1. Experimental Modal Analysis

2. Different Modal Analysis Techniques
f(t)
x(t) k c m
SDOF and MDOF Models
Different Modal Analysis Techniques
Exciting a Structure
Measuring Data Correctly
Modal Analysis Post Processing = +
+    +

3. Simplest Form of Vibrating System
Displacement
Displacement
d = D sinnt D
Time T 1 T
Frequency m
Period, Tn in [sec]
Mass and Spring
Once a (theoretical) system of a mass and a spring is set in motion it will continue this motion with constant frequency and amplitude. The system is said to oscillate with a sinusoidal waveform.
The Sine Curve
The sine curve which emerges when a mass and a spring oscillate can be described by its amplitude (D) and period (T). Frequency is defined as the number of cycles per second and is equal to the reciprocal of the period. By multiplying the frequency by 2 the angular frequency is obtained, which is again proportional to the square root of spring constant k divided by mass m. The frequency of oscillation is called the natural frequency fn. The whole sine wave can be described by the formula d = Dsin nt, where d = instantaneous displacement and D = peak displacement. k 1 Tn
Frequency, fn= in [Hz = 1/sec] k
n= 2  fn = m

4. Mass and Spring time m1 m Increasing mass reduces frequency
Increase of Mass
An increase in the mass of a vibrating system causes an increase in period i.e. a decrease in frequency.
Increasing mass
reduces frequency

5. Mass, Spring and Damper time Increasing damping reduces the amplitudek
c1 + c2
Increasing damping
reduces the amplitude
Mass, Spring and Damper
When a damper is added to the system it results in a decrease in amplitude with time. The frequency of oscillation known as the damped natural frequency is constant and almost the same as the natural frequency. The damped natural frequency decreases slightly for an increase in damping.

6. C = damping (force/vel.) K = stiffness (force/disp.)
Basic SDOF Model
f(t)
x(t) k c m
Acceleration Vector
Velocity Vector
Displacement Vector
Applied force Vector
M = mass (force/acc.)
C = damping (force/vel.)
K = stiffness (force/disp.)

7. SDOF Models — Time and Frequency Domain
H()
X() 
|H()| 1 k
 H() 0º
– 90º
– 180º
0 =  k/m
2m c
f(t)
x(t) k c m
The combined mass, spring and damping system results in a transfer function H: H=1/(w2m+jwc+k)
At low frequencies, the magnitude of the transfer function H is 1/k. (k>>w2m+jwc). The phase is 0
At high frequencies, the transfer function H is 1/-w2m (w2m>>jwc+k). The phase is -180º.
The undamped natural frequency is found where -w2m+k=However the magnitude is not infinity anymore due to damping in the system. The magnitude of the transfer function H=1/w0c at the resonance frequency w0.
0. w0=sqroot(k/m)

8. Modal Matrix
Modal Model
(Freq. Domain) X1 X2 X3 X4 F3
H21
H22

9. MDOF Model 1+2 d1+ d2 1 2 m d1 dF 2 1 1+2 Magnitude Frequency Phase
Combined responses
When considering real mechanical systems they will normally be more complex than the previous models. A simple example of two masses/springs/dampers is shown here.
In this system we will see the responses combined, and its frequency response function shows two resonance peaks corresponding to the two masse/spring/damper systems. 0°
-90°
-180° 2 1
1+2

10. Why Bother with Modal Models?
Modal Space = Simplicity
Physical Coordinates = CHAOS 1
Bearing
Rotor
Foundation 1 q1 2 q2 1 3 1 q3

11. Definition of Frequency Response Function
H(f)
X(f) F X f f f
The Frequency Response Function (FRF) is a model based in the frequency domain where the output spectrum is expressed as the input spectrum weighted by a system descriptor called the FRF.
This implies, that the FRF represents the complex ratio between the input and the output spectra which means that the FRF is described by means of a magnitude and a phase.
The physical interpretation of the FRF is that a sinusoidal input force at a specific frequency will produce a sinusoidal output at the same frequency. The output is the input with an amplitude multiplied by the magnitude of the FRF and a phase shifted by the phase of the FRF.
H(f) is the system Frequency Response Function
F(f) is the Fourier Transform of the Input f(t)
X(f) is the Fourier Transform of the Output x(t)

