Integrating Accelerometer Time History Signals

The scripts in integrate_th.zip integrate an acceleration time history to a velocity time history and/or a velocity time history to a displacement time history.

The main script is: integrate_th.m

The remaining scripts are supporting functions.

Integrating acceleration to velocity typically causes a spurious offset in the velocity signal, which in turn causes a “ski slope” effect in the resulting displacement signal.

So options are included for fading, trend removal, and mean filtering.  These options must be used with “engineering judgment.”

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Here is a script for differentiating a time history  differ.m

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- Tom Irvine

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SDOF Steady-State Response to a Sine Force or Base Excitation

These Matlab scripts calculate the steady-state response of a single-degree-of-freedom (SDOF) system to a sinusoidal force or base excitation:  steady.zip

- Tom Irvine

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Seismic Peak Ground Acceleration

The Iwate-Miyagi Nairiku earthquake struck northeast Honshu, Japan, on 14 June 2008.

This earthquake had a moment magnitude Mw 6.9 according to the USGS.

The peak ground acceleration (PGA) had a maximum vector sum (3 component) value of 4278 cm/sec^2 (4.36 G).

This is the highest ever recorded PGA, although other quakes have had higher moment magnitudes.  The Richter and moment magnitudes are a measure of the total energy released by a quake.

The PGA is measured at a point.  It depends on soil conditions, distance from the hypocenter, and other factors.

Reference:

Masumi Yamada et al (July/August 2010). “Spatially Dense Velocity Structure Exploration in the Source Region of the Iwate-Miyagi Nairiku Earthquake”. Seismological Research Letters v. 81; no. 4;. Seismological Society of America. pp. 597–604. Retrieved 21 March 2011.

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The highest PGA for earthquakes in the USA was 1.7 G for the 1994 Northridge, California quake, which had a 6.7 moment magnitude.

Reference:  Lin, Rong-Gong; Allen, Sam (26 February 2011). “New Zealand quake raises questions about L.A. buildings.” Los Angeles Times (Tribune). Retrieved 27 February 2011.

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The peak ground velocity (PGV) has a better correlation with structural damage according to some sources.

The largest recorded ground velocity from the 1994 Northridge earthquake, made at the Rinaldi Receiving station, reached 183 cm/sec (72 in/sec).

Reference:  USGS ShakeMap

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Further information is given at:  Vibrationdata Earthquake Engineering Page

- by Tom Irvine

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The Great Amplitude Format Debate

Shock and vibration test specifications for avionics and military equipment have almost always been specified in terms of acceleration. The main reason is that acceleration can easily be measured by accelerometers.

Velocity sensors are also available but are less common.

Gaberson, Chamblers et al, claim that pseudo velocity bests represents the damage potential of a shock or vibration event.  Pseudo velocity is the relative displacement multiplied by the natural frequency (rad/sec).  This assertion has merit.  I have written a paper on this subject at: sv_velocity.pdf

On the other hand, Steinberg gives empirical formulas for the fatigue potential for both shock and vibration for circuit boards in terms of relative displacement.  The formulas are given in Steinberg’s Book.

The shock response spectrum can be plotted in tripartite format, showing each of the three amplitude metrics as a function of natural frequency.  A good engineering practice is to review all three response parameters in this format for thoroughness.

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There are two pseudo velocity metrics.

Let omega be the natural frequency in (rad/sec)

The pseudo velocity shock spectrum (PVSS) is calculated by multiplying the relative displacement SRS value by omega.

The acceleration pseudo velocity shock spectrum (APVSS) is obtained by dividing each acceleration SRS value by omega.

Dr. Howard Gaberson has generated some examples which show that the PVSS and APVSS are nearly equal except at very low natural frequencies where the APVSS tends to be higher.

In addition, the true relative velocity can be calculated using the method in:  ramp_invariant_base.pdf

The two pseudo velocity metrics and the true relative velocity metric can be used somewhat loosely and interchangeably in regard to damage potential estimation.  Further experiments and research are needed to refine these concepts.

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See also:

Dr. Howard Gaberson’s Papers

SRS Tripartite

Stress-Velocity Relationship

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- Tom Irvine

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FAQ: Can Shock or Vibration be used to Cover Acceleration?

Introduction

Neither shock nor vibration response should be used to cover rigid-body acceleration because material limits depend on the strain rate and on the duration of the load.

Furthermore, static deflection shapes differ from dynamic mode shapes.

Material Stress Limits

The following is an excerpt from Reference 1 with some minor editing:

A material can sometimes sustain an important dynamic load without damage, whereas the same load, statically, would lead to plastic deformation or to failure.  Many materials subjected to short duration loads have ultimate strengths higher than those observed when they are static.

Hopkinson noted that copper and steel wire can withstand stresses that are higher than their static elastic limit and are well beyond the static ultimate limit without separating proportionality between the stresses and the strains.  This is provided that the length of time during which the stress exceeds the yield stress is of the order of 1 millisecond or less.

From tests carried out on steel (annealed steel with a low percentage of carbon) it was noted that the initiation of plastic deformation requires a definite time when stresses greater than the yield stress are applied.  It was observed that this time can vary between 5 milliseconds (under a stress of approximately 352 MPa) and 6 seconds with approximately 255 MPa; with the static yield stress being equal to 214 MPa).  Other tests carried out on five other materials showed that this delay exists only for materials for which the curve of static stress deformation presents a definite yield stress, and the plastic deformation then occurs for the load period.

