MEEN 67 Handou # MODAL ANALYSIS OF MDOF Sysems wih VISCOS DAMPING ^ Symmeic Moion of a n-dof linea sysem is descibed by he second ode diffeenial equaions M+C+K=F whee () and F () ae n ows vecos of displacemens and exenal foces, especively. M, K, C, ae he sysem (nxn) maices of mass, siffness, and viscous damping coefficiens. he soluion o Eq. () is deemined uniquely if vecos of iniial displacemens o and iniial velociies d = 0 () V = d o ae specified. Fo fee vibaions, he exenal foce veco F () =0, and Eq. () educes o M+C+K=0 A soluion o Eq. (2) is of he fom (2) = e α ψ (3) whee in geneal α is a complex numbe. Subsiuion of Eq. (3) ino Eq. (2) leads o he following chaaceisic equaion: 2 ( α ) α M + α C + K Ψ = f Ψ = 0 (4) MEEN 67 HD Modal Analysis of MDOF Sysems wih Viscous Damping L. San Andés 2008
whee f( α ) is a nxn squae maix. he sysem of homogeneous equaions (4) has a nonivial soluion if he deeminan of he sysem of equaion equals zeo, i.e. ( α ) Δ α = = = + α+ α + α + α 2 3 f 0 c0 c c2 c3... c (5) he oos of he chaaceisic polynomial Δ ( α ) given by Eq. (5) can be of hee ypes: a) Real and negaive, α < 0, coesponding o ove damped modes. b) Puely imaginay, α = ± iω, fo undamped modes. c) Complex conjugae pais of he fom, α = ζω ± iωd, fo unde damped modes. Clealy if he eal pa of any α > 0, i means he sysem is unsable. he consiuen soluion, eq. (3), is hen wien as he supeposiion of he oos α and is associaed vecos ψ saisfying Eq. (4), i.e. = C Ψ e α () Ψ = Ψ Ψ... Ψ nx2 n n (7) o leing [ ] [ 2 2 ] (6) wie Eq. (6) as Ψ (8) () = [ ]{ Ce α } Only if sysem is defined by symmeic maices. Ohewise, he complex oos may be no conjugae pais. MEEN 67 HD Modal Analysis of MDOF Sysems wih Viscous Damping L. San Andés 2008 2
Howeve, a ansfomaion of he fom, [ ] () 2 nx nx = Ψ q (9) is no possible since his implies he exisence of - modal coodinaes which is no physically appaen when he numbe of physical coodinaes is only n. o ovecome his appaen difficuly, efomulae he poblem in a slighly diffeen fom. Le Y be a - ows veco composed of he physical velociies and displacemens, i.e. Y=, and 0 Q= F () (0) be a modified foce veco. hen wie M+C+K=F () as 0 M -M 0 + = 0 M C 0 K o whee F (.a) AY+BY=Q 0 M -M 0 A=, B= M C 0 K (.b) (2) A and B ae x maices, in geneal symmeic if he M, C, K maices ae also symmeic. MEEN 67 HD Modal Analysis of MDOF Sysems wih Viscous Damping L. San Andés 2008 3
Fo fee vibaions, Q=0, and a soluion o Eq. (.b) is sough of he fom: =Y=Φ e α Subsiuion of Eq. (3) ino Eq. (.b) gives: (3) [ α A+B] Φ = 0 (4) which can be wien in he familia fom: whee DΦ= Φ α (5) - - M 0 0 M D=-B A= -, o 0 -K M C wih I as he nxn ideniy maix. Fom Eq. (5) wie 0 I D= - - -K M -K C (6) D Ι = = 0 α α Φ f Φ (7) he eigenvalue poblem has a nonivial soluion if ( α ) 0 Δ = = f α (8) Fom Eq. (8) deemine eigenvalues{ α }, =, 2.., and associaed eigenvecos{ Φ }. Each eigenveco mus saisfy he elaionship: DΦ = Φ (9) α MEEN 67 HD Modal Analysis of MDOF Sysems wih Viscous Damping L. San Andés 2008 4
and can be wien as Ψ Φ = 2 Ψ whee Ψ is a nx veco saisfying: 0 I Ψ Ψ = - - 2 2 -K M -K C Ψ α Ψ fom he fis ow of Eq. (20) deemine ha: 2 I Ψ = Ψ o α (20) 2 Ψ = α Ψ (2) and fom he second ow of Eq. (20) wih subsiuion of he elaionship in Eq. (2) obain α - - 2 -K M - K C -I Ψ =0 α (23) fo =, 2,.. Noe ha muliplying Eq. (23) by (-α K) gives + + = 2 2 Mα Cα K Ψ 0 (4) i.e. he oiginal eigenvalue poblem. Soluion of Eq. (23) delives he -eigenpais α Ψ Ψ (24) α ; Φ = =,2,.. In geneal, he j-componens of he eigenvecos numbes wien as Ψ ae complex Ψ = α + ib = δ j j j j i e φ j j=,2 n whee δ and φ ae he magniude and he phase angle. MEEN 67 HD Modal Analysis of MDOF Sysems wih Viscous Damping L. San Andés 2008 5
