Analysis Isolated Raft Systems Using 2 DOF Nonlinear Spring Mass Model

8/3/2019 Analysis Isolated Raft Systems Using 2 DOF Nonlinear Spring Mass Model

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Analysis of Isolated Raft SystemsUsing 2-DOF Nonlinear Spring-Mass Model

Frederick A. CostanzoCurtis B. Annibale

John X. Przybysz, Jr.Naval Surface Warfare Center Carderock Division, UERD

9500 MacArthur BoulevardWest Bethesda, MD 20817-5700

(301) 227-1650

NOMENCLATURE

M1 Upper MassM2 Lower MassX1, 1, 1 Displacement of Upper Mass and its DerivativesX2, 2, 2 Displacement of Lower Mass and its Derivatives T Time IncrementK Spring ConstantC Viscous Damper ConstantY Base Input DisplacementFNL Nonlinear Mount Force , Finite Difference Constantsb, c, d, e, f Mount Algorithm Polynomial Coefficientsu Mount Algorithm Coupled Variablerd Relative Displacement Across Mount

rv Relative Velocity Across Mountvsf Relative Velocity Normalization CoefficientF Internal Mount Forcefn Natural Frequency Damping Percentage (of Critical)

ABSTRACT

In support of a Navy R&D Program aimed at the development of shock isolationstrategies, a simple analytical tool was developed to aid in the performance of early-stageisolation system design iterations. This tool involves the idealization of an isolated raftsystem using a two degree-of-freedom (2DOF) spring-mass-damper system thatallows for nonlinear mount characteristics to be incorporated. For this system, thegoverning equations of motion are derived and solved using a finite differenceapproximation. After an analytical benchmark is performed, the resulting generalsolution is then applied to specific test cases involving isolated raft proof-of-concepttests using two different types of nonlinear mounts. The numerical solutions for thesetest cases are then compared directly to measured responses across and above theisolation mounts and evaluations of the accuracy of this simplified analytical tool aremade.


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INTRODUCTION

This paper documents the development of a simple analytical tool to be used in the early stages of thedesign of an isolation system. In the Navys shock isolation R&D Program, guidance was developed in the form ofa multi-step process that ultimately leads to an isolated raft design which provides the desired mitigatedenvironment for which Commercial-Off-The-Shelf (COTS) equipment can survive. A critical step in this processinvolves the isolator selection phase whereby a series of design pa rametric calculations are performedusing a simple 2DOF representation of the raft with mounted equipment and nonlinear isolators. The detailsof the development of this valuable analytical tool, which enables these design iterations to be made in an efficientmanner, are presented in the following sections of this paper.

The overall process is summarized in the collage shown in Fig. 1 below. The overall goal is to develop anisolated raft system that meets the COTS goals and can either be installed in a particular compartment onboardship, or, as illustrated in the lower right hand corner of Fig. 1, be placed in a test barge for proof-of-conceptdemonstration testing. Leading up to this point, however, are a number of steps which include analysis of adetailed finite element model of the final isolation design including a representation of the raft, installed COTSequipment items, and the isolators. Before this detailed analysis is possible, the selection of the particularisolator best suited for this isolation design must be made, which requires knowledge of the mounts dynamicforce-deflection-velocity relationship. To facilitate this selection, a simple 2DOF representation of the raft,mounted equipment and mounts, as shown in the upper left hand corner of Fig.1, provides a convenient way toefficiently investigate a variety of design options in a short period of time.

Fig . 1 Illustration of Isolated Raft Design Process


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Derivation of Equations of Motion for 2DOF System

For a 2DOF system, the derivation of the governing equations of motions begins with a study of aschematic of the idealized system, shown in Fig. 2. Here one can see that the idealized system includes two distinctmasses coupled together by a linear spring and damper, and the second mass, M 2, connected to the base

through a nonlinear mount interface. This nonlinear mount can be represented either by an analytic expression,lookup table or user supplied subroutine involving complex logic describing the behavior of the particularmount. A prescribed base motion drives the 2DOF system into motion.

