Final Exam Solution Dynamics :45 12:15. Problem 1 Bateau


 Tobias Atkins
 11 months ago
 Views:
Transcription
1 Final Exam Solution Dynamics :45 12:15 Problem 1 Bateau Bateau is a trapeze act by Cirque du Soleil in which artists perform aerial maneuvers on a boat shaped structure. The boat is modeled as a rigid uniform bar with mass and length. The boat is connected to the ceiling of the theater by means of two suspension bars with length that are considered massless. The gravitational constant is. The act consists of two phases: A and B, shown in the figure below. Phase A: The artists with combined mass are on top of the boat, which is in its equilibrium position. They exert a force on the boat to start up its motion. The generalized coordinate describing the motion of the boat is the angle between the suspension bars and the vertical. Phase B: The artists are suspended underneath the boat. This is modelled as if the artists form a pendulum of length. The generalized coordinate describing the motion of the artists is the angle between the artists and the vertical. Bateau Phase A Phase B a) Consider phase A. Derive the equation of motion of the system for arbitrary large angles using the force and moment balances method. In order to do so, make an accurate freebody diagram of the boat. Show that the linearized equation of motion is equal to (i). (i) First, the constraint equations are derived that are used to express the position, velocity and acceleration of the center of mass of the boat in terms of (t). Position: Velocity:
2 Acceleration: The freebody diagram of the boat in a positive deflected state must contain the gravity, the forces in the suspension bars and along the direction of the suspension bars and the external force. The force balances are: Substitution of the constraint equations: [ ] [ ] Multiply the first equation by and the second by : [ ] [ ] Adding yields the equation of motion for arbitrary large angles: Linearizing the equation of motion using and, yields (i): (i) Constraint equations: Freebody diagram: Force balances: Correct nonlinear equation: Correct linearization: 2p ( for the velocities, for the accelerations) 2p (1 for every missing or incorrect force) 2p ( for each correct balance) 3p 10p b) Assume that all artists are distributed evenly over the boat. Derive the expressions for the forces and in the suspension bars for arbitrary large motions of the boat. Prove that both forces are equal. Comment on the physical meaning of all terms in your result. 2
3 Moment balance about the center of gravity of the boat: From this it follows that both forces are equal: Now multiply the first equation of a) by and the second by : [ ] [ ] Subtract the first from the second: So: [ ] It can be seen that both suspension bars carry half of the total weight, the fictitious centripetal force and the component of the excitation force in the direction of the suspension bars. Forces equal by using : Solving from and : 2p Interpretation: 4p c) Calculate the steady state amplitude of the boat when the force applied by the artists is harmonic:. Sketch the magnitude plot of the frequency response function in which you indicate important characteristics. Write the equation of motion in standard form: Trial function and its second time derivative can be cosines because there is no damping: 3
4 Substitution and rewriting: Solving for : The magnitude plot of is equal to at and tends to zero for. The magnitude is equal to infinity for. EoM in standard form: Trial function equal to RHS: Solving amplitude : Correct plot: 2p (1 for every incorrect plot characteristic) 5p d) What is the expression of the frequency, with which the artists should apply the force, such that the amplitude of the boat increases the fastest? Derive the response of the boat for this specific case, using initial conditions. Prove that the response can be written as (ii). Sketch the response and indicate its important characteristics. (ii) The amplitude of the motion increases the fastest when the system is excited in its natural frequency, because in that case resonance will occur. The homogenous solution of the differential equation is simply: In resonance, the trial solution for the particular part must be multiplied by : Substitution in the equation of motion make that all terms multiplied by cancel: 4
5 From this it can easily be seen that and that for holds: So the total solution becomes: Apply the initial conditions: Response thus becomes equal to (ii): The plot of this response must clearly contain the linear increase in time. Fastest increase in resonance: Homogenous solution: Trial function with : Solving: 2p Initial conditions: Plot linear amplitude increase: 7p e) The system may be subjected to two types of damping: viscous damping or Coulomb damping (dry friction). Sketch the free vibration response of the system in both cases. Indicate and explain three important differences. In case of viscous damping, the decay of the amplitude in time is exponential. The frequency of the oscillation is in the case of underdamping and has no oscillation for critical of overdamping. The system will reach its equilibrium position for. In the case of Coulomb damping the decay of the amplitude in time is linear. The frequency of the oscillation is for all cases. The system will never reach its equilibrium position, but will come to rest as soon as the static friction force is larger than the restoring force due to gravity. Moreover, the response is only piecewise harmonic and not continuously harmonic due to the nonlinear nature of the EoM. Three correct differences: 3p ( for every difference that is also sketched) 3p 5
6 f) Consider case B. Derive the equations of motion of the system for arbitrary large angles and, using Lagrange s equation (iii). You may need the goniometric formula (iv). ( ) (iii) (iv) First, the constraint equations are derived that are used to express the position and velocity of the center of mass of the artists in terms of and (t). Position: Velocity: Kinetic energy: ( ) ( ) Substitution of the constraint equations: ( [ ]) Rewrite: Potential energy: No damping and no external forces so. 6
