Remote vibrometry recognition of nonlinear eigen-states for object coverage of randomly large size
11 Remote vibrometry recognitionof nonlinear eigen-states for object coverage of randomly large size
Michael C. Kobold (Indialantic, Florida)and Michael C. McKinley (Arlington, Texas)
Abstract
For objects of “large” vibration size such as waves on the sea surface, the choice of measurement method can create differentunderstandings of system behavior. In one case, laser vibrometry measurements of a vibrating bar in a controlled laboratory setting,variation in probe spot size can omit or uncover crucial structural vibration mode coupling data. In another case, a finite elementsimulation of laser vibrometry measures a nonlinearly clattering armor plate system of a ground vehicle. The simulation showsthat sensing the system dynamics simultaneously over the entire structure reveals more vibration data than point measurementsusing a small diameter laser beam spot, regardless of the variation of footprint (coverage) boundaries. Furthermore, a simulationmethod described herein allows calculation of transition probabilities between modes (change-of-state). Wideband results of bothcases demonstrate the 1/ f trend explained within – that the energy of discrete structural vibration modes tends to decrease withincreasing mode number (and frequency), and why. These results quantify the use of less expensive non-imaging classificationsystems for vehicle identification using the remote sensing of surface vibrations while mitigating spectral response distortion dueto coverage variation on the order of the structural wavelength (spectral reduction or elimination). Index Terms laser vibrometry, synchronization, coupled oscillators, image processing, nonlinear dynamics, Fourier analysis, statistics,probability, harmonics, diffraction, scattering, coherence, optics, optical refraction, pattern recognition.
I. I
NTRODUCTION
Remote vibrometry has many forms. The use of optics to sense vibration is a mature technology dates back to methods ofobservations with knife-edge imaging invented by Jean Bernard L´eon Foucault in 1858 that can be used to measure mirrorshape. The method was also used by T¨opler [1] in 1866 to measure phase and later used to remove unwanted optical phaseeffects: “. . . the Schlieren method, where all the spectra on one side of the central order are excluded.” [2] The
Schlieren method,a German word for streak , uses Foucault’s knife-edge in the “Fourier plane” at a distance of a focal length from a collectinglens. A related method, the streak tube, can optically process an entire slice of underwater scenery nearly instantaneously[3], [4], [5]. More recent physics discoveries use vibration to measure atomic size effects. Microscopy resolution smaller thanthe optical diffraction limit can be accomplished with the Nobel prize winning [6]) scanning tunneling microscope. Othermicroscopes image S-shells of large atoms and, in some atoms, their P-shells. The microscopic range, remote vibrometryvariant, the atomic force microscope (AFM) is able to detect separate molecules by measuring the difference in van der Waalsforce as a tiny vibrating cantilever sensitive to the atomic outer shell locations, scans over the sample. These are some examplesof remote vibrometry. This article describes how the nature of observations can provide an different pictures of the vibrationalmodes of common vehicles and other integrated structures built from plates, shells, and membranes (the “body-in-white”).
SUPPORTING RESULTS:
Three types of analysis provide support for this investigation of observation methods for structural surface waves. (i)Experimental results of a driven clamped bar with full fixity at both ends provides a laser vibrometry spectrum for bothsmall and large probe beam spot sizes. (ii) The author’s master’s thesis [7] contains calculation results of imaging a vehiclehull at 4 kilometers. Identification of the target vehicles based on spectral fingerprints formed from laser vibrometry. The targetarmor clattering against the vehicle hull provides a surface vibration that modulates the probe beam. (iii) Several analyticalcalculations using explicitly nonlinear mechanics validate the distinct clattering spectrum. These experimental, simulation, andanalytical results provide cross-supporting reinforcement for the three observational characteristics described in this work.
M. Kobold, P.E., 964 S. Shannon Ave. Indialantic, FL 32903 , worksite location: P O Box 32104 Panama City, FL 32407-8104.matadorsalsa @ runbox . com, michael.c.kobold @ navy . mil (Remove spaces to activate links and email addresses.)The helpful comments of Adam Al-Saleh of Panama City, Florida are gratefully appreciated.The insight and assistance of Dr. Pedro Encarnaci´on of Colorado Springs are appreciatively and always welcome.978-1-4799-7492-4/15/$31.00 ©2015 IEEE a r X i v : . [ phy s i c s . i n s - d e t ] F e b Fig. 1. The lowest frequency modes have high resonance quality Q compared to high frequency response, unless systems have large damping on these“lower” modes or frequency-dependent control systems. For adequate signal to noise ratios, these frequencies identify what are effectively discrete modes, asrepresented in the stem chart to the right. OBSERVATIONS:
This work defines three characteristics of composite structures such as vehicles. (1) The lower frequency modes are nearlydiscrete. (2) Energy from driven modes flows into other modes. And finally, (3) due to theory and observations defined herein,vibration strain-energy is higher in the fundamental mode, with energy decreasing as modal frequencies increase. The lattertwo characteristics can be considered [8] to be consequences [9] of the second law of thermodynamics [10].The probe beam spot size affects vibration modes that are observed with optics. Some vibration modes may not be obviousfrom return due to a probe beam with a small spot size, as discussed with Flight Lieutenant Ngoya Pepela’s thesis [11] in Figures5 and 6 for a vibrating bar. In the simulation of clattering armor plates, more mode information is acquired when viewingthe entire optical image and processing its spectrum at multiple locations. In spite of some interference, the non-imaging (spatially integrated radiant flux) spectrum of the probe return from the entire target’s surface provides more information thanthe small spot size in that phase information over time is embedded in the sensed signal [7]. On-average spectral elimination(or spectral reduction). [End of the table of contents. This will be removed in the published paper.] A. Three observational characteristics
Based on the three previously mentioned supporting results, the experiments, simulation, and analytical results, threeobservations become apparent: (1) the lower frequency modes have sharp resonances that are effectively discrete responses.Figure 1 is a sketch of a notional representation of this case. These modes can be modeled on center frequencies when dampingis limited to structural damping [12], as later discussed in Figures 5 and 6. (2) energy transfers from one structural componentand its vibration modes into the lower modes of nearby components through joints, including welds, which have nonlinearload-deflection curves. Each mode is comprised of a 3-D mode shape (deflection vectors) of the entire structure (Figure 2).The strain-energy (modal energy) of each mode couples to lower energy modes as the system transforms from transient tosteady state. Finally, (3) modal energy flows into the lowest frequencies first, saturating them (Figure 3), depleting the energyflow as the mode number increases from f , f , f , · · · and up through the closely-spaced high frequency modes that becomepractically infinitely dense. For discrete modes the finite element analysis (FEA) results are limited to the number of degreesof freedom (DOF) in the model.The energy at each of the modes shown in the notional sketch (Figure 1) transfers to lower energy modes if physical loadpaths exist that are conducive to mode coupling. These sketches summarize decades of trade secret testing. At lease oneexample appears in the literature for automotive modal analysis, ride and handling [13]. Helicopter data is available fromMIL-HDBK-810 [14]. A transfer of energy from a strongly driven mode to other modes appears in Figure 2. That drivermight be in a component located far from the receiving modes [15], but in ‘frequency space’ these modes might be coupled.Vibrations might be driven by the engine at one frequency, yet sensed in the back seat at another frequency. Nonlinearitiesin response can create harmonics of resonances that allow energy to couple with lower modes. The low energy tails of wideresonances (damped broadening) also “touch” other modes of lower frequencies. Another energy transfer mechanism occursin clattering plates where spatial distribution of mode amplitudes causes contact that excites different modes. This transfer of In practice, modal engineers see effectively discrete modes for the fundamental and other low frequency modes. Here fundamental means the lowestfrequency mode, rather than the music definition as the largest common integer-based factor. For 300 Hz beat with 500 Hz our modal fundamental modewould be at 300 Hz, not 100 Hz. The experience with effectively discrete modes occurs within trade secret production engineering analysis tasks such asautomotive ride and handling – one of the most guarded secrets. These frequencies and mode shapes are rarely published in the literature. Imaging systems require more hardware to calculate pixel values. Non-imaging systems integrate power incident on the entire optical aperture. This poweris termed the radiant flux, measured in watts. It is the flux of the Poyting vector (cid:126)S through an enclosing surface Φ E = (cid:72) (cid:126)S · ˆ ndA . The laser vibrometry industry coins the term spectral elimination to represent reductions to represent destruction or distortion of spectral ID features.Other industries use the same term for entirely different concepts.
Fig. 2. Vibrational strain-energy from different components transfers from a driven mode, the indicated most powerful response, to lower energy modes atdifferent frequencies. For example, the nonlinearity of all common, practical fasteners (bolted and riveted joints) allows energy transfer. In this notional case,a high- Q response at the driving frequency bleeds energy into other modes through friction or contact, a nonlinear process. If the response at the drivingfrequency, f driven , has a wider full width at half height, then energy would transfer faster for nonlinear systems. Some energy would then also transfer forlinear systems as well, more so modes with a smaller frequency difference ∆ f = | f i − f driven | .Fig. 3. As described by the Zienkiewicz quote in Section III, high frequency modes eschew energy that then tends to flow into lower frequency modes.These lower modes tend to “fill up” first and they retain more energy than higher frequency modes. There are at least two exceptions: the lack of vibrationcoherence can block the energy transfer, and interfaces that act like active systems can manipulate the energy or act like passive structural filters. energy is much similar to plucking a musical string at different locations to change the overall sound. Energy transfer betweenstates depends on the frequency of the mode, which is the square root of the eigenvalue in an analysis of normal modes.Figure 3 is a sketch where the final steady state of the vibrational system has energy ordered from strong resonances at thefundamental frequency to a paucity of energy at higher frequencies. A quote from Zienkiewicz, displayed later in Section III,provides an explanation for this common 1/ f dependence of modal energy. The exceptions noted also include strong dampingor added mass. The models, structures, and observational characteristics that comprise this article are summarized in the Conclusion in TableVI.
B. Physical model: symmetries, non-imaging, and coverage
By using a “pencil-thin” beam, conventional laser vibrometry systems can sometimes better identify modes in a simpleacademic structure such as the solitary vibrating bar in the following photos. However, a large spot size can adequatelymeasure those modes and more. This can be shown by simulation and analytical calculation. The simulation uses FEA forvibration deformation data that is handed-off to a set of MATLAB™functions that perform Fresnel propagation for the opticalsensing. The FEA shows the mode shapes and calculates the energy per mode which are the modal participation factors (MPFs)[16], [17]. The FEA surface vibration results are input for the MATLAB image propagation calculation (Fresnel propagation In 1990 a team worked with Walker Automotive for General Motors vehicle Production Engineering. Weights were placed along the tailpipes, mostlybefore the muffler, to quiet the vibration and acoustic noise – except in the case of the Corvette (a design reversal in response to customer complaints thatthey had become too quiet ). This was the testimony of one of the Walker engineers as a contrary example. Tuning the sound for deep rumble has becomewell-known recently as a public selling point for many sports vehicles.
