If the symmetry axes of a uniform symmetric body coincide with the coordinate axes, the products of inertia (Ixy etc.
|
|
- Rachel Booth
- 5 years ago
- Views:
Transcription
1 Prof. O. B. Wright, Autumn 007 Mechanics Lecture 9 More on rigid bodies, coupled vibrations Principal axes of the inertia tensor If the symmetry axes of a uniform symmetric body coincide with the coordinate axes, the products of inertia (Ixy etc.) are zero, and Ixx 0 0 I1 0 0 I 0 I 0 0 I 0. Example yy 0 0 I zz 0 0 I This choice of axes is special so we call them the principal axes. In fact in this example of a cylinder we are free to choose the directions of x and y for the principal axes provided that z is parallel to the cylinder axis. What about a body that is not symmetric? It turns out that we can always find principal axes for which I1 0 0 I 0 I I Remember that the values of I1, I and I are always positive (or some can be zero when point masses are used). In the principal axes, L1 I I11 L L 0 I 0 I. L 0 0 I I L is not parallel to in general. But if the body is spinning about a principal axis then L is parallel to. Can you see why from the above equation? 1
2 Prof. O. B. Wright, Autumn 007 Visualizing the inertia tensor Consider the equation I ijrr i j 1, where i, j x r y z, where r1 x, r y, r z. I x I y I z I xy I yz I zx 1 xx yy zz xy yz zx This is the equation for a surface in D, in fact an ellipsoid positioned at some angle to the axes. If we choose principal axes, the equation reduces to I x I y I z 1 1 This is an ellipsoid with axes parallel to x, y, z. With principal axes It can be shown* that if the vector cuts the ellipsoid at a point, then the direction of L is normal to the surface, as in this diagram shown for the principal axes. xy section of ellipsoid Remember L I means L is not parallel. *See Physical Properties of Crystals, J. F. Nye, Oxford University Press.
3 Prof. O. B. Wright, Autumn 007 Euler equations So far we have considered a rigid body that is spinning about a fixed axis. If the body is not symmetric then we have seen that torques must be exerted to keep the axis fixed (see last lecture). But what if the body is not symmetric and is spinning freely with no torques acting on it? In that case L is constant but the angular velocity vector will change with time because will change with time. However, if the body is spinning without external forces then it must spin with the centre of mass fixed (see Lecture ). It I is convenient to put the origin at the centre of mass because of this. The principal axes of the body are fixed with respect to the body, so we cannot choose a fixed set of axes that are principal axes. But we can define a set of axes that x, y, z rotate together with the body at the instant in time considered (see Lecture 8 for a similar treatment when the rotation is about a fixed axis). And let us assume that the axes point along the principal axes of the body. The x, y, z frame is a rest x, y, z frame in which no inertial forces occur. Therefore I1 0 0 I 0 I 0 in the x, y, z frame. 0 0 I Also L x I I11 L L 0 I 0 I in this frame. Alternatively, y L z 0 0 I I L = I1 1i + I j + I k, where i, j, k, and 1,, change with time. Note that the angular velocity of a body is a quantity that does not change direction in space or magnitude when we change the axes. (Please do not think that =0 in the frame of x, y, z!) Here we have chosen to express in the x, y, z axes. (Perhaps a better notation would have been L = I1 1i + I j + I k, but we miss off the for simplicity here.) At time t, we can use the equations we derived in Lecture 8:
4 Prof. O. B. Wright, Autumn 007 d a =a ω a for a vector a = a i + a j + a k in the dt x y z x, y, z frame. A similar argument can be used for the angular momentum vector L = L i + L j + L k L L i + L j+ L k where dl/ dt dl / dt 0. Therefore x y z x y z dl = dt 0 = LωL (1) where rotating L L L x i + L y j + L z k is the rate of change of L measured in the i, j, k x, y, z frame. L x I I11 L L Iω 0 I 0 I since y L z 0 0 I I x, y, z frame. I1 0 0 I 0 I 0 const. in the 0 0 I Expressing Eq. (1) in the x, y, z frame, we obtain I11 1 I11 0 L ω L I I. I I We can write the components of this equation as follows: I ( I I ) 1 1 I ( I I ) Euler equations 1 1 I ( I I ) 1 1 These equations determine the motion of the body seen in a frame fixed to the body 4
5 Prof. O. B. Wright, Autumn 007 (i.e. x, y, z ). The angular velocity 1 is calculated in the x, y, z frame. (To calculate it from the components of the angular velocity in the x,y,z frame we just need to use different basis vectors at a different angle: i.e. ω = i + j+ k ω = i + j + k, where I have chosen the subscript 0 to 0 x0 y0 z0 x y z refer to the x, y, z frame. This notation arises because we wanted to avoid using the for in the frame for simplicity.) x, y, z It is not always easy to use the Euler equations because we need to know (t). In fact the equations can be solved by expressing in terms of the angles that describe the orientation of the body. Stability of rotational motion Let us consider one application of the Euler equations. Suppose a body with I1, I and I all different is spinning about its rd