7 Lagrange s terms for : ( ) ( ) First equation of motion: Lagrange s terms for : ( ) ( ) Second equation of motion: Constraint equations: 2p ( for the position, for the velocity) Kinetic energy is : Rewrite kinetic energy: Potential energy: No damping and virtual work: Correct EoMs: 4p (2p for each correct EoM) 10p g) Show that the linearized equations of motion are equal to (v). Assume that and. Derive the expressions for the natural frequencies of the system. Prove that the corresponding natural mode shapes are (vi). [ ] { } [ ] { } { } (v) 7
8 { } { } { } { } (vi) Linearizing means, and neglecting higher order terms. By doing so, (v) is obtained. With the simplifications given, the system reduces to: [ ] { Rewrite: [ ] { Trial function: } [ ] { } { } } [ ] { } { } { } { } Eigenvalue problem: [ ] { } { } Characteristic equation is obtained by equating the determinant to zero: Roots are obtained using the quadratic formula: ( ) Since both roots are negative. This means that the natural frequencies are: First natural mode is obtained by back substitution of ( ) : [ ] { } { } 8
9 Using the first equation it follows that: ( ) ( ) So the first natural mode is: { } { } Second natural mode is obtained by back substitution of ( ) : [ ] { } { } Using the first equation it follows that: ( ) ( ) So the second natural mode is: { } { } Linearizing: Correct eigenvalue problem: Characteristic equation: Natural frequencies: Natural modes: 2p ( for each correct natural frequency) 2p ( for each correct natural mode) 7p h) Apply a coordinate transformation towards the natural coordinates and. Normalize the natural modes with respect to the mass matrix and derive the uncoupled equations of motion. Explain the benefit of the uncoupled equations of motion over the coupled equations of motion. Modal matrix: [ ] [ ] Coordinate transformation: [ ] [ ][ ]{ } [ ] [ ][ ]{ } { } 9
10 For the mass matrix: [ ] [ ][ ] [ [ ] [ ] [ ] ] [( ) ( ) ( ) ( ) ] [ ( ) ( ) ] Normalized if: For the stiffness matrix: [ ] [ ][ ] [ ] [ ] [ ] [ ] [ ] [ ] Substitution of the values of and : [ ] [ ][ ] [ ] [ ] The uncoupled equations of motion become: [ ]{ } [ ]{ } { } The advantage of uncoupling the equations of motion is that the response of the system to either initial conditions or harmonic excitations can be solved for each natural mode independently. This means that the response of the 2DOF system can be written as a combination of the solutions of two 1DOF problems for which the solution is obtained with relative ease. Modal matrix: Mass matrix: 2p Normalizing, and : Stiffness matrix: 2p Benefit of uncoupling: 7p 10
11 i) In the case that the artists make continuous loops at constant angular velocity, their motion can be seen as prescribed in time:. Apply this on the nonlinear equation of motion of the boat as derived in f). Prove that when and, the linearized equation of motion is equal to (vii). (vii) Recall the nonlinear equation of motion as determined in f). Apply : Rewrite : [ ] Simplified case: Rewrite: Linearizing for small angles this can be rewritten to (vii): (vii) Prescribing : Rewriting : Simplifying: Linearzing: 4p 11
12 j) Because of gravity, it is very hard to make loops at a constant angular velocity. This means that in practice the boat is not excited harmonically in this way, but periodic instead. Explain extensively how this influences the procedure required to obtain the solution as a result of this periodic excitation. Transform the excitation to a series of harmonic functions using a Fourier series. Then solve for all individual components. The response is the superposition of all individual responses. Fourier series: Procedure: 2p 3p Overview of points problem 1: a b c d e f g h i j tot
13 Problem 2 Railway track Some structures consisting of beams are supported by an elastic foundation. A good example is the railway track, which consists of steel rails that are connected to the fixed world by means of wooden or concrete crossbars (also known as ties or sleepers) approximately every meters. The crossbars and rails are supported by their foundation which is typically made up from crushed stone. The figure below shows a schematic side view of the railway track in between two crossbars. The rail is modeled as an uniform beam of length, linear density and bending stiffness which is simply supported (hinged) at the locations of both crossbars. The supporting effect of the foundation is modeled as a distributed series of translational springs, which have stiffness per unit beam length. The transverse deformation of the beam is denoted by. As a result of passing trains, the beam may be subjected to an external loading. The effect of gravity is neglected. a) Derive the partial differential equation of motion for transverse vibrations in beams on an elastic foundation (i) using Hooke s law for bending (ii). In order to do so, make an accurate freebody diagram of an internal element of the beam. (i) (ii) The freebody diagram must contain an internal element in positive deflected state, moment and, shear forces and, foundation force and external loading. Force balance in transverse direction and moment balance about the left end of the internal element: [ ] [ ] [ ] 13
14 Neglecting higher order terms in the moment balance yields: Substitution in the force balance and rewriting yields: Apply Hooke s law and rearrange the terms yields the equation of motion (i): Freebody diagram: Balances: Solving: Total 2p ( for every incorrect force or moment) 2p ( for each balance) 5p b) Derive the general solution of the equation of motion (i) for free standing vibrations in the beam. Use the technique of separation of variables. Introduce the wave number as (iii). What is the benefit of using separation of variables to solve the equation of motion? (iii) Free vibrations: Standing waves gives us the opportunity to separate variables: Substitution in the equation of motion: Divide by and, rearrange and introduce the separation constant : 14
