulum is a dynamic system that exhibits sensitive dependence upon initial conditions. This project explores the motion of
Linear Systems Lecture 3 1 problem Consider a non-linear time-invariant system of the form ( ẋ(t f x(t u(t y(t g ( x(t u(t (1 such that x R n u R m y R p and Slide 1 A: f(xu f(xu g(xu and g(xu exist and
Control Systems I Lecture 2: Modeling Suggested Readings: Åström & Murray Ch. 2-3, Guzzella Ch. 2-3 Emilio Frazzoli Institute for Dynamic Systems and Control D-MAVT ETH Zürich September 29, 2017 E. Frazzoli
Math 2250 Lab 4 Name/Unid: 1. (35 points) Leslie Leroy Irvin bails out of an airplane at the altitude of 16,000 ft, falls freely for 20 s, then opens his parachute. Assuming linear air resistance ρv ft/s
PHYSICS 210 SOLUTION OF THE NONLINEAR PENDULUM EQUATION USING FDAS 1. PHYSICAL & MATHEMATICAL FORMULATION O θ L r T m W 1 1.1 Derivation of the equation of motion O Consider idealized pendulum: Mass of
PY001/051 Compound Pendulum and Helical Springs Experiment 4 Physics 001/051 The Compound Pendulum Experiment 4 and Helical Springs Prelab 1 Read the following background/setup and ensure you are familiar
Lecture 17, Chapter 10: Rotational Energy and Angular Momentum 1 Rotational Kinetic Energy Consider a rigid body rotating with an angular velocity ω about an axis. Clearly every point in the rigid body
Oscillations Simple Harmonic Motion Lana Sheridan De Anza College Dec 1, 2017 Overview oscillations simple harmonic motion (SHM) spring systems energy in SHM pendula damped oscillations Oscillations and
Chapter 1 Lecture Notes Chapter 1 Oscillatory Motion Recall that when a spring is stretched a distance x, it will pull back with a force given by: F = -kx When the mass is released, the spring will pull
Computational Physics (6810): Session 8 Dick Furnstahl Nuclear Theory Group OSU Physics Department February 24, 2014 Differential equation solving Session 7 Preview Session 8 Stuff Solving differential
The dynamics of a Mobile Inverted Pendulum (MIP) 1 Introduction Saam Ostovari, Nick Morozovsky, Thomas Bewley UCSD Coordinated Robotics Lab In this document, a Mobile Inverted Pendulum (MIP) is a robotic
28. Pendulum phase portrait Draw the phase portrait for the pendulum (supported by an inextensible rod) θ + ω 2 sin θ = 0. Indicate the stable equilibrium points as well as the unstable equilibrium points.
083 Mechanical Vibrations Lesson Vibration Analysis Procedure The analysis of a vibrating system usually involves four steps: mathematical modeling derivation of the governing uations solution of the uations
Oscillator Homework Problems Michael Fowler 3//7 1 Dimensional exercises: use dimensions to find a characteristic time for an undamped simple harmonic oscillator, and a pendulum Why does the dimensional
Chaotic motion Phys 750 Lecture 9 Finite-difference equations Finite difference equation approximates a differential equation as an iterative map (x n+1,v n+1 )=M[(x n,v n )] Evolution from time t =0to
Lecture Notes for PHY 405 Classical Mechanics From Thorton & Marion s Classical Mechanics Prepared by Dr. Joseph M. Hahn Saint Mary s University Department of Astronomy & Physics September 1, 2005 Chapter
Section 7.4 Runge-Kutta Methods Key terms: Taylor methods Taylor series Runge-Kutta; methods linear combinations of function values at intermediate points Alternatives to second order Taylor methods Fourth
AP PHYSICS 1 BIG IDEAS AND LEARNING OBJECTIVES KINEMATICS 3.A.1.1: The student is able to express the motion of an object using narrative, mathematical, and graphical representations. [SP 1.5, 2.1, 2.2]
Vehicle longitudinal speed control Prof. R.G. Longoria Department of Mechanical Engineering The University of Texas at Austin February 10, 2015 1 Introduction 2 Control concepts Open vs. Closed Loop Control
Exam 3 Practice Solutions Multiple Choice 1. A thin hoop, a solid disk, and a solid sphere, each with the same mass and radius, are at rest at the top of an inclined plane. If all three are released at
Physics 2300 Spring 2018 Name Lab partner Project 3: Pendulum In this project you will explore the behavior of a pendulum. There is no better example of a system that seems simple at first but turns out
Stabilising A Vertically Driven Inverted Pendulum J. J. Aguilar, 1 C. Marcotte, 1 G. Lee, 1 and B. Suri 1 School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332, USA (Dated: 16 December
Torques and Static Equilibrium INTRODUCTION Archimedes, Greek mathematician, physicist, engineer, inventor and astronomer, was widely regarded as the leading scientist of the ancient world. He made a study
Annotated Answers to the 1984 AP Physics C Mechanics Multiple Choice 1. D. Torque is the rotational analogue of force; F net = ma corresponds to τ net = Iα. 2. C. The horizontal speed does not affect the
Multibody simulation Dynamics of a multibody system (Euler-Lagrange formulation) Dimitar Dimitrov Örebro University June 16, 2012 Main points covered Euler-Lagrange formulation manipulator inertia matrix
