PRACTICAL CONSIDERATIONS IN HUMAN-INDUCED VIBRATION Bars Erkus, 4 March 007 Itroducto Ths docuet provdes a revew of fudaetal cocepts structural dyacs ad soe applcatos hua-duced vbrato aalyss ad tgato of structures. Modal Aalyss of MDOF Structures Cosder a geeral DOF syste defed by Mx () t + Cx () t + Kx() t F() t () Where M, C ad K are the ass, dapg ad stffess atrces, F() t s the exteral exctato, ad x() t s the dsplaceet vector. Here, M ca be a dagoal atrx wth proper choces of the referece fraes. I practcal applcatos, M ad K atrces are coputed usg fte eleet (FE) procedures, C s predefed. It ca be show that the structural respose, x() t ca be wrtte ters of ay odal atrx, Φ ad the oral coordates, q() t as x() t Φq() t ad x () t Φq () t ad x () t Φq () t () where φ Φ φ φ φ φ ad φ (3) φ where φ s the th ode shape. Substtutg () to () ad pre-ultplyg by Φ T, aother for of the equato of oto s obtaed as Φ TMΦq () t + Φ T CΦq () t + Φ T KΦq() t Φ T F() t (4) It s coveet to defe ew structural atrces as M Φ T MΦ, C Φ T CΦ, K Φ T KΦ ad F() t Φ T F() t (5). Arup, Los Ageles; Bars.Erkus@arup.co
Practcal Cosderatos Hua Iduced Vbratos, Bars Erkus 4 March 007 where M, C ad K are called geeralzed ass, dapg ad stffess atrces, respectvely, ad F s the geeralzed exctato. Due to orthogoalty of the ode shapes, these ew structural atrces are dagoal,.e. c c k k M, C ad K (6) c k It should be oted that sce the dapg atrx C s predefed such a way that the geeralzed dapg atrx C has a dagoal for for ease of atheatcal coputatos. Ths type of dapg s kow as classcal dapg. Geeralzed atrces do ot have uch practcal eag ths for. However, f the odal atrx, Φ s selected a way that ode shapes are ass orthooralzed, the geeralzed structural atrces take a ore elegat ad useful for as M ζ λ ζ I, C λ λ ad K (7) ζ λ λ λ where ζ ad λ are the odal dapg rato ad the frequecy of the th ode. Therefore, whle a ass orthooralzed odal atrx yelds a detty geeralzed ass atrx, a radoly selected odal atrx yelds erely a dagoal geeralzed ass atrx. It s very portat to ote that odal dapg ratos ad frequeces are depedet of the selected odal atrx. Therefore, to characterze the behavor of a ode, oe oly eeds the odal dapg rato ad frequecy; geeralzed ass, s ot requred. To characterze the behavor of the overall MDOF syste, oe eeds the odal atrx that s used the egevalue aalyss of the equato of oto addto to the odal dapg ratos ad frequeces. Modal Te-Hstory Aalyss of MDOF Structures There are several uercal ethods to solve the atrx dfferetal equato (). Oe of ths ethod s to solve () drectly wth explct or plct uercal schees such as Newark s β ethod ad Ruge-Kutta ethod, whch ca be called te-hstory aalyss. Aother ethod s to solve the odal equatos that results fro the odal aalyss, ad the to cobe the odal resposes to fd the fal
Practcal Cosderatos Hua Iduced Vbratos, Bars Erkus 4 March 007 structural resposes. Ths ethod ca be called odal te-hstory aalyss. Modal te-hstory aalyss s cosderably te-cet ad ore frequetly used for hgher degree-of-freedo systes copared to te-hstory aalyss. However, a typcal odal te-hstory aalyss always requre a dagoal dapg atrx, whle regular te-hstory aalyss ca solve systes wth ay type of dapg atrx. I the followg, a revew of the odal te-hstory aalyss s gve. The odal equatos obtaed fro equato (4) ca be wrtte as st ode: q t + c q t + k q () t F () t d ode: q t + c q t + k q () t F () t -th ode: q t + c q t + k q () t F () t (8) Each of these equatos ca be solved usg regular te-hstory ethod, ad fal structural resposes ca be foud usg equatos (). Effectve Mass Ths secto provdes a geeral for of a ectve ass of a luped-ass MDOF syste that vbrates a predefed shape. Note that ths shape s arbtrary ad does ot eed to be a odal shape of the syste. Cosder the luped-ass, DOF syste show Fg., ad assue that t vbrates arbtrarly as show Fg. such that the deforato of the structure wll follow a fxed shape,.e. x () t ( costat)x () t, for all ad at te t (9) where x () t s the dsplaceet of ass at te t. The dsplaceet of ay ot ca be wrtte ters of a uque te-fucto, qt () ad a costat, a as x a qt (). Also, x a qt (). I ths case, the overall dsplaceet of the structure ca be wrtte a vector for as x x () t x () t a qt () a qt () or x aq() t where a x () t a qt () a (0) x () t a qt () a a a 3