15 For the time part: Solution: For the spatial part: Introduce the wave number as given: Solution: The benefit of separating the variables is that the temporal behavior and spatial behavior can be studied separately. Hence, in this we it is known exactly what the shape of the beam is and how this shape is behaving in time. Separation of variables: Correct time part: Correct spatial part: 2p Benefit separation of variables: Total 6p c) Determine the boundary conditions at and. Use them to determine the expressions for the natural frequencies and the corresponding natural mode shapes. How many natural frequencies does the system have? Hint: Look carefully at your boundary conditions. You can reduce the matrix problem easily to a matrix problem. At both ends, the displacement and bending moment are zero. For the following equations: this yields 15
16 At this yields the following equations: In matrix vector form: [ ] { } { } Note that in order to satisfy both the first and the second equation reduces the system to:, which [ ] { } { } Characteristic equation is obtained by equating the determinant zero: For nonzero values of, this is case for: So: Using the definition of the wave number (iii): Solving for the natural frequencies: Back substitution of in the matrix vector equation yields: [ ] { } { } 16
17 From this it follows that, which leaves for the natural mode shapes: Boundary conditions: 2p Eliminate time part: Creating matrix: reducing matrix: Characteristic determinant: Natural frequencies: Natural modes: 8p d) With the natural frequencies and natural mode shapes known, explain extensively the procedure with which you would determine the response of the system to an initial excitation. In this procedure, what determines the expression of the mode shape coefficients? The response of the system can be written as a linear combination of the motion of all natural modes over time: The initial excitation of all modes is determined by multiplying the above expression by and integrating over the length of the beam. This integral can be simplified using the orthogonality conditions. Such that equations in terms of natural coordinates are obtained: The solution of this is simply: The total solution is obtained by back substitution. The coefficients are determined such that the orthogonality conditions hold and that the integral encountered in the above derivation is equal to 1. is combination of modes: Initial excitation of all modes: Natural coordinates: Orthogonality conditions: Determination of : 5p 17
18 e) In the case that the system would have been so complicated that its natural frequencies cannot be obtained analytically, an estimation of the lowest natural frequency can be obtained using Rayleigh s quotient. Proof that in this case the expression for Rayleigh s quotient is equal to (iv). ( ) (iv) Look at the spatial part of the equation of motion and separate the parts with and without : Multiply by and integrate over the length of the beam: The first integral must be integrated by parts twice: [ ] [ ] [ ] ( ) In view of the boundary conditions, the terms between square brackets are zero and the equation reduces to: ( ) Solving for yields Rayleigh s quotient (iv): ( ) ( ) Multiply by, integrate: 2p Integration by parts: 3p BCs for [ ] terms Final expression: 7p 18
19 f) Two possible trial functions that can be used in Rayleigh s quotient are (v). Determine for both trial functions whether it is a comparison function or an admissible function. Based on this, explain which trial function you think will result in the best approximation of the lowest natural frequency. Suppose that you would calculate Rayleigh s quotient using both trial functions. How would you know which of the two estimation is best, without knowing the exact solution? (v) For the first trial function, consider the geometric boundary conditions: The first and second derivatives are: Because the second derivative is not zero at and, the trial function does not fulfil the natural boundary conditions. Hence, the first trial function is an admissible function. For the second trial function, consider the geometric boundary conditions: The first and second derivatives are: Because the second derivative is zero at and, the trial function does fulfil the natural boundary conditions. Hence, the second trial function is a comparison function. Since a comparison function is a better approximation of the real behavior of the beam than an admissible function, using this trial function results in a better approximation of the lowest natural frequency. After solving Rayleigh s quotient of both trial function, the lowest value is the best approximation. 19
20 Check geometric BCs: Check natural BCs: Conclusion comp. / admis.: Comparisson is better: Lowest value is best: 2p ( for each trial function) 6p g) Suppose the beam is given an initial deflection according to the first trial function of (v). Using the procedure referred to in d), it is possible to calculate the response of the system on this initial excitation (you do not need to do this!). Which of all natural mode shapes will be in visible in this response? Why? Since the trial function is a polynomial and the natural modes are sines, an infinite number of modes are excited. Because the function is symmetric, only the oddnumbered modes are excited. Hence, only the oddnumbered natural modes will be visible in the response. Polynomial vs sines: Infinte modes: Symmetry / Oddnumbered: 3p Overview of points problem 2: a b c d e f g tot
Introduction to Continuous Systems. Continuous Systems. Strings, Torsional Rods and Beams.
Outline of Continuous Systems. Introduction to Continuous Systems. Continuous Systems. Strings, Torsional Rods and Beams. Vibrations of Flexible Strings. Torsional Vibration of Rods. BernoulliEuler Beams.
More informationStructural Dynamics Lecture 4. Outline of Lecture 4. MultiDegreeofFreedom Systems. Formulation of Equations of Motions. Undamped Eigenvibrations.