principal axis. So, initially, 1= =0 and 0. The right hand side of all three Euler equations is zero. That means that (and so) must remain constant. The same argument holds if the body is spinning about the 1st or the nd principal axis. i.e. A body spinning about a principal axis will remain spinning about this axis. We can also see that if the body is spinning so that two or more values of 1, and are non-zero (i.e. when the angular velocity does not point along a principal axis), then does not remain constant. e.g. if 1 and 0, then from the rd Euler equation we can see that 0 (since I1 I). Let us see if a body with I1, I and I all different that is spinning about a principal axis is stable or not. Suppose that the body is spinning with 1=constant and= =0. Then let us assume that there is a small perturbation (e.g. we knock the body a bit) so that and 0 but and 1. 5
6 Prof. O. B. Wright, Autumn 007 The Euler equations tell us what should happen: I ( I I ) const I ( I I ) I ( I I ) ( I I )( I I ) I ( I I ) I I ( I1 I)( I1 I) 1 II (We can derive a similar equation for ). This is the equation for simple harmonic motion of provided that I1 is the largest or smallest moment of inertia (compared to I and I). This means that the rotation is stable, i.e. the system just oscillates about a single dominant value of, provided that we rotate about one of two axes. If on the other hand we rotate about the principal axis that has the intermediate value of moment of inertia, then the sign of the square bracket in the above equation is negative, so that the value of rapidly moves away from zero, and the motion is unstable. An illustration of which is the largest and smallest I component for a rectangular block this is shown in the figure above. Coupled vibrations Consider the coupled system of masses and springs below: mx kx k( x x ) a a b a mx kx k( x x ) b b b a 6
7 Prof. O. B. Wright, Autumn 007 Add and subtract these two equations to get: d m ( x ) ( ) a xb k xa xb dt d m ( x ) ( ) a xb k xa xb dt These equations are uncoupled equations with variables x a +x b and in x a -x b. The solutions are x a x b x 1 (t) Acos( 1 t 1 ), x a x b x (t) B cos( t ), 1 k / m k / m Solving for the positions of the masses: x a 1 Acos( 1t 1 ) 1 Bcos( t ) 1 (x 1 x ) x b 1 Acos( 1t 1 ) 1 Bcos( t ) 1 (x 1 x ) We call x 1 and x the normal coordinates of the system. If B=0 the masses oscillate in mode 1 with SHM at a single frequency 1, with x a = x b. If A=0 the masses oscillate in mode with SHM at a single frequency 1, with x a = -x b. If a single mode is present all masses pass through their equilibrium positions simultaneously. Each mode has its own 'shape'. Mode 1 and mode are called the normal modes of the oscillating system. The total number of modes (here ) is equal to the number of degrees of freedom (the number of coordinates) of the system. Here, x 1 corresponds to the motion of the centre of mass x 1 /=(x a +x b )/. x is the compression of the central spring, or, equivalently, the relative displacement of the two masses. If neither A or B are zero the motion is a combination of oscillations at both 1 and. Since we have derived the general solution, any motion can be expressed as a linear combination (a superposition) of the motion in the modes 1 and : 7
8 Prof. O. B. Wright, Autumn 007 xa 1 cos( 1t 1) 1 cos( t) xa xa A B x cos( t ) cos( t ) x x b 1 1 b 1 b This is a consequence of the principle of superposition that applies because the differential equation describing the motion is linear. Beats Beats arise where the motion of a part of an oscillating system is a superposition of two SHM at different frequencies. For example, the two SHM can be the two normal modes of system with degrees of freedom. For simplicity let us calculate the superposition of two harmonic oscillations with the same phase (=0) and amplitude: x 1 Acos 1 t, x Acos t x x 1 x Acos 1 t Acos t We can rewrite this in an interesting form. x Acos 1 t cos 1 t Now define the average angular frequency by av =( 1 + )/ and the 'modulation' angular frequency by mod =( 1 - )/, so x Acos mod tcos av t This can be thought of as an oscillation at angular frequency av, with an amplitude Acos mod t. If 1 then mod << av. This corresponds to nearly SHM at frequency av with slowly varying amplitude at frequency mod. See example below. 8
9 Prof. O. B. Wright, Autumn 007 Amod is a maximum twice every modulation cycle (one modulation cycle takes Tmod=/mod). We call the frequency beat=mod, the beat frequency. For the case of the system of two masses and three springs above, we can above the phenomenon of beating by starting the system of with the initial conditions: x A, x 0, x 0, and x 0. The solution is then a b a b x a Acos 1 t Acos t A cos mod tcos av t x b A cos 1 t Acos t Asin mod tsin av t where, in this case, av =1.7 k / m, mod =0.7 k / m. When the amplitude Acos mod t=0 the energy is nearly all in the mass b. When Asin mod t=0 the energy is nearly all in the mass a. The energy passes from mass a to b and back again to a in the time that cos mod t takes to go from 1 to 0 and back to 1, i.e. in a time / mod. This is equal to the beat period / beat. Matrix method for coupled vibrations We need a more rigorous method to find the normal modes. This can be done using matrices. For the case of the system of two masses and three springs above, 9
