MEEM 700 Mechancal Vbratons Mohan D. Rao Chuck Van Karsen Mechancal Engneerng-Engneerng Mechancs Mchgan echnologcal Unversty Copyrght 00 Lecture & MEEM 700 Multple Degree of Freedom Systems (ext: S.S. Rao,, Chapter: 6 Sectons:,,, 9, 0,, 4) opcs:.) Equatons of Moton.) Egenvalue Problem.) Orthogonalty 4.) Modal Space 5.) Solvng MDOF Problems n Smple Fashon Secton:,, 9 0 4 Lecture & MEEM 700
MDOF: Equatons of Moton Equatons of Moton Usng Newton s Second Law:.) Choose sutable coordnates for the pont masses and rgd r bodes n the system..) Determne statc equlbrum poston of the system. Measure moton from ths poston..) Draw free-body dagram of each mass or rgd body. Indcate all forces actng on the mass or rgd body when postve dsplacement or velocty s present. 4.) Apply Newton s Second Law: ranslaton: Rotaton: Lecture & MEEM 700 F x M θ mx J θ MDOF: Equaton of Moton [ m] [ c] [ k] {} x {} x {} x { F} General Form n-dof n System [ m]{ x} + [ c]{ x } + [ k]{ x} { F} n x n n x n n x n n x n x n x n x mass matrx vscous dampng matrx stffness matrx acceleraton vector velocty vector dsplacement vector externally appled force vector Lecture & MEEM 700 4
MDOF: General Soluton to Equaton of Moton.) Determne natural frequences (ω( ) and mode shapes {X} from undamped Free Vbraton. (egensoluton).) Use orthogonalty of mode shapes to transform Equatons of Moton nto MODAL SPACE..) Solve n sngle degree of freedom problems. 4.) Use mode shapes to transform back to physcal space. Lecture & MEEM 700 5 MDOF: ω and {X} from Egensoluton [ m]{ x} + [ k]{ x} { 0} For Harmonc Moton: [ k]{ x} ω [ m]{ x} x ω x ( ω [ m] + [ k] ){ x} { 0} classc egenvalue problem Computer or calculator soluton (MALAB, Mathematca,, etc.) Yelds n egenvalues and n egenvectors he natural frequences are the square roots of the egenvalues: ω ω ω for, n { X } for, n he mode shapes are the correspondng egenvectors. Lecture & MEEM 700 6
MDOF: ω and {X} from Egensoluton ( ω [ m] + [ k] ){ x} { 0} o solve, the characterstc determnant 0 [ k] ω [ m] 0 [ k] λ [ m] 0 results n an n order polynomal n n ( λ) ( λ) ( λ) a + a +... + a + a 0 n n 0 roots λ for, n natural frequences ω λ Substtute ω or λ nto orgnal equatons for mode shape soluton. Lecture & MEEM 700 7 MDOF: ω and {X} from Egensoluton Alternate Method λ [ I ]{ x} [ D]{ x} ( ω [ m] + [ k] ){ x} { 0} λ ω ( [ k] [ m] ){ x} { 0} λ ( [ ] λ k [ k] [ k] [ m] ){ x} {} 0 ( λ [ I] [ D] ){ x} { 0} [ I ] Identty Matrx [ D] [ k] [ m] Solutons are the egenvalues and egenvectors of [D]. λ for, n { X } for, n ω dynamcal matrx Lecture & MEEM 700 8 4
Orthogonalty of Mode Shape Vectors ( ω [ m] + [ k] ){ x} { 0} Usng the th natural frequency and mode shape [ m]{ X} [ k]{ X} ω Usng the jth natural frequency and mode shape [ m]{ X} [ k]{ X} ω j j j { X } and by { X } Pre-multply by { X} [ m]{ X} { X} [ k]{ X} ω j j { X} [ m]{ X} { X} [ k]{ X} ω 4 j j j j Lecture & MEEM 700 9 Orthogonalty of Mode Shape Vectors In general Because of the symmetry of [k], [m] { } X [ k]{ X} { X} [ k]{ X} j j { } X [ m]{ X} { X} [ m]{ X} j j ω { X } [ m]{ X} { X} [ k]{ X} j j { X } [ m]{ X} { X} [ k]{ X} ω j j j Subtract 4 ( ω ω ){ X} [ m]{ X} { 0} ω ω j j j j Also: j Modal vectors are orthogonal wth respect to [m], [k]. Lecture & MEEM 700 0 4 { X} [ m]{ X} { 0} { X} [ k]{ X} { 0} 5
