## West Seattle Bridge

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The West Seattle Bridge was built between 1981 and 1984 after the previous bridge was deemed inoperable as a result of being struck by the freighter Antonio Chavez in 1978. The bridge spans the east and west channels that form … Continue reading

## Relative Fatigue Damage Statistics for an SDOF System Subjected to Base Excitation

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This is a work-in-progress. More to come later… SDOF Response to White Noise Base Input Rainflow Statistical Study: white_noise_sdof_rainflow_study.pptx See also: Rainflow Cycle Counting Tom Irvine

## Interstate 40 Bridge Crack

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A crack in a steel beam forced the closure of the Interstate 40 Hernando DeSoto bridge that connects Arkansas and Tennessee over the Mississippi River at Memphis.   A critical beam was fractured to the point of being nearly severed. Photos … Continue reading

## Stress Concentration Factor & Crack Initiation

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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

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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

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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

## Another Method for Pyrotechnic-like Time History Synthesis for an SRS Specification

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I have added an exponential decay option to the wavelet synthesis function. This arranges the wavelets so that they will “resemble” a pyrotechnic-like time history. The function is included in the GUI package at: Matlab Link Note that the conservative … 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