linear dynamics for everyone: part 1 · 16 de apr2008 deskeng.com feature finite element analysis a...

16 DE Apr 2008 deskeng.com

F E A T U R E

FINITE ELEMENT ANALYSIS

Analysis work is rarely done becausewe have spare time or are just curi-ous about the mechanical behaviorof a part or system. It’s typically per-

formed because we are worried that thedesign might fail in a costly or dangerousmanner. Depending on the potential fail-ure mode our anxiety might not be toohigh, but given today’s demanding OEMsand litigious public, the task could involvehigh drama with your name written allover it.If you’ve done analysis, you’re com-

fortable with the concepts involved in sta-tic stress analysis; you define the loadingand boundary conditions, and identifysuccess with a model bathed in soothingtones of gray and blue with nary a red re-gion to be seen. However, in the back ofyour mind you might wonder about thatlarge vibrating motor or the plant ma-chinery that hums at a constant 12.5Hz.Alternatively, maybe you have an elec-tronics enclosure that is to be mounted on

the side of a building in an earthquake-prone region and your boss is question-ing your bracket design. Whatever thecase, you have the static world under con-trol. What about the rest?In this series of articles, we’ll briefly re-

view dynamic analysis fundamentals andsee how they can easily be applied tomake sure your design remains strong and

rock solid in the face of dynamic events,whether simple vibrations, earthquakes,or even rocket launches.

KEEPING IT SIMPLEStatic stress analysis is the proverbial“walk-in-the-park” for most people do-ing analysis work. It feels straightforward:we apply a fixed load and examine the re-

B Y G E O R G E L A I R D

Linear Dynamicsfor Everyone: Part 1> Why natural frequency analysis is good for you and your design.

Figure 1: First vibration mode shape for an NCAA aluminum baseball bat is shown here.

F E AT U R E



sulting static behavior (generally linear,given linear material behavior). We getback some nice clean stresses and deflec-tions that hopefully match our intuitionfor how our design should behave. Whilethere might be a few hiccups along theway, the end result usually appears logicalto our mechanical minds.The dynamic behavior of a structure

can also be viewed in the same light if wejust shift our perspective a bit and think interms of how our structure should natu-rally deform during a dynamic event.Whenever a structure is hit or given somesort of time-varying load (transient orsteady-state), it will respond to this loadwith a very characteristic behavior. If theload is not incredibly massive and thestructure doesn’t blow up or plasticallydeform as a result, then the dynamic re-sponse of your structure will most likelybe linear. That is to say, if the load is re-moved and the structure is given a chanceto calm down, then it will return to its un-deformed state. This is the same concept touse in linear static stress analysis: whenthe load is removed the stress in the struc-ture goes back to zero.What exactly do wemean by character-

istic dynamic behavior?All structures havenatural or characteristic modes of vibra-tion. The sound or note from a guitar stringis all about its natural frequency of vibra-tion.When a guitar string is plucked it will

vibrate at a certain note or tone. This noteis at the string’s characteristic frequency.Another example is aluminum baseball

bats. The best aluminum baseball bats aredesigned with characteristic vibrationsthat attempt to limit the sting that occurswhen you hit a ball outside the sweet spoton the bat. Each frequency creates a phys-ical deformation or shape, and the totaldynamic response of the bat is a combi-nation of all its characteristic mode shapes

(see Figures 1 and 2).In finite element analysis (FEA), these

natural frequencies are called eigenvaluesand their shapes are noted as eigenvec-tors or eigenmodes. This nomenclature isrooted in German and the word eigen de-notes “characteristic” or “peculiar to” andcame into common use withmid-19th cen-turymathematicians.With dynamic analy-ses, you’ll also see the terms normalmodes and normal modes analysis. Theuse of the word normal prior to mode isjust another way to say natural, charac-teristic, or eigen. When describing modeshapes, our preference is to just say normalmodes since they represent the inherentnatural response of the structure.