ISBN 978-93-84468-- Proceedings of 5 International Conference on Future Computational echnologies (ICFC'5) Singapore, March 9-3, 5, pp. 96-3 Dynamic Modeling of Rotary Double Inverted Pendulum Using Classical
BRAZOSPORT COLLEGE LAKE JACKSON, TEXAS SYLLABUS PHYS 2325 - MECHANICS AND HEAT CATALOG DESCRIPTION: PHYS 2325 Mechanics and Heat. CIP 4008015403 A calculus-based approach to the principles of mechanics
ENGI9496 Lecture Notes State-Space Equation Generation. State Equations and Variables - Definitions The end goal of model formulation is to simulate a system s behaviour on a computer. A set of coherent
Intro to Scientific Computing: How long does it take to find a needle in a haystack? Dr. David M. Goulet Intro Binary Sorting Suppose that you have a detector that can tell you if a needle is in a haystack,
Lecture 27. THE COMPOUND PENDULUM Figure 5.16 Compound pendulum: (a) At rest in equilibrium, (b) General position with coordinate θ, Freebody diagram The term compound is used to distinguish the present
International Journal of Electronics and Communication Engineering. ISSN 974-2166 Volume 5, Number 3 (212), pp. 363-37 International Research Publication House http://www.irphouse.com Design of Quadratic
Lab 1: Simple Pendulum 1 The Pendulum Laboratory 1, Physics 15c Due Friday, February 16, in front of Sci Cen 301 Physics 15c; REV 0 1 January 31, 2007 1 Introduction Most oscillating systems behave like
Physics A - PHY 2048C and 11/15/2017 My Office Hours: Thursday 2:00-3:00 PM 212 Keen Building Warm-up Questions 1 Did you read Chapter 12 in the textbook on? 2 Must an object be rotating to have a moment
Manifesto on Numerical Integration of Equations of Motion Using Matlab C. Hall April 11, 2002 This handout is intended to help you understand numerical integration and to put it into practice using Matlab
Quanser NI-ELVIS Trainer (QNET) Series: QNET Experiment #02: DC Motor Position Control DC Motor Control Trainer (DCMCT) Student Manual Table of Contents 1 Laboratory Objectives1 2 References1 3 DCMCT Plant
Numerical integration for solving differential equations After integration, it is natural to consider how to find numerical solutions to differential equations on the computer. Simple equations of motion
Feedback Basics David M. Auslander Mechanical Engineering University of California at Berkeley copyright 1998, D.M. Auslander 1 I. Feedback Control Context 2 What is Feedback Control? Measure desired behavior
LAST TIME: Simple Pendulum: The displacement from equilibrium, x is the arclength s = L. s / L x / L Accelerating & Restoring Force in the tangential direction, taking cw as positive initial displacement
Final Review: Chapters 1-11, 13-14 These are selected problems that you are to solve independently or in a team of 2-3 in order to better prepare for your Final Exam 1 Problem 1: Chasing a motorist This
An Overview of Fluid Animation Christopher Batty March 11, 2014 What distinguishes fluids? What distinguishes fluids? No preferred shape. Always flows when force is applied. Deforms to fit its container.
Extension A: Circles and Ellipses Algebra ; Pre-Calculus Time required: 35 50 min. Learning Objectives Math Objectives Students will write the general forms of Cartesian equations for circles and ellipses,
Friction Modeling, Identification, & Analysis Objectives Understand the friction phenomenon as it relates to motion systems. Develop a control-oriented model with appropriate simplifying assumptions for
Appendix 3B MATLAB Functions for Modeling and Time-domain analysis MATLAB control system Toolbox contain the following functions for the time-domain response step impulse initial lsim gensig damp ltiview
13 NUMERICAL SOLUTION OF ODE S 28 13 Numerical Solution of ODE s In simulating dynamical systems, we frequently solve ordinary differential equations. These are of the form dx = f(t, x), dt where the function
Human Arm Equipment: Capstone, Human Arm Model, 45 cm rod, sensor mounting clamp, sensor mounting studs, 2 cord locks, non elastic cord, elastic cord, two blue pasport force sensors, large table clamps,
1. Object Simple Harmonic Motion To determine the period of motion of objects that are executing simple harmonic motion and to check the theoretical prediction of such periods. 2. Apparatus Assorted weights
Adopted: 1980 Reviewed: 1985, 2007 Revised: 1990, 1998, 2001, 2008, 2011 Alabama Department of Postsecondary Education Representing Alabama s Public Two-Year College System Jefferson State Community College
Math 240: Spring-mass Systems Ryan Blair University of Pennsylvania Tuesday March 1, 2011 Ryan Blair (U Penn) Math 240: Spring-mass Systems Tuesday March 1, 2011 1 / 15 Outline 1 Review 2 Today s Goals
Ph1a: Solution to the Final Exam Alejandro Jenkins, Fall 2004 Problem 1 (10 points) - The Delivery A crate of mass M, which contains an expensive piece of scientific equipment, is being delivered to Caltech.
