Integration and Differentiation of Time Histories

1
Unit 21

• Integration and Differentiation of Time
  Histories

2
Accelerometer

• 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

3
Velocity Criteria

• 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 lt 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
  Vibration-Sensitive Equipment)

4
Velocity Sensor

• 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

5
Geophone
6
Laser Vibrometer
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
7
Scanning Laser Vibrometer

• 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

8
Displacement Sensor

• Dynamic displacement can be measured by a linear
  variable displacement transducer (LVDT)
• The frequency response is only suited for
  low-frequency measurements

LVDTs used to measure traffic-induced vibration
on underside of bridge
9
Old School Analog Method for Measuring Velocity
Displacement

• Measure vibration with charge mode piezoelectric
  accelerometer
• Analog signal goes through Bruel Kjaer 2635
  signal conditioner
• Select acceleration, velocity or displacement
  output with this knob
• Analog integration double integration applied
  for velocity displacement, respectively
• Highpass filtering needed to prevent spurious
  offsets, drifts, etc.
• Minimum highpass filtering frequencies
• 0.2 Hz for acceleration
• 1 Hz for velocity displacement

10
Typical Building Vibration Limits
Transportation Research Board Building Maximum Structure Vibration Criteria Transportation Research Board Building Maximum Structure Vibration Criteria Transportation Research Board Building Maximum Structure Vibration Criteria
Limiting Peak Particle Velocity 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
11
Hyatt Regency Hotel, Phoenix, Arizona
Typical Elevator Recommended Limits
Parameter Limit
acceleration/ deceleration lt 1.0 - 1.5 m/sec2
Speed lt 7.0 m/sec
Jerk rates lt 2.5 m/sec3
Sound lt 50 dBa
Ear-pressure change lt 2000 Pa
Fast elevator ride from ground floor to top
restaurant!
12
Accelerometer Measurement
Integrated Velocity
13
Hyatt Regency Elevator Accelerometer Measurement
Differentiated Jerk
14
Integration, Trapezoidal Rule
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.        
15
Differentiate, Matlab Function
functionvdifferentiate_function(y,dt) ddt12.
dt nlength(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(5n)8y(4(n-1))-8y(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
y input amplitude v output amplitude dt
time step
16
Sine Example
Generate sine function Amp 1 Dur 10
sec Freq 1 Hz Sample Rate 40 Hz (assume amp
unit G ) Save as sine_accel
17
Integrate from Acceleration to Velocity
Baseline shift Mean 61 in/sec
Vibrationdata gt Time History gt Integrate
Input File sine_accel Trend Removal None
(prior after) Output File
sine_vel
18
Integrate from Velocity to Displacement
Ski Slope Effect!
Vibrationdata gt Time History gt Integrate
Input File sine_vel Trend Removal None
(prior after) Output File
sine_disp
19
Integrate from Acceleration to Velocitywith Mean
Removal
Symmetric Oscillation about zero baseline
Vibrationdata gt Time History gt Integrate
Input File sine_accel
Trend Removal Prior None After Mean
Output File sine_vel
20
Integrate from Velocity to Displacementwith Mean
Removal
Stable oscillation about zero baseline But with
some distortion
Vibrationdata gt Time History gt Integrate
Input File sine_vel Trend Removal
Prior None After Mean Output
File sine_disp
21
Differentiate from Displacement to Velocity
Vibrationdata gt Time History gt Differentiate
Input File sine_disp


Output File sine_vel
22
Review Exercise, Sine Amplitude
Agrees with integration differentiation results
on previous slides
Vibrationdata gt Miscellaneous Functions gt
Steady-state Sine Amplitude
23
Launch Vehicle Separation Test
Filename pyro_test.txt
24
Integrate from Acceleration to Velocity
Vibrationdata gt Time History gt Integrate
Input File pyro_test.txt Trend Removal
None (prior after)
25
Integrate from Acceleration to Velocity with HP
Filtering
Vibrationdata gt Time History gt Integrate
Input File pyro_test.txt
Trend Removal Prior Highpass filter at 30 Hz
After none
26
Recall PSD Synthesis
27
PSD Synthesis Review

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.