RAYGH'S METHOD Revisio D B To Irvie Eail: toirvie@aol.co Noveber 5, Itroductio Daic sstes ca be characterized i ters of oe or ore atural frequecies. The atural frequec is the frequec at which the sste would vibrate if it were give a iitial disturbace ad the allowed to vibrate freel. There are a available ethods for deteriig the atural frequec. eaples are Soe. Newto s aw of Motio. Raleigh s Method. Eerg Method. agrage s Equatio Note that the Raleigh, Eerg, ad agrage ethods are closel related. Soe of these ethods directl ield the atural frequec. Others ield a goverig equatio of otio, fro which the atural frequec a be deteried. This tutorial focuses o Raleigh s ethod, which ields the atural frequec. Raleigh's ethod requires a assued displaceet fuctio. The ethod thus reduces the daic sste to a sigle-degree-of-freedo sste. Furtherore, the assued displaceet fuctio itroduces additioal costraits which icrease the stiffess of the sste. Thus, Raleigh's ethod ields a upper liit of the true fudaetal frequec. Defiitio Raleigh's ethod ca be suarized as where a PE a KE = total eerg of the sste () KE = kietic eerg PE = potetial eerg
Note that potetial eerg is also referred to as strai eerg for the case of certai sstes, such as beas. Equatio () ca ol be satisfied if the sste is vibratig at its atural frequec. Refereces. T. Irvie, Natural Frequecies of Bea Bedig Modes, Revisio E, Vibratiodata, 99.
APPENDIX A Pedulu Eaple Cosider a coservative sste. A eaple is the pedulu show i Figure A-. g Figure A-. The kietic eerg becoes zero whe the pedulu reaches its aiu agular displaceet. The kietic eerg reaches its aiu value whe the pedulu passes through =, which is also the "static equilibriu poit." O the other had, the potetial eerg reaches its aiu level as the pedulu reaches its aiu agular displaceet. The potetial eerg reaches its iiu value whe the pedulu is at its static equilibriu poit. For siplicit, the potetial eerg ca be cosidered as zero at the static equilibriu poit. et = pedulu ass = legth = agular displaceet Assue a sall agular displaceet.
The potetial eerg is PE g( cos ) (A-) The kietic eerg is Assue a displaceet equatio of KE ( ) (A-) ( t) si t (A-) a (A-) The velocit equatio is ( t) cos t (A-5) a (A-6) PE g( cos ) a Cosider the epasio (A-7) cos (A-)!! Cosider the aiu potetial eerg. Substitute equatio (A-) ito (A-7). PE a g (A-9)!! PE a g (A-)!!
PE a g (A-) Cosider the aiu kietic eerg. Substitute equatio (A-6) ito (A-). a (A-) KE g (A-) Siplifig, g (A-) The pedulu atural frequec is thus g (A-5) 5
APPENDIX B Catilever Bea with Ed Mass Cosider a ass outed o the ed of a catilever bea, as show i Figure B-. Assue that the ed-ass is uch greater tha the ass of the bea. Figure B-. E is the odulus of elasticit I is the area oet of iertia is the legth g is gravit is the ass is the displaceet The static stiffess at the ed of the bea is k (B-) Equatio (B-) is derived i Referece. The potetial eerg is PE (B-) The kietic eerg is KE (B-) Assue a ed displaceet of Asi t (B-) 6
The correspodig velocit is Acos t (B-5) The aiu displaceet is A. The aiu velocit is A. Thus, KEa A (B-6) PEa A (B-7) Applig Raleigh s ethod, A A (B-) The atural frequec of the ed ass supported b the catilever bea is thus (B-9) (B-) 7
APPENDIX C Catilever Bea with Iteral Distributed Mass Cosider a catilever bea with ass per legth. Assue that the bea has a uifor cross sectio. Deterie the atural frequec., Figure C-. The goverig differetial equatio is t (C-) The boudar coditios at the fied ed = are () = (zero displaceet) (C-) d (zero slope) (C-) The boudar coditios at the free ed = are d (zero bedig oet) (C-) d (zero shear force) (C-5) Propose a quarter cosie wave solutio.
9 o ( ) cos (C-6) d o si (C-7) d o cos (C-) d o si (C-9) The proposed solutio eets all of the boudar coditios epect for the zero shear force at the right ed. The proposed solutio is accepted as a approiate solutio for the deflectio shape, despite oe deficiec. Agai, Raleigh s ethod is used to fid the atural frequec. The total potetial eerg ad the total kietic eerg ust be deteried. The total potetial eerg P i the bea is P d (C-) B substitutio, P o cos (C-) P o cos (C-) P o cos (C-)
P o si (C-) P o (C-5) o 6 P (C-6) The total kietic eerg T is T (C-7) T o cos (C-) T o cos cos (C-9) T o cos cos (C-) T o cos cos (C-) T o cos cos (C-) T o si si (C-) T o (C-)
T o (C-5) Now equate the potetial ad the kietic eerg ters. 6 o o (C-6) 6 (C-7) 6 (C-) 6 / (C-9) f 6 / (C-) f 6 / (C-)
/ f (C-) f 66. (C-)
APPENDIX D Catilever Bea with Iteral Distributed Mass ad Ed Mass Figure D-. The followig is based o a quarter cosie wave solutio. The kietic eerg is o o T (D-) T o (D-) The potetial eerg is o 6 P (D-) Now equate the potetial ad the kietic eerg ters. o o 6 (D-),
6 (D-5) 6 (D-6) / 6 (D-7) / (D-) / f (D-9).6. f (D-)
APPENDIX E Raleigh s Quotiet Raleigh s ethod ca also be applied to ulti-degree-freedo-sstes, as follows. X T K X (E-) X T MX where K is the stiffess atri M is the ass atri X is a assued ode shape with arbitrar scale Equatio (E-) is essetiall a uerical approiatio. It overestiates the true fudaetal frequec. Thus, it should be used i a trial-ad-error aer. Note that the uerator i equatio (E-) is equal to twice the potetial eerg. The deoiator is equal to twice the kietic eerg if first ultiplied b. As a eaple, cosider the sste defied i Figure E- ad Table E-. 5
k k k Figure E-. Table E-. Paraeters Variable k k k. kg. kg Value N/ N/ N/ The hoogeeous equatio of otio is k k k k k k (E-) 6
The ass atri is M kg (E-) The stiffess atri is K N / 5 (E-) The atural frequecies ca be coputed usig the eigevalue ethod. The eigevalues are the roots of the followig equatio. det K M (E-5) Equatio (E-5) ca be solved eactl for sstes with up to four degrees-of-freedo. The first atural frequec is thus. rad /sec (E-6) The et task is to test the Raleigh quotiet ethod. Several cadidate ode shape are evaluated as show i Table E-. Note that X (E-7) 7
Table E-. Raleigh Quotiet Trials X.5.5 (rad/sec).6.6.6 6.5 5.77 Agai, the Raleigh quotiet overestiates the true fudaetal frequec. Thus, the best estiate for the atural frequec after five trials is.6 rad /sec (estiate) (E-) The estiated value is.% higher tha the eact value. The Raleigh quotiet ethod thus gives ver good results for this eaple.