Fig. 4. Coherent radiance undergoes a spatially harmonic phase modulation as a function of the probe beam location due to target vehicle surface skinstructural vibration. The Object modulates radiance from the outbound probe beam at a distance of up to several kilometers while the receiver system keepstrack of the phase of this exitance using the reference beam from articulating beam-splitter A and return mirror B, and appropriate delays (delay fiber or othersystems not shown) where appropriate. [18]). The physical schematic for the simulation appears in Figure 4. Laser vibrometers use many different methods that canhave an effect on the precision of the measurements. The laboratory measurements of the vibrating bar show the signal to noise ratio (SNR) is quite high for structural componentsand, unless the system is purposely damped, the resonances are sharp. These sharp resonances lead to the discrete mode picture(stem plot at right in Figure 1) that the field of modal engineering evolved to exploit. The literature provides the theoreticalreasons that strain-energy tends to eschew higher frequency modes and thus populate the lower modes. Years of experienceof vehicle design and analysis support these conclusions, as does the FEA of clattering armor plate reported here and indetail in a thesis [7]. The FEA results show that unsymmetrical modes of the clattering plates have higher frequencies thansymmetrical (in-phase) vibration of the parallel plates, and that these low frequency modes are discrete below 500 Hz, as isthe case with typical automobiles. These modes are usually easy for laser vibrometers to detect. The optical image propagationmodel shows that even a non-imaging spectrum of the surface of the clattering plates contains energy-ordered modes whosevibration strain-energy drops off with frequency, as described in the Zienkiewicz quotation provided here in Section III.Commercial laser vibrometers tend to use pencil-thin spot sizes on the object being measured. This spot’s cross-sectioncan be as small as a one pixel response. Large spot size imaging systems have far more information, much of which canhelp spectral identification (ID), such as the method developed herein. Further, non -imaging signals average responses fromall pixels for each time step. An optimum automated target recognition (ATR) system could use a combination of small spotsize and full size illumination. But a large spot size tends to help tracking in the paint-the-target phase, while still enhancingspectral ID. II. S TRUCTURAL NONLINEARITIES
Several different types of laser vibrometers can measure dynamical properties of vibrating surfaces. The type of laservibrometer is less important than the phenomenology of what can be measured remotely. A priori project restraints such asrequirements to calculate a transfer function, or to use pencil-thin probe beams, can limit the sensed spectra and thus omitor change perceived vehicle behavior(s) in a manner unrelated to errors in measurement or analysis. In many fields suchas acoustics, the modal results can be less accurate than a kinematic approach would provide. Nevertheless, especially forcomplicated nonlinear systems, it sometimes pays to ignore detailed kinematic analysis, which are often invalid for nonlinearsystems, in favor of modal engineering to tabulate the modes (deformed shapes for particular resonance frequencies), theirfrequencies, and their energy or modal participation factors [17].For over half a century modal engineers routinely used modal analysis for nonlinear vehicle structures. This work usesa structural model of a plate bolted to another plate on a base structure. The work also uses simplified and even “linear” This interferometric model allows analysis of the phase modulation of the exitance from the vehicle (the return ), hence its use for the optical analysis. Atsome level, even the laser vibrometers that use Doppler to measure vibration have images that are affected by phase modulation generated by vibration modeshapes. This is part of the mechanism of spectral elimination. Phase modulation is used here to determine the effect of mode shapes on the return for bothimaging and non-imaging systems. The one-dimensional model in the Appendix uses symmetric and antisymmetric 1-D modes. This solution applies to the symmetrical and unsymmetricalmodes resulting from the FEA for the tank hull and armor. The latter do not have full symmetry because the surfaces are never perfectly flat. Symmetricalbending ‘ ≈ ’ maintains the gap the same across the surface, while unsymmetrical ‘ (cid:39) ’ has both compressive and open gaps at different locations at the sametime with antisymmetrical (not mentioned) being alternately open and closed ‘ (cid:16) .’ In 3-D the average centroid surface is almost never perfectly flat and so asymmetrical mode cannot be geometrically symmetric, but a 1-D model of it can. See [7, p 31 fig 3]. component studies. These studies cannot deliver the behavioral metrics that the full system model provides. Component studiesdo verify limited aspects of the full nonlinear model, which is a crucial part of the overall analysis. In the full analysis, thestructure is a free-floating system using D’Alembert reactions, an analysis technique common to vehicle engineering. See, forexample, the numerous ‘quarter-panel models’ in the literature [19], [20], which are mostly meant to show how to model somefeatures for reasons stated previously. The sensor model is an optical model without the usual air turbulence analysis, except to check for Fante’s wavefrontcoherence breakup range [21] and a few other imaging issues [7] that are out of the scope of this paper.The physical model for the clattering armor assumes a linear superposition of vibration modes on a nonlinear structure.This is not a linear time-invariant (LTI) system. The details of removal of these typical LTI assumptions and its relationshipto stability, stabilization, and stabilizability, are in the thesis [7], available through DTIC dot mil. The analysis includes atreatment of how changes in fixity causes changes in the modes. Similar changes can model the aging of parts. Therefore,the modulation of the sensed optical radiant flux is related to the analytical nonlinear expressions in the Appendix that helpexplain the contact nonlinearity in the FEA simulation. The modeling complications related to fixity, nonlinearities, and modalparticipation factors [16] are dependent on control of the nonlinear solution per time step and over time using load increments,iterations, and other parameters. The analyst also needs to make sure the solution follows the millions of load deflection curves[22], [7] in the full system model.The finite element analysis (FEA) output represents the three-dimensional structural model of the clattering armor platesystem and provides the time history of a structural vibration that modulates the optical return recorded by the sensor. TheAppendix contains a set of single and two degree of freedom analyses that result in analytical expressions. The objective ofthe analyses is to use the optical images, or non-imaging radiant flux (watts), to determine the frequency of structural spectralmodes for target identification (ID).
A. Modal Participation Factors
Recent conversations with Navy scientists and engineers tend to evolve into studies of the role that the nature of systemobservation plays in the perception of the modal system, and how the low frequency strain-energy modes of the structureare effectively quantized. The discrete nature of the modes has been an underlying assumption of modal analysis and inits application to ride-and-handling for half a century. The modal participation factors [17] describe the dynamics of energyflow by using vectors of MPF’s that span the pertinent modes [16]. Each MPF is an element in vectors Φ [ f i ] that are arraysordered by mode number i , used in the FEA of vibrating structures such as vehicles, antennas, and spacecraft. For example,NASTRAN™may require ‘DMAP alters’ (macros) to read out some of the components of Φ [ f i ] .An example of the unintended consequences of design decisions for the 1990 era Corvette follows this detailed restatementof the three modal characteristics introduced earlier, which are essential to the correct system ID. They will be used in theexample. (1) The lower frequency modal states (eigenvectors) form discrete modes, even for the complicated system synthesismodels and the vehicles they represent. The work of vehicle design for these issues focuses on (2) transition probabilities and(3) the flow of strain-energy that preferentially fills the lower modes with energy, as discussed in the quoted passage in thenext section. B. Observational characteristics: Discrete, Transition, and Ordering Discrete modes are countable sets of distinct and high signal to noise ratio (SNR) response of bandwidth narrow enoughto be obviously a single mode. The strain-energy levels within each modal state are tabulated by the structural design team inorder to determine critical components. These critical regions absorb most of the ride-and-handling and structural engineeringeffort during vehicle design, resulting in a final list of lower frequency energy levels The lower modes relate to modes shapesthat are usually characteristic of the structure, such as bending, twisting, and stretching. A real-life example appears later inthis section.2.
Transitions between modal states , from one eigenvector of the structure to another, are often sigmoid in nature where theregion of rapid mode coupling depends on meeting strain-energy thresholds that allow more efficient coupling between modes.Most of the energy tends to flow to components with lower fundamental modes according to characteristic (3) below. Nearly As of 21 November 2019 only the first two hits for “quarter-panel dynamic response” were related to vehicles. No experimental results showed up in thetop (most pertinent) 10 pages of 10 hits each of 1,050 hits. Transition probabilities are related to structural coherence spectra. Modal engineers use them to reduce undesired vibration modes (resonances) in thestructure.[23], [24]. Acceleration spectra are measured in units-per-hertz such as strain (strain-energy amplitudes) or acceleration. In practice the “power” is displayed, signalprocessing “power” being the square, element-by-element, in g /Hz. However, the optically sensed units are dB/Hz, which for non-imaging is re radiant flux Φ e,ref in W/Hz). While there are systems where joints change behavior on a short-term basis during operation (e.g., magneto-rheological and similar semi-active systems),the sigmoid transition is usually driven by two factors: (A) energy thresholds (e.g., bolted joints have a bi-linear load curve [25]), and (B) design changesduring the drawing release phase of vehicle development. all joints are structurally nonlinear for the transfer of forces and reactions, a situation that provides substantial couplingnonlinearity.3. Ordering of modal participation:
The low frequency end of the spectrum tends to hoard the majority of the strain-energy,as the quote from Zienkiewicz explains. These characteristics are summarized in the table VI, the Conclusion. Analysis ofMPF’s (developed by characteristic 2) can determine state transition probabilities. However, modal engineers tend to be focusedon solving critical failure issues. Engineers might use similar terms for entirely different kinds of ‘modes’ such as ‘failuremode effects analysis’ (FMEA). However, a large part of modal engineering involves creation of a record of the order of thevibration modes, followed by an analysis of the tracks of the modes as the design variable change.A structural vibration example described below shows how a structural design change can de-couple modes in order toimprove ride-and-handling. However, this change has consequences other than aesthetic design constraints. The changes canincrease crashworthiness risk and other seemingly unrelated systems, and they can degrade previously adequate vibration andmodal engineering balances in the design. For example, stiffening one joint or component usually provides a much moreefficient flow of energy through that stiffened element of the structure. Strain-energy is drawn from the high frequency modesto the low modes in a surprisingly efficient manner once such a structural “channel” is open (components have structuralcoherence). Then the MPFs start to re-balancing as the entire integrated structure starts to equilibrate. So a stiffener can solveone problem and cause another. The engineering group that deals with braking and crash loads that transfer from the frontto rear bumper can have its margins go from adequate to negative due to a design change by the ride-and-handling team thatsolves a mode-coupling problem. This is similar to an old solution to convertible mode coupling:
C. Inter-organizational effects of design changes
The 1980 era Chevrolet Corvette convertibles an initial design had a fundamental bending mode just below 30 Hz, whichput it very close to the suspension mode. This is a overall vehicle bending mode that curves about a lateral axis, where thebumpers move vertically and together, both in direct opposition to the vertical motion of the seats. It is usually the lowestfrequency vehicle mode for a convertible (as if the vehicle is trying to do sit-ups). The Corvette designers needed to stiffenthis mode to reduce coupling to the nearby suspension modes, which were also near 30 Hz. The models in the late 1980’s hadvery tall rockers that satisfied this requirement. This was the most efficient structural fix because the stiffness of a beam tobending vertically is proportional to the cube of its section height [27]. Apart from the aesthetic issue with having to step highto get into the Corvette, these tall rockers also changed other load paths including a for-aft transmission of load, important tocrashworthiness, as well as making the vehicle slightly more heavy. Automotive is one of the few industries that has a priceon the engineering sufficient to remove a kilogram of mass, which was $50,000 in 1990. These and other costs were acceptedin order to move the fundamental bending mode away from the suspension mode to effectively eliminate coupling. In lateryears, other technologies helped solve this issue.With the exception of the Pininfarina chassis test results [13], few modal analyses such as these (chassis or body-in-white)are in the literature. To a scientist, the ride-and-handling issues and how they relate to the three observational characteristicsdescribed above might seem to be basic enough to warrant several scholarly articles. However, this is clearly the arena oftrade secrecy at such a high level of value that corporate lawyers would be unsurprised at the paucity of measured data availableto the public.