10 Prof. O. B. Wright, Autumn 007 mx kx k( x x ) kx kx a a b a a b mx kx k( x x ) kx kx b b b a a b m 0 xa k k xa 0 m x b k k xb or MxKx Notice that the matrix K is symmetric about its diagonal. This is not a coincidence. It is the result of Newton s rd law. Try x a it a e e b x b it m 0 a k k a 0 mb k k b or a K M b ( ) 0 This has a solution for non-zero a and b only if the matrix (otherwise we could multiply by the inverse matrix ( K M) ( K M) has no inverse 1 and then both a and b would be zero). The condition for no inverse matrix is that the determinant of ( K M) should be zero. That is, e f Det ( K M) =0, where Det eh gf g h. In our example k m k K M k m k 0 k k m So k m k k k,. m m k k m 10
11 Prof. O. B. Wright, Autumn 007 So we have found the possible vibrational frequencies. There are two of them. Finding the normal modes We need to find the possible values of a b in order to find the possible a e b it x. 1) For 1 k m k k and a 0. k k b, K 1 M K 1 M k k a 0 a b 0 a b. k k b A e A x 1 it, where A can be a complex number. x x A cos( t ) a b 1 1 Normal mode 1: ) For k m k k and a 0. k k b, K 1 M K 1 M k k a 0 a b 0 a b. k k b B e B x it, where B can be a complex number. x x Bcos( t ) a b Normal mode : 11
Lecture 38: Equations of Rigid-Body Motion
Lecture 38: Equations of Rigid-Body Motion It s going to be easiest to find the equations of motion for the object in the body frame i.e., the frame where the axes are principal axes In general, we can
More informationLecture 38: Equations of Rigid-Body Motion
Lecture 38: Equations of Rigid-Body Motion It s going to be easiest to find the equations of motion for the object in the body frame i.e., the frame where the axes are principal axes In general, we can
More information1/30. Rigid Body Rotations. Dave Frank
. 1/3 Rigid Body Rotations Dave Frank A Point Particle and Fundamental Quantities z 2/3 m v ω r y x Angular Velocity v = dr dt = ω r Kinetic Energy K = 1 2 mv2 Momentum p = mv Rigid Bodies We treat a rigid
More informationCP1 REVISION LECTURE 3 INTRODUCTION TO CLASSICAL MECHANICS. Prof. N. Harnew University of Oxford TT 2017
CP1 REVISION LECTURE 3 INTRODUCTION TO CLASSICAL MECHANICS Prof. N. Harnew University of Oxford TT 2017 1 OUTLINE : CP1 REVISION LECTURE 3 : INTRODUCTION TO CLASSICAL MECHANICS 1. Angular velocity and
More informationRotational Motion. Chapter 4. P. J. Grandinetti. Sep. 1, Chem P. J. Grandinetti (Chem. 4300) Rotational Motion Sep.
Rotational Motion Chapter 4 P. J. Grandinetti Chem. 4300 Sep. 1, 2017 P. J. Grandinetti (Chem. 4300) Rotational Motion Sep. 1, 2017 1 / 76 Angular Momentum The angular momentum of a particle with respect
More information27. Euler s Equations
27 Euler s Equations Michael Fowler Introduction We ve just seen that by specifying the rotational direction and the angular phase of a rotating body using Euler s angles, we can write the Lagrangian in
More informationNewton s Laws of Motion, Energy and Oscillations
Prof. O. B. Wright, Autumn 007 Mechanics Lecture Newton s Laws of Motion, Energy and Oscillations Reference frames e.g. displaced frame x =x+a y =y x =z t =t e.g. moving frame (t=time) x =x+vt y =y x =z
More information6. 3D Kinematics DE2-EA 2.1: M4DE. Dr Connor Myant
DE2-EA 2.1: M4DE Dr Connor Myant 6. 3D Kinematics Comments and corrections to connor.myant@imperial.ac.uk Lecture resources may be found on Blackboard and at http://connormyant.com Contents Three-Dimensional
More informationPhysics 106a, Caltech 4 December, Lecture 18: Examples on Rigid Body Dynamics. Rotating rectangle. Heavy symmetric top
Physics 106a, Caltech 4 December, 2018 Lecture 18: Examples on Rigid Body Dynamics I go through a number of examples illustrating the methods of solving rigid body dynamics. In most cases, the problem
More informationRotational & Rigid-Body Mechanics. Lectures 3+4
Rotational & Rigid-Body Mechanics Lectures 3+4 Rotational Motion So far: point objects moving through a trajectory. Next: moving actual dimensional objects and rotating them. 2 Circular Motion - Definitions
More informationNORMAL MODES, WAVE MOTION AND THE WAVE EQUATION. Professor G.G.Ross. Oxford University Hilary Term 2009
NORMAL MODES, WAVE MOTION AND THE WAVE EQUATION Professor G.G.Ross Oxford University Hilary Term 009 This course of twelve lectures covers material for the paper CP4: Differential Equations, Waves and
More informationManipulator Dynamics 2. Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA
Manipulator Dynamics 2 Forward Dynamics Problem Given: Joint torques and links geometry, mass, inertia, friction Compute: Angular acceleration of the links (solve differential equations) Solution Dynamic
More informationRigid bodies - general theory
Rigid bodies - general theory Kinetic Energy: based on FW-26 Consider a system on N particles with all their relative separations fixed: it has 3 translational and 3 rotational degrees of freedom. Motion
More informationGeneral Physics I. Lecture 10: Rolling Motion and Angular Momentum.
