## Stress Concentration Factor & Crack Initiation

### Gallery

This photo was taken outside of my former office in Huntsville, Alabama. As newly poured concrete hardens and dries, it shrinks. This is due to the evaporation of excess mixing water.  The cracks in the photo may have formed during this phase. … Continue reading

## United Airlines Flight 328

### Gallery

United Airlines Flight 328 was a scheduled domestic passenger flight from Denver to Honolulu, Hawaii on February 20, 2021. The Boeing 777-200 aircraft operating the route suffered engine failure shortly after takeoff The failure was “contained” but resulted in a … Continue reading

## k factor for one-sided normal tolerance limit

### Gallery

Matlab script: %% k_factor.m ver 1.0 by Tom Irvine%% k factor for one-sided normal tolerance limit%% p = probability percent% c = confidence percent% nsamples = number of samples%% k = tolerance factor% Zp = Z limit corresponding to probability … Continue reading

# Transforming Complex Mode Shapes into Real, Undamped Shapes

Modal test data processing methods perform curve-fitting on measured frequency response functions.  These are indirect methods for extracting natural frequencies and damping ratios.  The FRF methods also yield mode shapes which may be complex if the system has non-proportional damping.

Each complex mode shape can be multiplied by an independent complex scale factor to render it into an equivalent real, undamped form.  These transformed modes can then be compared to analytical modes from the generalized eigenvalue problem for the mass and stiffness matrices.

A method for doing this is the least-squares method given in:

Here is a Matlab code snippet for doing this for the case of a numerical simulation of a modal test with two-degrees-of-freedom.  The code should work for higher systems by making the appropriate substitutions.

The imaginary parts of the complex-valued modal coefficients are taken as the dependent variable and the real parts as the independent variable for the least squares calculation.

Note that a given mode’s imaginary coefficients may remain relatively high after the rotation process if the mode is a spurious computational one or if the coherence is poor at the corresponding frequency.

% CM = complex modes in column format
% RM = real modes (approximately)
%
% The least squares curve-fit uses the form: y = mx + b
%
% A(1) is the slope term m
%

%  disp(‘ Complex Modes ‘);

CM = [ 0.389-0.389i , 0.427+0.427i ;
0.59-0.591i , -0.564-0.563i ]

sz=size(CM);

num=sz(2);

RM=zeros(sz(1),sz(2));

for i=1:num

Y=imag(CM(:,i));
X(:,1)=real(CF(:,i));
X(:,2)=1;

XT=transpose(X);

A=pinv(XT*X)*(XT*Y);

theta=atan(A(1));

% Perform transformation via rotation

RM(:,i)=CM(:,i)*exp(-1i*theta);
end

%  disp(‘ Approximate Real Modes ‘);
RM

________________________________

– Tom Irvine

# Material Damping

Internal friction and heat generation cause a deviation from perfect elasticity, resulting in the hysteresis loop in the stress-strain curve.

The damping capacity  ψ is:

ψ =  ΔW/W

where W is the initial strain energy and ΔW is the energy lost during one cycle of oscillation.

Some damping capacities of common materials are:

There is some tendency for damping capacity to increase with at higher stresses. Damping also depends on temperature, strain rate and other parameters.

References:

Maringer, Damping Capacities of Materials

Luxfer, Magnesium Fixtures

–  Tom Irvine

# Shock & Vibration Test Fixture Design

Introduction

The best vibration fixtures are simple, rigid, lightweight with the minimum number of features.  Shaker and power amplifier systems are limited in terms of voltage, current and force pounds.  Lighter fixtures are need to achieve higher acceleration levels.

Materials

Aluminum alloy is a good choice for small to medium fixtures because it is cheaper than other materials such as magnesium.  Aluminum 6061-T6 & 7075-T6 are suitable alloys.

Magnesium is a better choice than aluminum in terms of weight to stiffness ratio and can provide higher damping.