D. Measured data: Vibrating clamped-clamped bar
In his thesis Fl. LT. Ngoya Pepela (he was an Australian Flight Lieutenant in 2006) showed that the modes are not justmaxima of a spectrum, but spikes with huge SNRs in the spectral response [11]. These are the “discrete” modes describedabove in ‘Observation 1.’ His Figures 5 and 6 each show a photo of the vibrating bar illuminated by a laser vibrometer, withthe corresponding spectral density plot are beneath the photos, plotted in the lower pane of each figure. The first (Figure 5)shows all the major modes whereas the second (Figure 6) is missing the modes at 1460 Hz. In Figure 6 the vibrating bar isseen to be illuminated by a laser beam with a large round spot size centered on the 1460 Hz node of the vibrating bar. InFigure 5 the right half of the laser beam of Figure 6 is blocked, illuminating only the left half of the symmetric 1460 Hz modeshape. The halved beam size (Figure 5) allows the laser vibrometry spectral analysis system to show a stronger 1460 Hz modereturn due to the elimination of spatial averaging. That averaging is introduce by phase-related destructive interference fromboth sides of the 1460 Hz mode shape in the full spot size result of Figure 5.Initially spectral elimination might appear to be a problem for the laser vibrometry industry for use with non-imaging sensorsif the spot size is large. However, Ngoya Pepela’s thesis [11] and another article by the author [28], with support from simulation Even spot welds have at least geometric nonlinearities in the Green’s strain tensor. This is due to large strain related to dissimilar stiffness and offset loadmoments (‘kick’s’ in aerospace jargon). Nonlinearities are usually due to frictional hold of bolted or riveted joints that “ must be ignored ” in most structuralanalyses [26]. Therefore, most system FEA cannot model MPFs unless the model is nonlinear and designed to develop MPFs. Physicists in Ann Arbor. mostly graduate students in 1998, were baffled, surprised that a wheelbase change of less than 2 cm would require over a yearof mechanisms analysis to re-balance just the toe, caster, and camber - in negotiations with competing requirements from groups that engineer tires, turnradius, vibration, and crashworthiness. Most vehicle designs barely consolidate the competing requirements, which is more difficult than it would seem.
Fig. 5. The image from a small spot size makes a spectrum with a f = 1460 Hz mode. Permission N. Pepela [11], [28]. and analytical calculations, show numerically how unlikely spectral elimination is for commonly manufactured items, even ifthe spot size encompasses the entire vehicle. Furthermore it is necessary, but not sufficient that super-symmetric structuresproduce spectral elimination in laser vibrometry [7]. The vibrating bar is the simplest form of structural super-symmetry thatresults in spectral elimination, but it is rarely the only structural component a laser vibrometer can use for identification of amanufactured structural system (vehicle), unless the target is the vibrating bar (a structural oscillator ) that requires identification.But even for such oscillators, the typical transducer is a piezoelectric system that is not super-symmetric, and thus not proneto spectral elimination; the source of oscillator energy is separate from the driven bar. Therefore, a vehicle that is purposefullyfitted with vibrating bars can still be identified with its spectral “fingerprint” in spite of possible spectral elimination fromoscillators. Pepela’s result [11] was that use of small spot sizes reduces spectral elimination from the laser vibrometry spectralresult (Figure 6 vice 5). E. The McKinley observation
In early 2018 Michael McKinley of Arlington, Texas, observed that Figure 5 contains modes (at approximately 2740 and2900 hertz) that are not visible in the small spot size collection [29] in Figure 6. While the small spot size is still large enoughto average out higher frequency (smaller vibration shape wavelength) modes, another spectral elimination that small spot sizevibrometers are susceptible to is the effect of the probe spot being on a Chladni line of nodes [30] of the vibration shapefor particular frequencies [31]. Detection of these modes at 2300, 2740, and 2900 Hz is a positive observational differenceprovided by using a large spot size, in this case.Therefore, spectral reduction and elimination (SR and SE) happens for all spot sizes, large and small, for super-symmetricstructures such as 1-D bars. The prior paragraph discusses why this is not a problem for manufactured vehicles; nature provides
Fig. 6. A larger laser spot size creates a laser vibrometry spectrum where the f = 1460 Hz mode phase-averages away. a mitigation (vibrometry is a useful classifier system for realizable structures) of the perceived problem that the McKinleyobservation clarifies (small spot sizes do not mitigate SE). F. Observational differences
The difference between small spot size (Figure 5), and larger spot size (Figure 6) comprises one dimension of observationvariation that can change the observer’s view of reality. Physical systems such as the Corvette design example can be usedto explain the existence of the three characteristics of modal systems listed above. Contact nonlinearity is the main physicalmodel of interest in this paper because contact is a relatively simple and ubiquitous form of nonlinearity, found in nearly allstructural joints. III. E
NERGY ORDERING
Vibration strain-energy transfers from one component to another, usually through joints that are necessarily nonlinear (evenspot welds), depending on the amount of ‘fixity’ of that interface or joint. An extensive treatment of how fixity changes themodes and therefore the sensed optical radiant flux is in the author’s thesis [7, B.2.3, p. 187], including nonlinear examplesthat compare to the contact nonlinearity of the FE model. Fixity is an engineering variable for a particular DOF that determinesthe ratio from zero to one (0% to 100%) of force, moment, or torsion [27] that will transfer from one component to anotherthrough the joint member for whom the fixity is defined [26].Even “simple” vehicle structures have complicated load paths that are nonlinear (mostly through joints) where forces fromone region of the vehicle affect parts elsewhere on the vehicle. These are the pathways for vibration energy flow that tendto dump energy into low frequency structural modes. Modal engineers use isolators and suppressors to stymie this natural tendency, but mostly just for critical components. Some components tend to have non-negligible vibration spectral energydown at these “fundamental” and near fundamental modes even without the nonlinear effects. Damping helps drive theenergy into the lowest modes because the resonance spreads (widens) to encompass a larger bandwidth.Zienkiewicz showed energy transfer using only viscous damping [22, 340-341] to form a ratio of damping to its criticalvalue, c i = 2 ω i c (cid:48) i depending on the modal frequency ω/ π , as quoted below. For vibration systems that are driven by aforcing function f , Zienkiewicz describes the differential equation (DE) M¨a + C ˙a + Ka + f = using a mass matrix ( M ),a damping matrix ( C ), and a stiffness matrix ( K ), acting on deflection and force vectors (cid:126)a and (cid:126)f . α and β are dampingparameters. Zienkiewicz quote: “ ... we have indicated that the damping matrix is often assumed as C = α M + β K . Indeed a form of this typeis necessary for the use of modal decomposition, although other generalizations are possible [references given in thebook]. From the definition of c (cid:48) i , the critical damping ratio [described above], we see that this can now be written as c (cid:48) i = ω i a Ti ( α M + β K ) a i = ω i ( α + βω i ) Thus if the coefficient β is of larger importance, as is the case with most structural damping, c (cid:48) i grows with ω i and at high frequency an over-damped condition will arise. This is indeed fortunate as, in general, an infinite numberof high frequencies exist which are not modeled by any finite element discretizations.” [22, 340, 341]Therefore, vibrational strain-energy tends to naturally migrate from higher to lower frequency modes. Some of the strain-energy can find its way down to the lower modes because over-damping causes more overlap of modal response resonances.Nonlinearities add to the methods of energy transport as discussed later.This results in a set of modes where the mode number increases monotonically with frequency and has an energy thatdecreases monotonically with mode number. Possible exceptions to this natural effect include artificial structures or temporarytransients such as might be introduced with magneto-rheological fluid or other systems controlled by magnetic or electric fieldvariations. The 1/ f phenomenon is found in many different fields of engineering. The local mode structure in Ngoya Pepela’s laboratory test measurement in Figures 5 and 6, which do not follow the 1/ f phenomenon within the displayed 3 kHz, appears to be a contrary example. This an example of not being able to ‘see thetrees for the forest’ within its part of the response spectrum of the very stiff clamped-clamped structure. For those results, thevibrating bar only had a half-dozen distinct modes across 0-3000 Hz, one of which was diminished in Figure 6. The utility ofthe structurally super-symmetric bar is that it shows discrete modes, and that the energy in some modes are spectrally reducedor eliminated by spatial “averaging” – integration over the area of a surface of variable phase that introduces varying levels ofcontinuous interference depending in large part on the limits of spatial integration [28].That the vibrating bar of Figures 5 and 6 is consistent with modal ordering becomes apparent by analysis of the physicsof a simple model of a point mass at the end of a spring of stiffness k . This explanation considers multi-variate non-uniformparametric change in the system. The maximum potential energy is approximately U max ≈ kx / where x is the largestextension of a simple spring. Assume that the frequency f = k/ (4 π m ) is increased, then the energy grows as the square ofthe frequency, parabolically as U ≈ π mf x . In practice, the deflection x decreases with frequency, as it must. However,there is a practical limit to reduction in deflection. When that limit is reached, while the frequency continues to increase,the potential energy must grow. This means that higher frequency modes require more energy, even for smaller deflection –especially when deflection is already small, when x (cid:28) U/ (2 π mf ) . The kinetic energy side of this simple calculation is Rayleigh’s method [32].It may help to see this from a force point of view using the gradient of U . If the restoring force is conservative, it is F = ∂U/∂x = 4 π mf x , and is thus a quadratic function of the frequency, f . It might be tempting to consider a “conjugateforce” for frequency, where the gradient ∂U/∂f = 4 π mx f (units of action : Energy × time, or momentum × distance) [33].However, the force ( F = ∂U/∂x ), cannot increase ad infinitum with f . At some point assumptions of linearity and structuralintegrity start breaking down. Combining the observations of this upper limit on the force, and the understanding that thespring extension is limited to x > , the energy required to have a particular mode increases approximately quadratically with f . This increase strain energy based on the modal displacement u (cid:48) i ≈ (cid:82) dxdyw ( x, y ) is “on average” energy in this sense –that the other variables are limited, and using experience that dynamical systems naturally tend to avoid populating the highermodes with energy. Many children experience this effect when they try to forcefully excite a higher frequency mode in a rope,only to fail unless they exert considerable effort to the point of overexertion.Using the observations above, a conjecture can be stated for the system restoring force of simple harmonic motion, basedon the gradient of potential energy, along with the FEA simulation included herein (Figure 7), including industrial experience,and the test data from the childhood of most of us. While system energy decays with frequency “on average,” a plot of the The ‘fundamental’ mode is the mode that has the lowest frequency, contrary to the definition for a musical fundamental frequency. Critically damped systems remove all oscillation. Examples of artificial exceptions to the 1/ f phenomenon could be exotic. Perhaps a feedback system might “manually” excite high harmonics withoutexciting lower harmonics, such tapping very near the base on a taut chord would tend to excite the higher frequency mode. One example is the spectrum of a surrogate missile plume. The power spectral density at low optical frequencies is large, decaying exponentially withhigher frequency in an approximately straight line for a log-log plot. Fig. 7. Spectra for armor-hull clattering systems, except for the symmetrical modes. Results use different armor-hull baffle stiffnesses (see legend). spectrum within its lowest half dozen modes might not show the 1/ f phenomenon, but that the full pattern does. Sometimesthe spectrum plot is ‘in the trees’ and the “forest” of the 1/ f phenomenon over the entire large bandwidth spectrum is not seenwithin the zoomed-in window of smaller width. For example, consider the ‘in the trees’ ranges of 150-250 Hz and 400-500Hz for the much lower fundamental frequency system of clattering armor in the FEA results of Figure 7.This figure plots several spectra of the simulated structure, 1 × different method of inter-modal energytransport. The 1/ f pattern shows up in its structural vibration spectrum. A collection of the deformed shapes for transientanalysis in the CSC thesis [7] shows how the transient result at each time step is a superposition of normal modes (eigenvectors)populated with energy according to the MPFs Φ( f ) . Those normal modes plots show how one mode at one frequency for oneplate excites another mode at a different frequency on the other plate because of deflections that line up along the surface tocontact the other plate at anti-nodes of a different mode. Again, the lower modes tend to be receiving modes.Using appropriate structural damping, the frequency response function in Figure 7 is the result of an impulse load thatrings all the modes of the FE model. It appears to be natural that MPFs decrease with the mode number of the mode theycharacterize; in a time-averaged sense, the vector is an array of monotonically decreasing participation values usually measuredas the strain-energy for each mode. This is part of the reason for the ubiquitous 1/ f phenomenon. At least for structural vibration[34], we can thank the Zienkiewicz section quoted above [22, 340, 341] provides an explanation of at least some of this 1/ f fall-off. IV. S IMPLE HARMONIC MOTION , NOT SO SIMPLE