To the right of the schematic in Fig. 2 is a free-body diagram of the idealized mount with an illustration of theforces that develop between the various elements. Notice the nonlinear force contribution indicated by the redcircle. Next, either Newtons Second Law or DAlemberts Principle is applied and the governing differentialequations of motion, given by Eq. (1) and Eq. (2), are derived. These equations represent a coupled set ofsecond order differential equations that are linear, except for the contribution of the nonlinear force term.

The next step is to develop a numerical solution to this coupled set of equations. The method selected was theapplication of Central Difference Approximations to the first and second derivatives. Central Differenceapproximations were selected in that they are convenient to apply, have no need for a starter method to begin

the solution, and lead to a set of explicit expressions for the propagation through time and thus generation of theoverall solution.

Fig. 2-Derivation of the Equations of Motion

( ) ( )

( ) ( )(2)Eq.

:

(1)Eq.

:M

112222

21212222

221111

21211111

NL

NL X

X

F X C KX X C KX X M

F X X C X X K X M F M

X C KX X C KX X M

X X C X X K X M F

+=++

+==

+=++

==

&&&&

&&&&

&&&&

&&&&


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( ) ( ){ }

( )( )

( )

2

(6)Eq. 2

2C-

2

12C

222

21

1,1

21

,11,2,2

21

1,1,122

1,222

,21,2

+

+

+

+

++

+

=

+

t C

t

M X

K t

M X

t X K X

t C

t

M t

F t

C X K X

t C

t

M X K

t

M X X

j

j j j

NL j j j j j

In developing the finite difference solution strategy, the Central Difference approximations for the first andsecond derivatives, given in Eq. (3) and Eq. (4), respectively, are substituted into the governing differentialequations. This process transforms the set of coupled second order differential equations into a set ofcoupled algebraic equations, which can be solved explicitly for the displacements of the masses M 1 and M 2 , which are represented by the variables X 1 and X 2 , respectively. This solution of the algebraic di fferenceequations leads to recurrence relations that are used to propagate forward in time in generating the numericalsolution.

[ ]

[ ] (4)Eq. 2)(

1

(3)Eq. 21

1,1,11,121

1,11,11

+

+

+

+

j j j

j j

X X X t

X

X X t X

&&

&

The next step in this solution strategy involves the application of the initial conditions. This establishes the ini tialstate of the variables found in the recurrence relations, thus enabling the start of the solution process. Theinitial conditions that apply to this problem are summarized in Eqs. (5) below.

X 1 (0) = 0

X 1 (0) = 0

X 2(0) = 0

X 2(0) = 0 . . Eqs . (5)

X 1 (0) = 0

X 1 (0) = 0

X 2(0) = 0

X 2(0) = 0 . . Eqs . (5)

Upon substitution of the Central Difference expressions into the governing differential equations of motion, andapplication of the associated initial conditions, the recurrence relations for X 2 and X 1 and their initial states arecompletely determined as Eq. (6) through Eq. (10) below:


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ANALYTICAL BENCHMARK OF NUMERICAL SOLUTION

Prior to applying this simple analytical tool toward the solution of a specific isolated raft problem, it was firstnecessary and instructive to benchmark this numerical strategy against a known solution to a test problem.Such a step helps verify that the numerical approximations were implemented correctly and thatresulting applications tend toward a convergent solution. This benchmark can either be performed with a

closed-form solution, or as done here, with a finite element solution.

For this benchmark example, the model shown in Fig. 3 was analyzed both with the numerical solutiondeveloped in this paper and with a finite element model. In this problem, the base excitation involves asinusoidal acceleration pulse wi th amplitude of 100 gs and a driving frequency of 15 Hz. The base motiondrives the rest of the 2DOF system through a 5 Hz linear spring. The values of M 1 and M 2 are 1000 lbs and2000 lbs, respectively, and these two masses are coupled together through a 10 Hz linear spring and alinear damper providing 5% of critical damping.

