Integration and Differentiation of Time Histories 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

Geophone

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 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 New residential structures 1.0 ~2.5 Industrial buildings 2.0 ~5.1 Bridges

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!

Accelerometer Measurement Integrated Velocity

Hyatt Regency Elevator Accelerometer Measurement Differentiated Jerk

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.

Differentiate, Matlab Function 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 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 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 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 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 with 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 Vibrationdata > Time History > Differentiate Input File: sine_disp Output File: sine_vel

Review Exercise, Sine Amplitude Agrees with integration & differentiation results on previous slides Vibrationdata > Miscellaneous Functions > Steady-state Sine Amplitude

Launch Vehicle Separation Test Filename: pyro_test.txt

Integrate from Acceleration to Velocity Vibrationdata > Time History > Integrate Input File: pyro_test.txt Trend Removal = None (prior & after)

Integrate from Acceleration to Velocity with HP Filtering Vibrationdata > Time History > Integrate Input File: pyro_test.txt Trend Removal Prior: Highpass filter at 30 Hz After: none

Recall PSD Synthesis

PSD Synthesis Review 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.