6.4 Methods for performing modal analysis
6.4.4 Data analysis
Before commencing the modal analysis of the measured transfer function data some sanity checks should be done. Transfer functions should show the mass and stiffness line asymptote behaviors already discussed in Section 6.3.2 and shown in Fig 6.8. If these inspections are passed then one can gain an initial overview of the investigated structure’s dynamic characteristics by calculating the summation and mode indicator functions: see Fig. 6.13.
The summation function is calculated by summing the amplitudes of all measured transfer functions, omitting the phase information, and then nor- malizing with the number of measured data sets. The resonance peaks are pronounced and one can get a good idea of frequencies that exhibit strong modes and areas of high modal density. For complex test objects the sum- mation functions for subsystems and components can also be calculated.
The overview is complemented by the mode indicator function that shows a sharp drop to zero where resonances are detected.
The next step is that of extracting the modal frequencies, modal damping and mode shape vectors from the matrix of transfer function data. This process is often called curve fi tting (Lembregts, 1989), which basically means that analytical functions are fi tted to the measured individual response functions such that the difference between the measured response function and the analytical expression is minimal. An example of this process will be shown below for a very simple case study, but for more complex systems a computer software package must be used. The modern
(Multiple) Force or pressure excitation
Test structure Acceleration and
pressure responses
Frequency response function matrix h11 h12 . . . h1j . . . h1N h21 h22 . . . h2j . . . h2N
. . . .
hi1 hi2 . . . hij . . . hiN
. . . .
hN1 hN2 . . . hNj . . . hNN
hjj= Drive point FRF
Frequency (Hz)
0 1000 2000
1.2
0.5
0.0 m/s2 N
6.12 Example of the data acquisition of transfer functions as required for modal analysis (courtesy of Ford of Europe, Germany).
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poly-reference algorithms implemented in these packages work in the time or frequency domain and are capable of fi tting multiple modes across mul- tiple response functions for a number of references.
If we assume damping to be proportional to the stiffness, then the for- mulation for the problem we are trying to solve is:
K C M x
[ ]+ [ ]− [ ] = { }⎛{ }f
⎝⎜ ⎞
⎠⎟
−
iω ω2
1
(6.25) For a single degree of freedom system the solution of this equation is simple, but for large multiple degree of freedom systems this becomes very ineffi cient since the inversion of large matrices need to be calculated.
Therefore a different formulation of this equation is used. The new formu- lation can be derived by pre- and post-multiplying Equation 6.25 by the eigenvector matrices [Φ]T and [Φ]. For the transfer function of a sdof system this then results in:
h c
r m
ω ω ω ω
( )=
(
−)
+ Φ22 2 i
(6.26)
584 1067 Mode
indicator function
Summation function
Frequency (Hz) (a)
(b) m/s2
N
1200 800
400 0
1.0
0.5
0.0
1.0 m/s2 N
0.5
0.0 1600 2000
6.13 (a) A mode indicator function dip towards zero indicates the presence of a mode. (b) The summation function enhances resonance peaks. Both give a fi rst impression of mode strength and distribution (courtesy of Ford of Europe, Germany).
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To illustrate the process of extracting the modal data we will apply a simple method that has been in use for a long time called the peak picking method:
see also Fig. 6.14. This method can be applied when the modes are clearly separated from each other and the damping is light, so that contributions from other modes are negligible. Very often this is not the case, but the method is adequate to get a fi rst estimate on the measured transfer function data. We are now looking for a solution to Equation 6.26. The unknowns in this equation are the modal frequency, the modal damping and the mode shape.
First a resonance peak from the plot of the measured transfer function is chosen and the corresponding resonance frequency is read from the frequency axis. Then the half power points to both sides of the chosen reso- nance peak are defi ned by following the amplitude of the plotted frequency response function downwards until it reaches an amplitude equal to the resonance peak amplitude value divided by 2. These points are called the half power points. Again the corresponding frequencies are read from the frequency axis. If the damping is light, i.e. below 3%, then we can calculate the damping factor to be:
ζ ω ω
= 2ω− 1
2 r (6.27)
With 2ζωr
c
=m we have a solution for the damping. The only unknown left, Φ2, can now be calculated directly from Equation 6.26. This factor is often represented by an A and is called the modal constant or residue. When the investigated system contains more than one mode, which will be the usual
- Circle fit 2
Fit one mode at a time:
- Peak picking
H
w
ω1
z =
ω2
ω2 – ω1
ωr
For light damping:
2·ωr
Hr
Hr
6.14 Example of the determination of the modal frequency and damping by the simple peak picking method.
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case for real structures, then the transfer function between two locations will have several single degree of freedom contributions: see Fig. 6.15. The governing equation then becomes
h
i c m
i j
i r j r
r r
r r
n ,
, ,
ω ω ω ω
( ) =
(
−)
+∑
= 2 Φ Φ2 1(6.28)
where r denotes the respective mode number.