Outline of MultiDegreeofFreedom Systems Formulation of Equations of Motions. Newton s 2 nd Law Applied to Free Masses. D Alembert s Principle. Basic Equations of Motion for Forced Vibrations of Linear
More information2.003 Engineering Dynamics Problem Set 10 with answer to the concept questions
.003 Engineering Dynamics Problem Set 10 with answer to the concept questions Problem 1 Figure 1. Cart with a slender rod A slender rod of length l (m) and mass m (0.5kg)is attached by a frictionless pivot
More informationChapter 11 Vibrations and Waves
Chapter 11 Vibrations and Waves 111 Simple Harmonic Motion 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.
More informationName: Fall 2014 CLOSED BOOK
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
More informationAA242B: MECHANICAL VIBRATIONS
AA242B: MECHANICAL VIBRATIONS 1 / 50 AA242B: MECHANICAL VIBRATIONS Undamped Vibrations of ndof Systems These slides are based on the recommended textbook: M. Géradin and D. Rixen, Mechanical Vibrations:
More informationAssignments VIII and IX, PHYS 301 (Classical Mechanics) Spring 2014 Due 3/21/14 at start of class
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,
More informationChapter 1. Harmonic Oscillator. 1.1 Energy Analysis
Chapter 1 Harmonic Oscillator Figure 1.1 illustrates the prototypical harmonic oscillator, the massspring system. A mass is attached to one end of a spring. The other end of the spring is attached to
More informationNormal modes. where. and. On the other hand, all such systems, if started in just the right way, will move in a simple way.
Chapter 9. Dynamics in 1D 9.4. Coupled motions in 1D 491 only the forces from the outside; the interaction forces cancel because they come in equal and opposite (action and reaction) pairs. So we get:
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 informationC. points X and Y only. D. points O, X and Y only. (Total 1 mark)
Grade 11 Physics  Homework 16  Answers on a separate sheet of paper, please 1. A cart, connected to two identical springs, is oscillating with simple harmonic motion between two points X and Y that
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 information2007 Problem Topic Comment 1 Kinematics Positiontime equation Kinematics 7 2 Kinematics Velocitytime graph Dynamics 6 3 Kinematics Average velocity
2007 Problem Topic Comment 1 Kinematics Positiontime equation Kinematics 7 2 Kinematics Velocitytime graph Dynamics 6 3 Kinematics Average velocity Energy 7 4 Kinematics Free fall Collisions 3 5 Dynamics
More informationMulti Degrees of Freedom Systems
Multi Degrees of Freedom Systems MDOF s http://intranet.dica.polimi.it/people/boffigiacomo Dipartimento di Ingegneria Civile Ambientale e Territoriale Politecnico di Milano March 9, 07 Outline, a System
More informationOscillatory Motion SHM
Chapter 15 Oscillatory Motion SHM Dr. Armen Kocharian 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
More informationCalculus/Physics Schedule, Second Semester. Wednesday Calculus
/ Schedule, Second Semester Note to instructors: There are two key places where the calculus and physics are very intertwined and the scheduling is difficult: GaussÕ Law and the differential equation for
More informationCHAPTER 11 TEST REVIEW
AP PHYSICS Name: Period: Date: 50 Multiple Choice 45 Single Response 5 MultiResponse Free Response 3 Short Free Response 2 Long Free Response DEVIL PHYSICS BADDEST CLASS ON CAMPUS AP EXAM CHAPTER TEST
More informationStatic & Dynamic. Analysis of Structures. Edward L.Wilson. University of California, Berkeley. Fourth Edition. Professor Emeritus of Civil Engineering
Static & Dynamic Analysis of Structures A Physical Approach With Emphasis on Earthquake Engineering Edward LWilson Professor Emeritus of Civil Engineering University of California, Berkeley Fourth Edition
More informationChapter 12 Vibrations and Waves Simple Harmonic Motion page
Chapter 2 Vibrations and Waves 2 Simple Harmonic Motion page 43845 Hooke s Law Periodic motion the object has a repeated motion that follows the same path, the object swings to and fro. Examples: a pendulum
More informationThe Torsion Pendulum (One or two weights)
The Torsion Pendulum (One or two weights) Exercises I through V form the oneweight experiment. Exercises VI and VII, completed after Exercises I V, add one weight more. Preparatory Questions: 1. The
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 informationCh 3.7: Mechanical & Electrical Vibrations
Ch 3.7: Mechanical & Electrical Vibrations Two important areas of application for second order linear equations with constant coefficients are in modeling mechanical and electrical oscillations. We will
More informationTranslational Mechanical Systems
Translational Mechanical Systems Basic (Idealized) Modeling Elements Interconnection Relationships Physical Laws Derive Equation of Motion (EOM)  SDOF Energy Transfer Series and Parallel Connections
More informationA Physical Pendulum 2
A Physical Pendulum 2 Ian Jacobs, Physics Advisor, KVIS, Rayong, Thailand Introduction A physical pendulum rotates back and forth about a fixed axis and may be of any shape. All pendulums are driven by
More informationIntroduction to Mechanical Vibration
2103433 Introduction to Mechanical Vibration Nopdanai Ajavakom (NAV) 1 Course Topics Introduction to Vibration What is vibration? Basic concepts of vibration Modeling Linearization SingleDegreeofFreedom
More informationKEELE UNIVERSITY PHYSICS/ASTROPHYSICS MODULE PHY OSCILLATIONS AND WAVES PRACTICE EXAM
KEELE UNIVERSITY PHYSICS/ASTROPHYSICS MODULE PHY10012 OSCILLATIONS AND WAVES PRACTICE EXAM Candidates should attempt ALL of PARTS A and B, and TWO questions from PART C. PARTS A and B should be answered
More informationResponse Spectrum Analysis Shock and Seismic. FEMAP & NX Nastran
Response Spectrum Analysis Shock and Seismic FEMAP & NX Nastran Table of Contents 1. INTRODUCTION... 3 2. THE ACCELEROGRAM... 4 3. CREATING A RESPONSE SPECTRUM... 5 4. NX NASTRAN METHOD... 8 5. RESPONSE
More informationLab 12. SpringMass Oscillations
Lab 12. SpringMass Oscillations Goals To determine experimentally whether the supplied spring obeys Hooke s law, and if so, to calculate its spring constant. To determine the spring constant by another
More informationPHYSICS. Course Structure. Unit Topics Marks. Physical World and Measurement. 1 Physical World. 2 Units and Measurements.
