Important design issues and engineering applications of SDOF system Frequency response Functions

Transcription

1 Impotnt design issues nd engineeing pplictions of SDOF system Fequency esponse Functions The following desciptions show typicl questions elted to the design nd dynmic pefomnce of second-ode mechnicl system opeting unde the ction of n extenl foce of peiodic ntue, i.e. F (t) cos(ωt) o F (t) sin(ωt) The system EOM is: M D K F t o cos Recll tht the system esponse is govened by its pmetes, i.e. stiffness (K), mss (M) nd viscous dmping (D) coefficients. These pmetes detemine the fundmentl ntul fequency, K n, nd viscous dmping tio, M D D c,with D c KM In ll design cses below, let ( Ω / n ) s the fequency tio. This tio (excittion fequency/system ntul fequency) lgely detemines the system peiodic foced pefomnce. Design Issues: SDOF FRF - Luis Sn Andés 03

2 PROBLEM TYPE Conside system excited by peiodic foce of mgnitude with extenl fequency Ω. ) Detemine the dmping tio needed such tht the mplitude of motion does not eve exceed (sy) twice the displcement ( s /K) fo opetion t fequency (sy) 0% bove the ntul fequency of the system (. n ). b) With the esult of (), detemine the mplitude of motion fo opetion with n excittion fequency coinciding with the system ntul fequency. Is this esponse the mximum eve expected? Explin. Recll tht system peiodic esponse is t () H cos( t ) s ( ) Solution. Fom the mplitude of FRF H 6 s H ( ) ( ) Amplitude, H 4 Set. nd / s H. Find the dmping tio fom the lgebic eqution: H ( ) ( ) H H fequency tio, Finlly, clculte the viscous dmping coefficient D D c Fo excittion t the ntul fequency, i.e., t esonnce, then, / s /() Q. Thus Q s Design Issues: SDOF FRF - Luis Sn Andés 03

3 The mximum mplitude of motion does not necessily occu t. In ctulity, the mgnitude of the fequency tio ( * ) s which mximizes the esponse, 0, is (fte some lgebic mnipultion): * ;nd s mx Coected /9/3 Note tht fo smll vlues of dmping s mx Design Issues: SDOF FRF - Luis Sn Andés 03 3

4 PROBLEM TYPE Conside system exited by n imblnce (u), giving n mplitude of foce excittion equl to M u Ω. Recll tht um e/m, whee m is the imblnce mss nd e is its dil loction M D K Mu cost Recll tht system peiodic esponse is () t uj cos( t ) ( ) ) Wht is the vlue of dmping necessy so tht the system esponse neve exceeds (sy) thee times the imblnce u fo opetion t fequency (sy) 0% below the ntul fequency of the system (0.9 n ). b) With the esult of (), detemine the mplitude of motion fo opetion with n excittion fequency coinciding with the system ntul fequency. Solution Fom the fundmentl FRF mplitude tio u J ( ) ( ) J Amplitude, J fequency tio, Set 0.9 nd /u J 3. Clculte the dmping tio fom the lgebic eqution. Design Issues: SDOF FRF - Luis Sn Andés 03 4

5 J Finlly, clculte the viscous dmping coefficient, D D c. Note tht fo foced opetion with fequency ntul fequency, i.e., t esonnce,, /u /() Q. Thus Q u The mximum mplitude of motion does not occu t. The vlue of fequency tio ( * ) which mximizes the esponse is obtined fom u 0 then * ;nd u mx coected /9/3 Note tht fo smll vlues of dmping u mx Design Issues: SDOF FRF - Luis Sn Andés 03 5

