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# Quasi Static Analysis

## Quasi Static Analysis

(OP)
Hi,

I have been reading through some of the threads here and technical papers on quasi static analysis and am trying to understand the procedure for definition.

If i understand it correctly, say I have a base excitation of 2g 11ms 1/2 sine. So, Fp = 1 /2(0.011) = 45.45Hz

So, if i now run a modal analysis of my structure and the first mode is above 250Hz (ie 5x base frequency), is it valid then to simply apply the 2g as a linear acceleration without any amplifaction factor?

The 5:1 ratio is taken from Harris Shock & Vibration Handbook  which states

"Any dynamic excitation at a frequency less than about 20 percent of the lowest normal mode (natural) frequency
of the equipment can be considered quasi-static"

Tom

### RE: Quasi Static Analysis

I could see that the procedure makes sense [b]in steady state sinusoidal excitation[/b[ far below resonances.  Then the system is "spring-controlled" and the masses make very little difference.

A half-sin pulse has more frequencies than just the sinusoid from which is carved.   It can be composed in the time domain as the multiplication of a rectangular pulse and a sinusoid.  So in the frequency domain, it is represented as a sinc function (the fourier transform of square pulse)  shifted left and right (representing convolution with the +f1 and -f1 frequencies of the sinusoid).  So the harmonics continue all the way to

That makes it a little tricker.  The error from neglecting dynamic/mass effects for the higher frequencies is bigger since some of them will be closer to resonance (and certainly some above resonance).

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### RE: Quasi Static Analysis

Whoops, forgot to finish my sentence (where's the edit button). "So the harmonics continue all the way to a frequency of infinity. although reducing in magnitude as frequency increases."

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### RE: Quasi Static Analysis

I had some errors in my post.  The masses are of course not irrelevant for quasi-static analysis of base excitation (I was thinking of a SDOF with force applied to the mass, not acceleration applied to the base).  But the comment about frequency content remains.

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### RE: Quasi Static Analysis

If your mode is above 250 Hz then the excitation could still exceed -14dB of the 0 Hz component, at around 270 Hz.

Cheers

Greg Locock

SIG:Please see FAQ731-376: Eng-Tips.com Forum Policies for tips on how to make the best use of Eng-Tips.

### RE: Quasi Static Analysis

Wel, twice that actually. I get -63 dB re 1g rms at 270 Hz

Cheers

Greg Locock

SIG:Please see FAQ731-376: Eng-Tips.com Forum Policies for tips on how to make the best use of Eng-Tips.

### RE: Quasi Static Analysis

(OP)

I saw a reference on Google books in de silvas shock handbook and I quote:

"If the longest natural period corresponding to the ?rst natural frequency of the structure is more than about twice the rise time, the loading should be classi?ed as shock or impact loading and transient dynamic analysis would be required. If the longest natural period of a system is less than about one third of the rise time, it would be suf?cient to perform static analysis and consider the loading to be quasi-static."

Tom

### RE: Quasi Static Analysis

http://home.comcast.net/~electricpete/eng-tips/sinpulsecompare3AQuasiStatic.pdf

Above is an analysis of a sdof system with half-sin pulse acceleration applied to the base.

The half pulse is carved from a sinusoid at frequency 50hz
and lasts 0.01 seconds.

The sdof system has m=1, k=0.247E7
This gives Fnat = sqrt(k/m)/(2*pi)= 250hz
(5 times as high)

At the end of the file you see displacements plotted (base and mass), velocities plotted (base and mass) and accelerations plotted (base and mass).

The displacements are pretty close, velocities a little further apart, and accelerations even further apart.  The peak acceleration for the mass looks is at least 8% higher on the base and the accel profile vs time substantially different.  (we would assume the accelerations the same under quasistatic analysis).

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### RE: Quasi Static Analysis

An typographical correction to the labeling of the curves at the end.  The red curves are for the mass and the blue curves are for the base.

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### RE: Quasi Static Analysis

Just for kicks, I repeated the same calculation where the system had a frequenecy 20 times higher than the sin associated with the base acceleration (instead of 5 times higher)

http://home.comcast.net/~electricpete/eng-tips/sinpulsecompare4a.pdf

It confirms that the solution for acceleration of the mass in this particular problem "oscillates" about the acceleration of the base with a frequency approx equal to the natural frequency of the system.

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### RE: Quasi Static Analysis

Wow, I knew there was a reason I don't use Maple. 6 pages!

There's quite a lot of the 250 hz in that bump, although since it is undamped that may be a bit unfair.