PHYSICS Course Structure Unit Topics Marks I Physical World and Measurement 1 Physical World 2 Units and Measurements II Kinematics 3 Motion in a Straight Line 23 4 Motion in a Plane III Laws of Motion
More informationLab 11. SpringMass Oscillations
Lab 11. SpringMass 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
More informationFor a rigid body that is constrained to rotate about a fixed axis, the gravitational torque about the axis is
Experiment 14 The Physical Pendulum The period of oscillation of a physical pendulum is found to a high degree of accuracy by two methods: theory and experiment. The values are then compared. Theory For
More informationTorsion Spring Oscillator with Dry Friction
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
More informationLab M5: Hooke s Law and the Simple Harmonic Oscillator
M5.1 Lab M5: Hooke s Law and the Simple Harmonic Oscillator Most springs obey Hooke s Law, which states that the force exerted by the spring is proportional to the extension or compression of the spring
More informationDamped Harmonic Oscillator
Damped Harmonic Oscillator Wednesday, 23 October 213 A simple harmonic oscillator subject to linear damping may oscillate with exponential decay, or it may decay biexponentially without oscillating, or
More informationSection 1 Simple Harmonic Motion. Chapter 11. Preview. Objectives Hooke s Law Sample Problem Simple Harmonic Motion The Simple Pendulum
Section 1 Simple Harmonic Motion Preview Objectives Hooke s Law Sample Problem Simple Harmonic Motion The Simple Pendulum Section 1 Simple Harmonic Motion Objectives Identify the conditions of simple harmonic
More informationPHY217: Vibrations and Waves
Assessed Problem set 1 Issued: 5 November 01 PHY17: Vibrations and Waves Deadline for submission: 5 pm Thursday 15th November, to the V&W pigeon hole in the Physics reception on the 1st floor of the GO
More informationcos(θ)sin(θ) Alternative Exercise Correct Correct θ = 0 skiladæmi 10 Part A Part B Part C Due: 11:59pm on Wednesday, November 11, 2015
skiladæmi 10 Due: 11:59pm on Wednesday, November 11, 015 You will receive no credit for items you complete after the assignment is due Grading Policy Alternative Exercise 1115 A bar with cross sectional
More informationAPPLIED MATHEMATICS AM 02
AM SYLLABUS (2013) APPLIED MATHEMATICS AM 02 SYLLABUS Applied Mathematics AM 02 Syllabus (Available in September) Paper I (3 hrs)+paper II (3 hrs) Applied Mathematics (Mechanics) Aims A course based on
More informationSection 1 Simple Harmonic Motion. The student is expected to:
Section 1 Simple Harmonic Motion TEKS The student is expected to: 7A examine and describe oscillatory motion and wave propagation in various types of media Section 1 Simple Harmonic Motion Preview Objectives
More informationSolution Derivations for Capa #12
Solution Derivations for Capa #12 1) A hoop of radius 0.200 m and mass 0.460 kg, is suspended by a point on it s perimeter as shown in the figure. If the hoop is allowed to oscillate side to side as a
More information1 2 Models, Theories, and Laws 1.5 Distinguish between models, theories, and laws 2.1 State the origin of significant figures in measurement
Textbook Correlation Textbook Correlation Physics 1115/2015 Chapter 1 Introduction, Measurement, Estimating 1.1 Describe thoughts of Aristotle vs. Galileo in describing motion 1 1 Nature of Science 1.2
More informationPhysics 326 Lab 6 10/18/04 DAMPED SIMPLE HARMONIC MOTION
DAMPED SIMPLE HARMONIC MOTION PURPOSE To understand the relationships between force, acceleration, velocity, position, and period of a mass undergoing simple harmonic motion and to determine the effect
More informationMeasuring the Universal Gravitational Constant, G
Measuring the Universal Gravitational Constant, G Introduction: The universal law of gravitation states that everything in the universe is attracted to everything else. It seems reasonable that everything
More informationUnforced Mechanical Vibrations
Unforced Mechanical Vibrations Today we begin to consider applications of second order ordinary differential equations. 1. SpringMass Systems 2. Unforced Systems: Damped Motion 1 SpringMass Systems We
More informationVIBRATION PROBLEMS IN ENGINEERING
VIBRATION PROBLEMS IN ENGINEERING FIFTH EDITION W. WEAVER, JR. Professor Emeritus of Structural Engineering The Late S. P. TIMOSHENKO Professor Emeritus of Engineering Mechanics The Late D. H. YOUNG Professor
More informationANSWERS 403 INDEX. Bulk modulus 238 Buoyant force 251
ANSWERS 403 INDEX A Absolute scale temperature 276 Absolute zero 276 Acceleration (linear) 45 Acceleration due to gravity 49,189 Accuracy 22 Actionreaction 97 Addition of vectors 67 Adiabatic process
More informationReduction in number of dofs
Reduction in number of dofs Reduction in the number of dof to represent a structure reduces the size of matrices and, hence, computational cost. Because a subset of the original dof represent the whole
More informationEQUIVALENT SINGLEDEGREEOFFREEDOM SYSTEM AND FREE VIBRATION