6 PROBLEM TYPE 3 Conside system excited by peiodic foce of mgnitude nd fequency Ω. Assume tht the sping nd dshpot connect to gound. ) Detemine the dmping tio needed such tht the tnsmitted foce to gound does not eve exceed (sy) two times the input foce fo opetion t fequency (sy) 75% of ntul fequency (0.75 n ). b) With the esult of (), detemine the tnsmitted foce to gound if the excittion fequency coincides with the system ntul fequency. Is this the mximum tnsmissibility eve? c) Povide vlue of fequency such tht the tnsmitted foce is less thn the pplied foce, iespective of the dmping in the system. Solution: Fom the fundmentl FRF mplitude fo bse foce excittion tnsmitted F K D F tnsmitted F o A T( ) ( ) Set A T nd 0.75, nd find the dmping tio. AT AT 0.86 * Amplitude fequency tio, Design Issues: SDOF FRF - Luis Sn Andés 03 6

7 Finlly, clculte the viscous dmping coefficient D D c At esonnce,, AT mgnitude of the tnsmitted foce.. Then clculte the Agin, the mximum tnsmissibility occus t fequency f * A which stisfies T 0 closed fom solution.. Pefom the deivtion nd find Recll tht opetion t fequencies, i.e. fo Ω.44 n, (4 % bove the ntul fequency) detemines tnsmitted foces tht e lowe thn the pplied foce (i.e. n effective stuctul isoltion is chieved). Design Issues: SDOF FRF - Luis Sn Andés 03 7

8 PROBLEM TYPE 4 Conside system excited by peiodic foce with mgnitude M cc (fo exmple) nd fequency Ω. ) Detemine the dmping tio ζ needed such tht the mximum cceletion in the system does not exceed (sy) 4 g's fo opetion t fequency (sy) 30% bove the ntul fequency of the system (.3 n ). b) With the esult of (), detemine the system cceletion fo opetion with n excittion fequency coinciding with the system ntul fequency. Explin you esult Recll the peiodic esponse is () t sh( ) cos( t ), then the cceletion of the system is t () H cos( t ) t () s ( ) Solution: Fom the mplitude of FRF F / K o n ( ) F / M o ( ) Follow simil pocedue s in othe poblems bove. OTHER PROBLEMS Think of simil poblems nd questions elted to system dynmic pefomnce. Design Issues: SDOF FRF - Luis Sn Andés 03 8

9 In pticul, you my lso "cook up" simil questions elted to the dynmic esponse of fist-ode systems (mechnicl, theml, electicl, etc). M V DV F t o cos Luis Sn Andés - MEEN 363/67 instucto The following woked poblems should tech you how to pply the fequency esponse function to esolve issues nd to design mny mechnicl systems Design Issues: SDOF FRF - Luis Sn Andés 03 9

10 P. Peiodic foced esponse of SDOF mechnicl system. DESIGN COMPONENT The signl lights fo il my be modeled s 76 lb mss mounted 3 m bove the gound of n elstic post. The ntul fequency of the system is mesued to be. Hz. Wind buffet genetes hoizontl hmonic F foce t Hz. The light filments will bek if thei pek cceletions exceed 5g. Detemine the mximum cceptble foce mplitude F when the dmping tio ζ0.0 nd 0.0. Full gde equies you to explin the solution pocedue with due ttention to physicl detils 3 m The excittion foce is peiodic, sy F(t)Fo sin(ωt). then the system esponse will lso be peiodic, Y(t), with sme fequency s excittion. Assuming stedy stte conditions: STEADY RESPONSE of M-K-C system to PERIODIC Foce with fequency ω ω Cse: peiodic foce of constnt mgnitude Define opeting fequency tio: ω n Ft () sin ωt whee: System peiodic esponse: Yt () δsh() sin ωt + Ψ δ H () s ( ) + ( ζ) tn Ψ K e () ζ ce with ngle, nge: 0 to -80deg Fom (), the cceletion is t () ω Yt () Asin ω t + Ψ 80 ( the mgnitude of cceletion is A F o ω o A K e ( ) + ( ζ) M e ( ) ζ + hence, define system mss A mx : M e : 0g 50lb mximum llowed cceletion of filment HZ : π s f n : HZ ntul fequency f : HZ f Let o : f n o The mximum foce llowed equls excittion fequency due to wind buffets F mx, ζ close to ntul fequency : A mx M e ( ) ζ +