12. Benefits of Frequency Response Function
H(f)
X(f)
Frequency Response Functions are properties of linear dynamic systems
They are independent of the Excitation Function
Excitation can be a Periodic, Random or Transient function of time
The test result obtained with one type of excitation can be used for predicting the response of the system to any other type of excitation
The FRF is a linear model which means, that it can be considered as a sum of sinusoids. The FRF therefore describes the dynamic properties of a system independent of the signal type used for the excitation.
This means that the choice of signal type for the input is free and can be optimized for the instrumentation and the structure under test.
Examples of signal types are Periodic, Random, Sine or Transient as functions of time.

13. Different Forms of an FRF
Compliance Dynamic stiffness
(displacement / force) (force / displacement)
Mobility Impedance
(velocity / force) (force / velocity)
Inertance or Receptance Dynamic mass
(acceleration / force) (force /acceleration)

14. Alternative Estimators
F(f)
H(f)
X(f)

15. Alternative Estimators+
F(f)
H(f)
X(f)
N(f)
SISO：
MIMO：
H1欠估计方法，输出端存在干扰噪声
响应的测量点数：n，激励力的点数：p，用不同激励力向量作s次试验

16. Alternative Estimators+
F(f)
H(f)
X(f)
M(f)
SISO：
MIMO：
H2过估计方法：输入端存在干扰噪声
响应的测量点数：n，激励力的点数：p，用不同激励力向量作s次试验

17. Which FRF Estimator Should You Use?
Accuracy
Definitions:
Accuracy for systems with: H1 H2 H3
Input noise - Best -
Output noise Best - -
Input + output noise - - Best
Peaks (leakage) - Best -
Valleys (leakage) Best - -
User can choose H1, H2 or H3 after measurement

18. Different Modal Analysis Techniques
f(t)
x(t) k c m
SDOF and MDOF Models
Different Modal Analysis Techniques
Exciting a Structure
Measuring Data Correctly
Modal Analysis Post Processing = +
+    +

19. Three Types of Modal Analysis
Hammer Testing
Impact Hammer ’taps’...serial or parallel measurements
Excites wide frequency range quickly
Most commonly used technique
Shaker Testing
Modal Exciter ’shakes’ product...serial or parallel measurements
Many types of excitation techniques
Often used in more complex structures
Operational Modal Analysis
Uses natural excitation of structure...serial or parallel measurements
’Cutting’ edge technique

20. Different Types of Modal Analysis (Pros)
Hammer Testing
Quick and easy
Typically Inexpensive
Can perform ‘poor man’ modal as well as ‘full’ modal
Shaker Testing
More repeatable than hammer testing
Many types of input available
Can be used for MIMO analysis
Operational Modal Analysis
No need for special boundary conditions
Measure in-situ
Use natural excitation
Can perform other tests while taking OMA data

21. Different Types of Modal Analysis (Cons)
Hammer Testing
Crest factors due impulsive measurement
Input force can be different from measurement to measurement (different operators, difficult location, etc.)
‘Calibrated’ elbow required (double hits, etc.)
Tip performance often an overlooked issue
Shaker Testing
More difficult test setup (stingers, exciter, etc.)
Usually more equipment and channels needed
Skilled operator(s) needed
Operational Modal Analysis
Unscaled modal model
Excitation assumed to cover frequency range of interest
Long time histories sometimes required
Computationally intensive