1 EQUIVALENT SINGLEDEGREEOFFREEDOM SYSTEM AND FREE VIBRATION The course on Mechanical Vibration is an important part of the Mechanical Engineering undergraduate curriculum. It is necessary for the development
More informationChap. 15: Simple Harmonic Motion
Chap. 15: Simple Harmonic Motion Announcements: CAPA is due next Tuesday and next Friday. Web page: http://www.colorado.edu/physics/phys1110/phys1110_sp12/ Examples of periodic motion vibrating guitar
More informationEquilibrium & Elasticity
PHYS 101 Previous Exam Problems CHAPTER 12 Equilibrium & Elasticity Static equilibrium Elasticity 1. A uniform steel bar of length 3.0 m and weight 20 N rests on two supports (A and B) at its ends. A block
More informationDifferential Equations
Differential Equations A differential equation (DE) is an equation which involves an unknown function f (x) as well as some of its derivatives. To solve a differential equation means to find the unknown
More informationBasic Energy Principles in Stiffness Analysis
Basic Energy Principles in Stiffness Analysis StressStrain Relations The application of any theory requires knowledge of the physical properties of the material(s) comprising the structure. We are limiting
More informationLab 1: damped, driven harmonic oscillator
Lab 1: damped, driven harmonic oscillator 1 Introduction The purpose of this experiment is to study the resonant properties of a driven, damped harmonic oscillator. This type of motion is characteristic
More informationApplications of SecondOrder Differential Equations
Applications of SecondOrder Differential Equations ymy/013 Building Intuition Even though there are an infinite number of differential equations, they all share common characteristics that allow intuition
More informationModeling and Experimentation: MassSpringDamper System Dynamics
Modeling and Experimentation: MassSpringDamper 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
More informationMarch 24, Chapter 4. Deflection and Stiffness. Dr. Mohammad Suliman Abuhaiba, PE
Chapter 4 Deflection and Stiffness 1 2 Chapter Outline Spring Rates Tension, Compression, and Torsion Deflection Due to Bending Beam Deflection Methods Beam Deflections by Superposition Strain Energy Castigliano
More informationFigure 5.16 Compound pendulum: (a) At rest in equilibrium, (b) General position with coordinate θ, Freebody
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
More informationWork sheet / Things to know. Chapter 3
MATH 251 Work sheet / Things to know 1. Second order linear differential equation Standard form: Chapter 3 What makes it homogeneous? We will, for the most part, work with equations with constant coefficients
More informationEndofChapter Exercises
EndofChapter Exercises Exercises 1 12 are conceptual questions that are designed to see if you have understood the main concepts of the chapter. 1. When a spring is compressed 10 cm, compared to its
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Exam Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) A 4.8kg block attached to a spring executes simple harmonic motion on a frictionless
More informationVibrations Qualifying Exam Study Material
Vibrations Qualifying Exam Study Material The candidate is expected to have a thorough understanding of engineering vibrations topics. These topics are listed below for clarification. Not all instructors
More informationIntroduction to Vibration. Mike Brennan UNESP, Ilha Solteira São Paulo Brazil
Introduction to Vibration Mike Brennan UNESP, Ilha Solteira São Paulo Brazil Vibration Most vibrations are undesirable, but there are many instances where vibrations are useful Ultrasonic (very high
More informationContents. Dynamics and control of mechanical systems. Focus on
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
More informationMath Methods for Polymer Physics Lecture 1: Series Representations of Functions
Math Methods for Polymer Physics ecture 1: Series Representations of Functions Series analysis is an essential tool in polymer physics and physical sciences, in general. Though other broadly speaking,
More informationCE 6701 Structural Dynamics and Earthquake Engineering Dr. P. Venkateswara Rao
CE 6701 Structural Dynamics and Earthquake Engineering Dr. P. Venkateswara Rao Associate Professor Dept. of Civil Engineering SVCE, Sriperumbudur Difference between static loading and dynamic loading Degree
More informationIntroduction to Finite Element Method
Introduction to Finite Element Method Dr. Rakesh K Kapania Aerospace and Ocean Engineering Department Virginia Polytechnic Institute and State University, Blacksburg, VA AOE 524, Vehicle Structures Summer,
More informationChapter 12. Static Equilibrium and Elasticity
Chapter 12 Static Equilibrium and Elasticity Static Equilibrium Equilibrium implies that the object moves with both constant velocity and constant angular velocity relative to an observer in an inertial
More informationSquarewave excitation of a linear oscillator
Squarewave excitation of a linear oscillator Eugene I. Butikov St. Petersburg State University, St. Petersburg, Russia Email: butikov@spb.runnet.ru Abstract. The paper deals with forced oscillations