Magnesium Zirconium K1A alloy in cast form has excellent hysteresis damping.

Magnesium Alloy AZ31B-Tool Plate also has very good damping.  It is a free machining material with good weldability properties.

Fixtures should have high damping in order to limit their resonant response.

Higher Damping also provides a transmissibility ratio relatively closer to unity gain across the entire frequency spectrum.

Joints

Some fixtures require joints. Bolted joints have more damping than welded ones. But welded joints have more rigidity if they are continuous.

Welded joints can be used with caution. They may have material distortion and high residual stress from thermal expansion and contraction during the welding process. After welding and before machining, the fixture should be left overnight in an oven at 250 °C to reduce warping and residual stress.

Welded joints may also have porosity, inclusions and microcracks. Some amount of microcracking is inevitable but needs to be managed via inspection and fracture calculations.

The quality of the weld is paramount. The weld joints should be continuous and smooth, with two or even three passes if possible.

References

Welded Joint Concerns for Shock, Vibration & Fatigue

Bruel & Kjaer Fixture Guidelines

– Tom Irvine

# Architectural Helmholtz Resonators

Tapiola Lutheran Church in Espoo, Finland, Opened in 1965

Helmholtz-type resonators are built into the walls of the church.  The slots between brick pairs and cavities behind them act as both absorbers and diffusors of sound.  The absorption reduces the reverberation time as desired for speech intelligibility during sermons.  Some reverberation is desirable, however, to enhance organ music.

A Helmholtz resonator is a volume of air which is enclosed in a container with at least one opening. It is also called a cavity resonator.  The air in the container’s neck acts as a mass. The air in the volume acts as a spring. The Helmholtz resonator thus behaves as a mechanical spring-mass system.

Courtroom in Bankstown, NSW, Australia, with slotted panels for reverberation reduction.  https://decorsystems.com.au/

Slotted block absorbers in a gymnasium.

The BT240 slotted modular bass trap panel provides excellent absorption down to 65 Hz.  https://soundacoustics.com.au/

– Tom Irvine

The area under each stress-strain curve represents the maximum energy that can be absorbed and is a measure of toughness.

Ductile materials undergo observable plastic deformation and absorb significant energy before fracture.  Brittle fracture is characterized by very low plastic deformation and low energy absorption prior to breaking.

Toughness is the ability of a material to absorb energy and plastically deform without fracturing. One definition of material toughness is the amount of energy per unit volume that a material can absorb before rupturing. This measure of toughness is different from that used for fracture toughness, which describes load bearing capabilities of materials with flaws.

Toughness requires a balance of strength and ductility. A material should withstand both high stresses and high strains in order to be tough. Generally, strength indicates how much force the material can support, while toughness indicates how much energy a material can absorb before rupturing.

Toughness (or, deformation energy, UT) is measured in units of joule per cubic metre (J·m−3) in the SI system and inch-pound-force per cubic inch (in·lbf·in−3) in US customary units.

_____________

Stress Strain Test Data Reference

As an example, consider that a plate made from A286 Stainless Steel must withstand shock loading defined by the following acceleration shock response spectrum (SRS) and its corresponding pseudo velocity response spectrum.  Note that A286 is a ductile material with excellent toughness.

The conversion formula is:    PV SRS (f) = scale*[Accel SRS (f) / (2 π f) ]

For English units:  scale = 386 in/sec^2 / G

The natural frequency of the plate is unknown.  So assume the peak pseudo velocity of 250 in/sec, which is the plateau.

The strain-velocity relationship for a plate is:

ε ~ 2 * ( PV / c ) = 2 * PV *sqrt( ρ / E )

c = sqrt ( E / ρ )

ε=strain
PV = pseudo velocity
c = speed of sound in the material
E = elastic modulus
ρ = mass/volume

The speed of sound in steel, aluminum and titanium is ~ 200,000 in/sec

The resulting strain for the plate is:

ε ~ 2 * ( PV / c )  ~ 2*(250 in/sec)/(  2e+05 in/sec) ~ 0.0025

This strain is approximately within the elastic range for the A286 stainless steel stress-strain curve.

