MECE 102 Engineering Mechanics Lab A First Year Course in Newtonian Mechanics, Experimentation, and Computer Tools Created by the Faculty of the Mechanical Engineering Department in the Kate Gleason College of Engineering at RIT

Week 11 Lecture Damped Harmonic Motion This week we will study: The transient response of a pendulum system Problem A: Analyze Pendulum without Friction (undamped) Problem B: Analyze Pendulum with Friction (damped) Use LabVIEW to aquire data and Matlab to simulate system

iCLICKER: Which assumption is incorrect when analyzing this week’s system? Select your Answer: A.Heat Transfer (Q) = 0 B.Work (W) = 0 C.Mass of rod = 0 D.Mass of bob = 0 E.None of the above

iCLICKER: Which assumption is incorrect when analyzing this week’s system? Select your Answer: A.Heat Transfer (Q) = 0 B.Work (W) = 0 C.Mass of rod = 0 D.Mass of bob = 0 E.None of the above

FORMULATE: State the Known and Desired Information

FORMULATE: Identify Assumptions

Week 11 Lab Experiment

CHART: Schematic Diagram Schematic Diagram of pendulum system FBD of Bob mass:

EXECUTE: Newton’s 2 nd Law Analysis From the Free Body Diagram we can sum forces in the tangential (or arc length “s”) direction and set them equal to the mass times the acceleration. Note that in the radial direction the Tension force in the rod will be balanced by the radial component due to gravity.

Select your Answer: A.Newton’s 2 nd Law B.Conservation of Energy C.Small angle approximation D.Impulse / Momentum E.None of the above

Select your Answer: A.Newton’s 2 nd Law B.Conservation of Energy C.Small angle approximation D.Impulse / Momentum E.None of the above

EXECUTE: Small angle simplification

Execute: Derived Equations The solution of the Ordinary Differential Equation (ODE) is: Where the initial displacement of the pendulum measured along its arc length, s, is: The period of the harmonic motion of the simple frictionless pendulum with small initial angular displacement is: The period of the pendulum is independent of the mass of the bob!

CHART: Example Plot of Undamped Pendulum Motion

CHART: Schematic Diagram Schematic Diagram of pendulum system FBD of Bob mass with Friction: B

FORMULATE: State the Known and Desired Information

iCLICKER: The Friction force associated with the pendulum is related to Select your Answer: A.The magnitude of the displacement. B.The magnitude of the velocity. C.The magnitude of the acceleration. D.The mass of the rod.

iCLICKER: The Friction force associated with the pendulum is related to Select your Answer: A.The magnitude of the displacement. B.The magnitude of the velocity. C.The magnitude of the acceleration. D.The mass of the rod.

FORMULATE: Identify Assumptions

EXECUTE: Apply and Simplify the Governing Equations Use the Work Energy Theorem to develop an expression for the decay in the Total Energy of the system. We will use the AVERAGE friction force acting during one period of oscillation to estimate the work done by the system to overcome friction.

EXECUTE: Apply and Simplify the Governing Equations The TOTAL amount of mechanical energy in the system at t = 0 is given by:

EXECUTE: Apply and Simplify the Governing Equations The average value of Kinetic Energy and Potential Energy during one period of oscillation is half of the Total Energy at the beginning of the period WE can use the average Kinetic Energy during one period of oscillation to estimate the average speed:

EXECUTE: Apply and Simplify the Governing Equations We need to approximate the instantaneous Friction which is directly proportional to the direction of velocity.

EXECUTE: Apply and Simplify the Governing Equations Now using the assumption of negligible Heat Transfer allows us to simplify the Work Energy Theorem to:

EXECUTE: Apply and Simplify the Governing Equations Taking the limit as  t  0, the ordinary differential equation describing the energy of a simple harmonic oscillator with friction is given by:

EXECUTE: Apply and Simplify the Governing Equations We know that the Potential Energy of the pendulum is proportional to the maximum angle of displacement. If the Potential Energy decays exponentially, then it the follows that the maximum angle will also decay exponentially.

CHART: Combining Harmonic motion with Frictional Damping

CHART: Angular position of pendulum versus time Natural Frequency is given by: Damping Ratio is given by:

Homework No Lab Report due tonight! Prior to LAB tomorrow Read section 11.2 of the textbook Watch LAB Videos Complete the on-line LAB quiz in myCourses Attempt to solve all assigned Homework problems in your logbook before RECITATION. WEEK 11 Problem Set: From Section 11.5: Problems 1, 2, 6

CHART: Plot of Pendulum System response versus time