More informationVibration Mechanics. Linear Discrete Systems SPRINGER SCIENCE+BUSINESS MEDIA, B.V. M. Del Pedro and P. Pahud
Vibration Mechanics Vibration Mechanics Linear Discrete Systems by M. Del Pedro and P. Pahud Swiss Federal Institute oftechnology, Lausanne, Switzerland SPRINGER SCIENCE+BUSINESS MEDIA, B.V. ISBN 9789401055543
More information18. FAST NONLINEAR ANALYSIS. The Dynamic Analysis of a Structure with a Small Number of Nonlinear Elements is Almost as Fast as a Linear Analysis
18. FAS NONLINEAR ANALYSIS he Dynamic Analysis of a Structure with a Small Number of Nonlinear Elements is Almost as Fast as a Linear Analysis 18.1 INRODUCION he response of real structures when subjected
More informationLecture 9: Harmonic Loads (Con t)
Lecture 9: Harmonic Loads (Con t) Reading materials: Sections 3.4, 3.5, 3.6 and 3.7 1. Resonance The dynamic load magnification factor (DLF) The peak dynamic magnification occurs near r=1 for small damping
More informationAP PHYSICS 1 BIG IDEAS AND LEARNING OBJECTIVES
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]
More informationNo Brain Too Small PHYSICS
MECHANICS: SIMPLE HARMONIC MOTION QUESTIONS SIMPLE HARMONIC MOTION (2016;3) A toy bumble bee hangs on a spring suspended from the ceiling in the laboratory. Tom pulls the bumble bee down 10.0 cm below
More informationStep 1: Mathematical Modeling
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
More informationChapter 12 Partial Differential Equations
Chapter 12 Partial Differential Equations Advanced Engineering Mathematics WeiTa Chu National Chung Cheng University wtchu@cs.ccu.edu.tw 1 2 12.1 Basic Concepts of PDEs Partial Differential Equation A
More informationSimple Harmonic Motion Investigating a Mass Oscillating on a Spring
17 Investigating a Mass Oscillating on a Spring A spring that is hanging vertically from a support with no mass at the end of the spring has a length L (called its rest length). When a mass is added to
More informationMath Assignment 5
Math 2280  Assignment 5 Dylan Zwick Fall 2013 Section 3.41, 5, 18, 21 Section 3.51, 11, 23, 28, 35, 47, 56 Section 3.61, 2, 9, 17, 24 1 Section 3.4  Mechanical Vibrations 3.4.1  Determine the period
More informationSIR MICHELANGELO REFALO CENTRE FOR FURTHER STUDIES VICTORIA GOZO
SIR MICHELANGELO REFALO CENTRE FOR FURTHER STUDIES VICTORIA GOZO HalfYearly Exam 2013 Subject: Physics Level: Advanced Time: 3hrs Name: Course: Year: 1st This paper carries 200 marks which are 80% of
More informationTHEORY OF VIBRATION ISOLATION
CHAPTER 30 THEORY OF VIBRATION ISOLATION Charles E. Crede Jerome E. Ruzicka INTRODUCTION Vibration isolation concerns means to bring about a reduction in a vibratory effect. A vibration isolator in its
More informationIndex. Index. More information. in this web service Cambridge University Press
Atype elements, 4 7, 18, 31, 168, 198, 202, 219, 220, 222, 225 Atype variables. See Across variable ac current, 172, 251 ac induction motor, 251 Acceleration rotational, 30 translational, 16 Accumulator,
More informationChapter 16 Waves in One Dimension
Lecture Outline Chapter 16 Waves in One Dimension Slide 161 Chapter 16: Waves in One Dimension Chapter Goal: To study the kinematic and dynamics of wave motion, i.e., the transport of energy through a
More informationVibrations and Waves Physics Year 1. Handout 1: Course Details
Vibrations and Waves JanFeb 2011 Handout 1: Course Details Office Hours Vibrations and Waves Physics Year 1 Handout 1: Course Details Dr Carl Paterson (Blackett 621, carl.paterson@imperial.ac.uk Office
More information本教材僅供教學使用, 勿做其他用途, 以維護智慧財產權
本教材內容主要取自課本 Physics for Scientists and Engineers with Modern Physics 7th Edition. Jewett & Serway. 注意 本教材僅供教學使用, 勿做其他用途, 以維護智慧財產權 教材網址 : https://sites.google.com/site/ndhugp1 1 Chapter 15 Oscillatory Motion
More informationPhysics 2001/2051 The Compound Pendulum Experiment 4 and Helical Springs
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
More informationIntroduction to Finite Element Method. Dr. Aamer Haque
Introduction to Finite Element Method 4 th Order Beam Equation Dr. Aamer Haque http://math.iit.edu/~ahaque6 ahaque7@iit.edu Illinois Institute of Technology July 1, 009 Outline EulerBernoulli Beams Assumptions
More informationCHAPTER 4 TEST REVIEW
IB PHYSICS Name: Period: Date: # Marks: 74 Raw Score: IB Curve: DEVIL PHYSICS BADDEST CLASS ON CAMPUS CHAPTER 4 TEST REVIEW 1. In which of the following regions of the electromagnetic spectrum is radiation
More informationRigid Body Kinetics :: Force/Mass/Acc
Rigid Body Kinetics :: Force/Mass/Acc General Equations of Motion G is the mass center of the body Action Dynamic Response 1 Rigid Body Kinetics :: Force/Mass/Acc Fixed Axis Rotation All points in body
More informationPhysics 202 Homework 1
Physics 202 Homework Apr 3, 203. A person who weighs 670 newtons steps onto a spring scale in the bathroom, (a) 85 kn/m (b) 290 newtons and the spring compresses by 0.79 cm. (a) What is the spring constant?