A BEAM AS ONE EXAMPLEIf we picture a simply supported beam(fixed at one end), its natural mode shapesare determined by its geometry while itsfrequency of motion is fixed by its stiff-ness and density. Got all of that? Take alook at the graphic of our beam for its firstthree modes (see Figures 3 and 4). The firstthree modes of the beam are well-definedbut come in pairs to cover all permissibleranges of motion for that beam. In 3D, thefirst mode can oscillate within a 360-de-gree envelope around its longitudinal axis.Numerically, the eigen solution processjust gives us the two orthogonal modes,but it implies the full 360-degree envelope.All structures have a nearly infinite

number of permissible shapes or eigen-values/eigenmodes. Fortunately, only the

Figure 2: Second vibration mode shape is shown here for an NCAA aluminum baseball bat.

Figure 3: Undisturbed simple beam,plus two of the first vibration mode shapes (two directions of motion).

F E A T U R E


deskeng.com Apr 2008 DE 19

lower frequencies dominate the responseof the structure so we can typically ignorethe higher frequencies. A rule of thumb isthat the first three modes capture the ma-jority of the response of the structure andtherefore one can safely ignore the higherfrequencies. (The reasoning for this state-ment will be given in Part II of this series).The frequency of these modes or their

eigenvalues is dependent upon the stiff-ness and the density of the beam. The fre-quency equation for structures can thusbe written as:

� = K / m

where K is the stiffness of the structureand m is the mass. This wonderfully sim-ple equation represents a great deal of in-formation about the system. The classicway to graphically describe this equationis with a mass suspended by a spring,where the mass block can only move upand down or has one degree of freedom(DOF) in FEA parlance. The eigenmodeof this system is up and down.

PAPER MILL DESIGN EXAMPLEIn commercially interesting structures, thesame equation holds. The eigenvalue ofthe structure is still determined by

� = K / m

For example, consider a forming boardused within a paper mill. The structure is10 meters long andmade of stainless steel.The paper mill has an operating frequen-cy of around 9Hz. If the structure’s nat-ural frequency is near this operating fre-quency, it will quickly resonate and tearitself apart. More importantly, it will alsotake the multi-million dollar paper millalong with it (see Figures 5 and 6).The parameters of the original design

placed the first mode at 8.4Hz, whichwould have been a disaster. The formingboard is manufactured from 9.5mm-thickstainless-steel plates, so our first designinclination was to simply increase thethickness of the plates. We pursued thisapproach for several days but as we in-creased the thickness, the mass of thestructure also increased almost in lock-step with the stiffness (see above equa-tion). At the end of all this head banging,we got a marginal improvement (~11Hzresonance) with 25mm-thick plates, butit was going to cost a fortune to manu-facture.

At this point we stepped back fromour rush to find a solution and thoughtabout how stiffness is developed in longslender structures. We realized that wehad very little shear transfer between thetop and bottom surfaces of the formingboard. This insight led us to add diagonalsteel rods that would connect the top andbottom planes and allowed us to keep

the thickness of the plates at 9.5mm. Thenew design tested out on the computerwith a first mode frequency of 13Hz. Withthe eigenvalue of the forming board nowsignificantly higher than the operatingfrequency of the mill, resonance is im-possible and the system is dynamicallystable. Additionally, the thinner plates(9.5mm instead of 25mm) meant it was

more than half the cost of the first, mar-ginal redesign.

DYNAMIC LOAD CONSIDERATIONSWhen a structure is loaded in a transientor time-varying fashion (e.g., when anelectric motor creates a constant, sinu-soidally varying load), if the eigenvalueof the structure is lower or higher thanthe excitation frequency, the structure willjust behave as if the load was applied sta-tically. Let us say that we have this struc-ture with an eigenvalue at 10Hz and it iswhacked by a transient (e.g., half sine-wave with frequency of 10Hz), we wouldexpect the structure to vibrate subsequentto the hit and then gradually return to itsstatic zero-stress condition.However, if the structure's dynamic

load is time-varying (e.g., sine wave at10Hz), the structure will resonate. If littledamping is present (think metal or stiffplastic structures), then we may see theclassic harmonic resonance that causedthe collapse of the Tacoma NarrowsBridge in 1940. What kills structures isresonance, and the worst kind of reso-nance occurs when the structure sees theexcitation load over and over again. Themost effective way to eliminate this wor-ry is to design your structure to have low-er or higher natural frequencies than itsoperational frequency; this goal is thedominant reason for performing an eigenanalysis.

