Analytical and Experimental Modal Analysis
20
Signal Processing: Modal Analysis Analytical and Experimental Modal Analysis
Transcript of Analytical and Experimental Modal Analysis
Signal Processing: Modal Analysis
Analytical and Experimental
Modal Analysis
Signal Processing: Modal Analysis
Signal Processing: Modal Analysis
Signal Processing: Modal Analysis
21222312222 f)xx(kxk)xx(cxm =−++−+ &&&&
1212112121111 f)xx(kxk)xx(cxcxm =−++−++ &&&&&
1222212112111 fxkxcx)kk(x)cc(xm =−−++++ &&&&
2121223213222 fxkxcx)kk(x)cc(xm =−−++++ &&&&
)f()x](k[)x](c[)x](m[ =++ &&&
+−
−+=
+−−+
=
=
322
221
322
221
2
1
00
kkkkkk
]k[ccc
ccc]c[
mm
]m[
=
=
=
=
2
1
2
1
2
1
2
1
ff
)f(xx
)x(xx
)x(xx
)x(&&
&&&&
&
&&
)N(f)N(x)NN(c)NN(m 11 ××××
General:
Signal Processing: Modal Analysis
Undamped system:
[c] = 0
)tjexp()()x( ωψω−= 2&&
02 =ψ+ψω− )](k[)](m[
0=ψ+ψλ− )](k[)](m[
{ } 01 =ψλ−− )(]I[]k[]m[
eigenvalueiλ
eigenvectori)(ψ
Signal Processing: Modal Analysis
m1= 5 kg m2 = 10 kg k1 = k2 = 2N/m k3 = 4N/ m
01001
6224
10005 1
=
λ−
−
−
−
052515354 =−−−λ−λ− )/)(/()/)(/(
sec]/rad[sec]/rad[)/(/ /
115252
22
2111
=ω=λ=ω=λ
=
ψψ
−−
−−=ω=λ
00
525351525254
5221
11211 .//
////
ψψ
=
ψψ
=ψ
ψ=ψ=ψ−ψ
11
11
21
111
2111
2111 05252
)(
//
ψ−
ψ=ψ=ω=λ
12
12
2222
211 )(
Signal Processing: Modal Analysis
Signal Processing: Modal Analysis
0=ψψλ≠λ jTiji )](m[)(
0=ψψ jTi )](k[)(
iiriji M)](m[)( =ψψλ=λ
iiTi K)](k[)( =ψψ
=ψψ
\M
\]][m[][ i
T
=ψψ
\K
\]][k[][ i
T
Ki/Mi =2iω
Weighted Orthogonality relations
Generalized mass and stiffness
Signal Processing: Modal Analysis
=ψ=ψ=ω
11
152 11121
1 )(set)/( /
6
1511
10005
11
1211
1
=ω=
==
MK
MT
=
=
=ω=
==
−
−
−=ψ=ψ=ω
215006
2150015
215
215211
10005
211
211
11
2222
2
2122
/]K[
/]M[
/MK
/M
)(Set
T
Signal Processing: Modal Analysis
22iiii MK ω=ω=
Mi = 1
−=φ
==
ΦΦ
φφ
=ω
=φ
φ=φ
==
φφ
φφ
=ω
15250152
110005
1
151151
110005
52
2
222
12
22
122
1
2111
121
11
21
11211
/./
)(
M
//)(
M)/(
T
T/
ii
i )(M
)( ψ=φ1
Mass normalized eigenvectors
Signal Processing: Modal Analysis
)f()(fqq
)](m[)()f()(M/fqq
f)f()(qKqM
)f(][)q](K[)q](M[)f(][)q](][k[][)q](][m[][
)f()x](k[)x](m[
)x(][qq][)x(])()()[(][
Tiiiii
iTi
Ti
iiiii
iiiiii
T
TTT
n
φ==ω+
ψψψ
==ω+
=ψ=+
ψ=+
ψ=ψψ+ψψ
=+
ψ=ψ=
ψψψ=ψ−
2
2
1
21
&&
&&
&&
&&
&&
&&
KK Generalized coordinates
Signal Processing: Modal Analysis
−=ψ
211
11]
=ψψ
−=ψψ
−=
=ψ=
015006
210
015211
11
2
1
2
1
][]k[][
]][m[][
xx
q][)x(
T
T
2215
215
615
211
11
215006
2150015
2121
2111
2
1
2
1
2
1
ffqq
ffqq
ff
/
−=+
+=+
−=
+
+
&&
&&
&&
&&
Uncoupling of equations
Signal Processing: Modal Analysis
{ }
Tr
rTTT
][\
\][]H[
\
\]][m[][]][k[][][]H[][
]m[]k[]H[
)f](H[)x()f()x(]H[
)f()x(]m[]k[
fx)mk()tjexp(fkxxm
φ
ω−ωφ=
ω−ω=φφω=φφ=φφ
ω−=
==
=ω−
=ω−
ω=+
−
−
−
−
1
22
2221
21
1
2
2
&&
Transfer Function approach
Signal Processing: Modal Analysis
=
0
00
kff M
=
φφ
φφφφ
ω−ω
φφ
φφφφ
=
−
0
00
2
1
12
121111
22
1
21
11211
M
M
M
L
M
L
f\)(
\)x(
NNN
N
r
NNN
N
k
kN
k
k
r
NNN
N
N
f\)(
\
x
xx
φ
φφ
ω−ω
φφ
φφφ
=
−
MM
M
L
M
2
11
22
1
11211
2
1
k
kN
k
riNiii f\)(
\(x
φ
φ
ω−ωφφφ=
−
MK
1
1
2221
Signal Processing: Modal Analysis
221 ω−ω
φφ=∑
= r
krirN
rikH
kririkr
rrr
ikv
rrrr
krirN
rik
A
jA
jH
φφ=
ωωε+ω−ω=
ωωε+ω−ωφφ
= ∑∑=
222221 2
Summary:
Signal Processing: Modal Analysis
Signal Processing: Modal Analysis
Signal Processing: Modal Analysis
Signal Processing: Modal Analysis
Signal Processing: Modal Analysis