A spring-damper-mass single
DOF system (SDOF) provides a one dimensional model of the clattering armor plate whichcan provide analytical solutions to the vibration DEs. Assume ‘small deflection,’ where strain-energy density is low enoughfor linear elasticity assumptions (lack of permanent set in the structure being modeled). Also assume that modes are nearlymonochromatic functions of sines and cosines.In a more formal development of the transfer of energy from one mode to another, Lord Rayleigh points out that pure sineor cosine waves do not exist in reality. He derived the differential equations for Newton’s theory for isothermal compressive- vacuum vibration resulting in this second order form [35]. The footnote describes these generic variables in Equation 1 fora wave in y with respect to x and time t . These concepts are covered in engineering vibration textbooks [36, p. 30]. (cid:18) dydx (cid:19) d ydt = dpdρ d ydx (1)Lord Rayleigh then comments on the ability of simple harmonic motion to maintain shape in nature. The extent to whichthe mode shapes are not harmonic introduces a possible corruption of the pure energy-per-mode concept assumed by MPFs.In the scientific literature a ‘mode’ is derived from the parameters used for the study of statistics which include the median,mean, and the mode. In a field where Gaussian distributions abound thanks to the physics described by the Central LimitTheorem, the mode is merely the location on the abscissa, in terms of the frequency for a spectrum, at the maximum of theresponse. In this sense the typical emphasis on energy-in-mode used by modal engineers is appropriate [23] to identify thedeformed shape of the eigenvector for that approximate center frequency. However, transfer functions are inappropriate forthese nonlinear systems. If transfer functions are modified as Bendat does for simple solitary nonlinear systems [15], which isout of the scope of this work, there may be some consistency between the methods that would be useful to develop, separately. A. Contact nonlinear response
The author’s CSC thesis [7, App F] describes and plots the control law as a sigmoid for the stiffness, which is the slope ofthe load-deflection curve. The control law is a mathematical representation of the nonlinear stiffness associated with the contactstate between armor and hull. This classifier is an arctangent function, a smooth version of the typical bilinear gap elementload-deflection table. As the armor and hull make and break contact, the stiffness switches between high and low values ofstiffness, respectively. When the armor and hull are in contact, negative deflection (compressive penetration) results and thereis a large stiffness due to the large slope of the deflection curve. When the armor has separated from the hull, the nonlinearcontact stiffness becomes a deflection curve of small slope. This small value of extra stiffness provides computational stability.The author’s thesis [7] contains a description of the stability and application of the control law for this case, demonstrating thatthe damping DOF and frequency DOFs are no longer related. The following analysis expands on one of Dr. Winthrop’s controllaws listed in his 2004 AFIT dissertation [42]. His is a different control law, but the analysis methods are complementary.Phase space (state space) plots [7, Fig 12] validate this model’s behavior. The arctangent function that does the switchingin this simple closed form model appears in state space formulation as show below in Equation 2, which has the constantslisted as Equation 3. The state space description for the single mass speed, v , and acceleration, ˙ v , shown in Equation 2 is afunction of the open and closed stiffnesses, k open and k closed , for a damping coefficient of ζ = d/m in units of hertz, where m is a point mass. ˙ x = v − ˙ v = ω o x ( t ) − π tan − k x x ( t ) + k open k closed + dm v (2)This model of a “welded-together” panel system uses the parameters in Equation 3. A “manual,” analytical calculation [7,App C] helped validate the FEA results for the simulated vibration created before application of the laser vibrometry model.Ngoya Pepela’s optically measured modes provided qualitative support for the application of the laser vibrometry model thatused the FEA results to modulate the probe beam. This section and the Appendix describes the former, a structural SDOFmodel. ω o = 2 π . , k open k closed = 0 . , dm = 0 . , k x = 100 (3)In the arctangent gap model [7, Fig 11] (not plotted here), the stiffness will transition at contact, x = 0 . Contact surfaces areimperfect due to microscopic protuberances that comprise the roughness of the surfaces. As the two plates come into contact( x ≤ ) the roughness of the surfaces deform, compressing the protuberances, and a transition from low to high stiffnessoccurs rapidly in order to match the surrounding material. For a “low” damping system, the damping is µ = 0 . kg/s and theopen to closed dimensionless stiffness ratio is k open /k closed = 0 . . The Appendix describes the dimensioned and dimensionlessmodels. A plot of speed versus gap opening in the thesis [7, Fig 12] reveals that the high stiffness during compression of “Since the relation between the pressure [ p ] and the density [ ρ ] of actual gases is not that expressed in [ p = const − ( u o ρ o /ρ ) ], we conclude that aself-maintaining stationary aerial wave is an impossibility, whatever may be the velocity u o of the general current, or in other words that a wave cannot bepropagated relatively to the undisturbed parts of the gas without undergoing an alteration of type. Nevertheless when the changes of density concerned aresmall, [ p = const − ( u o ρ o /ρ ) ] may be satisfied approximately; and we can see from [ dp/dρ = ( u o ρ o /ρ ) ] that the velocity of stream necessary to keepthe wave stationary is given by [ u o = (cid:112) dp/dρ ] which is the same as the velocity of the wave estimated relatively to the fluid.” [35] The extent to which the literature supports the Central Limit Theorem can be found in papers from 1937 through 2003 with Uspensky [37], Landon andNorton [38], Khinchin [39], North [40], and Le Cam [41]. The term welded assumes the plate deflections at both points are the same using an infinitely stiff connection. The closed gap stiffness used in this workis more like hard rubber [7, App B.2.5], but the effect is the same. the base is a shallow orbit in x versus ˙ x phase space. Changes in damping lead to changes in the range of the orbits duringstabilization. These control law formulations were implemented using a MATLAB system[43].Lyapunov function analysis shows the contact control law is asymptotically stable. The Lyapunov function in this case isthe total energy with simple damping loss. In rare cases vibration energy gain to the hull upon which the armor plate clattersmatches the cycle’s damping energy losses and is in synch with plate contact. Persistent energy loss is due to structuraldamping, material heat losses due to bending, and fastener frictional losses. The energy gain per cycle is less than the energyloss due to damping [7, App F]. If the losses where small enough to just equal the gains, the system would be ‘stable in thesense of Lyapunov.’This analysis above summarizes a clattering system. Details of the physics of boundary conditions, initial conditions, andoperating states are found in the thesis [7, App F]. B. Laser vibrometry return from the probe beam
The noise floor did not rise in Figure 6 but rather, the surrounding spatial areas had phase differences within the large spotsize that, due to spatial averaging, underwent destructive interference in a bandwidth that removed most of the 1460 hertzmode. This interference is a combination of optical phase shifts along-range caused by reflection from deflection amplitudesthat are related to the structural wavelength along the bar for each mode.The purpose of the vibrating bar measurement was to show spectral elimination, but it also shows the discrete nature ofhigh quality (low damping) structural modes. The FEA of this system [7] complemented this laboratory measurement [11]by showing that, while spectral elimination can occur with structures that might be constructed in the lab (one dimensionalmodes shapes are the simplest super-symmetric structures), it is impractical to build commercial structures that are substantiallysuper-symmetric.The main result of the CSC simulations was that academic models can produce reductions of vibration sensitivity thattheoretically verify part of Ngoya Pepela’s spectral elimination thesis [11], within meaningful assumptions. The single and twoDOF models described here show why this is the case, theoretically. A separate 2014 article [28] describes the lack of spectralelimination (or spectral reduction) for vehicle and other types of structures that are manufactured economically.The lab vibration measurement shared some spectral reduction features seen in the nonlinear cross-spectral covariancesimulation [7]. The transient vibration results for the surface of the 3-D armor plate were the input for a MATLAB systemthat forms an image similar to one that a laser vibrometer would detect. Commercial laser vibrometers typically only provideimages processed with their proprietary system. For the purpose of simulating the image of the vibrating plate, the opticalmodel for this work used conventional Fresnel diffraction in a MATLAB system of functions. The Fresnel propagation methodintroduced by Goodman [18] provides a computationally adequate method to propagate an image of the return from the targetback to the detector. In this sense, Pepela’s laboratory measurement provided a validation for the FEA and vice versa.The laboratory measurements in Figures 5 and 6 use a clamped-clamped bar that has a high frequency fundamental mode(compared to vehicle fundamental modes that are below 50 Hz). The spectrum has not dropped off with frequency by 3 kHzbecause this is still the “lower frequency region” for the structure, as discussed in the section III. It has a stiff fixity meant totest the laser vibrometer for structural wavelengths that would undergo a difference in phase across the beam spot size.The spectral elimination effect, seen mostly in(Figure 6), provided laser vibrometer manufacturers information on what spotsize to suggest or program for their probe beams. They want to avoid such ‘spectral elimination.’ The simulation and analysis inthe CSC thesis [7] shows users and manufacturers that, except for 1-D structures (bars), even specially manufactured structuresare difficult to create in the super-symmetric form, and thus do not exhibit spectral elimination.A detailed study is available that has comparisons of both structural and optically sensed CSCs, to imaging versus non-imaging returns [7]. The latter choice of observation type, non-imaging, analyzes a scalar signal composed of the spatial averageof the entire image per time step. The simulation analyzes the surface of homogeneous rolled armor (HRA) using only onescalar metric, radiant flux that is the spatial integration of the irradiance reflected from the HRA. These coupled FEA-opticsresults, using non-imaging return, identify the modes on the target (HRA clattering on a hull), nearly as well as the imagingversion of the CSC.