THE FINAL MATHIn our prior discussion we haven’t men-tioned anything about the magnitude ofan eigenmode. That is to say, we have dis-cussed its frequency and its shape but leftout any description of its magnitude. Ineigen analysis (normal mode analysis) noload is applied to the structure. Without aload (e.g., a force or pressure), a predic-tion of the actual eigenmode is impossi-

ble. The extraction of the eigenmode (theshape of the permissible deformationmode) involves a fancy piece of math thatis commonly available in a multitude oftextbooks. The core thought is that we aresolving the dynamic equation:

{f(t)} = [m]{ x’’(t)} + [C] {x’(t)} + [K] {x(t)}

If damping [C] is ignored (a good as-sumption for a lot of designs) and the ap-plied force f(t) is set to 0.0, the equationreduces to this more manageable formula:

[m]{ x’’(t)} + [K] {x(t)} = 0

This is the key equation for eigen analysisand states that only the mass and the stiff-ness of the structure control its naturalmodes.To solve this equation see your favorite

math handbook. The gist of the discus-sion is that the eigenvalue of the structureboils down into this elegant formula:

� = K / m

And since no forces are used in thecalculation of the eigenvalue, its associat-ed eigenmode is dimensionless. Your FEAprogram then scales the eigenmode suchthat the maximum displacement withineach mode shape is near 1.0 or some rela-tive value tied to the mass of the struc-ture. When these eigenmodes are dis-


FINITE ELEMENT ANALYSISF E AT U R E

Figure 5: Mode 1 of vibrating paper-mill forming board

Figure 4: Second and third vibration modal-shape pairs for a simple supported beam.

played within an FEAprogram, we see animaginary magnitude; this visual can be

problematic for many initiates who arefirst venturing into the eigen world of dy-

namic analysis, but we will discuss theimplications in Part II of this series.

MODE ANALYSIS ESSENTIAL CHECKLISTDetermine what type of loading you mayhave on your structure and whether ornot that loading might set up a resonantcondition. Try to determine your loadingfrequencies and ensure that they fall out-side of the eigenvalues of your structure.Run an eigen analysis and look at the

first three normal mode frequencies. See ifthey fall within your danger zone.If the normal mode frequencies are out-

side your loading frequencies then stop.You are done and all is good.If your normal mode frequencies are

within your range of interest and you can’tredesign around them, then stay tuned forour future articles. We will show thatmaybe it isn’t that bad after all. �

George Laird, Ph.D., P.E. is a mechanical engi-neer with PredictiveEngineering.com and can bereached at [email protected]. Sendyour comments about this article to [email protected].

F E A T U R E


deskeng.com Apr 2008 DE 21

Figure 6: Mode 2 of vibrating paper-mill forming board

REPRINTS

[email protected]

Your article reprints can be personalized with your companyname, logo, or product photos. PDF versions are available.

Your coverage.Enhanced.Your coverage.Enhanced.

Contact Jeanne DuVal at

603.563.1631 x 274 or

[email protected]

for info and a quote.

Use Reprints for:> Trade show handouts

> Press kits

> Direct-mail campaigns

> Sales call leave-behinds

> Product enclosures

> Take-away for lobby displays

F E AT U R E

62 DE May 2008 deskeng.com

FEA/VIBRATION ANALYSIS

In part I of this series (DE April 2008,p. 16), we explained the concept thatevery structure has natural frequenciesof vibration (eigenvalues) and that these

natural frequencies have specific deforma-tion shapes (eigenmodes or normalmodes).We also took a swipe at how one woulduse this information in the structural de-sign world by noting that excitation fre-quencies outside of a structure’s first coupleof eigenvalues means it will behave stati-cally stable.We nowwant to expand uponthis theme and demonstrate how this sim-ple form of analysis can be leveraged touncover how your structuremight behaveunder dynamic loading.