22. Frequency Response Function
FFT
Frequency Domain
Output
Input
Time Domain
Inverse
FFT
FFT

23. Hammer Test on Free-free Beam
Roving hammer method:
Response measured at one point
Excitation of the structure at a number of points
by hammer with force transducer
FRF’s between excitation points and measurement point calculated
Modes of structure identified
First
Mode
Amplitude
Second
Mode
Distance
Third
Mode
Beam
To understand what modal testing is, consider a simple free-free beam where the acceleration in one point on the beam is measured with an accelerometer. By exciting the beam with a modal hammer equipped with a force transducer in a number of point, it is possible to calculate the responses between the excitation points and the mesurement point.
If the imaginary part of the response (i.e. motion, be it acceleration, velocity or displacement) is calculated as a function of frequency, a set of curves would be obtained as shown in the 3D plot in the diagram.
The first three natural (resonance) frequencies, where the amplitudes start building up, are clearly seen along the frequency axis. By connecting the peaks at each of the natural frequencies the corresponding mode shapes can be traced out. Furthermore, if the 3 dB bandwidths at each of the resonances are determined, the loss factor can be calculated which gives the amount of modal damping for each of the modes.
From the 3D plot it can be seen that information about the dynamic properties of the structure lies at the natural frequencies and in their vicinity. Thus by determining the 3 above mentioned parameters experimentally for the first 10 modes, for example, a mathematical model can be built up in a computer.
Frequency
Force
Acceleration
Force
Force
Force
Force
Force
Force
Force
Frequency Domain View
Force
Force
Domain
Modal
View
Force
Force
Press anywhere
to advance animation

24. Measurement of FRF Matrix (SISO)
One row
One Roving Excitation
One Fixed Response (reference)
SISO
X1 H11 H12 H13 ...H1n F1
X2 H21 H22 H23 ...H2n F2
X3 H31 H32 H33...H 3n F3
: : :
Xn Hn1 Hn2 Hn3...Hnn Fn =

25. Measurement of FRF Matrix (SIMO)
More rows
One Roving Excitation
Multiple Fixed Responses (references)
SIMO
X1 H11 H12 H13 ...H1n F1
X2 H21 H22 H23 ...H2n F2
X3 H31 H32 H33...H 3n F3
: : :
Xn Hn1 Hn2 Hn3...Hnn Fn =

26. Shaker Test on Free-free Beam
Shaker method:
Excitation of the structure at one point
by shaker with force transducer
Response measured at a number of points
FRF’s between excitation point and measurement points calculated
Modes of structure identified
Amplitude
First
Mode
Distance
Second
Mode
Third
Mode
Beam
Another method, which will give the same result, is to excite the structure in one point using a modal shaker and measure the acceleration in a number of points on the structure, either as shown here by mounting an accelerometer in each point and measure all the responses simultaneously or by moving the accelerometer from point to point and measure the responses one by one.
Frequency
Acceleration
Frequency Domain View
Domain
Modal
View
White
noise
excitation
 Force
Press anywhere
to advance animation

27. Measurement of FRF Matrix (Shaker SIMO)
One column
Single Fixed Excitation (reference)
Single Roving Response SISO or
Multiple (Roving) Responses SIMO
Multiple-Output: Optimize data consistency
X1 H11 H12 H13 ...H1n F1
X2 H21 H22 H23 ...H2n F2
X3 H31 H32 H33...H 3n F3
: : :
Xn Hn1 Hn2 Hn3...Hnn Fn =

28. Why Multiple-Input and Multiple-Output ?
Multiple-Input: For large and/or complex structures more shakers are required in order to:
get the excitation energy sufficiently distributed
and
avoid non-linear behaviour
Multiple-Output: Measure outputs at the same time in order to optimize data consistency
i.e. MIMO

29. Situations needing MIMO
One row or one column is not sufficient for determination of all modes in following situations:
More modes at the same frequency (repeated roots), e.g. symmetrical structures
Complex structures having local modes, i.e. reference DOF with modal deflection for all modes is not available
In both cases more columns or more rows have to be measured - i.e. polyreference.
Solutions:
Impact Hammer excitation with more response DOF’s
One shaker moved to different reference DOF’s
MIMO
Repeated roots 重根
Polyreference 多参考点

30. Measurement of FRF Matrix (MIMO)
More columns
Multiple Fixed Excitations (references)
Single Roving Response MISO or
Multiple (Roving) Responses MIMO
X1 H11 H12 H13 ...H1n F1
X2 H21 H22 H23 ...H2n F2
X3 H31 H32 H33...H 3n F3
: : :
Xn Hn1 Hn2 Hn3...Hnn Fn =