Some caution is needed in applying this method too far into the plastic range due to the complexities of nonlinear behavior.  Further research and material testing is needed.

___________________

Notes:

This approach is a rough approximation given that the shock response spectra curves assumed a linear model.

The elastic modulus is considered to be constant during plastic deformation as modeled by the stain hardening effect.

The yield point for engineering materials in general is defined as defined as the amount of stress that will result in a plastic strain of 0.2%, as indicated in point 4 in the above figure.

The speed of sound in the material has a very small change due to both elastic and plastic deformation.   A number of papers have been written on the “acoustoelastic effect,” but they tend to theoretical.

____________________________________________________________________________________________

#### 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.

Annealed Steel Test Results
Parameter

Stress
(MPa)

Stress
(ksi)

Static Yield Stress

214

31.1

6 sec for plastic deformation onset

255

37.0

5 msec for plastic deformation onset

352

51.1

#### Dynamic Strengthening of Materials

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.

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

#### 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 23.1.

Stress Strain Velocity Relationship Paper

Stress Velocity Relationship

– Tom Irvine

# Golden Gate Bridge Singing

Some San Francisco Bay Area residents from Marin County to the Presidio have noticed a sustained, series of high-pitched tones.  The sound reached a new peak volume, and recordings of the eerie noise spread across social media in early June,.

The sound is due to high northwest winds blowing through the slats of the Golden Gate bridge’s newly-installed sidewalk railing.   The new slats were thinner than the ones in the previous railing.   The purpose of the new design is to make the span more aerodynamically stable on gusty days.

The sound is most likely an Aeolian tone, a noise produced when wind blows over a sharp edge, resulting in tiny harmonic vortices in the air.

The modification of the Golden Gate Bridge railing is the most recent and most audible element of a multi-phase retrofit that has been underway since 1997. Following the magnitude 6.9 Loma Prieta Earthquake in 1989, The Golden Gate Bridge, Highway, & Transportation District began to prepare the iconic bridge for the wind and earthquake loads that it may encounter in the future.  The design maximum wind speed is 100 mph.

A sample sound file of the Bridge’s tones is taken from:  https://youtu.be/8PnuOf33jN8

It was converted to mp3 format using an online utility.  Golden_Gate_singing.mp3

The mp3 file was then called into Matlab and processed using the Vibrationdata tools.

Spectral peaks occur at 354, 398, 439 & 481 Hz.  The spacing is nearly uniform with an average separation of 42.3 Hz.   The nearest musical notes are F, G, A & B, respectively.

The maximum individual peak occurs at 439 Hz. Higher frequency peaks occur at 880, 1051 & 1160 Hz.

Note that these frequencies should vary with wind speed per the Strouhal number.  The formation of the “vortex street” also depends on the Reynold’s number.

____

Vortex shedding behind a circular cylinder. In this animation, the flow on the two sides of the cylinder are shown in different colors, to show that the vortices from the two sides alternate. Courtesy, Cesareo de La Rosa Siqueira.

____

– Tom Irvine

# PSD Fatigue Damage Severity Criteria

PV = pseudo velocity

The following method is intended for components which are to be tested to power spectral density (PSD) specifications on shaker tables. The method can be applied to sine-on-random specifications as well.

The purpose of this document is to recommend velocity severity categories for PSD base inputs in terms of Stress-Velocity Relationship (SVR) fatigue damage.

The categories can then be used to plan the design and analysis efforts required to ensure that a component will pass its PSD test, or at least mitigate risk of failure with associated cost and schedule delays.

The method draws from Gaberson, Steinberg, Morse and Dirlik.

Slides:  psd_severity_revA.pptx

Matlab Scripts: Vibrationdata Signal Analysis Package

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