THE DOMINATORS: MODES WITH MASSAn interesting fact about normal modesanalysis is that we can associate a percent-age of the structure’s mass to each mode.With enoughmodes, you get 100 percent ofthe mass of the structure, though for com-plex structures this canmean hundreds ofmodes. The common thought is that if youcapture 90 percent of themass of the struc-ture that will be good enough. For now,we’ll start classically and then show whatthis concept means in a real-world engi-neering situation.We use the supported beam because it is

simple to visualize, simple to formulate,and best of all, simple to draw on a whiteboard. In Figure 1 (next page), we show aquick example of the first threemodes of asimple supported beam alongwith the per-centage of mass associatedwith themode.The first mode dominates with 82 percentof the mass of the beam swinging up and

down. The second and third modes con-tribute a little bit of mass but nothing likewhat we saw in the first mode.All of thesemodes operate in one direc-

tion. In the real world, the mass fraction isassociated with all six degrees of freedomwithin a particular natural frequency.Whatthismeans is that if we excite this firstmodein the vertical direction (the direction of themass fraction), then the structurewill movewith 82 percent of its mass behind thismode. If we think like Newton and realizethat F=m*a, thenwe can visualize the dom-inance of this mode and the huge forcesthat can be generated at resonance.

PEA POD TRANSPORTLet’s leverage this information in a coupleof typical engineering problems. Manu-facturers use vibrating conveyors to movematerials ranging from pea pods to lumpsof coal. One such vibrating conveyor isshown above. It moves pea pods within a

food-processing plant using a vibratorymotor that creates a sinusoidally varyingforce that is aligned down the axis of theconveyor (y-axis). This force causes the con-veyor to swing forward and up on its fiber-glass laminate springs.When operating at its resonant fre-

quency, the conveyor tosses the pea podsforward and upward in a gentle swing-ing fashion. The material transport rate isdetermined by its operating frequency andthe length and angle of the fiberglassspring laths.Afundamental problemwith this type of

conveyor is that during startup as the vi-bratory motors spin up to speed, nonop-erating modes get excited, often causingthe conveyor to tear itself apart before itreaches the target operating frequency. Oureigen analysis of the conveyor shows that ithas to pass through three modes beforereaching its operating frequency at 18Hz.Table 1 shows a brief summary of the data

Themotormount for this vibrating conveyor is themass of flexiblemetal plates hanging off the endof the conveyor. Yellow elements are fiberglass laminate springs; themotor is not shown.


Linear Dynamicsfor Everyone: Part 2>Vibration analysis can show detailed structual behavior under dynamic loading.

F E AT U R EFEA/VIBRATION ANALYSIS

deskeng.com May 2008 DE 63

harvested from this analysis.Along with this hard data we have the

shapes of these four modes shown in Fig-ure 2. We now have a complete picture ofthe dynamic mechanical behavior of theconveyor system.Although the vibratorymotor has to dri-

ve through threemodes to reach the targetfrequency of 18Hz, it has three things in itsfavor: the applied force is along the y-axis;the mass fractions of the first three modesare small (less than 6 percent); and the dom-inant directions of the first threemodes arenot alignedwith the forcing function. Thus,with a basic eigen analysis, one can ap-prove the design of a very complex engi-neering system.

MAKING YOUR RIDE AS SMOOTH AS SILKEver wonder what makes a quiet ride ina motor vehicle? It has to do with avoid-ing modes that might be driven to reso-nance; that is to say, keeping the struc-ture dynamically static in its mechanicalbehavior.In an analysis of amodernmotor home,

imagine the FEAmodel is highly idealizedusing beam elements for the small struc-tural tubes, plate elements for themain lon-gitudinal beams, and lots ofmass elementsto represent the engine, air conditioners,water and diesel tanks, and passengers.Af-ter a normal modes analysis, we have 45modes ranging from 2.3Hz to 15Hz.

Trying to figure out which mode willcause trouble is essentially impossiblewith-out knowing something about the massparticipationwithin eachmode. To help ussort through this mess, we can graphicallyshow themass participation sums for the x,y, and z directions.Ride smoothness in many cases is just

the “hop” in the structure or the bounce inthe y-direction. The biggest bounce of in-terest occurs at mode 21 where the mass

participation jumps from around 23 per-cent to 43 percent (20 percent of themass ismoving upward at mode 21). If we inves-tigate thismode a little deeper, wewill findout that the entire coach frame is buckingupward at a frequency of around 10Hz.Wenow have a pretty good picture of what toavoid— anything around 10Hz.Luckily, standard road-noise rarely ex-

ceeds anything higher than 5Hz. Given ourcurrent knowledge of the mode behaviorand its mass participation fractions, we arein good shape for a smooth and stable ride.

