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.
B.
C.
D.
E.
Heat Transfer (Q) = 0
Work (W) = 0
Mass of rod = 0
Mass of bob = 0
None of the above
iCLICKER:
Which assumption is incorrect when analyzing this
week’s system?
• Select your Answer:
A.
B.
C.
D.
E.
Heat Transfer (Q) = 0
Work (W) = 0
Mass of rod = 0
Mass of bob = 0
None of the above
FORMULATE: State the Known and Desired Information
FORMULATE: Identify Assumptions
Week 11 Lab Experiment
CHART: Schematic Diagram
FBD of Bob mass:
𝑟
𝜽
𝑠
𝜽
𝜽
Schematic Diagram of pendulum system
EXECUTE: Newton’s 2nd 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.
iCLICKER:
The relationship sin 𝜃 ≈ 𝜃 is based on
• Select your Answer:
A.
B.
C.
D.
E.
Newton’s 2nd Law
Conservation of Energy
Small angle approximation
Impulse / Momentum
None of the above
iCLICKER:
The relationship sin 𝜃 ≈ 𝜃 is based on
• Select your Answer:
A.
B.
C.
D.
E.
Newton’s 2nd Law
Conservation of Energy
Small angle approximation
Impulse / Momentum
None of the above
EXECUTE: Small angle simplification
Execute: Derived Equations
The solution of the Ordinary Differential Equation (ODE) 11.18 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
B
𝑟
FBD of Bob mass
with Friction:
𝜽
𝑠
𝜽
𝜽
Schematic Diagram of pendulum system
FORMULATE: State the Known and Desired Information
iCLICKER:
The Friction force associated with the pendulum is
related to
• Select your Answer:
A.
B.
C.
D.
The magnitude of the displacement.
The magnitude of the velocity.
The magnitude of the acceleration.
The mass of the rod.
iCLICKER:
The Friction force associated with the pendulum is
related to
• Select your Answer:
A.
B.
C.
D.
The magnitude of the displacement.
The magnitude of the velocity.
The magnitude of the acceleration.
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 Dt  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