31. Operational Modal Analysis (OMA): Response only!
Modal Analysis (classic): FRF = Response/Excitation
Time(Response) - Input
Working : Input : Input : FFT Analyzer
40m
80m
120m
160m
200m
240m
-80
-40 40 80
[s]
[m/s²]
FFT
Frequency Domain
Time Domain
Output
Autospectrum(Response) - Input
Working : Input : Input : FFT Analyzer
200
400
600
800 1k
1,2k
1,4k
1,6k 1m
10m
100m 1 10
[Hz]
[m/s²]
Inverse
FFT
Input
Frequency Response H1(Response,Excitation) - Input (Magnitude)
Working : Input : Input : FFT Analyzer
200
400
600
800 1k
1,2k
1,4k
1,6k
100m 10
[Hz]
[(m/s²)/N]
Impulse Response h1(Response,Excitation) - Input (Real Part)
Working : Input : Input : FFT Analyzer
40m
80m
120m
160m
200m
240m
-2k
-1k 1k 2k
[s]
[(m/s²)/N/s]
Autospectrum(Excitation) - Input
Working : Input : Input : FFT Analyzer
200
400
600
800 1k
1,2k
1,4k
1,6k
100u 1m
10m
100m 1
[Hz]
[N]
Natural Excitation
Frequency
Response
Function
Impulse
Response
Function
FFT
Time(Excitation) - Input
Working : Input : Input : FFT Analyzer
40m
80m
120m
160m
200m
240m
-200
-100
100
200
[s]
[N]
Output
Vibration
Response
H(w) = = =
Input
Force
Excitation

32. Different Modal Analysis Techniques
f(t)
x(t) k c m
SDOF and MDOF Models
Different Modal Analysis Techniques
Exciting a Structure
Measuring Data Correctly
Modal Analysis Post Processing = +
+    +

33. The Eternal Question in Modal…
To Shake!
To Tap, or...
Measurement Configuration for a Mechanical System
In order to measure the characteristics of a system, a controlled and measurable input must be used.
An example of a measurement setup for a mechanical system is shown. The system can be excited in two different ways, using a shaker or using an impact hammer. The input force is measured by means of a force transducer. In the first case, the transducer is mounted at the end of the push-rod (stinger) between the structure and the shaker, in the second case at the tip of the hammer. The output acceleration at a given point and in a given direction is measured by means of an accelerometer. F1 a F2

34. Measuring one row of the FRF matrix by moving impact position
Impact Excitation
Measuring one row of the FRF matrix by moving impact position
H11() H12()       H15()
    
# 5
# 4
Accelerometer
# 3
# 2
# 1
Measurements
All possible different Hij found by combinations of excitation/response points, can be assembled in the frequency response function matrix [M]. The row index indicates the response point, and the column index indicates the excitation point. Either of two methods is usually used to measure the frequency response functions, shaker excitation or transient excitation.
Shaker Excitation may be performed by attaching a shaker at one point and moving the accelerometer to each of the other points, yielding one column of [H].
Transient Excitation or impact testing is normally done by attaching an accelerometer at one fixed point and exciting each other point with a hammer, yielding one row of [H].
Impact
Hammer
LAN
Force
Transducer

35. Impact Excitation Magnitude and pulse duration depends on:
Weight of hammer
Hammer tip (steel, plastic or rubber)
Dynamic characteristics of surface
Velocity at impact
Frequency bandwidth inversely proportional to the pulse duration
Impact Excitation
Impact excitation by means of a hammer is a very convenient technique. The magnitude of the force pulse during the impact can be calculated from Newton’s second law:
| F | = | m dv/dt | ~ 2v0m/td,
where m is the mass and v is the velocity of the hammer, v0 is the velocity just before contact and td is the pulse duration.
The pulse duration is given by td ~ 2 m/k, where k is the stiffness of the system consisting of hammer plus structure, and the magnitude of the force is thus given by | F | ~ v0 mk.
Thus both the duration and the magnitude of the pulse depend on the weight of the hammer, the stiffness of the hammer tip used and of the surface. The magnitude of the force also depends on the velocity at impact. The bandwidth of the force spectrum is inversely proportional to the pulse duration.
a(t)
GAA(f) 1 1 2 2 t f

