Mechanical Vibrations Chapter 4
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1 Mechaical Vibraios Chaper 4 Peer Aviabile Mechaical Egieerig Deparme Uiversiy of Massachuses Lowell Mechaical Vibraios - Chaper 4 1 Dr. Peer Aviabile Modal Aalysis & Corols Laboraory
2 Impulse Exciaio Impulsive exciaios are geerally cosidered o be a large magiude force ha acs over a very shor duraio ime The ime iegral of he force is Fˆ F() d (4.1.1) Whe he force is equal o uiy ad he ime approaches zero he he ui impulse exiss ad he dela fucio has he propery of δ ( ξ) for ξ Mechaical Vibraios - Chaper 4 2 Dr. Peer Aviabile Modal Aalysis & Corols Laboraory
3 Impulse Exciaio Iegraed over all ime, he dela fucio is δ( ξ)d 1 < ξ < (4.1.2) If his fucio is muliplied imes ay forcig fucio he he produc will resul i oly oe value a ξ ad zero elsewhere f () δ( ξ)d f ( ξ) < ξ < (4.1.3) Mechaical Vibraios - Chaper 4 3 Dr. Peer Aviabile Modal Aalysis & Corols Laboraory
4 Impulse Exciaio Cosiderig impac-momeum o he sysem, a sudde chage i velociy is equal o he acual applied ipu divided by he force. Recall ha he free respose due o iiial codiios is give by x x() & si ω ω + x() cosω Mechaical Vibraios - Chaper 4 4 Dr. Peer Aviabile Modal Aalysis & Corols Laboraory
5 Impulse Exciaio The he velociy iiial codiio yields Fˆ x si ω Fˆ h() mω (4.1.4) ad i ca be see ha he soluio icludes h() h() 1 mω si ω (4.1.5) Mechaical Vibraios - Chaper 4 5 Dr. Peer Aviabile Modal Aalysis & Corols Laboraory
6 Impulse Exciaio Whe dampig is cosidered i he soluio, he free respose is give as (x()) x x()e & ω ζω 1 ζ si ω which ca be wrie as Fˆ x e 2 mω 1 ζ 2 1 ζ ζω 2 si ω x()e & ω d σ 1 ζ 2 si ω d (4.16) or as x Fˆ mω d e σ si ω d Fˆ h() Mechaical Vibraios - Chaper 4 6 Dr. Peer Aviabile Modal Aalysis & Corols Laboraory
7 Arbirary Exciaio Usig he ui impulse respose fucio, he respose due o arbirary loadigs ca be deermied. The arbirary force is cosidered o be a series of impulses Mechaical Vibraios - Chaper 4 7 Dr. Peer Aviabile Modal Aalysis & Corols Laboraory
8 Arbirary Exciaio Sice he sysem is cosidered liear, he he superposiio of he resposes of each idividual impulse ca be obaied hrough umerical iegraio x() f ( ξ)h( ξ)dξ This is called he superposiio iegral. Bu i is also referred o as he Covoluio Iegral or Duhammel s Iegral (4.2.1) Mechaical Vibraios - Chaper 4 8 Dr. Peer Aviabile Modal Aalysis & Corols Laboraory
9 Sep Exciaio Deermie he idamped respose due o a sep. For he udamped sysem, 1 h() si ω mω which is subsiued io (4.2.1) o give F x() si ω d mω x() F k ( 1 cos ) ω ( ξ) ξ (4.2.2) Mechaical Vibraios - Chaper 4 9 Dr. Peer Aviabile Modal Aalysis & Corols Laboraory
10 Sep Exciaio This implies ha he peak respose is wice he saical displaceme x() F k ( 1 cos ) ω (4.2.2) 2 Displaceme versus Time Dis placeme Time Noe: Force seleced such ha F/k raio is Mechaical Vibraios - Chaper 4 1 Dr. Peer Aviabile Modal Aalysis & Corols Laboraory
11 Sep Exciaio Whe dampig is icluded i he equaio, he h() e mω ζω 1 ζ 2 si ω 1 ζ 2 (4.2.2) ad x() F k 1 e mω ζω 1 ζ 2 cosω 1 ζ 2 ψ (4.2.3) Mechaical Vibraios - Chaper 4 11 Dr. Peer Aviabile Modal Aalysis & Corols Laboraory