DESIGNING THE STIFFER STRUCTUREOne of the realities of a normal modesanalysis is that you don’t get any informa-tion about the magnitude (deformation orstress) of the actual response. This is dueto the fact that you are not applying a loadto the structure.While this poses some lim-itations, we can also use something calledthe strain energy density to estimatewherethe structure is the most flexible or the“weakest.”The mode shape of the structure repre-

sents the permissible deformed shape,which directly correlates to the strain en-ergy pattern. Elements with large valuesof strain are those that most directly affectthe natural frequency of that mode. If youcan lower the strain energy, you’ll increasethat frequency.Figure 3 (bottom page 66, left) shows an

electronics enclosure that is attached to acouple of brackets. The strain energy den-sity for the first mode is contoured overthe brackets. In this design, the bracketsare bolted onto the C-channel used as theattachment point to the building. The de-sign goal is to survive a rather severe earth-quake (GR-63-Core Zone 4 specification).To simulate the earthquake the structure

is shaken in all three axes. The first mode at7Hz has 45 percent of the mass swingingback and forth in the Z-direction as corre-lated by the high strain energy shown in

Figure 1: This illustrates a vibration analysis of a simply supported beam. From left, the normal mode shows a specific deformation shape where 82percent of the mass of the beam swings up and down; the second and third modes of the beam only occur at much higher frequencies. In the second,only 10 percent of the mass deforms this way, and in the third, only 3 percent. These modes have very little effect on the overall behavior.

Continued on page 66

Figure 2: Four vibrational modes of conveyor belt pictured from the left as it ramps up in frequency to final operating value. Mode 1 shape at 3.5Hz hasa mass fraction of 5% along the x axis; Mode 2 shape at 13Hz has a mass fraction of 5.8% along the z axis; Mode 3 shape at 16Hz has a mass fraction of1% in a y axis rotation; and the Mode 4 shape at 18Hz has a mass fraction of 60% along the y axis.

Mode Eigenvalue MassFraction

DominantDirection ofMass Fraction

1 3.5Hz 5% x-axis2 13Hz 5.8% z-axis3 16Hz 1 % y-axis rotation4 18Hz 60% y-axis

Table 1: Summary of Eigen Analysis Results

Figure 3 ( above center) at the flex points ofthe bracket. To improve this design, weonly need to address the high-strain ener-gy locations. This is done by capping theends of the bracket. With the bracket rein-forced, the strain energy is reduced (asshown in Figure 3, above right) and the fre-quency jumps to 10Hz.

ADVANCED MODAL ANALYSIS CHECKLISTDon’t panic when you have eigenvaluesright on top of your operating frequencies.Only natural frequencies with significantmass participation factors are important.Eigenmodes have directions as do their

mass participation fractions. Investigatethese directions and see if they corre-spond to your forcing-function direction.If they don’t (let’s say they’re orthogo-nal), then the structure will remain dy-namically stable.If you need to stiffen up your structure,

look at the modes where the mass partici-pation is high and then investigate theirstrain energy density. Modify your struc-ture to lower the strain energy in high-en-ergy sections and you’ll see a significantincrease in your eigenvalues.Be methodical and look at your struc-

ture from all directions. The secret to mak-

ing a dynamically stable structure is to tieeverything together: eigenvalue (themodefrequency); mode direction (i.e., modeshape); mass participation fraction; massparticipation direction; and strain energydensity. If you remain cognizant of all ofthese factors, you will have a good degreeof success in not being surprisedwith aber-rant or disastrous dynamic behavior inyour structure.�

George Laird, Ph.D., P.E. is a mechanical engineerwith PredictiveEngineering.com and can be reachedat [email protected]. Send commentsabout this article to [email protected].

F E AT U R EFEA/VIBRATION ANALYSIS

66 DE May 2008 deskeng.com

Continued from page 63

Figure 3: Left) Strain energy density of initial design of electronics housingmounting brackets indicating areas most likely to affect natural frequency offirst mode; Center) Bracket subjected to simulated earthquake (GR-63-Core Zone 4 spec) showing high strain energy of first mode at 7Hz at flex pointsof bracket; Right) Revised bracket design with capped ends displays much lower strain energy and pushes first natural frequency to 10Hz.