C. Finite element modeling and analysis
The FEA produces a surface of vibration that is curved in 3-D. The vibrational deflections are the input for the scriptsand functions run within MATLAB that produce an optical simulation of the target’s image. Unlike the small and large spotsizes (Figures 5 and 6) in Ngoya Pepela’s measurement [11], the remote sensing of the armor-plate clattering uses a completecoverage large spot size such that all points on the armor modulate the probe beam. Assume the surrounding clutter to betime-gated or otherwise removed. The input that the MATLAB script uses is the set of displacements to simulate images of thetransient dynamical response of the surface of the clattering armor for hundreds of time steps in a duration of a few seconds.One such ‘vibration deformed shape’ is seen in Figure 8. Due to the location of Chladni lines of nodes for the 1460 Hz mode, the return contained Doppler shifts from different phases of the vibration for thatmode. Pepela’s laser vibrometer [11] used these Doppler shifts from different locations on the object that destructively interfered. Other laser vibrometersmeasure speed with two pulses or use other techniques. Fig. 8. FEA of the deformed shape (green) for the clattering armor-on-hull model. Exaggerated micron deflections would not be visible compared to thelarge physical geometry of the plates. There is a 14.24 ms time difference (many time steps) between these two output time frames. The views are into thefore-shortened long direction (1 m) of the 0.5 m wide plate. The un-deformed shape is a light tan color.
The FEA representation of the vibration shape was applied to the MATLAB model of the probe beam as a phase modulation.Using Fresnel diffraction [18], the return was imaged onto the detector 4 km away. A 10 µ m probe wavelength satisfied thenumerical requirements for results shown in Figure 9.A spectrum of the time sequence of sensed images, such as the two shown in Figure 9, appears in Figure 7. More detailedplots are available [7]. V. N ONLINEAR EIGEN - STATES
Creating a simple model for a clattering structure may seem easy, until the fact that the eigenvalues must be nonlinear complexfunctions becomes apparent. A two degree of freedom (DoF) model developed in the Appendix herein was meant to validatethe FEA. This FEA validates a major dynamical 2 DOF analysis result: Antisymmetrical and unsymmetrical modes increase infrequency for increases in contact stiffness. Investigation of these closed form simple damped sprung mass systems providesnot only insight, but qualitative numerical validation. Whether an eigenvalue increases for anti-symmetric and un-symmetricalmodes when gap-closed stiffness increases. This simple model result provides another analytical basis for results seen in theFEA. Additionally, the behavior of the modes indicates that symmetrical modes are better target identification features thanun-symmetrical modes. Hence, there is a need to analyze the energy balance and stability [7, App F] to validate the simpleone and two DOF models as is summarized in the Appendix herein, including the time variation of amplitude and phase.Using the SDOF model as a basis, the Appendix uses Equation 4 to provide the 2DOF solution. Equation 4 may appeardeceptively simple because the time variation of the amplitude and phase is not apparent until explicitly formed. Eigenvalues For the computers at that time, even when running MATLAB simultaneously on several machines on the Air Force Institute of Technology cluster,there was a delicate balance between adequate structural grid densities and optical grid densities for the spatial Fourier transform that performs the Fresnelpropagation of the return. (It is more of a numerical modeling issue than a matter of compute power.) The thesis[7] defines several forms of symmetry. These symmetries include structural, clattering (as opposed to synchronized hull and armor), and modessimilar to the deformed shape of the components to inertial loads. The latter mode is similar to a mode shape (eigenvector) where the hull and armor movesynchronized together, w hull ( x, y ) = w MRA ( x, y ) from which comes the 1 degree of freedom (DOF) and 2 DOF models where the plates are averaged over x and y into two point masses. The 1 DOF model grounds the “hull” mass point to zero displacement, so it is removed in the 1 DOF model. ‘Sprung mass’ systems are an automotive term for a systems where vibration is being isolated or suppressed. At an academic or ‘free body diagram’level, they can sometimes be approximated by spring-mass-damper systems. Fig. 9. Simulated magnitude of radiant flux image of the return after modulation by the vibrating plate in Figure 8. The image horizontal width appearsrotated 90 degrees from the long fore-shortened axis in the prior image in order to better display the 20 ×
40 mm detector grid shape. The maximum deflection“hill” along the 1 m length shows up as a vertical maximum magnitude in the lower (later) pane of both figures. λ i are complicated functions of natural frequencies of individual modes which require ‘mode tracking’ [44] with respect tostiffnesses [45], due to their transient nature in reality [46]. Figure 10 plots the real part of the dimensionless eigenvalues ofEquation 4 as a function of dimensionless frequency versus dimensionless closed stiffness k closed , scaled by the hull stiffness k for a closed stiffness (cid:15) (cid:48) ≡ /(cid:15) = k open /k closed [7]. Two models appear in Figure 10. The hull system is eight times stifferthan the “armor” system. m d xdt + 2 µ dxdt + ω sys (cid:18)
12 + k open k closed − tan − xπ (cid:19) = 0 (4)From a chaos point of view, nonlinear attractors exist for the system defined by Equation 4. We know this because thesystems exist in reality and their vibrations modes do fluctuate, albeit not monotonically. The nonlinear attractors are related tothe underlying hull forcing functions (D’Alambert’s forces due to vehicle inertial loads), the timing and shape of the clattering,the plate stiffnesses, the structural and added damping (e.g., washers or armor-hull batting), and the fixity of the joints.Initially the two eigenvalues for the 2DOF system are equal. As the closed stiffness increases, a critical value of stiffness, k crit, is reached where the armor can no longer follow the hull. It starts to clatter in an unsymmetrical mode. A 3-D contactsurface cannot be anti-symmetric unless it is axisymmetric (1-D), as are the SDOF and 2DOF models. When the stiffnessexceeds k crit the two curves become distinct, branching into two separate paths. This can be see for the plots of two modelswhere the base stiffness is 128 and 1024 N/mm. In the region before clattering, (cid:15) < k crit, the slope of eigenvalue curves, forthe × k closed higher base stiffness curve, decreases eight-fold compared to the smaller base stiffness. At this higher 1024 N/mmbase stiffness there is also an eight-fold increase in k crit. The intercept also increases 8-fold. Derivations of these conceptsand further results, including how the imaginary parts of the frequencies (not shown, Figure 10 is the real part) contribute tothe transfer of energy between the modes are developed in the Appendix.VI. A NALYSIS AND CONCLUSION
A system model of the structural vibration of a contact-plate system was shown to have surface waves with lowfrequency modes that are effectively discrete modes of the PSD [17]. This clattering armor system provided an example Fig. 10. Real parts of the eigenvalues for the dimensionless DE show why higher-energy antisymmetric modes are less likely to be excited. Above a k critthe symmetric (lower frequency branch) and antisymmetric (higher frequency branch) modes [36, 167] for this two DOF problem break out into modes ofwell separated energy. Table VI: Comparison of
Observational Characteristics related to spectral elimination (SE)Characteristic 1.