12 Sep Exciaio This ca be simplified as h() σ e mω d si ω d x() F k 1 σ e mω d cosω d ψ C C.1 M1 ; K2 C.5 C Mechaical Vibraios - Chaper 4 12 Dr. Peer Aviabile Modal Aalysis & Corols Laboraory
13 Base Exciaio For base exciaio, m & x k(x y) c(x& y) & (3.5.1) z x y (3.5.2) he equaio of moio is expressed as zx-y ad will resul i & z + 2ζω z& + ω Noice ha he F/m erm is replaced by he egaive of he base acceleraio (ie, Fma) 2 z & y (4.2.4) Mechaical Vibraios - Chaper 4 13 Dr. Peer Aviabile Modal Aalysis & Corols Laboraory
14 Base Exciaio For ad udamped sysem iiially a res, he soluio for he relaive displaceme is z() 1 && y( ξ)si ω d ω ( ξ) ξ (4.2.5) Mechaical Vibraios - Chaper 4 14 Dr. Peer Aviabile Modal Aalysis & Corols Laboraory
15 Ramp Exciaio ad Rise Time This soluio mus always be cosidered i wo pars - he ime less ha ad greaer ha 1 The ramp of he force is f () F 1 ad h() for he covoluio iegral is h() 1 mω si ω ω k si ω (4.4.1) (4.4.1) Mechaical Vibraios - Chaper 4 15 Dr. Peer Aviabile Modal Aalysis & Corols Laboraory
16 Ramp Exciaio ad Rise Time The respose for he firs par of he ramp is x() F k ω F k 1 1 ξ si ω si ω ω 1 ( ξ)dξ < 1 (4.4.2) ad he respose of he sep porio afer 1 is ( ) F si ω x () > k ω Mechaical Vibraios - Chaper 4 16 Dr. Peer Aviabile Modal Aalysis & Corols Laboraory
17 Ramp Exciaio ad Rise Time The superposiio of he wo pieces of he soluio gives he oal respose due o he force as ( ) F si ω si ω x () > k ω1 ω1 (4.4.3) Mechaical Vibraios - Chaper 4 17 Dr. Peer Aviabile Modal Aalysis & Corols Laboraory
18 Recagular Pulse The recagular pulse is he sum of wo differe sep fucios - oe posiive ad oe egaive shifed i ime Sep Up Sep dow Combied kx() F kx() F kx() F kx() F ( 1 cosω) < 1 ( 1 cosω ( 1) ) < 1 ( 1 cosω () ) ( 1 cosω ( 1) ) < 1 ( cosω () ) + ( cosω ( 1) ) < 1 (4.4.4) (4.4.5) (4.4.6) Mechaical Vibraios - Chaper 4 18 Dr. Peer Aviabile Modal Aalysis & Corols Laboraory
19 MATLAB Examples - VTB3_1 VIBRATION TOOLBOX EXAMPLE 3_1 >> m1; c.1; k2; f1; F1; >> vb3_1(m,c,k,f,f) >>.8 Displaceme versus Time.6.4 Displaceme Time Mechaical Vibraios - Chaper 4 19 Dr. Peer Aviabile Modal Aalysis & Corols Laboraory
20 MATLAB Examples - VTB3_2 VIBRATION TOOLBOX EXAMPLE 3_2 >> m1; c.1; k2; f1; F1; >> VTB3_2(m,c,k,F,f) >> >> 1 Displaceme versus Time Displaceme Time Mechaical Vibraios - Chaper 4 2 Dr. Peer Aviabile Modal Aalysis & Corols Laboraory
21 MATLAB Examples - VTB1_4 VIBRATION TOOLBOX EXAMPLE 1_4 pulse >> clear; p1:1:1; ppp./p; ppp[(p./p-p./p) pp (p./p-p./p) (p./p-p./p)]; >> x; v; m1; d.5; k2; d.1; 4; >> uppp; [x,xd]vtb1_4(,d,x,v,m,d,k,u); >> :d:*d; plo(,x);plo(,x); >> Mechaical Vibraios - Chaper 4 21 Dr. Peer Aviabile Modal Aalysis & Corols Laboraory
22 MATLAB Examples - VTB1_4 VIBRATION TOOLBOX EXAMPLE 1_4 ramp up (basically a saic problem) >> clear; x; v; m1; d.5; k2; d.1; 3; >> :d*1:; u./3; [x,xd]vtb1_4(,d,x,v,m,d,k,u); >> :d:*d; plo(,x); >> Mechaical Vibraios - Chaper 4 22 Dr. Peer Aviabile Modal Aalysis & Corols Laboraory
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