Transcript Unit_21_integrate_differentiate

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
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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)
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A peak velocity of 50 in/sec is sometimes considered as the shock severity
threshold for military components
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Allowable building floor vibration limits are typically < 2.0 in/sec
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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
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The laser is expensive and requires a direct line of sight
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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
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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.
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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;
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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
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Integrate from Acceleration to Velocity
Baseline shift
Mean 61 in/sec
Vibrationdata > Time History > Integrate
Trend Removal = None (prior & after)
Input File: sine_accel
Output File: sine_vel
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Integrate from Velocity to Displacement
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Ski Slope Effect!
Vibrationdata > Time History > Integrate
Trend Removal = None (prior & after)
Input File: sine_vel
Output File: sine_disp
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Integrate from Acceleration to Velocity
with Mean Removal
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Symmetric Oscillation
about zero baseline
Vibrationdata > Time History > Integrate
Trend Removal Prior = None After = Mean
Input File: sine_accel
Output File: sine_vel
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Integrate from Velocity to Displacement
with Mean Removal
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Stable oscillation about
zero baseline
But with some distortion
Vibrationdata > Time History > Integrate
Trend Removal Prior = None After = Mean
Input File: sine_vel
Output File: sine_disp
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Differentiate from Displacement to Velocity
Vibrationdata > Time History > Differentiate
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Input File: sine_disp
Output File: sine_vel
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Review Exercise, Sine Amplitude
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Agrees with integration &
differentiation results on
previous slides
Vibrationdata > Miscellaneous Functions > Steady-state Sine Amplitude
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Launch Vehicle Separation Test
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Filename:
pyro_test.txt
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Integrate from Acceleration to Velocity
Vibrationdata > Time History > Integrate
Trend Removal = None (prior & after)
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Input File: pyro_test.txt
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Integrate from Acceleration to Velocity
with HP Filtering
Vibrationdata > Time History > Integrate
Trend Removal Prior: Highpass filter at 30 Hz
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Input File: pyro_test.txt
After: none
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Recall PSD Synthesis
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PSD Synthesis Review
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1. Generate acceleration white noise
2. Manipulate the time history via FFTs and inverse FFTs so that its
satisfies the PSD specification
3. Integrate resulting acceleration time history to velocity
4. Integrate resulting velocity time history to displacement
5. Remove third-order polynomial trend from displacement
6. Apply tapering using half-cosine function to beginning and end of
displacement
7. Differentiate displacement to velocity and again to acceleration
Steps 3 through 7 allow the set of acceleration, velocity and displacement
time histories to each have zero mean values.
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