General Physics I Lecture 10: Rolling Motion and Angular Momentum Prof. WAN, Xin (万歆) 万歆 ) xinwan@zju.edu.cn http://zimp.zju.edu.cn/~xinwan/ Outline Rolling motion of a rigid object: center-of-mass motion
More informationClassical Mechanics. Luis Anchordoqui
1 Rigid Body Motion Inertia Tensor Rotational Kinetic Energy Principal Axes of Rotation Steiner s Theorem Euler s Equations for a Rigid Body Eulerian Angles Review of Fundamental Equations 2 Rigid body
More information9 Kinetics of 3D rigid bodies - rotating frames
9 Kinetics of 3D rigid bodies - rotating frames 9. Consider the two gears depicted in the figure. The gear B of radius R B is fixed to the ground, while the gear A of mass m A and radius R A turns freely
More informationPhys 7221 Homework # 8
Phys 71 Homework # 8 Gabriela González November 15, 6 Derivation 5-6: Torque free symmetric top In a torque free, symmetric top, with I x = I y = I, the angular velocity vector ω in body coordinates with
More informationPHYS 705: Classical Mechanics. Euler s Equations
1 PHYS 705: Classical Mechanics Euler s Equations 2 Euler s Equations (set up) We have seen how to describe the kinematic properties of a rigid body. Now, we would like to get equations of motion for it.
More informationPart 8: Rigid Body Dynamics
Document that contains homework problems. Comment out the solutions when printing off for students. Part 8: Rigid Body Dynamics Problem 1. Inertia review Find the moment of inertia for a thin uniform rod
More informationRotational motion of a rigid body spinning around a rotational axis ˆn;
Physics 106a, Caltech 15 November, 2018 Lecture 14: Rotations The motion of solid bodies So far, we have been studying the motion of point particles, which are essentially just translational. Bodies with
More informationPhysics 351, Spring 2018, Homework #9. Due at start of class, Friday, March 30, 2018
Physics 351, Spring 218, Homework #9. Due at start of class, Friday, March 3, 218 Please write your name on the LAST PAGE of your homework submission, so that we don t notice whose paper we re grading
More informationLecture 9: Eigenvalues and Eigenvectors in Classical Mechanics (See Section 3.12 in Boas)
Lecture 9: Eigenvalues and Eigenvectors in Classical Mechanics (See Section 3 in Boas) As suggested in Lecture 8 the formalism of eigenvalues/eigenvectors has many applications in physics, especially in
More informationPeriodic Motion. Periodic motion is motion of an object that. regularly repeats
Periodic Motion Periodic motion is motion of an object that regularly repeats The object returns to a given position after a fixed time interval A special kind of periodic motion occurs in mechanical systems
More information12. Rigid Body Dynamics I
University of Rhode Island DigitalCommons@URI Classical Dynamics Physics Course Materials 015 1. Rigid Body Dynamics I Gerhard Müller University of Rhode Island, gmuller@uri.edu Creative Commons License
More informationArtificial Intelligence & Neuro Cognitive Systems Fakultät für Informatik. Robot Dynamics. Dr.-Ing. John Nassour J.
Artificial Intelligence & Neuro Cognitive Systems Fakultät für Informatik Robot Dynamics Dr.-Ing. John Nassour 25.1.218 J.Nassour 1 Introduction Dynamics concerns the motion of bodies Includes Kinematics
More informationSOLUTIONS, PROBLEM SET 11
SOLUTIONS, PROBLEM SET 11 1 In this problem we investigate the Lagrangian formulation of dynamics in a rotating frame. Consider a frame of reference which we will consider to be inertial. Suppose that
More informationProblem 1: (3 points) Recall that the dot product of two vectors in R 3 is
Linear Algebra, Spring 206 Homework 3 Name: Problem : (3 points) Recall that the dot product of two vectors in R 3 is a x b y = ax + by + cz, c z and this is essentially the same as the matrix multiplication
More informationTranslational and Rotational Dynamics!
Translational and Rotational Dynamics Robert Stengel Robotics and Intelligent Systems MAE 345, Princeton University, 217 Copyright 217 by Robert Stengel. All rights reserved. For educational use only.
