Chapter 4 Multiple-Degree-of-Freedom (MDOF) Systems. Packing of an instrument
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1 Chaper 4 Mulple-Degree-of-Freedom (MDOF Sysems Eamples: Pacg of a srume Number of degrees of freedom Number of masses he sysem X Number of possble ypes of moo of each mass Mehods: Newo s Law ad Lagrage s equao * More ha oe aural frequeces ad mode shapes Three-sory buldg
2 Lagrage s Mehod: a alerave o usg Newo s law whe he defcao of forces ad momes (Geeralzed forces s o obvous, whch s powerful for Sysems wh more ha oe degree of freedom Sysems wh comple elemes Geeralzed coordaes (q : depede coordaes requred for descrpo of moo of all he compoes of he sysem
3 Lagrage s mehod Vrual wor: δw Q δq ocoservave forces Q δw δq Lagrage s equaos: (T s ec eergy ad U s poeal eergy d d T T ( q q U q Q Alerave formas: Lagraga Fuco: LT-U, d d L L ( q q Q Whe dampg s volved, Raylegh dsspao fuco s defed as he geeralzed force for vscous dampg s derved as, Vscous dampg force: Lagrage s equaos: d d Q F F q D L L r s c rs (,... F q q * ( Q q q q r s * Q : Nocoservave forces whou vscous dampg forces
4 Eample: Usg Lagrage s mehod o derve he goverg equaos for he followg mahemacal model of a vbrao sysem (-DOF sysem.
5 Eample: The vbrao model of a smple mache par s show below, here ess a appled mome M( ad he ds roaes whou raslao. Derve he moo equaos of he sysem usg Lagrage equao.
6 Free Vbrao of Two-Degree-of-Freedom Sysems (Udamped ( ( m m Coupled goverg equaos: 0 K M 0 0 m m M K Vecor equao: Resposes: m 9 g, m g, 4 N/m, 3 N/m, (0 mm, (00, ad v (0v (00
7 Modal Aalyss: Usg approprae rasformaos o rasform he vbrao problem bewee dffere coordae sysems (see eqs. (4.73-(4.76 ad Wdow 4.4 for he procedure, whch s a mpora cocep for vbrao aalyss, desg ad esg Calculag he soluo of a mulple-degree-of-freedom vbrao problem by performg a umber of mar calculaos (ca be easly performed by a compuer code Decoupled he complcaed goverg equaos for MDOF vbrao problem physcal coordaes o SDOF vbrao problem he modal coordaes whch ca be easly solved
8 Modal Aalyss Procedure:
9 Eample: Calculae he soluo of he hree-degree-of-freedom sysem of he fgure below by modal aalyss. Use he values of m m m 3 4 g ad 3 4 N/m, ad he al codo (0 m wh all oher al dsplacemes ad veloces zero.
10 Modal Aalyss Mode Summao Mehod Seps for Compug he Respose By Mode Summao: Wre he equaos of moo mar form, defy M ad K Calculae M -/ (or L Calculae ormalzed mass sffess mar Compue he egevalue problem for he mar K Trasform he al codos o q 0 M (0 ad Calculae he modal epaso coeffces ad phase cosas φ Assemble he me respose for T v q(0 a, T v q(0 d q Trasform he soluo o physcal coordaes?k M KM ~ ad ge ad v ( q (0 M (0 T v q(0 sφ ( q( d s φ v ( M q( d s ( φ u
11 Mode Summao Eample (4.4.3: The four-sory buldg subjeced o wd force wh a al dsplaceme of (00.05 m, (00.0 m, 3 (00.0 m ad 4 (00.00 m. Deerme each floor respose. Mode shapes: Respose:
12 Nodes of a Mode: a zero ery he mode shape, whch meas f he sysem s eced by a al codo o vbrae oly a specfc aural frequecy, he coordae wh zero ery wll o move ode u Grea moug pos for mache The sesor placed o he ode ca o measure he vbrao a he specfc aural frequecy Rgd-Body mode: Sysem raslaes (or roaes from s equlbrum poso oe coordae, bu vbraes he oher coordaes
13 Vscous Dampg MDOF Sysems Two basc choces for cludg dampg Modal Dampg Arbue some amou o each mode based o eperece,.e., a arful guess or Esmae dampg due o vscoelascy usg some appromao mehod (MSE Model he dampg mechasm drecly (hard ad sll a area of research Modal Dampg Mehod Solve he udamped vbrao problem followg Wdow 4.4 Decouple sysem based o M ad K,.e., use he udamped modes Arbue some ζ o each mode of he decoupled sysem (a guess: r ( ζ r ( r ( 0,,... Use modal rasform o oba modal al codos ad ge soluos for r ( use he verse rasform o recover he physcal soluo * The Values of ζ are assged based o eperece or measureme
14 Modal Dampg by Mode Summao modes are from udamped sysem ( (0 (0 (0 a ad s (0 ad, where s ( q v q v q v q v d v v KM M v e d q T T T d T d d ζ φ φ ζ φ ζ Compue ad rasform bac o he physcal coordaes q( d u e d s( ( φ ζ
15 Lumped Dampg Models I some cases (FEM, mache modelg, he dampg mar s deermed drecly from he equaos of moo. The our aalyss mus sar wh: 0 0 ad subjec o, ( ( ( 0 K C M 0 0 m m M c c c c c C K * I geeral, hese equaos ca o be decoupled sce K ad C ca o be dagoalzed smulaeously K M C β α proporoal dampg r r r..., 0, ( ( ( ζ β α ζ I desg cases, α ad β ca be chose o ge he desred modal dampg rao
16 Eample: The puch press ca be modeled for vbrao aalyss he dreco as dcaed by he hree-degree-of-freedom sysem below. Calculae he respose f he floor s subjec o a force of 0 s ( Newos. m 400 g, m 000 g, m g, N/m, N/m ad N/m
17 Eample (4.6.: A smple wo-degree-of-freedom sysem wh harmoc ecao. I hs sysem, le m9 g, m g, 4 N/m ad 3 N/m ad he dampg s assumed o be proporoal wh α0 ad β0.. Calculae he seady-sae respose.
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