18 DE June 2008 deskeng.com

F E AT U R EFEA/VIBRATIONANALYSIS


Linear Dynamicsfor Everyone> Part 3: Extracting real quantitative data toanticipate everything from earthquakes to rocket launches.

If you’ve kept up with this series of ar-ticles, you now know more about thedynamic behavior of structures than95 percent of your peer group within

the design and engineering world. Andafter reading about how vibration analysisreveals key information about structuralbehavior (see DE, April and May 2008),the terms “natural frequencies, normalmode shapes, mass participation factors,and strain energy” have become integralto your vocabulary.

Up to this point the discussion has cen-tered on qualitative terms about the me-chanical response of structures due to dy-namic loading. In this last part, we'll showhow to extract real quantitative data (i.e.,displacements and stresses) from a sim-ple normal-modes analysis.

DOING IT ON THE CHEAPThe dynamic response of a structure is de-rived from its individual normal modes. Ifyou hit your structure, its dynamic re-sponse is formed by the summation of itsindividual modes. Mathematically, weknow that each one of our normal modeshas a frequency, a mode shape, and a bit ofmass associated with that shape (a massparticipation factor). All of this data is de-rived from the basic equation of motion:

ma+kx = 0From this equation, the standard lineardynamics solution can be derived as:

v = K / mwhere v is the frequency or eigenvalue ofthe system. Since no forces are involvedin this equation we can't have any real dis-

placements or stresses.If wewant real data, we need real forces

as in: (ma+kx = F).The brute-force approach is to solve the

model in the time domain. A time-baseddisplacement, force, or acceleration load isinserted into the model and then the com-puter makes a few thousand solves. Atsome later date, we then wade throughpiles of output data (remember, we are

doing a complete solve at each time step)to figure out what went wrong, when, andwhere. This can be a daunting task and isoften just plain impractical.

If the loads are frequency-based (dis-placement, force, or acceleration as a func-tion of frequency), then the door is wideopen to all sorts of very numerically effi-cient solution strategies. First you performa normal modes analysis and then apply a

This is a satellite FEAmodel showing instrumentation attachment points (black squares) foridealized mass elements as defined by a center of gravity connected by rigid links (orange lines).The model is driven with an input power spectral density (PSD) function.

de0608 Laird 2.4.qxd:Layout 1 5/22/08 1:08 PM Page 18

frequency-based load. A linear solve ismade only at each normal mode or eigen-value. The resulting displacements andstresses can be viewed individually orsummed in some fashion to arrive at acombined damage response to the load-ing. The technique is extremely useful be-cause you have reduced the number ofsolves from thousands to just a handful.Although we can't use this technique forevery type of structure (linear behavioronly), when you can use it, you have theability to quickly gain insight into the dy-namic behavior of the structure on thecheap (minutes versus days).

MODAL FREQUENCY SWEEP (MFS)Frequency-based loads are more commonthan you might think. Our previous ex-ample was of a vibratory conveyor. Sincethat was rather straightforward, let's lookat something a bit more complex.

Figure 1 shows a high-precision opti-cal-mirror assembly that will be attachedto an airplane or helicopter. Airbornestructures provide a vibration-rich envi-

ronment with their high-speed gearbox-es, jet turbines, rotating blades, etc. To as-sess the robustness of the design, you canperform a virtual shaker-table experiment.Our desired output is the deflection re-sponse at the focal point of the mirror un-der severe vibration.

Without having to build themirror, youcan input a sin sweep of 1g from 200 to3000Hz to the model, and see what getsamplified or harmonically driven in thestructure. Figures 2a and 2b show the out-put graph of acceleration and displacementas a function of frequency for the sin sweep.

The mirror system has eigenvalues at710, 1292, 1570, 1996Hz, etc. But as shownin Figures 2a and 2b, only the mode at 710Hz creates any real sympathetic acceler-ations and displacements. This is logical

since this mode is aligned with the forcingfunction and has a mass participation fac-tor of 35 percent in the direction of theforcing function. In a way, we knewwhatthe results would be before we ran theanalysis. But now we have real numbers.