More informationLecture 37: Principal Axes, Translations, and Eulerian Angles
Lecture 37: Principal Axes, Translations, and Eulerian Angles When Can We Find Principal Axes? We can always write down the cubic equation that one must solve to determine the principal moments But if
More informationEngineering Mechanics Prof. U. S. Dixit Department of Mechanical Engineering Indian Institute of Technology, Guwahati Introduction to vibration
Engineering Mechanics Prof. U. S. Dixit Department of Mechanical Engineering Indian Institute of Technology, Guwahati Introduction to vibration Module 15 Lecture 38 Vibration of Rigid Bodies Part-1 Today,
More informationSeminar 6: COUPLED HARMONIC OSCILLATORS
Seminar 6: COUPLED HARMONIC OSCILLATORS 1. Lagrangian Equations of Motion Let consider a system consisting of two harmonic oscillators that are coupled together. As a model, we will use two particles attached
More informationTorque and Rotation Lecture 7
Torque and Rotation Lecture 7 ˆ In this lecture we finally move beyond a simple particle in our mechanical analysis of motion. ˆ Now we consider the so-called rigid body. Essentially, a particle with extension
More informationTheory & Practice of Rotor Dynamics Prof. Rajiv Tiwari Department of Mechanical Engineering Indian Institute of Technology Guwahati
Theory & Practice of Rotor Dynamics Prof. Rajiv Tiwari Department of Mechanical Engineering Indian Institute of Technology Guwahati Module - 5 Torsional Vibrations Lecture - 4 Transfer Matrix Approach
More informationOscillations. Phys101 Lectures 28, 29. Key points: Simple Harmonic Motion (SHM) SHM Related to Uniform Circular Motion The Simple Pendulum
Phys101 Lectures 8, 9 Oscillations Key points: Simple Harmonic Motion (SHM) SHM Related to Uniform Circular Motion The Simple Pendulum Ref: 11-1,,3,4. Page 1 Oscillations of a Spring If an object oscillates
More informationLecture 41: Highlights
Lecture 41: Highlights The goal of this lecture is to remind you of some of the key points that we ve covered this semester Note that this is not the complete set of topics that may appear on the final
More informationTheory and Practice of Rotor Dynamics Prof. Dr. Rajiv Tiwari Department of Mechanical Engineering Indian Institute of Technology Guwahati
Theory and Practice of Rotor Dynamics Prof. Dr. Rajiv Tiwari Department of Mechanical Engineering Indian Institute of Technology Guwahati Module - 2 Simpul Rotors Lecture - 2 Jeffcott Rotor Model In the
More information!T = 2# T = 2! " The velocity and acceleration of the object are found by taking the first and second derivative of the position:
A pendulum swinging back and forth or a mass oscillating on a spring are two examples of (SHM.) SHM occurs any time the position of an object as a function of time can be represented by a sine wave. We
More informationLecture 35: The Inertia Tensor
Lecture 35: The Inertia Tensor We found last time that the kinetic energy of a rotating obect was: 1 Trot = ωω i Ii where i, ( I m δ x x x i i, k, i, k So the nine numbers represented by the I i tell us
More information2.003 Engineering Dynamics Problem Set 6 with solution
.00 Engineering Dynamics Problem Set 6 with solution Problem : A slender uniform rod of mass m is attached to a cart of mass m at a frictionless pivot located at point A. The cart is connected to a fixed
More informationPhysics 106b/196b Problem Set 9 Due Jan 19, 2007
Physics 06b/96b Problem Set 9 Due Jan 9, 2007 Version 3: January 8, 2007 This problem set focuses on dynamics in rotating coordinate systems (Section 5.2), with some additional early material on dynamics
More informationExam III Physics 101: Lecture 19 Elasticity and Oscillations
Exam III Physics 101: Lecture 19 Elasticity and Oscillations Physics 101: Lecture 19, Pg 1 Overview Springs (review) Restoring force proportional to displacement F = -k x (often a good approximation) U
More informationDiagonalization of Matrices
LECTURE 4 Diagonalization of Matrices Recall that a diagonal matrix is a square n n matrix with non-zero entries only along the diagonal from the upper left to the lower right (the main diagonal) Diagonal
More informationChapter 12. Recall that when a spring is stretched a distance x, it will pull back with a force given by: F = -kx
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
More informationLecture 10. Central potential
Lecture 10 Central potential 89 90 LECTURE 10. CENTRAL POTENTIAL 10.1 Introduction We are now ready to study a generic class of three-dimensional physical systems. They are the systems that have a central
More informationCourse Name: AP Physics. Team Names: Jon Collins. Velocity Acceleration Displacement
Course Name: AP Physics Team Names: Jon Collins 1 st 9 weeks Objectives Vocabulary 1. NEWTONIAN MECHANICS and lab skills: Kinematics (including vectors, vector algebra, components of vectors, coordinate
More informationPhysics 312, Winter 2007, Practice Final
Physics 312, Winter 2007, Practice Final Time: Two hours Answer one of Question 1 or Question 2 plus one of Question 3 or Question 4 plus one of Question 5 or Question 6. Each question carries equal weight.