Discrete modes Energy transitions Energy ordered by mode
Concept sketches Figure 1 ( Σ Nj E j ) Figure 2 ( E j (cid:42)(cid:41) E j +1 ) Fig. 3 ( E j ⇐ = E j +1 )Dependency High Q resonances Clattering FEA model Same deflection ∝ ω at high f requires E j (cid:37) Spectral effect Figure 5 and 6 Figure 7 Damping, lim f (cid:37) c ( f ) ∝ f eschews E strain [22]Character Low frequency Nonlinear joints Steady state vibration energy E i ∝ /f Small spot size Sees all modes Blind to many ∆ E jk No Spectral elimination (SE)Large probe spot SE in bars Sees more ∆ E jk Some SE (spectral reduction)Super-symmetric Some missed modes Misses most E j (cid:42)(cid:41) E j +1 Provides th. of SE [28]. Target rarely super-sym.Manufactured parts Sees all modes, Fig 7 FRF, coh( f ), MPFs Typ. Figure 7 (super-sym. target is rare )of nonlinear response. Both SDOF and 2DOF models derived in the Appendix have qualitatively verified the symmetricalversus unsymmetrical modes, and the higher frequency of the unsymmetrical clattering modes. The full finite element modeltransient results also showed the tendency for vibration strain-energy to collect in the lower frequency modes, as most modelengineers have seen [13], and as was calculated analytically in the classical literature [22, 340, 341].(1) The lab tests on the doubly clamped bar show experimentally that high SNR modes are effectively discrete. In the absenceof substantial damping they are not just the maxima of a spectrum, but spikes in the spectral response [11]. The system inFigure 5 has a signal-to-noise-ratio that is huge. These massive SNRs show that the lower modes of structure, that are fairlyhigh in frequency for this case due to the double clamped nature of the bar, can easily be considered discrete. Hence, quoderat demonstrandum (Q.E.D.), the observational characteristic (1) is demonstrated – lower frequency modes are essentiallydiscrete.This analysis of nonlinear eigenvalues provides useful results for theory and simulation for common nonlinear structures.Symmetrically moving parallel plates have more strain (and thus more strain-energy) than clattering plates where theunsymmetrical motion interrupts their nearly sinusoidal in time out-of-plane-motion to spew strain-energy into acoustics and even permanent set (deformation of the material). Through simulation and by experience the observational characteristic(1) appears true, that lower modes are effectively discrete for high quality systems (low damping), and that, when there arepathways (nonlinear joints) that allow energy transfer, (2) and (3) are in effect; damping and clattering help the energy flow intothe lower modes from higher frequency modes – although overall energy is lower as damping increases (see Figure 7). Mostvehicle components have bolted or riveted joints that allow such energy transfer [15] as can be seen with plots of coherencespectra, coh( f ) [47].(2) The tools in industry that provide MPFs include mature engineering methods [17] from the 1980’s such as the FEAtools put in place, for example, by MacNeal-Schwendler Corporation engineers in NASTRAN [16]. These MPFs stored inthe FEA vector Φ can measure the energy transmitted between states (between eigenvectors of the system) where the state(mode) i changes it energy with mode j. ∆Φ t ( f i ) = Φ t ( f i ) − Φ t − ( f i ) = − ∆Φ t ( f j ) . For surface waves on the ocean [48]this observation would include wave energy of large ocean swells driving up capillary waves and eventually creating foam asseen in large storms. Energy transfers from one type of wave to another, Rossby-Kelvin, gravity-capillary, internal-surface, ...,and direction variations thereof, may prove useful.Modal engineering for industry, especially for vehicles (i.e., ride-and-handling) is a trade secret endeavor. Vibration spectralplots for vehicles found in the literature, such as the Pininfarina paper [13] are rare. It makes use of analysis tools in order tocalculate modal participation factors [16]. These MPFs show the ‘participation’ (energy per mode [17]) using modes developedfrom normal modes that are FEA-produced eigenvectors. The participation flows from energetic modes to lower energy modessimilar to how large swells on the ocean in a storm are accompanied by unsettled surfaces, rather than smooth large wavesthat a tsunami has before it nears the shore [48]. In the latter case, not enough time has passed to transfer the energy to othermodes until the wave crashes on shore where sufficient coupling to other modes exists due to the constraints of the shorestructures.Therefore, due to (3) the energy ordering of modal states for complicated system synthesis models and the ordinary vehiclesthey represent, the work of vehicle design for these issues focuses on (2) transition probabilities and (1) energy levels Φ( f i ) .There are usually other types of oscillator interactions that produce an ordering of energy levels similar to that described byZienkiewicz [22, 340-341]. These concepts are summarized in Table VI.Pencil-thin probe beams like that used in the lab measurement have observational characteristics quite different from thelarge spot size used for the FEA-MATLAB model of fully illuminated clattering armor. The former can be less susceptible tospectral elimination. However, Mr. McKinley’s observations discussed in section II-D on page 6 show different modes appearand disappear for either change in spot size. The latter low fidelity beam method is adequate for spectral identification ofeconomically manufactured vehicles. Such full coverage probe beams are less likely to be subject to spatial coherence issuesor illuminate solely a node of the Chladni zone [31]. Pencil-thin beam returns fail to convey vibration modulation in thismanner [30]. For the large (full coverage) spot size, some of the beam will nearly always get through.While these results are generic for plate structures, application to waves on the surface of a volume appear to fulfill similarbehavior of (1) discrete modes, (2) modal participation dissipation, and (3) higher energy in lower frequency modes. A tsunamihas a much lower spatial frequency and temporal frequency than typical 2-4 meter gravity waves. However, the larger Rossbyand Kelvin waves are also candidates for the study of the extreme low frequency application of the principles observed in thisarticle. VII. A PPENDIX – O
NE AND T WO DOF
CONTACT EIGENVALUES
This Appendix summarizes the nonlinear dynamics of a simplified one-dimensional (1-D) and 2-D forms of a structuralcontact system, its relationship to the full 3-D FEA, and in the end, test measurements. The 2-D form provides for a differencein foundation stiffness, and transition to 3-D.
A. Closed-form SDOF nonlinear contact response
This subsection displays the Mathcad™output for the closed form solution to the damped SDOF oscillator meant to representa lumped mass model of the HRA-hull contact vibration system. The mathematical derivations are based on the nonlinearsolutions [42] with a simplification of the control law, u ( x ) , that models simple contact along the x -axis.The SDOF dimensionless system describes the effect of nonlinear contact stiffness from a solution composed of symmetricand antisymmetric one dimensional modes. This SDOF solution applies to the symmetrical and unsymmetrical modes of the3-D FE model for the HRA-hull system, respectively. In 3-D the unsymmetrical modes are non-uniform, occurring when someparts of the armor is moving opposite to the hull (i.e. clattering). For clarity 1-D modes are distinguished as symmetric or Figure 7 shows that decreasing baffle stiffness between the plates (see its legend) drives unsymmetrical resonances lower. Transition probabilities are related to CSCs [7] and the coherence spectrum [47]. antisymmetric, while 3-D systems are labeled symmetrical and unsymmetrical. Application of derivatives to the dimensionlesssystem is first made under the assumption that all variables are nonlinear functions of time. Then after starting with a restrictedcase, the variables are brought into explicit nonlinear use, one at a time, to refine the calculation.The state-space representation [49] is shown here in its phase space form (location x ( t ) and derivatives). With the definition cos φ ψβ ≡ cos (cid:104) ( ˙ ψt + ψ o ) t + ( ˙ βt + β o ) (cid:105) , x ( t ) = ( ˙ a ( t ) t + a o ( t )) e − ( ˙ µ ( t ) t + µ o ( t )) t cos φ ψβ (5)Even the first derivative of Equation 5, is non-trivial. Mathcad allows factoring in several ways. Collecting on cos φ andthen on sin φ , Equation 6 provides a compact expression for the undamped speed in Equation 6. ˙ x ( t ) e ( ˙ µt + µ o ) t = (cid:0) a ˙ µt + (2 a o ˙ µ + ˙ aµ o ) t + a o µ o − ˙ a (cid:1) × cos (cid:16) ˙ ψt + ( ψ o + ˙ β ) t + β o ) (cid:17) − (cid:16) (2 ˙ a ˙ ψt + (2 a o ˙ ψ + ˙ aψ o + ˙ a ˙ β ) t + a o ψ o − a o ˙ β (cid:17) × sin (cid:16) ˙ ψt + ( ψ o + ˙ β ) t + β o (cid:17) (6)With more assumptions restricting the nonlinearity of the solution where appropriate, the acceleration is also collected on cos φ and then sin φ in Equation 7 for this simplified expression: ¨ x = (cid:0) − ˙ aψ o t + [ ˙ aµ o − a o ψ o ] (cid:1) cos( ψ o t + β o ) − ( ˙ aµ o ψ o t + a o µ o ψ o ) sin( ψ o t + β o ) ∀ ˙ µ = 0 , ˙ β = 0 , ˙ ψ = 0 (7)The first order nonlinear solution in Equation 7 uses a constant amplitude, a = constant. While not explicitly a functionof time, a has a constant time rate of change, ˙ a . A further order of nonlinearity to allow the amplitude change rate to be afunction of time would follow the above assumption with the use of ˙ ψ , the time rate of change of a dimensionless frequency.The following subsections describe how Figure 10 shows that while the stabilizing stiffness increases above the critical lift-off frequency, symmetric modes remain at a constant frequency while antisymmetric modes increase in frequency. This 1-Dbehavior implies the same effect for 3-D modes, symmetrical and unsymmetrical, as is seen in the FEA results [7]. B. Closed-form Mathcad 2 DOF contact
Mathcad symbolic equations in the following calculations comprise the closed form solutions for a damped two DOF (2DOF)contact vibration model. Stiffness k x is the “rate” that defines the sharpness (hardness) of the contact. C. Two DOF DE’s and solutions
Equation 8 is a dimensioned form of the two DOF damped oscillator DE where P , m , k , d , (cid:15) , ξ , and t are applied force,outboard mass, foundation stiffness, damping, control law stiffness, surface displacement, and time. The symmetrical 3-D mode shows up in frequency response curves [7] but it is not perfectly synchronous across the surface, thus it is not a perfectlysymmetric mode like the 1-D model. It exists in 3-D the same way quantum mechanical modes exits - because of their energies - spikes in the spectra atthe correct frequency, not because we saw them optically. Although with FEA plenty of studies on modal decomposition are in the literature to reinforce thecombination of modes that modal engineers use to analyze vibration. The symmetric (1-D) and symmetrical (3-D) modes do not clatter while the other modesdo.