The equivalent units are as follows

Table 1.  Annealed Steel Test Results
Parameter	Stress  (MPa)	Stress (ksi)
5 msec for plastic deformation onset	352	51.1
6 sec for plastic deformation onset	255	37.0
Static Yield Stress	214	31.1

Dynamic Strength

Reference 2 notes:

As far as steels and other metals are concerned, those with lower yield strength are usually more ductile than higher strength materials.  That is, high yield strength materials tend to be brittle.  Ductile (lower yield strength) materials are better able to withstand rapid dynamic loading than brittle (high yield strength) materials.  Interestingly, during repeated dynamic loadings low yield strength ductile materials tend to increase their yield strength, whereas high yield strength brittle materials tend to fracture and shatter under rapid loading.

Reference 2 includes the following table where the data was obtained for uniaxial testing using an impact method.

Dynamic Strengthening of Materials

Material	Static Strength (psi)	Dynamic Strength (psi)	Impact Speed (ft/sec)
2024 Al (annealed)	65,200	68,600	>200
Magnesium Alloy	43,800	51,400	>200
Annealed Copper	29,900	36,700	>200
302 Stainless Steel	93,300	110,800	>200
SAE 4140 Steel	134,800	151,000	175
SAE 4130 Steel	80,000	440,000	235
Brass	39,000	310,000	216

Shock vs. Acceleration

The following paragraph is taken from Reference 3.

Acceleration loads are expressed in terms of load factors which, although dimensionless, are usually labeled as “g” loads. Shock environments (methods 516.5 and 517) are also expressed in “g” terms. This sometimes leads to the mistaken assumption that acceleration requirements can be satisfied by shock tests or vice versa. Shock is a rapid motion that excites dynamic (resonant) response of the materiel but with very little overall deflection (stress). Shock test criteria and test methods cannot be substituted for acceleration criteria and test methods or vice versa.

References

1.  C. Lalanne, Sinusoidal Vibration (Mechanical Vibration and Shock), Taylor & Francis, New York, 1999.

2.  R. Huston and H. Josephs, Practical Stress Analysis in
Engineering Design, Dekker, CRC Press, 2008.  See Table 13.1.

3.  MIL-STD-810F, Method 513.5, Section 1.3.3 Acceleration versus shock

Further information is given at:  Vibrationdata Acceleration Page

- by Tom Irvine

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Some Earthquake Engineering Terminology

Response Spectrum

Response spectrum is a plot of the maximum responses (acceleration, velocity, or displacement) of idealized single-degree-of-freedom oscillators as a function of the natural frequencies of the oscillators for a given damping value. The response spectrum is calculated for a specified vibratory motion input at the oscillators’ supports.

Operational Basis Earthquake (OBE)

OBE is a ground motion with 10% probability of exceedance within 50 year period (475 years return period).  The facility is expected to remain operational after the OBE event without damage.  Reference:  NFPA 59A.

Safe Shutdown Earthquake (SSE)

SSE is a Maximum Considered Earthquake (MCE) ground motion with 2% probability of exceedance within 50 year period. Plastic behavior and significant finite movements and deformations are permissible. The facility is not required to remain operational after the SSE event.   Reference:  NFPA 59A.

SSE & OBE Relationship

SSE is the maximum potential earthquake of the selected site, and the OBE will be decided accordingly.

The OBE acceleration is one-half of the SSE value in some regulatory standards.

The OBE acceleration may be less than one-third the SSE value in other references.

Damping

The U.S. Nuclear Regulatory Commission, Regulatory Guide 1.61 gives different damping values for SSE & OBE analysis.  The damping values are higher for the SSE case.

Note that damping is non-linear for seismic response due to joint slipping and other factors.  Higher damping is expected for higher input excitation.

Further information is at:  Vibrationdata Earthquake Engineering Page

- Tom Irvine

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Modal Transient Analysis Numerical Engine

Modal Transient Analysis of a System Subjected to an Applied Force via a Ramp Invariant Digital Recursive Filtering Relationship: force_ramp_invariant.pdf

Reference Papers:

Smallwood

Irvine General Coordinate

Irvine Impulse Response Function 1

Irvine Impulse Response Function 2

Irvine SRS

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SDOF Matlab Script: arbit_force.m

MDOF Matlab Script: mdof_modal_arbit_force_ri.m

Matlab Supporting Functions:

ramp_invariant_filter_coefficients.m

enter_time_history.m

fix_size.m

Generalized_Eigen.m

mdof_plot.m

idof_plot.m

ODE_force_input.m

progressbar.m

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Python versions of the programs are being made available at:

Python Digital Recursive Filtering Page

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The same method can be applied to a multi-degree-of-freedom system with enforced motion on specified dof.

Note that the enforced motion method can also be used for base excitation if a seismic mass is inserted into the system model.  The seismic mass value may be arbitrary.

The method is given in the paper:  modal_enforced_motion_ramp_invariant.pdf

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Matlab script for enforced acceleration: mdof_modal_enforced_acceleration_ri.m

Matlab script for enforced displacement: mdof_modal_enforced_displacement_ri.m

Supporting functions:

ODE_acceleration_input.m

ODE_displacement_input.m

partition_matrices.m

(Some of the previously listed functions are also required.)

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See also:

Vibrationdata Modal Transient Matlab Page

Vibrationdata Modal Transient C/C++ Page

Enjoy,

Tom Irvine

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