Modal Mass, Stffness, & Dampng { } [ ]{ } X m X M { } [ ]{ } X k X K Modal Mass Modal Stffness [ ] α[ ] β[ ] f c m + k Proportonal Dampng { } [ ]{ } X c X C Modal Dampng Lecture & MEEM 700 Modal Mass, Stffness, & Dampng Defne [ X] { X} { X} { X} m 0 n [ X ] [ m][ X] [ M] 0 m nn Modal Matrx Modal Mass Matrx k 0 [ X ] [ k][ X] [ K] 0 k For proportonal dampng: c nn 0 [ X ] [ c][ X] [ C] 0 c nn Modal Stffness Matrx Modal Dampng Matrx Lecture & MEEM 700 6
Example [ m] 0 0 0 0 5 0 0 0 0 [ c] 0 0 4 [ k] 000 000 0 000 000 000 0 000 4000 [ ]{ } [ ]{ } From ω m X k X.00 ω 9.5 { X }.6.00.00 ω 4.4 { X } 0.00 0.50.00 ω 6.76 { X } 5.6.00 Lecture & MEEM 700 Example (contnued) [ X ].00.00.00.6 0.00 5.6.00 0.50.00 6.75 5.00 6.5 [ M] [ X] [ m][ X].08.00 6.9 [ C] [ X] [ c][ X] 080 000 690 [ K] [ X] [ k][ X] Lecture & MEEM 700 4 7
Modal Space Orgnal Equatons of Moton: [ m]{ x} + [ c]{ x } + [ k]{ x} { F} Defne a new coordnate system { x} [ X]{ q} { } Modal Matrx Substtute nto equaton of moton Pre-multply by [ X ] q Modal Space prncpal coordnates [ m][ X]{ q } + [ c][ X]{ q } + [ k][ X]{ q} { F} [ X ] [ m][ X]{ q } + [ X] [ c][ X]{ q } + [ X] [ k][ X]{ q} [ X] { F} Lecture & MEEM 700 5 Modal Space [ M ]{ q } + [ C]{ q } + [ K]{ q} [ X] { F} Uncoupled equatons of moton n the form of sngle degree of freedom systems. M q + C q + K q X F + X F + + X F n n + + + + + n M q C q K q X F X F X Fn M q + C q + K q X F + X F + + X F n n n nn n nn n nn n n n Solve Solve each SDOF system ransform back to physcal space for total soluton ( ) x t q t x t q t [ X ] x t q t n n Lecture & MEEM 700 6 8
Example ( ) ( ) F t 5sn 0t k k k k4 m m m c c c c4 x x x m 0 kg k 000 c 0 N m m 5 kg k 000 c 0 N m m 5 kg k 000 c 0 N m k 500 c 5 N 4 m 4 N sec m N sec m N sec m N sec m Determne the total soluton f the system starts from rest. Lecture & MEEM 700 7 Example Equatons of Moton 0 x 0 0 0 x 000 000 0 x 0 5 x 0 0 0 x 000 000 000 + + x 5sn ( 0t) 5 x 0 0 5 x 0 000 500 x 0 Determne the mode shapes and natural frequences [ k]{ x} ω [ m]{ x} mode mode mode ω 6.6 ω 7. ω.4 rad sec rad sec rad sec [ X ].00.00.00.8 0.00 0.78.00.00.00 Lecture & MEEM 700 8 9
Example Uncouple the Equatons of Moton (transform to Modal Space) { x} [ X]{ q} [ X ] [ m][ X]{ q } + [ X] [ c][ X]{ q } + [ X] [ k][ X]{ q} [ X] { F} ( ( t) ) 9.6 q 7.6 q.8 5sn 0 77 q 0 q 90 q 9000 + + q 0 4.4 q 0. q,0 q 0.78( 5sn ( 0t) ) Lecture & MEEM 700 9 Example Solve each SDOF system ( ) 9.6q + 7.6q + 77q sn 0t 7.6 ωn ω 6.6 ζ 0.0 9.6( 77) ω 6.60 ( 0.0( 6.6) ) t ( ( ) ( )) ( ) q t e Acos 6.6t + Bsn 6.6t + Qsn 0t φ 0 XK r.5 from ( r, ζ ) 0.78 6.6 F 0 d QK ( ) 0.78 F 0.78 Q 0.044 77 0 φ ζ r r tan π Lecture & MEEM 700 0 0
Example ( ) ( ) 0.8t ( ) ( ) q t e Acos 6.6t + Bsn 6.6t + 0.044sn 0t π Intal condtons: { x( 0) } [ X] { q( 0 )} { x ( 0) } [ X] { q ( 0) } { q( 0) } [ X] { x( 0 )} { q ( 0) } [ X] { x ( 0) } For a system at rest: q( ) { 0 } { 0 }, { q ( 0) } { 0} ( ) ( )( ) ( π ) 0.8t 0.8t ( ( )) ( ( )) + ( ) ( )( ) ( )( ) 0.8t q t e 0.0sn ( 6.6t) + 0.044sn 0t π 0 q 0 A+ 0 + 0.044sn A 0 q t e 6.6Bcos 6.6t 0.8e Bsn 6.6t 0.44cos 0t π 0 q 6.6B 0 0.44 B 0.0 ( ) ( ) Lecture & MEEM 700 In a smlar manner: ( ) ( ) ( ) 0q + 90q + 9000q 0 5sn 0t q ( t ) ( ) 4.4q + 0q +,0q 9.5sn 0t { } [ ]{}.t ( ) ( ) 0 ( ) ( ) q t e.5sn t 0.005sn 0t ( t) t.00.00.00 q X X q q t.8 0.00 0.78.00.00.00 q ( ) ( ) + ( ) + ( ).8 0.78.00 + x t q t q t q t x t q t q t x t q t q t q t Lecture & MEEM 700