X2

1000#

2000#10 Hz5%

linear 5 Hz

Base

X1

Y

X2

1000#

2000#10 Hz5%

linear 5 Hz

Base

X1

Y

BASE INPUT

NSWCCD-UERD 10/18/00

Time (msec)

A c c e l e r a t i o n ( g ' s )

Max100.0

-100.Min

-120

-90

-60

-30

0

30

60

90

120

0 50 100 150 200 250 300

Fig. 3 Analytical Benchmark Problem

When this problem is solved using the analytical tool developed in this paper, as well as with the finite elementcode, CSA/NASTRAN, the results shown in Fig. 4 were obtained. From both the comparisons of the computedresponses for acceleration of M 1 and the relative displacement response across the linear 5 Hz spring, it is clearthat the two solutions are in exact agreement. This result gave confidence that the finite difference strategywas properly implemented, and thus at this point, this 2DOF analytical tool is now ready to be applied to anactual isolated raft problem.


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Payload Response

NSWC/CD-UERD 10/18/00

Shock Mount NASTRAN

Time (msec)

A c c e l e r a t i o n ( g ' s )

Max53.97

-60.1Min

-90

-60

-30

0

30

60

0 50 100 150 200 250 300

Mount Response

NSWC/CD-UERD 10/18/00

Shock Mount NASTRAN

Time (msec)

R e l a t i v e D i s p l a c e m e n t ( i n )

Max19.10

-20.3Min

-30

-20

-10

0

10

20

0 50 100 150 200 250 300

Fig. 4 Response Comparisons for Acceleration of M 1 and for Relative Displacement Across 5 Hz Spring

SPECIFIC NONLINEAR MOUNT MODELS

The next step is the application of this tool to a real problem. Two candidate mounts will be analyzed for theillustration in this paper. However, this method has been applied with numerous algorithms from the existingmount database at UERD. The top half of Fig. 5 contains photographs of both mount Type A and mountType B. Both of these isolators have undergone detailed characterization tests involving a large test massinstalled on a floating shock platform, subjected to underwater explosion (UNDEX) testing. The bottom halfof Fig. 5 illustrates the mount characterization test mass installed on the inner bottom of a floating shock platformfor one of the tests. During a series of UNDEX tests, dynamic response measurements were made below, aboveand across these mounts. From a comprehensive analysis of this data performed the Navy, the dynamicresponse characteristics of these mounts were identified and cast into the form of convenient algorithms fordescribing their responses under dynamic motions.

One of the initial steps in the mount characterization effort was to determine the linear stiffness and dampingtrends of the candidate isolators. This step results in useful insight as to the degree of linearity associated withthese particular isolators, as well as provides linearized forms of the respective mount algorithms thatsubsequently can be used for a first cut analysis using a linear analysis tool. In order to determine theselinear trends, regression planes are determined through statistical analyses of the mount characterizationdata, and from these identified planes the l inear stiffness and damping can be readily determined from therespective slopes of these curves. Fig. 6 illustrates the identification of such linear trends for mount Type A.

However, in most cases a detailed nonlinear characterization of isolators is required and thus a moredetailed analysis of the characterization data is necessary. Generally, nonlinear mount characterizationalgorithms can be found in the form of functional descriptions, lookup tables, or a combination of analytic

functions with nested logic in a user defined subroutine. For mount Types A and B discussed here, thefunctional form which best suited the characteriza tion was tha t of a fifth degree polynomial. Thisrepresentation is based on an independent variable, u, which is a linear combination of the relativedisplacement and relative velocity across the mount. The functional form and best-fit coefficients for mountsType A and Type B for both the axial and transverse directions are presented in Fig. 7.