More information3-D Kinetics of Rigid Bodies
3-D Kinetics of Rigid Bodies Angular Momentum Generalized Newton s second law for the motion for a 3-D mass system Moment eqn for 3-D motion will be different than that obtained for plane motion Consider
More informationPhysics 351, Spring 2015, Final Exam.
Physics 351, Spring 2015, Final Exam. This closed-book exam has (only) 25% weight in your course grade. You can use one sheet of your own hand-written notes. Please show your work on these pages. The back
More informationGame Physics. Game and Media Technology Master Program - Utrecht University. Dr. Nicolas Pronost
Game and Media Technology Master Program - Utrecht University Dr. Nicolas Pronost Rigid body physics Particle system Most simple instance of a physics system Each object (body) is a particle Each particle
More informationLECTURE 14: DEVELOPING THE EQUATIONS OF MOTION FOR TWO-MASS VIBRATION EXAMPLES
LECTURE 14: DEVELOPING THE EQUATIONS OF MOTION FOR TWO-MASS VIBRATION EXAMPLES Figure 3.47 a. Two-mass, linear vibration system with spring connections. b. Free-body diagrams. c. Alternative free-body
More informationChapter 13 Lecture. Essential University Physics Richard Wolfson 2 nd Edition. Oscillatory Motion Pearson Education, Inc.
Chapter 13 Lecture Essential University Physics Richard Wolfson nd Edition Oscillatory Motion Slide 13-1 In this lecture you ll learn To describe the conditions under which oscillatory motion occurs To
More informationRotational Kinetic Energy
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
More informationCS-184: Computer Graphics
CS-184: Computer Graphics Lecture #25: Rigid Body Simulations Tobias Pfaff 537 Soda (Visual Computing Lab) tpfaff@berkeley.edu Reminder Final project presentations next week! Game Physics Types of Materials
More informationAssignment 6. Using the result for the Lagrangian for a double pendulum in Problem 1.22, we get
Assignment 6 Goldstein 6.4 Obtain the normal modes of vibration for the double pendulum shown in Figure.4, assuming equal lengths, but not equal masses. Show that when the lower mass is small compared
More informationRotational Motion. PHY131H1F Summer Class 10. Moment of inertia is. Pre-class reading quiz
PHY131H1F Summer Class 10 Today: Rotational Motion Rotational energy Centre of Mass Moment of Inertia Oscillations; Repeating Motion Simple Harmonic Motion Connection between Oscillations and Uniform Circular
More informationFaculty of Computers and Information. Basic Science Department
18--018 FCI 1 Faculty of Computers and Information Basic Science Department 017-018 Prof. Nabila.M.Hassan 18--018 FCI Aims of Course: The graduates have to know the nature of vibration wave motions with
More information(Refer Slide Time: 1:58 min)
Applied Mechanics Prof. R. K. Mittal Department of Applied Mechanics Indian Institution of Technology, Delhi Lecture No. # 13 Moments and Products of Inertia (Contd.) Today s lecture is lecture thirteen
More informationAn introduction to Solid State NMR and its Interactions
An introduction to Solid State NMR and its Interactions From tensor to NMR spectra CECAM Tutorial September 9 Calculation of Solid-State NMR Parameters Using the GIPAW Method Thibault Charpentier - CEA
More informationCHAPTER 12 OSCILLATORY MOTION
CHAPTER 1 OSCILLATORY MOTION Before starting the discussion of the chapter s concepts it is worth to define some terms we will use frequently in this chapter: 1. The period of the motion, T, is the time
More informationRigid body dynamics. Basilio Bona. DAUIN - Politecnico di Torino. October 2013
Rigid body dynamics Basilio Bona DAUIN - Politecnico di Torino October 2013 Basilio Bona (DAUIN - Politecnico di Torino) Rigid body dynamics October 2013 1 / 16 Multiple point-mass bodies Each mass is
More informationGeneral Physics I Spring Oscillations
General Physics I Spring 2011 Oscillations 1 Oscillations A quantity is said to exhibit oscillations if it varies with time about an equilibrium or reference value in a repetitive fashion. Oscillations
More informationVideo 2.1a Vijay Kumar and Ani Hsieh
Video 2.1a Vijay Kumar and Ani Hsieh Robo3x-1.3 1 Introduction to Lagrangian Mechanics Vijay Kumar and Ani Hsieh University of Pennsylvania Robo3x-1.3 2 Analytical Mechanics Aristotle Galileo Bernoulli
More informationModal Analysis: What it is and is not Gerrit Visser
Modal Analysis: What it is and is not Gerrit Visser What is a Modal Analysis? What answers do we get out of it? How is it useful? What does it not tell us? In this article, we ll discuss where a modal
More informationStructural Dynamics Prof. P. Banerji Department of Civil Engineering Indian Institute of Technology, Bombay. Lecture - 1 Introduction
Structural Dynamics Prof. P. Banerji Department of Civil Engineering Indian Institute of Technology, Bombay Lecture - 1 Introduction Hello, I am Pradipta Banerji from the department of civil engineering,
More informationThe object of this experiment is to study systems undergoing simple harmonic motion.