More informationLab 10  Harmonic Motion and the Pendulum
Lab 10 Harmonic Motion and the Pendulum L101 Name Date Partners Lab 10  Harmonic Motion and the Pendulum L (measured from the suspension point to the center of mass) Groove marking the center of mass
More informationPREMED COURSE, 14/08/2015 OSCILLATIONS
PREMED COURSE, 14/08/2015 OSCILLATIONS PERIODIC MOTIONS Mechanical Metronom Laser Optical Bunjee jumping Electrical Astronomical Pulsar Biological ECG AC 50 Hz Another biological exampe PERIODIC MOTIONS
More informationHS AP Physics 1 Science
Scope And Sequence Timeframe Unit Instructional Topics 5 Day(s) 20 Day(s) 5 Day(s) Kinematics Course AP Physics 1 is an introductory firstyear, algebrabased, college level course for the student interested
More informationReview of Strain Energy Methods and Introduction to Stiffness Matrix Methods of Structural Analysis
uke University epartment of Civil and Environmental Engineering CEE 42L. Matrix Structural Analysis Henri P. Gavin Fall, 22 Review of Strain Energy Methods and Introduction to Stiffness Matrix Methods
More informationDynamic Loads CE 543. Examples. Harmonic Loads
CE 543 Structural Dynamics Introduction Dynamic Loads Dynamic loads are timevarying loads. (But timevarying loads may not require dynamic analysis.) Dynamics loads can be grouped in one of the following
More informationWheel and Axle. Author: Joseph Harrison. Research Ans Aerospace Engineering 1 Expert, Monash University
Wheel and Axle Author: Joseph Harrison British MiddleEast Center for studies & Research info@bmcsr.com http:// bmcsr.com Research Ans Aerospace Engineering 1 Expert, Monash University Introduction A solid
More informationOscillations. Simple Harmonic Motion (SHM) Position, Velocity, Acceleration SHM Forces SHM Energy Period of oscillation Damping and Resonance
Oscillations Simple Harmonic Motion (SHM) Position, Velocity, Acceleration SHM Forces SHM Energy Period of oscillation Damping and Resonance 1 Revision problem Please try problem #31 on page 480 A pendulum
More informationChapter 7 Hooke s Force law and Simple Harmonic Oscillations
Chapter 7 Hooke s Force law and Simple Harmonic Oscillations Hooke s Law An empirically derived relationship that approximately works for many materials over a limited range. Exactly true for a massless,
More informationSquarewave External Force in a Linear System
Squarewave External Force in a Linear System Manual Eugene Butikov Annotation. The manual includes a description of the simulated physical system and a summary of the relevant theoretical material for
More informationName: AP Homework 9.1. Simple Harmonic Motion. Date: Class Period:
AP Homework 9.1 Simple Harmonic Motion (1) If an object on a horizontal, frictionless surface is attached to a spring, displaced, and then released, it will oscillate. If it is displaced 0.120 m from its
More informationAAPT UNITED STATES PHYSICS TEAM AIP 2008
8 F = ma Exam AAPT UNITED STATES PHYSICS TEAM AIP 8 8 F = ma Contest 5 QUESTIONS  75 MINUTES INSTRUCTIONS DO NOT OPEN THIS TEST UNTIL YOU ARE TOLD TO BEGIN Use g = N/kg throughout this contest. You may
More informationExam 3 Review. Chapter 10: Elasticity and Oscillations A stress will deform a body and that body can be set into periodic oscillations.
Exam 3 Review Chapter 10: Elasticity and Oscillations stress will deform a body and that body can be set into periodic oscillations. Elastic Deformations of Solids Elastic objects return to their original
More information