The state-space definition of the system combines the state x ( t ) with its output y ( t ) and changes in state ( ∂x/∂t ). These matrix relationships [50] ofthe state change vector ( ˙ x ) and system output and its state and input ( u ( t ) ) are one of many sets of A, B, C , and D matrices, easily confused with otherfield’s ABCD systems such as optical ABCD ray matrices for laser resonator orientation [51], [52]. ˙ x = Ax + Bu using the state and input matrices y = Cx + Du via output and feed-through matricesAll matrices and variables can be functions of time. Some definitions distinguish state space as discrete compared to continuous phase space. However thefield of linear systems often uses continuous output and even state variables and the resulting Kalman filter is nearly ubiquitous [49]. ψ and β are canonical rate terms for frequency and phase. Their subscripted ‘o’ terms are the initial frequency and phase values. The ‘naught’ termshave units of hertz and radians while the un-subscripted terms are time derivatives of frequency and phase. (cid:18) P ( t )0 (cid:19) = (cid:18) m m (cid:19) d dt (cid:18) ξ ξ (cid:19) + (cid:18) d d (cid:19) ddt (cid:18) ξ ξ (cid:19) + (cid:34) (cid:18) (cid:19) + (cid:18) (cid:15) (cid:15) (cid:19) (cid:18) − π (cid:18) arctan( rξ ) 00 0 (cid:19) (cid:19)(cid:35) × (cid:18) k − k − k k + k (cid:19) (cid:18) ξ ξ (cid:19) (8)The dimensionless ratio k = ( k + k ) /k is the stiffness from the second oscillating point mass (the 1-D representation ofthe armor) to the base of the system as a whole, which includes the hull for this SDOF system. The hull is freed to oscillate inthe 2DOF system. The dimensioned variable ξ is the gap opening in the dimensionless x direction. The dimensioned controllaw for structural contact for this two DOF problem is the displacement u ( −→ ξ ) of Equation 9. The dimensionless stiffness fromthe first oscillator to ground is unity. ξ indicates the dimensioned form of gap opening in the dimensionless x direction. The dimensioned control law for structuralcontact for this two DOF problem is the displacement u ( ξ ) of Equation 9.Dimensioned u ( −→ ξ ) = (cid:34) − π (cid:18) arctan( k x ξ ) 00 arctan( k x ξ ) (cid:19) (cid:35) (9)The use of Winthrop’s method [42] on Equation 8 produces the dimensionless DE in Equation 10. The dimensionless appliedload is F applied and the dimensionless stiffness only applies to DOF one: −→ (cid:15) = [ (cid:15) ; 0] .Dimension less (cid:18) F applied ( t )0 (cid:19) = (cid:18) (cid:19) ¨ −→ x + 2 (cid:18) µ µ (cid:19) ˙ −→ x + (cid:18) (cid:15) u ( x ) − − k (cid:19) −→ x (10)Equation 11 uses the vector −→ x = [ x x ] T , the dimensionless location of the two masses where x − x is the gap opening.Continuing with Winthrop’s assumptions [42] assumes a straightforward solution. Equation 11 shows all the variables that varywith time. x i = a i ( t ) e µ i ( t ) t cos( ψ i ( t ) t + β i ( t )) (11)Substituting assumed solutions from Equation 11 into Equation 10 results in a Special Eigenvalue Problem (SEVP). Thedimensionless system frequencies ψ i are functions of the individual frequencies f i = ω i / π from ω i = ( k i /m i ) − ζ i where ζ i = d i / m i from Equation 8. Two assumptions help make the solutions tractable. First, the removal of the driving loadprovides the homogeneous solution. Second, variables dependent on time and space (uniformity with respect to location) varydifferently. To first order the time variation of the dimensionless frequency, ψ , is the derivative in this nonlinear system thathas the largest effect on the solution [42]. Equation 12 summaries some of the simplifications: a ( t ) = a ( t ) = constant = aµ ( t ) = µ ( t ) = constant = µβ ( t ) = β ( t ) = constant = βF applied ( t ) = 0 (12)In words, these simplifications are: • Constant amplitude, a , cancels out of the DE. • Uniform damping, µ , is for simplicity. • But uniform phase, β , is realistic. • Solve for free vibration first (homogeneous solution)Using a new variable for dimensionless phase, φ = ψt + β , to simplify the state space gap opening (Equation 5 simplified inEquation 11), the solution starts with a definition of the speed (Equation 13) and acceleration (Equation 14), where coefficientsare maintained as variables of time in the derivatives. ˙ x = − (cid:104) aµe − µt cos φ + aψe − µt sin φ (cid:105) (13)Maintain the coefficients as variables of time in the subsequent derivative: ¨ x = a ( µ − ψ ) e − µt cos φ + a ( µψ + µψ ) e − µt sin φ (14)Using Equations 13 and 14, time derivatives formed below come from application of these differential operators: Π i ≡ ( µ i − ψ i ) + 2 µ i ψ i arctan φ i Ω i ≡ µ i (cid:20) − µ i − ψ i arctan φ i (cid:21) (15)From Equation 10 the resulting dimensionless DE in Equation 16 below starts to take the form of an SEVP (Equation 17),easily solved by eigenvalue methods. The time derivative operators Π and Ω (defined above) allow the vector −→ x = [ x x ] T tofactor out. The damping constant, µ , and dimensionless frequency, ψ , remain in the calculation until the “nonlinear” eigenvaluesand eigenvectors are developed in the formal SEVP solution. A purely imaginary frequency exp( if (cid:48) t ) = exp( i [ iµ ] t ) is in theexponential term in exp( − µt ) where the sign of µ indicates damping. The ensuing 1-D analysis provides insight into thedynamics and validates the transient ‘nonlinear’ modes output by the 3-D FEA, modes that exist in reality. (cid:18) Π
00 Π (cid:19) (cid:18) x x (cid:19) + (cid:18) Ω
00 Ω (cid:19) (cid:18) x x (cid:19) ++ (cid:18) (cid:15) u ( x ) − − k (cid:19) (cid:18) x x (cid:19) = (cid:18) (cid:19) (16)These relations use a different dimensionless stiffness, (cid:15) ≡ (cid:15) = k closed /k open where k closed = k and k open (cid:28) k for numericalstability. Equation 16 is of the SEVP format as shown below in Equation 17. (cid:18) A , A , A , A , (cid:19) (cid:18) x x (cid:19) = (cid:18) (cid:19) (17)Equation 18 shows the format of the system matrix A for submission to an eigen-solver. Some of the terms in A , and A , were kind enough to cancel. A = (cid:18) A , A , A , A , (cid:19) = (cid:18) − µ − ψ + 1 + (cid:15)u ( x ) − − − µ − ψ + k (cid:19) (18)Solution of the characteristic polynomial, | [ A ] − λ [ I ] | = 0 , provides the eigenvalues of the system where both DOF haveequal displacements. Using a Mathcad ‘97 symbolic solver two solutions are available: (1) Free classical vibration, and (2)a ”welded” system vibration solution. The latter welded components (the two masses) is equivalent to being epoxied withhard rubber [7]. As defined in Equation 9, the welded model reduces to free vibration for slight contact, u = 0, just beforecompression when u turns negative. The system dynamics are revealed via the welded solution as the control law is varied inthe range ≤ u ≤ . The switching control law, u ( x ) , determines which frequency is active, switching from one frequencyvalue for the open gap state to another for closed gap state. The nonlinear FEA results for 3-D show that both modes areactive at the same time at different locations on the surface. Spectral energy flows into and out of both ‘open gap’ and ‘closedgap’ modes when averaged over many cycles of time. The contact gap system is a time composite system with dynamics thatare more easily modeled using modal analysis techniques. The noise and vibration industry developed these techniques withnonlinearities such as contact in mind [24], [53], [54], [55].States evolve in an oscillatory manner. This energy flow between alternately created and destroyed states might be a usefulmodel for other systems where the physics appears to forbid continuous energy level ordering. For vehicle structures, experienceindicates that the oscillation period is usually less than an hour, on the order of minutes. D. Two DOF nonlinear contact eigenvalues
Eigenvalues in Equation 19 are a dual set for the control law, u ( x ) , for an open and closed gaps. In the dimensionless system for the open gap state the stiffness for DOF 1 is normalized to unity and DOF 2 has a stiffness of k = 1 + ( k /k ) . When thesystem goes into contact u ( x ) adds (cid:15) to DOF 1 which becomes (cid:15) , as seen in Equation 10 and 16. This addition uses theFEA maxim that “stiffnesses add” [56]. The total closed stiffness is (cid:15) + k . The derivation of the complete dimensionless The 1997 version of Mathcad has a small Maple kernel that is easier to use than recent versions. “eigenvalues” λ i in Equation 19 are Mathematica results for the solution of the eigenvalue problem described in Equations 17and 18 above [7]. (cid:18) λ λ (cid:19) = (cid:18) (1 + (cid:15) + k ) − µ + ψ ) − √ κ (1 + (cid:15) + k ) − µ + ψ ) + √ κ (cid:19) κ ≡ − (cid:15) + (cid:15) + 2 k − (cid:15)k − k ( (cid:15) ≡ (cid:15) of Eq. 16 for 18 ) (19)For subsequent nonlinear calculations assume u = 1 so that contact is active; there is no gap between the HRA and the hull.Assume (cid:15) > k (dimensionless Equation 16) and for stability (cid:15) k > (cid:15) k = k base (dimensioned 2 DOF Equation 8). Note thatall epsilons are dimensionless but they also are a part of the control law that modulates the dimensioned system, Equation 8.Figure 10 on page 15 plots the real part of values of the eigenvalues. DOF 1 and 2 represent the hull and armor, respectively,in the 2DOF system. After a critical (cid:15) , the higher frequency mode (the clattering, antisymmetric mode) separates from thesymmetric mode. When the hull has a stiffness that sufficiently exceeds the critical stiffness, the symmetric mode settles intoa constant eigenvalue with respect to (cid:15) regardless of the dimensionless stiffness between the two masses. ( ∂λ /∂(cid:15) → and ∂λ /∂k → but not for λ .) From the 3-D FEA results in Figure 7 on page 10, the higher frequency clattering mode is thelower energy mode. This is consistent with experience and the Zienkiewicz quote on page 9 [22, 340-241]. The excitationan antisymmetric mode requires more energy for the same deflection amplitude as an equivalent symmetric mode. If thereis an avenue for vibration strain and strain energy to flow into a lower frequency mode the energy “conduit” can be a jointor structural connection (nonlinear structural components). Plucking a taut cord close to the held end, to vibrate it at higherfrequency will similarly excite the lowest frequency mode given enough time.Application of the SDOF model over all DOF’s using a theory of linear structural response gives a relationship for a lineartransfer function [47] (the Fourier transform of the impulse response [49]). Each of the many DOFs in a linear time-invariant(LTI) system has a relationship that has the same form as shown in a transfer function [7, p. 81]. But these functions areoften inapplicable to such nonlinear systems. Common vehicles are filled with nonlinear joints, all of which can be adequatelymodeled in FEA, with some effort to avoid misapplication of St. Venant’s principle [57], [58], [59]. For the ensuing 2 DOFsystem, the results of the prior subsections and the 2 DOF extension of the SDOF model are formulated with Mathematica. E. Eigenvalues: Damped 2 DOF sprung mass
Repeated roots occur for the eigenvalues λ ( (cid:15), k ) plotted in Figure 10 on page 15 when the dimensionless closed stiffnessis less than a critical stiffness, (cid:15) < k crit . Repeated roots indicate the symmetric and antisymmetric modes have the samefrequency. However, since there are real and imaginary parts to the eigenvalues [36, 171], that there is a growth in energy forone mode and a decrease in energy for the other. For most joints that undergo cyclic loads the contact frictional footprint areaalso oscillates, which causes the stiffness to oscillate. Therefore, the imaginary parts of the eigen-frequencies act to re-balancethe strain-energy according to the control law u ( t ) . Over time energy will move from the higher energy antisymmetric modeto the lower frequency symmetric mode, unless the lower mode is suppressed.The repeated roots also show that when k base is small enough, the two masses that represent the HRA-hull system vibrate as ifthey were attached and in free space. In this case the two masses vibrate together, toward and away from the base, with negligiblerelative motion. For this small k base trivial solution the center of mass is oscillates according to f CG = k base / π [ m hull + m MRA ] .This is the symmetric mode. Alternately, they can vibrate apart and together with their center of mass remaining stationary. Forsuch antisymmetric modes the increase in the eigenvalue that occurs when gap stiffness increases provide a basis for similarresults seen in the unsymmetrical FEA modes for stiffer bolts and batting material. The hull and armor 2DOF model willallow DOF 1 and 2 to have different unsymmetrical frequencies sin in general m hull (cid:54) = m MRA . Where the deflection at higherfrequencies does not modulate of the probe beam as well as lower frequencies, this nonlinear eigenvalue behavior has theresult that symmetrical modes are better target identification features than the unsymmetrical modes.