Practcal Cosderatos Hua Iduced Vbratos, Bars Erkus 4 March 007 Jot Jot Jot Jot FIG. A geeral luped-ass DOF syste Now cosder the followg proble: Fd a realzato of the DOF syste gve Fg. such that the equvalet syste s characterzed wth the vbrato of oly oe DOF (stead of DOFs) ad a ectve ass s attached to that ot. Such a realzato s show Fg. 3. To fd the ectve ass,, the ketc eergy of the orgal syste s equated to the ketc eergy of the equvalet syste at te t. The ketc eergy of the orgal syste s gve by E orgal -- ( x () t ) + -- ( x () t ) + + -- ( x () t ) -- ( a q () t ) + -- ( a q () t ) + + -- ( a q () t ) -- ( a + a + + a )( q () t ) -- ( q () t ) k a k k () The ketc eergy of the equvalet syste s FIG. Arbtrary vbrato 4
Practcal Cosderatos Hua Iduced Vbratos, Bars Erkus 4 March 007 FIG. 3 A equvalet syste wth sgle dsplaceet E equv -- -- ( x () t ) a ( q () t ) () Equatg the ketc eerges yelds k k a k --------------------- a (3) Equato (3) ca be used to fd the ectve asses of varous vbrato shapes. Effectve Mass for a Mode Shape I ths secto, frst, a revew of ode shapes ad geeralzed asses s gve. The, a ectve ass for a gve ode shape s foud. I the followg, ectve asses for two cases are derved: (a) a radoly selected odal atrx ad (b) a ass orthooralzed odal atrx. Cosder the proble defed the prevous secto. Let a DOF structure vbrate ts ode shape stead of a arbtrary predefed shape,.e., a or a k φ k for all k. Please ote that the selected ode atrx Φ s radoly selected ad s ot ecessarly a ass orthooralzed ode atrx. It ca be show wth soe atrx algebra that the geeralzed ass of the by φ ode (6) s deed gve 5
Practcal Cosderatos Hua Iduced Vbratos, Bars Erkus 4 March 007 k k ( φk ) (4) Therefore, the ectve ass for the ode ad dsplaceet s gve by th, ----------- ( φ ) (5) If the ode shapes are selected such that the odal atrx s ass orthooralzed as gve (7), the (6) ad the ectve ass for ths specal case s gve by ----------- ( ), φ (7) Ths copletes the dervato of the ectve asses. I the followg secto, ectve asses are derved thorugh the equato of oto of the equvalet syste. Respose to a Exctato I ths secto, a geeralzed for of the ectve ass coputato s gve whe a DOF, luped-ass structure vbrates a predefed arbtrary shape ad excted by a arbtrary force at a gve DOF. The, a specal case s cosdered, where the vbrato shape s take as oe of the a ode shapes of the structure stead of a arbtrary shape. Also gve for ths specal case s the peak resposes whe the exctato s a susodal fucto. Let the DOF structure show Fg. vbrate a arbtrary shape defed by equatos (0) wth a exteral force F(t) appled at the drecto of x. The goverg equato of oto of ths vbrato ca be foud easly by applyg vrtual eergy prcples as DOF as show Fg. 4. Note that the force F(t) s excted exactly --- a k a k q ---a a T + Caq + ---a a T Kaq Ft () k (8) whch ca be wrtte as 6
Practcal Cosderatos Hua Iduced Vbratos, Bars Erkus 4 March 007 Jot Jot FIG. 4 Arbtrary vbrato wth a exctato k a k q + a T Caq + a T Kaq a Ft () k (9) where C ad K are the dapg ad stffess atrces of the DOF syste, respectvely. Now, cosder a syste that s equvalet to the DOF syste, whch s gve Fg. 4, such that a ectve ass s attached to the DOF, ad a exteral force s appled at the DOF. Ths equavalet th syste s show Fg. 5. The goverg equato of oto of the equvalet syste ca also be derved usg vrtual eergy prcples as a ---- q ---a T + Caq + ---a T Kaq Ft () a a a (0) whch ca be wrtte as a q + a T Caq + a T Kaq a Ft () () th Therefore, the ectve ass at the DOF for a forced vbrato at the DOF s gve by k k a k --------------------- a () The respose of the overall structure ca be foud by two ethods. I the frst ethod, the equato () s solved for qt (), or equvaletly, the equato 7