36. Weighting Functions for Impact Excitation
Criteria
How to select shift and length for transient and exponential windows:
Transient weighting of the input signal
Exponential weighting of the output signal
Weighting Functions for Impact Excitation
A transient window should be used for the input signal in order to improve the signal to noise ratio. In order to select the optimum shift and length it is a good idea to expand the y-scale so that the noise floor appears. On the figure the maximum y-value on the graph is 1 unit whereas the maximum value of the signal itself is 183 units.
For the output signal exponential weighting will often improve the analysis by reducing leakage due to truncation. To set up the optimum window one should here look at the magnitude of the weighted signal. The length of the exponential window (the time constant) should be selected so that the signal is attenuated into the noise floor at the end of the time record or at least 40 dB. The starting point of the window should be the same as for the input signal (unless there is a delay in the system).
Leakage due to exponential time weighting on response signal is well defined and therefore correction of the measured damping value is often possible

37. Compensation for Exponential Weighting
b(t)
With exponential weighting of the output signal, the measured time constant will be too short and the calculated decay constant and damping ratio therefore too large
Window function 1
Original signal
Time
Weighted signal
shift
Length =
Record length, T
Correction of decay constant s and damping ratio z:
Correct
value
Measured
value

38. Range of hammers Description Application Building and bridges
12 lb Sledge
3 lb Hand Sledge
Large shafts and larger machine tools
1 lb hammer
Car framed and machine tools
General Purpose, 0.3 lb
Components
Mini Hammer
Hard-drives, circuit boards, turbine blades

39. Impact hammer excitation
Advantages:
Speed
No fixturing
No variable mass loading
Portable and highly suitable for field work
relatively inexpensive
Disadvantages
High crest factor means possibility of driving structure into non-linear behavior
High peak force needed for large structures means possibility of local damage!
Highly deterministic signal means no linear approximation
Conclusion
Best suited for field work
Useful for determining shaker and support locations

40. Measuring one column of the FRF matrix by moving response transducer
Shaker Excitation
Measuring one column of the FRF matrix by moving response transducer
H11()               
H21()                
H51()               
Accelerometer
# 5
# 4
# 3
# 2
# 1
Force
Transducer
Measurements
In Modal Analysis, the test structure is stimulated with a known excitation, and the associated response, is measured.
The ratio of the Fourier Transforms of these two quantities is the frequency response function;
This describes the response at point i when excited by a unit force at point j.
In a structure where we want to describe the dynamic behaviour with n points, nn different frequency response functions can be measured. Each measurement represents one degree of freedom. To completely describe the behaviour of the structure, it might be necessary to measure in the x, y, and z directions, in which case for n measurement points there will be 3n degrees of freedom, and 3n x 3n different frequency response functions.
LAN
Vibration
Exciter
Power
Amplifier

41. Attachment of Transducers and Shaker
Force
Transducer a F
Accelerometer mounting:
Stud
Cement
Wax
(Magnet)
Shaker
Accelerometer
Properties of Stinger
Axial Stiffness: High
Bending Stiffness: Low
Force Transducer and Shaker:
Stud
Stinger (Connection Rod)
Advantages of Stinger:
No Moment Excitation
No Rotational Inertia Loading
Protection of Shaker
Protection of Transducer
Helps positioning of Shaker
Different methods exist for transducer mounting each optimized for the practical situation.
Stud mounting is the best method and should be used whenever possible. The steel stud gives the highest possible mounted resonance for the accelerometer extending the effective measurement range to its maximum for that particular accelerometer.
One version of the stud is the clips mounting used for the dedicated B&K modal testing accelerometers 4506/7/8.
Cementing the studs is an optimal solution where it is not possible to drill fixing holes in the device under test.
Wax is a very popular fixing method when high temperature is not a problem.
Fixing the shaker is often the most cumbersome job in a modal test. The Stinger method is definitely the most optimal for most modal testing jobs. The advantages are mentioned in the slide.