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Fig. 5 Mount Characterization Approach


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Fig. 6 Example of Linear Stiffness/Damping Factor Trend

F b*u + c*u 2 + d*u 3 + e*u 4 + f*u5 u = rd + (rv/vsf)

Coefficient/Parameter Type A Type A Type B Type B

axial transverse axial transverse

b 5465.75 1216.65 10477.1 9387.9c 631.39 39.81 663.31 -186.88

d 993.37 62.76 -1918.24 -684.21

e -79.01 -2.42 -309.66 206.89

f -78.44 271.63 261.22

vsf 135 90.64 184.7 331.9

Limits of Validity

rd_lim_comp (inches) -1.9 -4 -1.8 -2.1

rd_lim_tens (inches) 1.4 3 1.1 1.4

rv_lim_comp (inches/second) -160 -150 -115 -150

rv_lim_tens (inches/second) 60 70 70 70

Fig. 7 Nonlinear Mount Algorithms for Mounts Type A and Type B

Type A, All Three Masses, AxialType A, All Three Masses, Axial


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One important point that must be made is that when applying numerical algorithms to shock isolation problems,one must be aware of the limits of validity of these algorithms. Such limits as determined for the Type A and TypeB mounts are presented in Fig. 7 for both the axial and radial directions. The ranges of application of thealgorithms must be kept within these limits to ensure reasonable accuracy and to avoid the generation oferroneous results. The limits of validity are related to the goodness of fit of the selected functional form, as well asto the ranges at which the mounts were exercised during their characterization testing. For the applications of this2DOF tool to actual isolated raft responses, only the analyses involving these two isolators for the verticaldirection will be presented in this paper. For this direction, the limits of validity for relative displacement for the

Type A and Type B mounts are 1.9 in. and 1.8 in. respectively.

APPLICATION OF 2DOF MODEL TO AN ISOLATED RAFT PROBLEM

In order to demonstrate the value and usefulness of the 2DOF analysis tools, application of this tool will nowbe made to an isolated raft system. The specific system for this demonstration involves a prototype raftfashioned for an actual shipboard space, as part of a Navy R&D Program proof-of-concept demonstrationtesting effort. This raft and simulated equipment arrangement are illustrated in Fig.8 below.

Fig. 8 Raft and Simulated Equipment Arrangement

The next step is to characterize the raft with a two-degree of freedom model. This idealization is performedaccording to the steps outlined in the Navys draft design guidance. The sketches in Fig. 9 illustrate how thisis accomplished for the particular raft under consideration. First of all, the total mass of the raft structure,supported equipment items, and half the weight of the mounts are computed to be 14,240 lbs., as noted inthe figure. This total mass is supported by four mounts located near the corners of the raft. For thisapplication, the assumption is made that this mass is evenly distributed amongst the four mounts. Thisquartering of the total mass is illustrated by the region shaded in green in Fig. 9, and amounts to 3,560 lbs. permount. A further partitioning of this mass is achieved by determining, usually from an accompanying finiteelement model of the entire raft system, the fundamental mode of flexural vibration and associated modaleffective weight. For this raft system, the fundamental vibration frequency (1 st mode vertical bending) wasdetermined to be 13.5 Hz, and the associated modal effective weight was 2,206 lbs.


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Fig. 11- Base Input Motion and Idealized 2DOF System With Type A Mount

Comparisons of the computed responses for the position on the raft near the mount location (M 2) aresummarized by the series of plots presented in Fig. 12. The upper left hand plot shows the computedacceleration for M 2 (red curve) compared with the corresponding measured acceleration (blue curve) for thislocation during the test. This comparison indicates that the 2DOF system correlates fairly well with the overallvariation of acceleration in time, bu t misses some o f the measured peak values. The associated velocityhistori es, obta ined th rough integration of these respective accelerations, are shown in the upper right handcurve and indicate a much more favorable comparison. The corresponding computed and measured shockresponse spectra are also in close agreement and are shown in the lower figure along with a notionalCOTS threshold, developed from a half-sine pulse. From this plot, one would draw the same conclusionfrom both the measured and computed curves that the isolated environment at the raft position directly abovethe mount falls below this threshold.