11-2 A General Method, and Rolling without Slipping Let s begin by summarizing a general method for analyzing situations involving Newton s Second Law for Rotation, such as the situation in Exploration
Nonlinear Analysis: Modelling and Control, 2010, Vol. 15, No. 4, 451 458 Modeling nonlinear systems using multiple piecewise linear equations G.K. Lowe, M.A. Zohdy Department of Electrical and Computer
Lecture 7: Newton s Laws and Their Applications 1 Chapter 4: Newton s Second Law F = m a First Law: The Law of Inertia An object at rest will remain at rest unless, until acted upon by an external force.
Rotational Motion Equipment: Capstone, rotary motion sensor mounted on 80 cm rod and heavy duty bench clamp (PASCO ME-9472), string with loop at one end and small white bead at the other end (125 cm bead
Phys 270 Final Exam Time limit: 120 minutes Each question worths 10 points. Constants: g = 9.8m/s 2, G = 6.67 10 11 Nm 2 kg 2. 1. (a) Figure 1 shows an object with moment of inertia I and mass m oscillating
Your Comments I love physics as much as the next gal, but I was wondering. Why don't we get class off the day after an evening exam? What if the ladder has friction with the wall? Things were complicated
page 104 104 CHAPTER 1 First-Order Differential Equations 16. The following initial-value problem arises in the analysis of a cable suspended between two fixed points y = 1 a 1 + (y ) 2, y(0) = a, y (0)
3_Model Systems HarmonicOscillators.nb Chapter 3 Harmonic Oscillator - Model Systems 3.1 Mass on a spring in a gravitation field a 0.5 3.1.1 Force Method The two forces on the mass are due to the spring,
MATH 35, by T. Lakoba, University of Vermont 60 7 Pendulum. Part II: More complicated situations In this Lecture, we will pursue two main goals. First, we will take a glimpse at a method of Classical Mechanics
General Physics I Lecture 8: Rotation of a Rigid Object About a Fixed Axis Prof. WAN, Xin ( 万歆 ) email@example.com http://zimp.zju.edu.cn/~xinwan/ New Territory Object In the past, point particle (no rotation,
Chapter 4 Oscillatory Motion 4.1 The Important Stuff 4.1.1 Simple Harmonic Motion In this chapter we consider systems which have a motion which repeats itself in time, that is, it is periodic. In particular
Chapter 3 Numerical Methods Part 3 3.4 Differential Algebraic Systems 3.5 Integration of Differential Equations 1 Outline 3.4 Differential Algebraic Systems 3.4.1 Constrained Dynamics 3.4.2 First and Second
16 CHAPTER 2 One Dimensional Dynamical Systems We begin by analyzing some dynamical systems with one-dimensional phase spaces, and in particular their bifurcations. All equations in this Chapter are scalar
Lecture 9 - Rotational Dynamics A Puzzle... Angular momentum is a 3D vector, and changing its direction produces a torque τ = dl. An important application in our daily lives is that bicycles don t fall
Physics 6A Torque is what causes angular acceleration (just like a force causes linear acceleration) Torque is what causes angular acceleration (just like a force causes linear acceleration) For a torque
58:080 Experimental Engineering OBJECTIVE Lab 1g: Horizontally Forced Pendulum & Chaotic Motion The objective of this lab is to study horizontally forced oscillations of a pendulum. This will be done trough
Lab 11. Spring-Mass Oscillations Goals To determine experimentally whether the supplied spring obeys Hooke s law, and if so, to calculate its spring constant. To find a solution to the differential equation
Name: Fall 2014 1. Rod AB with weight W = 40 lb is pinned at A to a vertical axle which rotates with constant angular velocity ω =15 rad/s. The rod position is maintained by a horizontal wire BC. Determine
10.3 Static Equilibrium and Torque SECTION OUTCOMES Use vector analysis in two dimensions for systems involving static equilibrium and torques. Apply static torques to structures such as seesaws and bridges.
Three-Tank Experiment Overview The three-tank experiment focuses on application of the mechanical balance equation to a transient flow. Three tanks are interconnected by Schedule 40 pipes of nominal diameter
Chapter 4 Non-Linear Oscillations and Chaos Non-Linear Differential Equations Up to now we have considered differential equations with terms that are proportional to the acceleration, the velocity, and