Practcal Cosderatos Hua Iduced Vbratos, Bars Erkus 4 March 007 FIG. 5 Equvalet SDOF syste for the forced vbrato q ---- a T + Caq + ---- a T Kaq a a a ---- Ft () a (3) s solved for q. The, the structural resposes are gve by x aq,.e. x l a l q for ay DOF l. I the secod ethod, the goverg equato of oto of the equvalet SDOF syste s wrtte as x a T Ca a T Ka + -------------x + -------------x a a a a ---F() t (4) ad solved for x ad the respose of structure s the foud as x a l ---ax,.e. x l ---x (5) a a Next, the equvalet SDOF s foud for a specal case where the arbtrary shape s set to oe of the ode shapes of the structure. Note that the ode shape s ot ecessarly ass or dsplaceet oralzed. Let the ode shape cosderato be the th ode,.e. a. The equato of oto of the DOF structure becoes k ( φk ) q + φ T C q + φ T K q φ Ft () k (6) whch ca be splfed to q + ζ ω q + ω q φ Ft () (7) 8
Practcal Cosderatos Hua Iduced Vbratos, Bars Erkus 4 March 007 the equatos defed above. I addto, peak structural resposes ca be coputed usg the sae equa- where, ζ are ω gve by (7). Slarly, the equato of oto of the equvalet SDOF syste becoes Coparg equatos (7) ad (8), the ectve ass that s attached to the whch s vbratg the ( φ ), th q + ζ ω q + ω q φ Ft () ode s foud to be th (8) DOF of the structure,, ------------- ( φ ) (9) As explaed before, the structural resposes ca be foud two ways. I the frst ethod, the equato of oto of the equvalet SDOF s wrtte ters of q as follows:, q +, ζ ω q +, ω q -------------F() t ( φ ) (30) ad the structural resposes are foud by solvg (30) for q ad are gve by x q,.e. x l φ l q for ay DOF l. I the secod ethod, equato (8) s rewrtte ters of x as, x +, ζ ω x +, ω x ----- Ft () (3) ad the structural resposes are foud by solvg (3) for x ad are gve by x φl ------ x,.e. x l ------ x (3) It s easy to ote that the two ethods preseted above are detcal except that forer oe uses oralzed odal resposes whle the latter uses the structural resposes the equato of oto, whch results dfferet forcg fucto cocets, ad therefore sae structural resposes. Now, cosder a specal case where the exctato s a susodal fucto ad s defed by Ft () Ps( ω t) (33) ω th where s the atural frequecy of the ode. The structural resposes ca coveetly foud by 9
Practcal Cosderatos Hua Iduced Vbratos, Bars Erkus 4 March 007 tos. As a exaple, cosder equato (30) ad the peak structural accelerato. The peak structural accelerato correspods to the peak of q, whch ca be foud as q peak P ------------- ( φ ) ------------------------, ζ (34) whch s splfed usg (9) to q peak φ P ----------------- ζ (35) ad the peak structural resposes are gve by x peak φ q peak peak,.e. x l φ l q peak l φ P -----------------. (36) ζ Also cosder the equato (3). The peak structural accelerato x peak s gve by x peak P ------ ------------------------ φ, ζ (37) whch splfes to x peak ----------------- P ζ (38) ad the peak structural acceleratos are gve by x peak ------ peak φ,.e. (39) φ x x l peak φl peak ------ φ x l φ P ----------------- ζ whch s detcal to (36). These results show that peak structural resposes are depedet of the ectve ass the case of a specal vbrato, where the structure deforato s defed by oe of ts odes. Also ote that f the odal shapes, φ above equatos. are selected such that they are ass orthooralzed, the the Oe portat observato s that the oly requred forato to copute peak acceleratos s the ode shapes ad correspodg geeralzed asses. Dfferet oralzato ethods do ot provde ay sgfcat coveece. 0
Suary Ths docuet provdes the coputato of ectve asses for a geeral -DOF, luped-ass syste whe t vbrates a predefed shape. Specal cases are cosdered whe the predefed shape s selected as oe of the ode shapes of the structure ad a exctato s appled at a pot the drecto of the correspodg DOF. Above dervatos are straghtforward for cotuous systes. The results are suarzed Table. TABLE Effectve asses, geeralzed asses ad peak acceleratos for dfferet vbrato shapes Vbrato Characterstcs Vbrato Vector Geeralzed Mass Effectve Mass Peak Accelerato a Arbtrary a k a k a k k Solve (3) or (4) k k --------------------- a th ode k ( φ k ) (ot oralzed), k φ ------------- ( ) x l peak l φ P ----------------- ζ th ode φ (dsplaceet oralzed) b k ( φ k ), k th ode φ (ass oralzed) c, ------------- ( φ ) x l peak l φ P -------- ζ φ a. For exctato Ft () Ps( ω t) appled at DOF b. Ths s the stadard procedure GSA c. Ths s the stadard procedure SAP000 ad alterate procedure GSA ------------- ( ) x l peak l φ P ----------------- ζ