Transcript webinar_natural_frequencies

Dynamic Concepts, Inc.
Huntsville, Alabama
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THE NASA ENGINEERING & SAFETY CENTER (NESC)
SHOCK & VIBRATION TRAINING PROGRAM
By Tom Irvine
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Dr. Curtis Larsen
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Dr. Curtis E. Larsen is the NASA Technical Fellow for Loads
and Dynamics
He is the head of the the NASA Engineering & Safety Center
(NESC) Loads & Dynamics Technical Disciplines Team (TDT)
Thank you to Dr. Larsen for supporting this webinars!
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NASA ENGINEERING & SAFETY CENTER (NESC)
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• NESC is an independently funded program with a dedicated
team of technical experts
• NESC was Formed in 2003 in response to the Space Shuttle
Columbia Accident Investigation
• NESC’s fundamental purpose is provide to objective
engineering and safety assessments of critical, high-risk
NASA projects to ensure safety and mission success
• The National Aeronautics and Space Act of 1958
• NESC is expanding its services to benefit United States:
Military
Government Agencies
Commercial Space
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NESC Services
• NESC Engineers Provide a Second Pair of Eyes
• Design and Analysis Reviews
• Test Support
• Flight Accelerometer Data Analysis
• Tutorial Papers
• Perform Research as Needed
• NESC Academy, Educational Outreach
http://www.nasa.gov/offices/nesc/academy/
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Preliminary Instructions
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• You may ask questions during the presentation
• Otherwise set your phones to mute
• These presentations including your questions and comments are being recorded
for redistribution
• If you are not already on my distribution list, please send and Email to:
[email protected]
• You may also contact me via Email for off-line questions
• Please visit:
http://vibrationdata.wordpress.com/
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Unit 1A
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Natural Frequencies:
Calculation, Measurement, and Excitation
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Measuring Frequency
TUNING FORK
SOUND PRESSURE
0.4
A note
44 peaks / 0.1 seconds = 440 Hz
0.2
0
-0.2
-0.4
0
0.02
0.06
0.04
0.08
0.10
TIME (SEC)
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Basic Definitions
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Natural Frequency
The natural frequency is the frequency at which a mass will vibrate if it is given
an initial displacement and then released so that it may vibrate freely.
This free vibration is also called "simple harmonic motion, " assuming no
damping.
An object has both mass and stiffness.
The spring stiffness will try to snap the object back to its rest position if the
object is given an initial displacement. The inertial effect of the mass, however,
will not allow the object to stop immediately at the rest position. Thus, the
object “overshoots” its mark.
The mass and stiffness forces balance out to provide the natural frequency.
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Basic Definitions (continued)
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Damping
Consider a mass that is vibrating freely. The mass will eventually return to
its rest position. This decay is referred to as "damping.“
Damping may be due to
viscous sources
dry friction
aerodynamic drag
acoustic radiation
air pumping at joints
boundary damping
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Basic Definitions (continued)
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Single-degree-of-freedom System (SDOF)
A single-degree-of-freedom system is a system which only has one natural
frequency. Engineers often idealize complex systems as single-degree-offreedom systems.
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Multi-degree-of-freedom System (MDOF)
A multi-degree-of-freedom system is a system which has more than one natural
frequency.
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Earth
EARTH'S NATURAL FREQUENCY
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The Earth experiences seismic
vibration.
The fundamental natural frequency
of the Earth is 309.286 micro Hertz.
This is equivalent to a period of
3233.25 seconds, or approximately
54 minutes.
Reference: T. Lay and T. Wallace,
Modern Global Seismology,
Academic Press, New York, 1995.
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Golden Gate Bridge
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Steel Suspension Bridge
Total Length = 8980 ft
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Golden Gate Bridge
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In addition to traffic loading, the Golden Gate Bridge must withstand the
following environments:
1. Earthquakes, primarily originating on the San Andreas and Hayward
faults
2. Winds of up to 70 miles per hour
3. Strong ocean currents
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The Golden Gate Bridge has performed well in all earthquakes to date,
including the 1989 Loma Prieta Earthquake. Several phases of seismic
retrofitting have been performed since the initial construction.
Note that current Caltrans standards require bridges to withstand an
equivalent static earthquake force (EQ) of 2.0 G.
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Golden Gate Bridge Natural Frequencies
Mode Type
Period of
vibration
(sec)
Natural
Frequency
(Hz)
Transverse
Vertical
Longitudinal
Torsional
18.2
10.9
3.81
4.43
0.055
0.092
0.262
0.226
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SDOF System Examples - Pendulum
g
L

