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 9 Lecture
Simple Harmonic Motion
• This week we will study:
• The transient response of a spring – mass system
• Use LabVIEW and the Ultrasonic transducer to acquire
data
• The relationships between
• Gravitational PE of the Mass
• Kinetic Energy of the Mass
• Elastic Potential Energy of the Spring
iCLICKER:
The week 9 Lab Experiment studies Simple
Harmonic Motion. Which one of the following
correctly identifies the system being analyzed?
• Select your Answer:
A.
B.
C.
D.
E.
Mass
Spring
Mass and Spring
Mass, Spring and Support structure
None of the above
iCLICKER:
The week 9 Lab Experiment studies Simple
Harmonic Motion. Which one of the following
correctly identifies the system being analyzed?
• Select your Answer:
A.
B.
C.
D.
E.
Mass
Spring
Mass and Spring
Mass, Spring and Support structure
None of the above
FORMULATE: State the Known Information
• Given the mass, m, of a dead weight suspended from a
linear elastic spring.
• Given the spring constant, K, of the spring, and the initial
displacement , z1, of the mass, m, from the neutral
position.
• Find the transient response of the vertical position, z(t), of
the mass, m, as a function of time, t.
FORMULATE: Identify the Desired Information
FORMULATE: Identify Assumptions
CHART: Schematic Diagram
Note the Neutral Position of
the spring, with no external
mass applied, is different than
the Equilibrium Position, z0,
associated with the spring and
mass.
Figure 9.1: Schematic Diagram of a mass suspended from a spring with a
transducer oriented to measure oscillation.
CHART: Schematic Diagram
Figure 9.1: Schematic Diagram of a mass suspended from a
spring with a transducer oriented to measure
oscillation (zoomed-in).
CHART: Free Body Diagram
Figure 9.2: Free Body Diagrams of a mass and spring,
constrained to move in tandem.
CHART: Data Table
• Each row in the STATE TABLE corresponds to one unique instant of
time for the spring – mass system.
CHART: Example Data Plot
iCLICKER:
Which of the following statements is FALSE?
• Select your Answer:
A. When the spring-mass system passes through the
equilibrium position, the KE of the mass is at its maximum.
B. The Mass changes direction at the minimum and maximum
peak values of elastic spring PE.
C. All of the Energy plots are in phase.
iCLICKER:
Which of the following statements is FALSE?
• Select your Answer:
A. When the spring-mass system passes through the
equilibrium position, the KE of the mass is at its maximum.
B. The Mass changes direction at the minimum and maximum
peak values of elastic spring PE.
C. All of the Energy plots are in phase.
EXECUTE: CHART: Example Data Plot
Period = T
Examples of in-phase and out-of-phase data
Example Problem
A 200 [gram] mass is suspended from a spring that has a spring
constant of 5 [N/m]. The mass is displaced slightly and released
to induce vertical motion.
Calculate the Period , T, of the oscillating spring-mass system
[sec].
Example Problem
A 200 [gram] mass is suspended from a spring that has a spring
constant of 5 [N/m]. The mass is displaced slightly and released
to induce vertical motion.
Calculate the Period , T, of the oscillating spring-mass system
[sec].
To calculate the Period of a simple spring-mass system:
𝑚
𝑇 = 2𝜋
𝑘
Example Problem
A 200 [gram] mass is suspended from a spring that has a spring
constant of 5 [N/m]. The mass is displaced slightly and released
to induce vertical motion.
Calculate the Period , T, of the oscillating spring-mass system
[sec].
To calculate the Period of a simple spring-mass system:
𝑇 = 2𝜋
𝑚
𝑘
= 2
0.2 𝑘𝑔
5 𝑁/𝑚
= 2
0.2 𝑘𝑔
𝑚
𝑠
5 (𝑘𝑔· 2 )/𝑚
T = 1.26 sec
Make sure you are using CONSISTENT SI UNITS!!
Can you think of any practical
everyday situations where
“out of phase” signals are useful?
Can you think of any practical
everyday situations where
“out of phase” signals are useful?
•
Noise-cancelling Headphones
Can you think of any practical
everyday situations where
“out of phase” signals are useful?
•
•
Noise-cancelling Headphones
Earthquake “resistant” structures
Can you think of any practical
everyday situations where
“out of phase” signals are useful?
•
•
•
Noise-cancelling Headphones
Earthquake “resistant” structures
Spoons!
http://news.cnet.com/8301-11386_3-57599437-76/smart-spoon-helpsstabilize-parkinsons-tremors/
EXECUTE: Recall the Governing Equations
EXECUTE: Apply and Simplify the Governing Equations
EXECUTE: Apply and Simplify the Governing Equations
Also recall the initial condition of the mass – spring system:
position
EXECUTE: Apply and Simplify the Governing Equations
Eqn. 9.28 is not a standard Algebraic equation.
What “type” of equation is equation 9.28?
EXECUTE: Apply and Simplify the Governing Equations
Eqn. 9.28 is not a standard Algebraic equation.
Eqn. 9.28 is known as a non-homogenous linear ordinary differential
equation.
EXECUTE: Apply and Simplify the Governing Equations
Eqn. 9.28 is not a standard Algebraic equation.
Eqn. 9.28 is known as a non-homogenous linear ordinary differential
equation.
To approximate solution to 9.28:
- Analyze raw data from Lab experiment  Studio
- Numerical simulation of system  Studio
- Analytical Mathematical solution  “Ordinary Diff Eq” course
EXECUTE: Iterate
• Multiple trials will be conducted in LAB.
• Each group member will conduct their own, unique trial.
• Test parameters that should be varied include:
1.
2.
3.
Amount of Mass hanging from spring
Unique initial displacement values, z1
Conduct at least one trial with z1 > 0 and at least one trial with z1 < 0
Week 9 Lab Experiment
Homework
• Prior to LAB tomorrow
• Read section 9.2 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 9 Problem Set:
• From Section 9.5: Problems 2, 3, 4, 7
• Lab Report is due Monday by 6pm!
• The Scribe must upload the LAB REPORT to the myCourses Dropbox.