Transcript No Slide Title

Vibrationdata
Unit 21
Integration and Differentiation of
Time Histories
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Accelerometer
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Mechanical vibration is usually characterized in terms of acceleration
•
The main reason is that acceleration is easier to measure than velocity or
displacement
•
Acceleration can be measured with a piezoelectric, piezoresistive or variable
capacitance accelerometer
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Velocity Criteria
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Hunt, Gaberson, Bateman, et al, have published papers showing that dynamic
stress is directly proportional to modal velocity (future webinar)
•
A peak velocity of 50 in/sec is sometimes considered as the shock severity
threshold for military components
•
Allowable building floor vibration limits are typically < 2.0 in/sec
•
Colin Gordon has established a generic vibration criteria for building floor
vibration in terms of velocity (see ISO Generic Vibration Criteria for VibrationSensitive Equipment)
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Velocity Sensor
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Velocity measurements require a Doppler laser or a geophone
•
The laser is expensive and requires a direct line of sight
•
The geophone is bulky and is intended for seismology measurements
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Geophone
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Laser Vibrometer
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Advantage
No mass loading effect from laser
on object.
Disadvantage
Laser system actually measures
relative velocity between laser
source and object, so laser source
must be kept still.
A single point laser vibrometer is used to
compare the vibration of two similar guitars
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Scanning Laser Vibrometer
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A Scanning Laser Vibrometer measurement
shows the velocity profile of a vibrating
turbine blade
•
The measurement grid has been tailored to
match the specific shape of the blade
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Displacement Sensor
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• Dynamic displacement
can be measured by a
linear variable
displacement transducer
(LVDT)
• The frequency response is
only suited for lowfrequency measurements
LVDTs used to measure traffic-induced vibration on underside of bridge
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Old School Analog Method for
Measuring Velocity & Displacement
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Measure vibration with charge mode
piezoelectric accelerometer
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Analog signal goes through Bruel &
Kjaer 2635 signal conditioner
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Select acceleration, velocity or
displacement output with this knob
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Analog integration & double
integration applied for velocity &
displacement, respectively
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Highpass filtering needed to prevent
spurious offsets, drifts, etc.
•
Minimum highpass filtering
frequencies:
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0.2 Hz for acceleration
1 Hz for velocity & displacement
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Typical Building Vibration Limits
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Transportation Research Board Building Maximum Structure Vibration Criteria
Limiting Peak Particle Velocity
Structure and Condition
(in/sec)
(cm/sec)
Historic buildings,
Certain other old buildings
0.5
~1.3
Residential structures
0.5
~1.3
New residential structures
1.0
~2.5
Industrial buildings
2.0
~5.1
Bridges
2.0
~5.1
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Hyatt Regency Hotel, Phoenix, Arizona
Typical Elevator
Recommended Limits
Parameter
Limit
acceleration/
deceleration
< 1.0 - 1.5 m/sec^2
Speed
< 7.0 m/sec
Jerk rates
< 2.5 m/sec^3
Sound
< 50 dBa
Ear-pressure
change
< 2000 Pa
Fast elevator ride from ground
floor to top restaurant!
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Accelerometer
Measurement
Integrated
Velocity
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Hyatt Regency
Elevator
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Accelerometer
Measurement
Differentiated
Jerk
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Integration, Trapezoidal Rule
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The integration of a time history is carried out on a “running sum”
basis.
Let the acceleration time history be represented by
a1, a2, a3, . . . , an.
The velocity time history is calculated as follows.
Δt
v1   a1
2
v2  v1  Δ t  a2 
v3  v2  Δ t  a3 
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Differentiate, Matlab Function
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function[v]=differentiate_function(y,dt)
%
ddt=12.*dt;
%
y = input amplitude
v = output amplitude
dt = time step
n=length(y);
%
v(1)=( -y(3)+4.*y(2)-3.*y(1) )/(2.*dt);
v(2)=( -y(4)+4.*y(3)-3.*y(2) )/(2.*dt);
v(3:(n-2))=(-y(5:n)+8*y(4:(n-1))-8*y(2:(n-3))+y(1:(n-4)))/ddt;
v(n-1)=( +y(n-1)-y(n-3) )/(2.*dt);
v(n) =( +y(n-1)-y(n-2) )/dt;
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Sine Example
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Generate sine
function:
Amp = 1
Dur = 10 sec
Freq = 1 Hz
Sample Rate = 40 Hz
(assume amp unit: G )
Save as: sine_accel.txt
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Integrate from Acceleration to Velocity to Displacement
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Time History Analysis > Integrate & Differentiate > Double Integrate
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Integrate from Acceleration to Velocity to Displacement
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Mean Velocity
= 61 in/sec
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Integrate from Acceleration to Velocity to Displacement with Mean Removal
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Integrate from Acceleration to Velocity to Displacement with Mean Removal
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Save displacement as: sine_disp.txt
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Differentiate from Displacement to Velocity
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Time History Analysis > Integrate & Differentiate > Differentiate
Input File: sine_disp.txt
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Review Exercise, Sine Amplitude
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Agrees with integration &
differentiation results on
previous slides
Miscellaneous > Sine Amplitude Conversion
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Double Integrate Pyrotechnic Shock Ground Test Data
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Filename: pyro_test.txt
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Experiment with Different Trend Removal Options
1st order trend removal used on all signal for this example
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