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Unit 21. Integration and Differentiation of Time Histories. Accelerometer. 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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Unit 21
Unit 21

Integration and Differentiation of Time Histories


Accelerometer
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


Velocity criteria
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 < 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)


Velocity sensor
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



Laser vibrometer
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


Scanning laser vibrometer
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


Displacement sensor
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


Old school analog method for measuring velocity displacement
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



Hyatt regency hotel phoenix arizona
Hyatt Regency Hotel, Phoenix, Arizona Displacement

Typical Elevator Recommended Limits

Fast elevator ride from ground floor to top restaurant!


Unit 21

Accelerometer Measurement Displacement

Integrated Velocity


Unit 21

Hyatt Regency Elevator Displacement

Accelerometer Measurement

Differentiated Jerk


Integration trapezoidal rule
Integration, Trapezoidal Rule Displacement

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.


Differentiate matlab function
Differentiate, Matlab Function Displacement

function[v]=differentiate_function(y,dt)

%

ddt=12.*dt;

%

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;

y = input amplitude

v = output amplitude

dt = time step


Sine example
Sine Example Displacement

Generate sine function:

Amp = 1

Dur = 10 sec

Freq = 1 Hz

Sample Rate = 40 Hz

(assume amp unit: G )

Save as: sine_accel


Integrate from acceleration to velocity
Integrate from Acceleration to Velocity Displacement

Baseline shift

Mean 61 in/sec

Vibrationdata > Time History > Integrate Input File: sine_accel

Trend Removal = None (prior & after) Output File: sine_vel


Integrate from velocity to displacement
Integrate from Velocity to Displacement Displacement

Ski Slope Effect!

Vibrationdata > Time History > Integrate Input File: sine_vel

Trend Removal = None (prior & after) Output File: sine_disp


Integrate from acceleration to velocity with mean removal
Integrate from Acceleration to Velocity Displacementwith Mean Removal

Symmetric Oscillation about zero baseline

Vibrationdata > Time History > Integrate Input File: sine_accel

Trend Removal Prior = None After = Mean Output File: sine_vel


Unit 21

Integrate from Velocity to Displacement Displacementwith Mean Removal

Stable oscillation about zero baseline

But with some distortion

Vibrationdata > Time History > Integrate Input File: sine_vel

Trend Removal Prior = None After = Mean Output File: sine_disp


Unit 21

Differentiate from Displacement to Velocity Displacement

Vibrationdata > Time History > Differentiate Input File: sine_disp

Output File: sine_vel


Review exercise sine amplitude
Review Exercise, Sine Amplitude Displacement

Agrees with integration & differentiation results on previous slides

Vibrationdata > Miscellaneous Functions > Steady-state Sine Amplitude


Launch vehicle separation test
Launch Vehicle Separation Test Displacement

Filename:

pyro_test.txt


Integrate from acceleration to velocity1
Integrate from Acceleration to Velocity Displacement

Vibrationdata > Time History > Integrate Input File: pyro_test.txt

Trend Removal = None (prior & after)


Integrate from acceleration to velocity with hp filtering
Integrate from Acceleration to Velocity Displacementwith HP Filtering

Vibrationdata > Time History > Integrate Input File: pyro_test.txt

Trend Removal Prior: Highpass filter at 30 Hz After: none


Recall psd synthesis
Recall PSD Synthesis Displacement


Psd synthesis review
PSD Synthesis Review Displacement

Generate acceleration white noise

Manipulate the time history via FFTs and inverse FFTs so that its satisfies the PSD specification

Integrate resulting acceleration time history to velocity

Integrate resulting velocity time history to displacement

Remove third-order polynomial trend from displacement

Apply tapering using half-cosine function to beginning and end of displacement

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.