Similar comparisons for this same idealized system with the Type A mount, but at a position near the center of theraft corresponding to M 1, are presented in Fig. 13. Here it can be seen that the 2DOF model correlates verywell in both the acceleration and velocity domains, tracking extremely well with the peaks and time variation in bothcases. In addition, the corresponding shock response spectra also agree well, and both the measured andcomputed spectra for this mid-raft location fall below the notional COTS threshold.

The measured and computed nonlinear mount force-displacement histories are compared for the Type Amount in Fig. 14. From the shapes and extents of these respective hysteresis curves it is observed that themount algorithm for the Type A mount, in conjunction with this 2DOF idealization, produces a reliablerepresentation of the across-mount dynamic response characteristics. Also, from this plot, it can be seen that themaximum excursion across the Type A mount is approximately 1.7 in., which falls within the earlier stated rangeof applicability for the mathematical algorithm developed for this mount.

Next, a similar set of comparisons is made for calculations performed for the Type B mount. The base inputmotion used in the 2DOF analysis for this mount is shown in Fig. 15, along with the idealized 2DOF systemwith the Type B mount. Comparisons of the computed responses for the position on the raft near the mountlocation (M 2) are illustrated in Fig. 16 for this mount. As was the case for the Type A mount, this comparison forthe Type B mount indicates that the 2DOF system also correlates well with the overall variation of acceleration intime, but underestimates some of the measured peak values. The associated velocity histories are shownin the upper right hand curve and indicate a much better agreement. The corresponding computed andmeasured shock response spectra are also in close agreement and are shown in the lower figure along with a

Type A MountA7001V TestFixture Input


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Fig. 12 Comparison of Measured and Computed Responses (Type A Mount) Near Mount Location

Fig. 13 Comparisons of Measured and Computed Responses (Type A Mount) Near Center of Raft

COTS Threshold A4022V 2DOF Raft

COTS Threshold A4004V 2DOF Payload


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Fig. 14 Type A Mount Nonlinear Mount Performance Computed vs. Measured

Fig. 15 Base Input Motion and Idealized 2DOF System With Type B Mount

Type B Mount

A7001V TestFixture Input


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Fig. 16 Comparisons Between Measured and Computed Responses (Type B Mount)Near Mount Location

notional COTS threshold. Again, as with the Type A comparisons one would draw the same conclusion from boththe measured and computed Type B response curves that the isolated environment at the raft position directlyabove the mount falls below this threshold at a location near the mount location.

Comparisons for this same idealized system with the Type B mount at a position near the center of the raftcorresponding to M 1, are presented in Fig. 17. Here excellent correlation between the measured responses andthose computed with this simple 2DOF model are observed for both the acceleration and velocity domains.This agreement holds true for both the comparisons of the peaks and time variations. In addition, thecorresponding shock response spectra also agree well, and both the measured and computed spectra for thismid-raft location for the Type B mount fall below the notional COTS threshold at all frequencies, except for aslight violation at the mount natural frequency.

The measured and computed nonlinear mount force-displacement histories for the Type B mount are presented inFig. 18. Again good agreement is observed for both the extents and shapes of these respective hysteresiscurves, and, as was the case for the Type A mount, it is concluded here that a simple 2DOF modelproduces a reliable representation of the across-mount dynamic response characteristics for the Type B mount.Also, from this plot, it can be seen that the maximum excursion across the Type B mount is approximately 1.5 in.,which falls in the earlier stated range of applicability for the mathematical algorithm developed for this mount.

COTS Threshold A4022V 2DOF Raft


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Fig. 17- Comparisons Between Measured and Computed Responses (Type B Mount) Near Center of Raft

Fig. 18 Type B Mount Nonlinear Mount Performance Computed vs. Measured

COTS Threshold A4004V 2DOF Payload


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Analysis Isolated Raft Systems Using 2 DOF Nonlinear Spring Mass Model

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