42. Connection of Exciter and Structure
Force Transducer
Measured structure
Exciter
Slender
stinger
Accelerometer
Force and acceleration measurements unaffected by stinger compliance, but ...
Minor mass correction required to determine actual excitation Fm Fs
As the force transducer has to be placed between the stinger and the device under test, the measurement of the force is unaffected by the dynamics of the stinger, the shaker and the shaker support. Note, however, that minor corrections are to be made due to the mass of the tip.
As the tip in the test situation adds mass to the structure under test there may be a need for compensating it. The compensation expressions in the slide are relevant for the hammer but in general a calibration with a known mass will give the compensation needed.
This calibration method is described later on.
Structure
Tip mass, m
Piezoelectric
material
Shaker/Hammer
mass, M

43. Shaker Reaction Force Reaction by external support Reaction by
exciter inertia
Example of an
improper
arrangement
Structure
Suspension
Structure
Suspension
Exciter
Suspension
Exciter
Support
Three different ways of shaker mounting are used.
The figure to the left gives the reaction from the external support. It is mostly used for smaller structures where it is possible to support the structure by “soft” means. The problem is that the exciter support must be resonance free.
The middle figure gives reaction excitation by the exciter inertia. This of course limits the force but additional masses may be added to the exciter.
In general the above setups work only when the suspension frequencies are much lower than the lowest frequencies of interest for the modal test.
The example to the right is not recommended, but is often seen where huge shakers are to be used. This should in general not be necessary for modal testing.
One good question is “how big a shaker is needed” for modal. Often large shakers are seen for exciting small structures. They are needed for qualification or fatigue testing, where large frequency ranges need to be excited. For modal testing all the required information is around the resonance peaks where the dynamic mass is small. This puts much less requirement to the amount of force needed for the modal test than for e.g. fatigue testing. Even for a structure like a car body, 100 N is enough.

44. Sine Excitation Crest factor
RMS
Crest factor
Time
A(f1)
B(f1)
For study of non-linearities, e.g. harmonic distortion
Sine Excitation
Here, the crest factor of a sine wave is given. The low value means that sine excitation is a good choice in situations where high peaks might cause problems.
It is also indicated how sine excitation can be used to study non-linearities of a structure. For a given frequency a number of measurements are made, all with different input levels.
To cover a broad frequency range a sweeping technique must be used. The sweep rate must be sufficiently low to allow the system time to respond to the changing frequencies.
For broadband excitation:
Sine wave swept slowly through the frequency range of interest
Quasi-stationary condition

45. Swept Sine Excitation Advantages Low Crest Factor
High Signal/Noise ratio
Input force well controlled
Study of non-linearities possible
Disadvantages
Very slow
No linear approximation of non-linear system
Swept Sine Excitation
The advantages and disadvantages of the swept sine excitation are listed here. The most important advantage of the technique is the fact that it makes the study of non-linearities such as harmonic distortion possible. Sine excitation is the only technique which can be used for this purpose.

46. Random Excitation Random variation of amplitude and phase
Time
Random variation of amplitude and phase
 Averaging will give optimum linear
estimate in case of non-linearities
System
Output
GAA(f), N = 1
B(f1)
GAA(f), N = 10
Random Excitation
A random signal is a signal where the amplitude and the phase of the instantaneous spectrum at a given frequency vary at random from record to record. Therefore, averaging will give a linear approximation to the frequency response function of a non-linear system excited by a random signal. This is important in applications based on linear models, for instance Modal Analysis.
A(f1)
Freq.
Freq.
System
Input

47. Random Excitation Random signal:
Characterized by power spectral density (GAA) and amplitude probability density (p(a))
a(t)
p(a)
Time
Can be band limited according to frequency range of interest
GAA(f)
Baseband
GAA(f)
Zoom
Random Excitation
Apart from its autospectrum, a random signal is characterized by its amplitude probability density. The amplitude probability has a Gaussian distribution.
A random signal can easily be band limited to the frequency range of interest to obtain an optimum dynamic range. A Hanning weighting function should be used to minimize leakage in the analysis.
Freq.
Freq.
Frequency
range
Frequency
range
Signal not periodic in analysis time  Leakage in spectral estimates

48. Random Excitation Advantages Best linear approximation of system Zoom
Fair Crest Factor
Fair Signal/Noise ratio
Disadvantages
Leakage
Averaging needed (slower)
Random Excitation
The advantages and disadvantages of random excitation are listed here. The most important advantage of random excitation is the fact that it gives the best linear approximation of a non-linear system. The crest factor is typically 3 to 3.5. A random signal excites all frequencies of interest simultaneously and therefore this technique is relatively fast. However, contrary to pseudo random signals, averaging is necessary.