Noe: fo viscous damped sysems, no only he ampliudes bu also he phase angles ae abiay. Howeve, he aios of ampliudes and phase diffeences ae consan fo he elemens of he eigenvecos i.e. ( δ δ ) ( φ φ ) / = cons and = cons fo j, k =, 2,..N j k jk j k jk Ψ A consiuen soluion of he homogeneous equaion (fee vibaion poblem) is hen given as: = Y= C e α Φ = (25) Le he (oos) α be wien in he fom α = ζ ω + iω (26) and wie Eq. (25) as d = ( + i ) Y= = C e ζ ω ω Φ d (27) and since Φ α Ψ = Ψ, he veco of displacemens is jus = Ψ = ( + i ) C e ζ ω ω d (27) ORHOGONALIY OF DAMPED MODES he eigenvalues α and coesponding eigenvecos Φ saisfy he equaion: MEEN 67 HD Modal Analysis of MDOF Sysems wih Viscous Damping L. San Andés 2008 6
α ΑΦ + B Φ = 0 (28) Conside wo diffeen eigenvalues (no complex conjugaes): { αs; s} and { αq; q} Φ Φ, hen if is easy o demonsae ha: and infe (29) and Α = Α B=B (a symmeic sysem), i ( αs αq) Φ A Φ = 0 Φs AΦ q=0 ; Φ fo s ΒΦ q=0 αs αq s q A his ime, consuc a modal damped maix Φ (x) fomed by he columns of he modal vecosφ, i.e. [..... ] Φ= Φ Φ Φ Φ Φ (30) 2 n And wie he ohogonaliy popey as: ΦΑΦ=σ Φ ΒΦ=β (3) Whee σ and β ae (x) diagonal maices. Now, in he geneal case, he equaions of moion on he physical coodinaes ae of second ode and given by: M+C+K=F () MEEN 67 HD Modal Analysis of MDOF Sysems wih Viscous Damping L. San Andés 2008 7
Wih he definiion Y =, Eqs. () ae conveed ino fis ode diffeenial equaions: 0 Α Y+Β Y=Q= F () (32) Whee 0 M -M 0 A=, B= M C 0 K (2) o uncouple he se of fis-ode equaions (32), a soluion of he following fom is assumed: = Y = Φ z = ΦZ () () () (33) Subsiuion of Eq. (33) ino Eq. (32) gives: ΑΦΖ+ ΒΦΖ=Q (34) Pemuliply his equaion by he damped modes o ge: Φ Α Φ Ζ+ ΦΒΦΖ=Φ Q σζ+β Ζ=G o = Φ and use he ohogonaliy popey 2 of Φ Q Eq. (36) epesens a se of uncoupled fis ode equaions: (35) (36) MEEN 67 HD Modal Analysis of MDOF Sysems wih Viscous Damping L. San Andés 2008 8 σ z + β z = g 2 he esul below is only valid fo symmeic sysems, i.e. wih M,K and C as symmeic maices. Fo he moe geneal case, see he exbook of Meiovich o find a discussion on LEF and RIGH eigenvecos.
σ 2 2 2 2 2 σ z + β z = g (37) z + β z = g () 2N 2N 2N 2N 2N σ = Φ ΑΦ β = Φ B Φ = α σ, =, 2..2N whee ; since α ΑΦ + Β Φ = 0. In addiion, g = Φ Q () α = β / σ (38) (39) Iniial condiions ae also deemined fom Y o o o = wih he ansfomaion σζ =Φ o ΑYo (40.a) Ζ = o σ Φ ΑY o =, 2, (40.b) he geneal soluion of he fis ode equaion σ z + β z = g, wih iniial condiion z( 0) = z = o, is deived fom he Convoluion inegal α α ( τ) = 0 ( τ ) τ (4) z z e g e d o σ wih α = β / σ MEEN 67 HD Modal Analysis of MDOF Sysems wih Viscous Damping L. San Andés 2008 9
Once each of he z() soluions ae obained, hen eun o physical coodinaes o ge: Y = = Φ z = ΦZ = () () () (33=43) and since given by: Φ α Ψ = Ψ, he physical displacemen dynamic esponse is = Ψ = z () () (44) and he velociy veco is coespondingly equal o: = α Ψ = z () () (45) See he accompanying MAHCAD wokshee wih a deailed example fo discussion in class. MEEN 67 HD Modal Analysis of MDOF Sysems wih Viscous Damping L. San Andés 2008 0