Chapter 9 Simple Harmonic Motion 9.1 Purpose The object of this experiment is to study systems undergoing simple harmonic motion. 9.2 Introduction This experiment will develop your ability to perform calculations
More information1 f. result from periodic disturbance same period (frequency) as source Longitudinal or Transverse Waves Characterized by
result from periodic disturbance same period (frequency) as source Longitudinal or Transverse Waves Characterized by amplitude (how far do the bits move from their equilibrium positions? Amplitude of MEDIUM)
More informationSelected Topics in Physics a lecture course for 1st year students by W.B. von Schlippe Spring Semester 2007
Selected Topics in Physics a lecture course for st year students by W.B. von Schlippe Spring Semester 7 Lecture : Oscillations simple harmonic oscillations; coupled oscillations; beats; damped oscillations;
More information5.80 Small-Molecule Spectroscopy and Dynamics
MIT OpenCourseWare http://ocw.mit.edu 5.80 Small-Molecule Spectroscopy and Dynamics Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 5.80 Lecture
More informationPhysical Dynamics (PHY-304)
Physical Dynamics (PHY-304) Gabriele Travaglini March 31, 2012 1 Review of Newtonian Mechanics 1.1 One particle Lectures 1-2. Frame, velocity, acceleration, number of degrees of freedom, generalised coordinates.
More informationRotations and vibrations of polyatomic molecules
Rotations and vibrations of polyatomic molecules When the potential energy surface V( R 1, R 2,..., R N ) is known we can compute the energy levels of the molecule. These levels can be an effect of: Rotation
More informationChapter 3 Higher Order Linear ODEs
Chapter 3 Higher Order Linear ODEs Advanced Engineering Mathematics Wei-Ta Chu National Chung Cheng University wtchu@cs.ccu.edu.tw 1 2 3.1 Homogeneous Linear ODEs 3 Homogeneous Linear ODEs An ODE is of
More information28. Pendulum phase portrait Draw the phase portrait for the pendulum (supported by an inextensible rod)
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.
More informationImportant because SHM is a good model to describe vibrations of a guitar string, vibrations of atoms in molecules, etc.
Simple Harmonic Motion Oscillatory motion under a restoring force proportional to the amount of displacement from equilibrium A restoring force is a force that tries to move the system back to equilibrium
More informationSCHEME OF BE 100 ENGINEERING MECHANICS DEC 2015
Part A Qn. No SCHEME OF BE 100 ENGINEERING MECHANICS DEC 201 Module No BE100 ENGINEERING MECHANICS Answer ALL Questions 1 1 Theorem of three forces states that three non-parallel forces can be in equilibrium
More informationLecture II: Rigid-Body Physics
Rigid-Body Motion Previously: Point dimensionless objects moving through a trajectory. Today: Objects with dimensions, moving as one piece. 2 Rigid-Body Kinematics Objects as sets of points. Relative distances
More informationLecture 32: Dynamics of Rigid Bodies
Lecture 3: Dynamics of Rigid Bodies Our next topic is the study of a special ind of system of particles a rigid body The defining characteristic of such a system is that the distance between any two particles
More information(Refer Slide Time: 0:36)
Engineering Mechanics Professor Manoj K Harbola Department of Physics Indian Institute of Technology Kanpur Module 04 Lecture No 35 Properties of plane surfaces VI: Parallel axis transfer theorem for second
More informationThe... of a particle is defined as its change in position in some time interval.
Distance is the. of a path followed by a particle. Distance is a quantity. The... of a particle is defined as its change in position in some time interval. Displacement is a.. quantity. The... of a particle
More informationChapter 4 The Equations of Motion
Chapter 4 The Equations of Motion Flight Mechanics and Control AEM 4303 Bérénice Mettler University of Minnesota Feb. 20-27, 2013 (v. 2/26/13) Bérénice Mettler (University of Minnesota) Chapter 4 The Equations
More informationLecture AC-1. Aircraft Dynamics. Copy right 2003 by Jon at h an H ow
Lecture AC-1 Aircraft Dynamics Copy right 23 by Jon at h an H ow 1 Spring 23 16.61 AC 1 2 Aircraft Dynamics First note that it is possible to develop a very good approximation of a key motion of an aircraft
More informationChapter 14 Oscillations. Copyright 2009 Pearson Education, Inc.