F. 1-D eigenvectors compared to 3-D FEA
Equations 16 and 17 introduced the SDOF DE and its ‘special’ eigenvalue problem (SEVP). In this SEVP the system matrix [ A ] models a simplified 1-D nonlinear contact system. The derivation of the system matrix [ A ] , from [ A ] is on page 19 inequations 16-18. The solution to the SEVP [ A ] Λ = 0 provides two eigenvectors −→ Λ i from Λ .This is the solution of the dynamical DE in Equation 10. It uses four main assumptions [7] to model the contact point for1-D as a surface of slideline contacts in NASTRAN™: (1) The damping is uniformly constant per unit area and equivalentto the 1-D model, (2) for both the foundation base to the hull as with the hull to the HRA armor. (3) The combinationof low but sufficient base stiffnesses (“open” stiffness) and high “closed” stiffness is stable. (4) For the analytical results, aseemingly severe assumption sets dimensionless frequencies to be equal, ψ = ψ . The FEA more properly analyzes dampingand simulates these results via transient nonlinear time-integrated results using Newmark-Beta methods for algorithmic stability Bolted joints do not have the same load paths as the usual easy way to stitch an FE model, welded axles (rotation only). This common mistake is usuallydriven by schedule concerns but it provides answers that are orders of magnitude in error, even for “loads models.” [22], [56], for part of the solutions for the proper 3-D dimensioned DEs for all modes. Therefore, the millions of 3-D DOFin the FEA remove this last assumption.In 1-D each of the two DOFs obey the assumed nonlinear solution in Equation 20 where the amplitude is x max which isscaled by a characteristic length L ∗ providing a dimensionless solution x ( t ) . L ∗ is best chosen by using the Buckingham π theorem [56], [60] could range between the average of the microscopic roughness height (microns) to the gap averaged overthe surface area in the 3-D model where some part of the HRA-hull system is barely touching. The latter is most easily beaccomplished using the normal modes along with a few pertinent static deformed shapes. x ( t ) = x max L ∗ (cid:18) · · · nonlinear terms · · · (cid:19) e jψt + φ = a ( t ) e − µ i t cos( φ i t + β ( t )) (20)The single DOF solution in equation 20 applies to both DOF’s in matrix form, with cross-terms, in the dimensionless systemmatrix from Equations 16-18 on page 19 as shown in Equation 21 below. [ A ] = (cid:18) − µ − ψ + 1 + (cid:15) − − − µ − ψ + 1 + k open (cid:19) (21) The dimensionless frequencies, ψ , in this system matrix occur in the same form as an eigenvalue λ from [ A ] = | [ A ] − λ [ I ] | . This form where ψ = ψ = λ i are the eigenvalues, only applies to the above assumptions of (2) equaldamping and (4) equal frequencies for the two masses which are necessary for a practical nonlinear solution. There is aphysical superposition of both symmetric and antisymmetric modes at the same time, resulting in hull – HRA resonation atboth ψ = ±√ λ and ψ = ±√ λ , in a linear combination of modes. Each of the two Λ i modes will have both ψ and ψ active for that one mode. Therefore, ψ and ψ represent the same energy for λ (symmetric vibration), but ψ and ψ produce different energy for λ (unsymmetric vibration).The FEA eigenvalues (i.e., values of the diagonal NASTRAN Λ matrix) are related to their eigenvectors −→ u iT , as plotted inthe thesis [7, Fig21-23]. The FEA modal frequencies are the square roots of the eigenvalues, summarized therein [7, Table 7,Fig 20]. Those frequencies and plots of −→ u iT represent eigenvalues λ i within [Λ] and eigenvectors Γ i within [Σ] , in analogy tothe 2DOF model, where the columns of [Σ] are the eigenvectors. As discussed above, the similar-frequency argument makesphysical sense, but the similar damping assumption is an artifice used to obtain a practical, simplified solution that is usefulfor comparisons. The simplification allows the extra ψ terms to cancel. Experience and the FEA results both validate that thistechnique is adequately appropriate for use in this particular case.The eigenvalues in matrix form are [Λ] = [ λ ,
0; 0 , λ ] but it is more convenient to display them in the vector form (Equation22), where the parameter κ was defined in Equation 19. (cid:18) λ λ (cid:19) = 12 (cid:18) (cid:15) + k − √ κ − µ − ψ (cid:15) + k + √ κ − µ − ψ (cid:19) (22)The 2 DOF eigenvectors in Equation 23 are one dimensional mode shapes for displacement of the hull, u , and the armor, v , for modes 1 and 2: Σ = (cid:18) u v u v (cid:19) = 12 (cid:18) − − (cid:15) + k + √ κ − − (cid:15) + k − √ κ (cid:19) T (23) G. Analysis of the 2 DOF SEVP DE
The eigenvalues λ i = ω i for each column Γ of the eigenvector matrix ( Σ ) are the result of solving Equation 21. Equation24 represents an extreme fixity of this Mathematica™solution, a low frequency “DC” limit ψ = 0 . lim ψ =0 −→ λ = 12 (cid:18) (cid:15) + k − µ − √ κ (cid:15) + k − µ + √ κ (cid:19) (24)The parameter κ , defined in 19, in this first order correction to the linear eigenvalues (defined for Equation 22) is only afunction of the stiffnesses, the square of the base and hull stiffnesses. Compared to prior Mathematica™results, this ψ = 0 model provides a “DC” eigenvalue solution that is otherwise not available. Σ T = 12 (cid:0) −→ Γ −→ Γ (cid:1) T = 12 (cid:18) − − (cid:15) + k + √ κ − − (cid:15) + k − √ κ (cid:19) (25) This relation is only valid for the assumptions described earlier: (2) µ = µ and (4) ψ = ψ . Otherwise morenonlinear terms remain and the system is not susceptible to SEVP solution for modes in Equation 25. H. Synthesis of the 2 DOF SEVP DE
This subsection describes the use of Mathematica™to synthesize the eigenvalues and eigenvectors from the system matrix[A]. Since the kernel of the SEVP is [ A ] = | [ A ] − λ [ I ] | , and the eigenvalue solutions are the dimensionless frequenciessquared λ i = ψ i , the approximate system matrix [A] derives from [ A ] , as defined for Equation 22. This is only approximatebecause of the many combinations of nonlinear and approximately linear variables (e.g. “linear” ψ ) selected in [42] and usedthe “Mathematical Preliminaries” section of [7]. Adding a diagonal of λ i [ I ] = ψ i [ I ] to the matrix [ A ] allows the extractionof [ A ] from [ A ] = | [ A ] − λ [ I ] | as shown in Equation 26. [ A ] = (cid:18) − µ + (cid:15) − − − µ + k open (cid:19) (26)Equation 26 represents a 2 DOF system with springs of stiffnesses k and (cid:15) , and uniform damping of the same magnitudefor both DOFs. The eigenvectors and eigenvalues are most easily recognized in relation to the standard EVP in Equation 27assuming ( λ i = ψ ∀ i ∈ [1 , ). (cid:18) − µ + (cid:15) − − − µ + k open (cid:19) × −→ Γ i = λ × −→ Γ i (27) I. Synthesis of unmatched DE
For clarification and to bring us back to the full nonlinear DE, stipulate that the eigenvalues for each DOF are the same foreach mode λ ≡ λ , that the physical argument leading to Equation 20 holds the frequencies equal. Therefore, the frequencysubscript matches the subscript for the eigenvector, rather than matching the damping modes as was done in the prior subsection;the combinations of frequency and damping are ( ψ k , µ i ) ∀ i (cid:54) = k .The control law, u ( ξ ) of Equation 28 is the nonlinear stiffness [7, Fig 11], based on the relative displacement, ξ = L ∗ x .The arctangent switch changes the stiffness between the hull lumped point mass and the HRA plate lumped point mass, k open ↔ k closed . u ( ξ ) = 12 − arctan( k x ξ ) (28)The phase φ k = ψ k t + β ( t ) ≈ ψ k t relates to the eigenvector whose temporal dynamics λ i describes both DOFs, the hull andHRA point masses. Here the frequencies ψ k are not matched to the damping µ i . Redefining Π i and Ω i for this mixed-indexformat: ∀ Π i,k ≡ ( µ i − ψ k ) + 2 µ i ψ k arctan φ k Ω i,k ≡ µ i (cid:20) − µ i − ψ k arctan φ k (cid:21)(cid:18) Π ,k
00 Π ,k (cid:19) × (cid:18) Λ Λ (cid:19) k + (cid:18) Ω ,k
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