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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
- Acceleration can be measured with a piezoelectric, piezoresistive or variable capacitance accelerometer

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 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

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

- 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

- 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

- 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

Typical Building Vibration Limits Displacement

Hyatt Regency Hotel, Phoenix, Arizona Displacement

Typical Elevator Recommended Limits

Fast elevator ride from ground floor to top restaurant!

Accelerometer Measurement Displacement

Integrated Velocity

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 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 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 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 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 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

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

Differentiate from Displacement to Velocity Displacement

Vibrationdata > Time History > Differentiate Input File: sine_disp

Output File: sine_vel

Review Exercise, Sine Amplitude Displacement

Agrees with integration & differentiation results on previous slides

Vibrationdata > Miscellaneous Functions > Steady-state Sine Amplitude

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 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 Displacement

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.

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