POWER SPECTRAL DENSITY (PSD) ANALYSISSatellites are expensive and failure is evenmore expensive. During launch (Figure 3a)they get pounded by a broad and chaoticspectrum of vibrations (accelerations) fromthe rocket motor, stage separation, acousti-cal noise, etc. No single acceleration fre-quency dominates and there are multiplelayers of noisy events that occur randomly.

To numerically model such loading, astatistical approach is used where acceler-ation measurements are converted into apower spectral density (PSD) functionwithunits of acceleration squared against fre-quency (Figure 3b). Once again, we have a


F E AT U R EFEA/VIBRATIONANALYSIS

Figures 2a and 2b: Virtual shaker-table results of acceleration and displacement at the focal pointof the large airborne-mountedmirror showing acceleration in cm per second on the y-axis of2a and displacment in cm on the y-axis in 2b.

Figure 3a: A rocket launch and other chaoticevents create vibration (acceleration) spectrathat are best idealized in a statistical sensevia power spectral density functions.

Continued on page 70

1) If you review the normal mode re-sponse, itsmodes and itsmass participa-tion factors, you might not need thesemore advanced techniques.2) If you go aheadwith this analysis, con-vert all loads into the frequency domain.3) Results from modal frequency analy-ses are in the frequencydomain.Theypro-vide a virtual snap shot of your structureoperating at steady state at the forcingfrequency of interest.4) Results from power spectral densityanalyses are averages of the forced re-sponse.They are estimates of the highestpossible displacements and stresses.

ANALYSISCHECKLIST

Figure 3b: The rocket acceleration measure-ments are converted into a power spectraldensity function with units of accelerationsquared against frequency. The appliedload is based on frequency.

Figure 1: This is an image of an advanced opti-cal platform for an airborne system placedon a virtual shaker table.



loading curve where the applied load is based on frequency.The statistical nature of this calculation comes from the con-

version of the acceleration time history data to the PSD func-tion. The PSD amplitudes are actually root mean square (RMS) ac-celeration values that are fitted to a standard statisticaldistribution where the mean value is zero and what is plotted isthe standard deviation versus frequency. As non-intuitive as thismay sound, it is a very effective way to convert chaotic, randomnoise into a numerically useful load-function.

Another unique aspect of the PSD analysis is that all of themodes of the structure are assumed to be vibrating or excited bythe PSD function simultaneously. Somewhat like a bell beingrung, the output response is a summation of the amplitudes of allof the frequencies of the structure within the range of interest.

As an example, an FEAmodel of a satellite (see page 18) has var-ious instrument payloads represented as mass elements. Thesepayloads are attached to themain structure of themodelwith rigidlinks. Inmany cases, the utility of a PSD analysis is to determine thetransfer function of the structure or how the satellite frame willtransmit acceleration into the instrumentation packages.

The resulting transfer function is just another PSD. We thenuse this output to perform amore detailed PSD analysis on the in-strumentation package to determinewhether it will survive launch.

The dominant mode of the satellite is around 70Hz (Figure 4)and the output PSD function reflects this fact with a huge spikeat this frequency (marked with a star). If we knew that our in-strumentation package was susceptible to frequencies at 70Hz,our design solution would be to develop a stiffer satellite framethat would not have a dominant normal mode at 70Hz. Evenwithout the PSD analysis, a clear understanding of the normalmode frequencies, their mode shapes, and corresponding mass-participation factors allows you to make valid predictions.

Now you have the concepts, the vocabulary, and the big picturefor operating in the world of linear dynamics. A simple modalanalysis can put you well on the way to successfully meetingyour structural engineering challenges. Why not give it a try?�

George Laird, Ph.D., P.E. is a mechanical engineer with PredictiveEngineer-ing.com and can be reached at [email protected]. You can sendcomments about this article to [email protected].

Continued from page 20

F E AT U R EFEA

Figure 4: The output spectral density plot of the satellite analysis indi-cates the location of the first normal mode with the star marking 70Hz.The chart shows frequency on the x axis and PSD (g-2/Hz) on the y axis.


linear dynamics for everyone: part 1 · 16 de apr2008 deskeng.com feature finite element analysis a...

Documents