11 : without ny dmping with dmping F mx o, ξ ξ : 0. Fo the foce found the mplitude of cceletion is F mx ( o, 0) 33.7 lbf F mx ( o, ξ) lbf A(, ζ) : M e ( ) ζ Note the impotnce of dmping tht leds to substntil incese in foce llowed + F mx o, ξ F mx o, 0 53 Amplitude of cc/g 3 6 ff n A mx g f n s ξ 5 GRAPH NOT FOR EAM fequency (d/s) with dmping without dmping SInce o~, simple engineing fomul gives A mx M e ξ 300 lbf which gives vey good estimtion of the mximum wind foce llowed b) system with dmping ξ0. will poduce 55 % incese in llowble foce Hence, the il lightsystem will be moe elible, lsting longe. F mx o, ξ F mx o, 0 53 c) Posts e usully hollow fo the cbles to be outed. These posts hve lyes of elstomeic mteil (~ubbe-like) inside to incese thei stuctul dmping. Moden posts e wound up fo composites tht integte dmping lyes. Clely, dding "tue" dshpot is not cost-effective

12 EAMPLE - EAM TYPE: Dynmic mesuements wee conducted on mechnicl system to detemine its FRF (fequency esponse function). Focing functions with multiple fequencies wee exeted on the system nd digitl signl nlyze (FFT) ecoded the mgnitude of the ACCELERATION/FORCE ([m/s]/n) Fequency Response Function, s shown below. Fom the ecoded dt detemine the system pmetes, i.e. ntul fequency (wn:d/s) nd dmping tio (z), nd system stiffness (K:N/m), mss (M:kg), nd viscous dmping coefficient (C:N.s/m). Explin pocedue of ANALYSIS/INTERPRETATION of test dt fo full cedit. Acceletion/Foce (m/s^/n) Solution: Ax x 50 fequency (Hz) Mgnitude of FRF fo mechnicl system Test dt showing mplitude of (cceletion/foce) Recll tht fo n imposed extenl foce of peiodic fom: the system esponse Y(t) is given by: whee the mplitude of motion (Yop) nd phse ngle (Y) e defined s: Ft () sin ωt Yt () Y op sin ωt + ψ [] [] Y op () [3] fom [], we find tht the cceletion is given by: K ( ) + ζ Ψ tn ζ [3b] with ω ω n [4] Y () t ω Y op sin( ωt + ψ) op sin ωt + Ψ 80 [5] whee: op () M ( ) + ζ [6] since: K ω ω M ω n K

13 thus, the mgnitude of mplitude of cceletion ove foce mplitude follows s: op () Fo excittion t vey high fequencies, >>.0 Fom the gph (test dt): ( ) + ζ M 0. m s N M [7] M Thus The units of this expession e /kg m op () M : 0kg s N The system ppes to hve little dmping, i.e. mplitude of FRF ound fequency of 50 Hz is the lge nd vying pidly ove now fequency nge. Thus, tke the ntul fequency s expessed in d/s s: We cn estimte the stiffness (K) fom the fundmentl eltionship: K : fo excittion t the ntul fequency (), the tio of mplitude of cceletion to foce educes to op f o M ζ fom the gph (test dt), the tio is ppoximtely equl to one (/kg). Thus. the dmping tio is detemined s ζ :.0 M ζ 0.05 kg ω n M f n : 50Hz ω n : f n π ω n 349 d s K N m Tht is, the system hs dmping tio equl to 5%. This esult could hve lso been esily obtined by studying the tio of (mplitude t the ntul fequency divided by the mplitude t vey high fequency, i.e.) 0 ζ 0. Once the dmping tio is obtined, the dmping coefficient cn be esily detemined fom the fomul: C : ζmω n C 349 N s m The numbe of clcultions is miniml. One needs to intepet coectly the test dt esults, howeve.