Chapter 14 Oscillations 14-1 Oscillations of a Spring If an object vibrates or oscillates back and forth over the same path, each cycle taking the same amount of time, the motion is called periodic. The
More informationFinite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras. Module - 01 Lecture - 11
Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras Module - 01 Lecture - 11 Last class, what we did is, we looked at a method called superposition
More informationOscillations. Oscillations and Simple Harmonic Motion
Oscillations AP Physics C Oscillations and Simple Harmonic Motion 1 Equilibrium and Oscillations A marble that is free to roll inside a spherical bowl has an equilibrium position at the bottom of the bowl
More informationFIRST YEAR MATHS FOR PHYSICS STUDENTS NORMAL MODES AND WAVES. Hilary Term Prof. G.G.Ross. Question Sheet 1: Normal Modes
FIRST YEAR MATHS FOR PHYSICS STUDENTS NORMAL MODES AND WAVES Hilary Term 008. Prof. G.G.Ross Question Sheet : Normal Modes [Questions marked with an asterisk (*) cover topics also covered by the unstarred
More informationMultiple Integrals and Vector Calculus: Synopsis
Multiple Integrals and Vector Calculus: Synopsis Hilary Term 28: 14 lectures. Steve Rawlings. 1. Vectors - recap of basic principles. Things which are (and are not) vectors. Differentiation and integration
More information7.4 Rigid body simulation*
7.4 Rigid body simulation* Quaternions vs. Rotation Matrices - unit quaternions, a better way to represent the orientation of a rigid body. Why? (1) More compact: 4 numbers vs. 9 numbers (2) Smooth transition
More informationDamped harmonic oscillator
Prof. O. B. Wright, Mechanics Notes Damped harmonic oscillator Differential equation Assume the mass on a spring is subject to a frictional drag force -'dx/dt. (This force always points in the opposite
More informationRigid body simulation. Once we consider an object with spatial extent, particle system simulation is no longer sufficient
Rigid body dynamics Rigid body simulation Once we consider an object with spatial extent, particle system simulation is no longer sufficient Rigid body simulation Unconstrained system no contact Constrained
More informationTorque and Simple Harmonic Motion
Torque and Simple Harmonic Motion Recall: Fixed Axis Rotation Angle variable Angular velocity Angular acceleration Mass element Radius of orbit Kinematics!! " d# / dt! " d 2 # / dt 2!m i Moment of inertia
More informationPhysical Dynamics (SPA5304) Lecture Plan 2018
Physical Dynamics (SPA5304) Lecture Plan 2018 The numbers on the left margin are approximate lecture numbers. Items in gray are not covered this year 1 Advanced Review of Newtonian Mechanics 1.1 One Particle
More informationIn this section of notes, we look at the calculation of forces and torques for a manipulator in two settings:
Introduction Up to this point we have considered only the kinematics of a manipulator. That is, only the specification of motion without regard to the forces and torques required to cause motion In this
More informationSimple Harmonic Motion
Physics 7B-1 (A/B) Professor Cebra Winter 010 Lecture 10 Simple Harmonic Motion Slide 1 of 0 Announcements Final exam will be next Wednesday 3:30-5:30 A Formula sheet will be provided Closed-notes & closed-books
More informationAP Pd 3 Rotational Dynamics.notebook. May 08, 2014
1 Rotational Dynamics Why do objects spin? Objects can travel in different ways: Translation all points on the body travel in parallel paths Rotation all points on the body move around a fixed point An
More informationHB Coupled Pendulums Lab Coupled Pendulums
HB 04-19-00 Coupled Pendulums Lab 1 1 Coupled Pendulums Equipment Rotary Motion sensors mounted on a horizontal rod, vertical rods to hold horizontal rod, bench clamps to hold the vertical rods, rod clamps
More informationChapter 14 Oscillations. Copyright 2009 Pearson Education, Inc.
Chapter 14 Oscillations Oscillations of a Spring Simple Harmonic Motion Energy in the Simple Harmonic Oscillator Simple Harmonic Motion Related to Uniform Circular Motion The Simple Pendulum The Physical
More informationChapter 14 Oscillations
Chapter 14 Oscillations If an object vibrates or oscillates back and forth over the same path, each cycle taking the same amount of time, the motion is called periodic. The mass and spring system is a
More information13. Rigid Body Dynamics II
University of Rhode Island DigitalCommons@URI Classical Dynamics Physics Course Materials 2015 13. Rigid Body Dynamics II Gerhard Müller University of Rhode Island, gmuller@uri.edu Creative Commons License
More informationRigid Body Rotation. Speaker: Xiaolei Chen Advisor: Prof. Xiaolin Li. Department of Applied Mathematics and Statistics Stony Brook University (SUNY)
Rigid Body Rotation Speaker: Xiaolei Chen Advisor: Prof. Xiaolin Li Department of Applied Mathematics and Statistics Stony Brook University (SUNY) Content Introduction Angular Velocity Angular Momentum
More informationProblem Set 2 Due Tuesday, September 27, ; p : 0. (b) Construct a representation using five d orbitals that sit on the origin as a basis: 1
Problem Set 2 Due Tuesday, September 27, 211 Problems from Carter: Chapter 2: 2a-d,g,h,j 2.6, 2.9; Chapter 3: 1a-d,f,g 3.3, 3.6, 3.7 Additional problems: (1) Consider the D 4 point group and use a coordinate
More information(Refer Slide Time: 2:08 min)
Applied Mechanics Prof. R. K. Mittal Department of Applied Mechanics Indian Institute of Technology, Delhi Lecture No. 11 Properties of Surfaces (Contd.) Today we will take up lecture eleven which is a
More information