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Finally, the nratrix of coeflicients [lJ usui,illycontains the ;xrrameterr thatexpres how the parts of fhe systernirte ract or are coupled Consequently, Eq ( I I 4) mi-qht h reexpressed as : {s IInteractionsl{response} timuli} As we know from plevious chapters, there are a variety of ways to solveEq(114) Howeverusing the matrix inverseyielclsatparticularlyinteresting resultThe fbrmal solution can be expressed as
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or (r'ecalling detlnitionof ntatrixmultiplication our from Section 2) 8I
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Thus, we lind that the invertedmatrix itself',asidetiom providing a solution, exhas trernelyuseful properties That is, eachof its elementsrepresents response asingle the of part of the system a unit stimulus any otherpart of the system to of Notice that theselbrmulationsare Iinearand, theretbre, superposition proportionand (the ality hold Superltositirn meansthat if a systemis subjectto severaldifTerent stimuli b's), the responses be corr-rputed individuallyand the resultssummed obtainatotal can to response roportionalin' meansthat multiplying the stimuli by a quantityresults the P in re sponse thosestimuli beingmultipliedby the sarne to quantity ThLrs, coefficientour the is a p r o p o r t i o n a l i t y c o r l s t a n t t h a t g i v e s t h e v a l ud u etr -a u n i t l e v e l o f l r s T h i s r e s u l t r s eof o 1 independentof the eft-ects b2 and 6: on ,r1, which are retlected in the coefficients of no' trnda,', respectively Therefore, can draw the general we conclusion theelement that a,,toftheinvertedmatrixrepresentsthevalueofx,cluetoaunitquantityofb, Usin-e exanrple the structure, the of elementrr,l' of the matrix inversewouldrepresent the fbrce in member i due to a unit externalforce at node1 Even for sntall systems, such behavior individualstimulus-response of interactions would not be intuitivelyobvious As suchthe matrix inverseprovidesa powerful technique understanding interrelation1br the shipsof component partsof complicated systems
EXAMPLE2 I I
A n o l y z i n g t h e B u n g e eJ u m p e rP r o b l e m Problem Stotement At the beginning chap 8, we set up a probleminvolvingrhree of individualssuspended vertically connected bungeecordsWe deriveda s)'stem linear by of algebraicequations basecl fbrce balances eachjuniper, on tilr
| L 0
I - rso -roo -o 0 r00 rsO s I
--50 s0 I |
I,I _
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-ss86 I
In Example 82, we used MATLAB to solve this system for the vertical positionsof the jumpers(ther's) In the present example, MATLAB to compute matrixinverse use the and
I I 2 E R R O A N A L Y S IA N D S Y S T E M O N D I T I O N R S C
Solution Start up MAILAB and enterthe coefficienrmatrix: >> K t150 -100 0; 100 150 50;O -50 501;
The inversecan then be computedas
>> KI Kf = = inv(K)
00200 00200 00200
00200 00300 00300
00200 00300 00500
Each elementof the inverse, I of the invertedmatrix represents vertical change k, the in position (in meters)of jumper I due to a unit changein fbrce (in Newtons) applied to Jumper/ First, observethat the numbersin the first column ("1: l) indicatethat the positionof all threejumpers would increaseby 002 m if the force on the first jumper was increased by 1 N This makessense, because additionalforce would only elongatethe first cord the by that amount ln contrast,the numbersin the secondcolumn (j :2) indicatethat applying a force jumper would move the first jumper down by 002 m, but the second of I N to the second and third by 003 m The 002-m elongationof the first jumper makes sensebecause the first cord is subjectto an extra I N regardless whetherthe force is appliedto the first or of secondjumper However, for the secondjumper the elongationis now 003 m because along with the first cord, the secondcord alsoelongates due to the additionalforceAnd of course,the third jumper showsthe identicaltranslationas the second jumper as thereis no additionalforce on the third cord that connects them As expected,the third column (j:3) indicatesthat applying a force of I N to the jumpers moving the samedistances occured third jumper resultsin the first and second as jumper However,now because the additional when the force was appliedto the second of elongationof the third cord, the third jumper is moved fartherdownward Superposition and proportionalitycan be demonstrated using the inverseto deterby mine how much fartherthe third jumper would move downwardif additionalforcesof 10, 50, and 20 N were appliedto the first, second, and third jumpers,respectively This can be done simply by using the appropriate elements the third row of the inverseto compute, of A r - j : / 3 r r A F r+ k r j a f ' 2 + k ; r A F r : 0 0 2 ( 1 0 ) 0 0 3 ( 5 0 ) 0 0 5 ( 2 0 ) 2 J m : + +
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