Cheers

Greg Locock

SIG:Please see FAQ731-376: Eng-Tips.com Forum Policies for tips on how to make the best use of Eng-Tips.

### RE: Quasi Static Analysis

Good point that damping should reduce those higher-frequency components.   If there is only structural damping involved,  my gut feel is that  it would have a large effect on that first acceleration peak since that peak occurs less than one cycle of 250hz into the transient and the relative velocity is still fairly low at that time.   But without having the computer solve a specific case, I'm not sure.

Don't blame the long ugly output on Maple... blame it on me.  I could easily have supressed the output of those long equations.  Using ":" at the end of a command line supresses Maple response to a command, while using ";" at the end of a line displays Maple's response to the command.    Somehow I had the vague  idea that including the intermediate results would be valuable since it would make it easier to validate these results by manual application of Laplace transform method if someone were inclined to do so.  Riiiight!  That was a little bit silly, in retrospect.

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### RE: Quasi Static Analysis

Sorry, a minor correction in bold:
"...my gut feel is that  it would not have a large effect on that first acceleration peak..."

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### RE: Quasi Static Analysis

(OP)
electricpete,

Thanks for the analysis The peak acceleration increase of about 8-10% is what I get when I plot an SRS curve for undamped SDOF assuming 250Hz natural frequency

Tom

(OP)
Well guessed!

Cheers

Tom

### RE: Quasi Static Analysis

Quasi-static analysis is indeed valid for shock events (half sine pulse) regardless of frequency ratios between structure and shock pulse.   The real determining factor is the octave rule.   If the first two modes are seperated by at least one octave, the system can be assumed to be a 1DOF system. Response curves to a sine pulse for a 1DOF system can be found in any vibrations text or by doing an internet search.  I have said curve in front of me, and for a frequency ratio of 5 the Amplification factor for a lightly damped system (less than 10%) is about 1.1.  So you take your 2G load and multiply by 1.1 and apply a 2.2G body load to your structure.  I suppose if your structure is large and R>5, I just apply the G load directly since the difference between 2G and 2.2G won't make enough diffenrece in stresses to write home about.

You can use the same octave rule for sinusoidal vibrations as well.  As long as you can approximate the system as 1DOF just look up the amplification (or attenuation) on the transmisibility curve.

True, if you look at the complete transient response of your system to a sine pulse you might see some frequency content of higher modes, but it will usually be insignificant compared to the response of the fundamental and won't contribute to the stresses.

Now if your modes are less than one octave apart, all bets are off because dynamic coupling of the modes will lead to amplifications that can be orders of magnitude higher than those given for a 1DOF system.

One octave is simply a doubling of frequency, so if your 2nd mode is above 500Hz, have at it with the static analysis.   If you want to prove it to yourself, run a simple FEA model of the structural response to a half-sine pulse.   Set your model up in such a way that the frequencies are an octave apart.   Next, run a static analysis of the same structure with a body acceleration (remember to use the correct amplification factor) and I bet you will find that the stresses and displacements are pretty darn close.

### RE: Quasi Static Analysis

(OP)
Spongebob,

Thanks for the very informative post

Can I just clarify.

(1). Make sure the 2nd mode is at least double the fundermental to classify the structure as 1DOF - Then use dynamic load factor curve to give amplification factor

(2). It doesnt matter what ratio the shock pulse to fundamental frequency is if (1) above is not valid. So if R>10 for example you still have to treat as a transient analysis, unless you have this octave doubling

Tom

### RE: Quasi Static Analysis

I would say that if the first two modes are less than an octave apart, I would be very hesitant to treat the system as 1DOF even if the first mode is well above the frequecy of the shock pulse.

I think that there is still a chance of some degree of amplification due to dynamic coupling.   I also think the degree of coupling is going to depend on the effective modal mass of the higher modes.

Let me try to present an intuitive way to think about it rather than doing some actual math.  Imagine a machine mounted on a foundation that can be considered as a 2DOF system.  The foundation mass is large in relation to the machine and therefore the foundation has the lowest natural frequency.   If you apply an acceleration to the foundation it will respond at its natural frequency.   If the resonance of the machine is closer than an octave away, the frame could actually amplify the input acceleration seen by the machine.   For a lightly damped system, this can lead to amplifications of almost 50X.   For moderatley damped systems, it might not be a problem.

You could try running some simple models to test this theory out.

In practice I don't use quasi-static analysis too often.   If I am going through the trouble to make an FEA model to find the natural frequencies anyway, it is not too much more of an effort to just go ahead and solve for the dynamic response.   Although if you have one of those low end FEA programs that only has modal analysis then you simply can't do that.

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