m
m = mass
L = length
g = gravity
 = angular displacement
The natural frequency ωn for a pendulum is
ωn 
g
L
The natural frequency has dimensions of radians/time. The
typical unit is radians/second.
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SDOF System
Spring-Mass System
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X
m
k
m = mass
k = spring stiffness
c = damping coefficient
X = displacement
c
The natural frequency for a spring-mass system is
ωn 
k
m
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SDOF System Examples
Cantilever Beam with End Mass
EI, 
m
L
E is the modulus of elasticity
I is the area moment of inertia
L is the length
fn  1
2π

m
is the beam mass per length
is the end mass
3 EI
0.2235 ρL  m  L3
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Circuit Board Natural Frequencies
Circuit Boards are often Modeled as Single-degree-of-freedom Systems
Component
CEP
PSSL
MUX
PDU
PCM Encoder
TVC
Fundamental
Frequency (Hz)
65
210
220
225
395
580
Average = 328 Hz
Std Dev = 203 Hz
Range = 65 Hz to 600 Hz
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More Formulas
ωn  2 π fn
fn  ωn
2π
The variable is the natural frequency in cycles/time. The typical unit
is cycles/second, which is called Hertz. The unit Hertz is abbreviated
as Hz.
Note that the period T is the period is the time required for one
complete cycle of oscillation
T 1
fn
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Recommended Text
Dave S. Steinberg
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SDOF System
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M = 0.71 kg
K = 350 N/mm
fn = 111.7 Hz
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SDOF Animation. File: sdof_fna.avi
(click on image)
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fn = 111.7 Hz
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Two DOF System
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M2 = 0.71 kg
K2 = 175 N/mm
M1 = 0.71 kg
K1 = 350 N/mm
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Two DOF System Animation
Files: tdofm1.avi & tdofm2.avi
(click on images)
Mode 1
f1 = 60.4 Hz
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Mode 2
f2 = 146 Hz
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Astronaut
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Spring-loaded chair device for measuring astronaut's mass
The chair oscillates at a natural frequency which is dependent on the astronaut's
mass.
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Resonance
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Resonance occurs when the applied force or base excitation frequency coincides
with the system's natural frequency.
As an example, a bulkhead natural frequency might be excited by a motor
pressure oscillation.
During resonant vibration, the response displacement may increase until the
structure experiences buckling, yielding, fatigue, or some other failure
mechanism.
The Tacoma Narrows Bridge failure is often cited as an example of resonant
vibration. In reality, it was a case of self-excited vibration.
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Excitation Methods
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There are four methods by which a structure's natural frequency may be excited:
1. Applied Pressure or Force
Hammer strikes mass
Modal Test
Bat hits baseball, exciting bat’s natural frequencies
Airflow or wind excites structure such as an aircraft
wing
Ocean waves excite offshore structure
Rotating mass imbalance in motor
Pressure oscillation in rocket motor
2. Base Excitation
Vehicle traveling down washboard road
Earthquake excites building
A machine tool or optical microscope is excited by floor excitation
Shaker Table Test
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Excitation Methods (Continued)
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3. Self-excited Instability
Airfoil or Bridge Flutter
4. Initial Displacement or Velocity
Plucking guitar string
Pegasus drop transient
Accidental drop of object onto floor
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Base Excitation
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Courtesy of UCSB and R. Kruback
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1989 Loma Prieta Earthquake
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LOMA PRIETA EARTHQUAKE (continued)
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The earthquake caused the Cypress Viaduct to collapse, resulting in 42 deaths. The
Viaduct was a raised freeway which was part of the Nimitz freeway in Oakland, which
is Interstate 880. The Viaduct had two traffic decks.
Resonant vibration caused 50 of the 124 spans of the Viaduct to collapse. The
reinforced concrete frames of those spans were mounted on weak soil. As a result,
the natural frequency of those spans coincided with the forcing frequency of the
earthquake ground motion. The Viaduct structure thus amplified the ground motion.
The spans suffered increasing vertical motion. Cracks formed in the support frames.
Finally, the upper roadway collapsed, slamming down on the lower road.
The remaining spans which were mounted on firm soil withstood the earthquake.
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Pegasus Vehicle
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Pegasus
Drop Video
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(click on image)
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Pegasus
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Pegasus Drop Transient
Fundamental Bending Mode
PEGASUS REX2 S3-5 PAYLOAD INTERFACE Z-AXIS
5 TO 15 Hz BP FILTERED
2.5
y=1.55*exp(-0.64*(x-0.195))
Flight Data
2.0
1.5
ACCEL (G)
1.0
0.5
0
-0.5
-1.0
fn = 9.9 Hz
damp = 1.0%
-1.5
-2.0
-2.5
0
0.5
1.0
1.5
2.0
TIME (SEC)
2.5
3.0
3.5
4.0
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Boeing 747
Wind Tunnel Test
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Boeing 747 – Flutter_747.avi
(click on image)
Flutter – combined bending and torsional motion.
(Courtesy of Smithsonian Air & Space. Used with permission.)
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More Flutter Videos
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(Courtesy of Smithsonian Air & Space. Used with permission.)
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Tacoma
Narrows Bridge
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Torsional Mode at 0.2 Hz - Aerodynamic Self-excitation
Wind Speed = 42 miles per hour. Amplitude = 28 feet
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Tacoma
Narrows Bridge Failure
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November 7, 1940
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Helicopter Ground Resonance
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f(t)
x
m
m
k
c
A new design undergoing testing may encounter severe vibration while it is on the
ground, preparing for takeoff.
As the rotor accelerates to its full operating speed, a structural natural frequency of
the helicopter may be excited.
This condition is called resonant excitation.
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TH-55 Osage, Military Version of the
Hughes 269A
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Guidance Systems
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Consider a rocket vehicle with a closed-loop guidance system.
The autopilot has an internal navigation system which uses accelerometers and
gyroscopes to determine the vehicle's attitude and direction.
The navigation system then sends commands to actuators which rotate the
exhaust nozzle to steer the vehicle during its powered flight.
Feedback sensors measure the position of the nozzle. The data is sent back to
the navigation computer.
Unfortunately, the feedback sensors, accelerometers, and gyroscopes could be
affected by the vehicle's vibration. Specifically, instability could result if the
vibration frequency coincides with the control frequency.
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SHOCK PULSE
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Response Spectra Concept
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Soft
Hard
Natural Frequencies (Hz):
0.063
0.125
0.25
0.50
1.0
2.0
4.0
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Unit 1A Exercise 1
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
A particular circuit board can be modeled as a single-degree-of-freedom system.

Its weight is 0.1 pounds.
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Its stiffness is 400 pounds per inch.

Calculate the natural frequency using Matlab script:
vibrationdata > miscellaneous functions > Structural Dynamics >
SDOF System Natural Frequency
Script is posted at:
http://vibrationdata.wordpress.com/2013/05/29/vibrationdata-matlab-signal-analysis-package/
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Unit 1A Exercise 2
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A rocket vehicle is carried underneath an aircraft. It experiences an initial
displacement because gravity causes it to bow downward while it is attached to the
aircraft. It is suddenly released and allowed to vibrate freely as it falls. It continues
falling for about 5 seconds prior to its motor ignition, as a safety precaution.

An acceleration time history of the drop is given in file: drop.txt.
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Plot using script: vibrationdata > Statistics

Determine the natural frequency by counting the peaks and dividing the sum by time.

Estimate damping using script: sinefdam.m
http://vibrationdata.wordpress.com/2013/04/26/curve-fitting-one-or-more-sine-functions/
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Unit 1A Exercise 3
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A flagpole is made from steel pipe.
The height is 180 inches.
The pipe O.D. is 3 inches.
The wall thickness is 0.25 inches.
The boundary conditions are fixed-free.
Determine the fundamental lateral frequency.
Use script: vibrationdata > miscellaneous functions > Structural Dynamics >
Beam Natural Frequency & Base Excitation Response
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First Three Modes of Flagpole
Mode 1
Mode 2
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Mode 3
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Unit 1A Exercise 4 Tuning Fork
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Determine the natural frequency of the tuning fork.
The file is: tuning_fork.txt
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