49. Burst Random Characteristics of Burst Random signal :
Gives best linear approximation of nonlinear system
Works with zoom
a(t)
Time
Advantages
Best linear approximation of system
No leakage (if rectangular time weighting can be used)
Relatively fast
A burst random signal is a random signal weighted with a transient time window.
The transient window must be synchronised with the analysis trigger.
The length of the burst must be made so, that when it stops there must be time enough to let the structure ring out. This is to prevent leakage.
The advantages of burst random is that:
1. Due to the random nature of the signal you get the best linear approximation of the system.
2. Leakage free, the signal is limited in time.
3. It is as fast as the random test.
Disadvantages:
1. Crest factor is higher that in random test, as the contribution to the RMS value is limited due to the time limitation of the signal.
2. Due to low damping of the structure a time weighting may be required.
Disadvantages
Signal/noise and crest factor not optimum
Special time weighting might be required

50. Pseudo Random Excitation
Pseudo random signal:
Block of a random signal repeated every T
a(t)
Time T T T T
Time period equal to record length T
Line spectrum coinciding with analyzer lines
No averaging of non-linearities
System
Output
GAA(f), N = 1
Pseudo Random Excitation
A pseudo random signal is a purely deterministic signal made by repeating a segment of a random signal. The period is equal to the record length in the analyzer. This implies that the signal only contains frequencies coinciding with the lines computed in the analyzer. Normally it is designed to have the same level in each line. At a given frequency a system will always be excited at the same level since each recording contains the same information. Therefore no averaging of non-linearities can be obtained.
GAA(f), N = 10
B(f1)
A(f1)
Freq.
Freq.
System Input

51. Pseudo Random Excitation
Pseudo random signal:
Characterized by power/RMS (GAA) and amplitude probability density (p(a))
a(t)
p(a)
Time T T T T
Can be band limited according to frequency range of interest
GAA(f)
Baseband
GAA(f)
Zoom
Pseudo Random Excitation
The autospectrum of a pseudo random signal is measured in power or RMS. The signal is also characterized by its amplitude probability which is not as smooth as the amplitude probability density for a random signal.
However, like random signals, pseudo random signals can be shaped to fit the frequency range of interest. The fact that the signal is periodic within the record length also has the implication that leakage can be avoided using Rectangular weighting.
Freq.
Freq.
Freq. range
Freq. range
Time period equal to T
 No leakage if Rectangular weighting is used

52. Pseudo Random Excitation
Advantages
No leakage
Fast
Zoom
Fair crest factor
Fair Signal/Noise ratio
Disadvantages
No linear approximation of non-linear system
Pseudo Random Excitation
The advantages and disadvantages of pseudo random excitation are listed here. The most important advantage is the fact that leakage in the analysis can be avoided by using Rectangular weighting.
Since the signal is periodic and since all frequencies will be excited at a sufficiently high level in each record, averaging is only needed if the signal to noise ratio is poor. This makes pseudo random excitation a faster technique than random excitation.

53. Multisine (Chirp) For sine sweep repeated every time record, Tr Tr
A special type of pseudo random signal where the crest factor has been minimized (< 2)
It has the advantages and disadvantages of the “normal” pseudo random signal but with a lower crest factor
Additional Advantages:
Ideal shape of spectrum: The spectrum is a flat magnitude spectrum, and the phase spectrum is smooth
Applications:
Measurement of structures with non-linear behaviour

54. Periodic Random
A combined random and pseudo-random signal giving an excitation signal featuring:
No leakage in analysis
Best linear approximation of system A A A B B B C C C
Pseudo-random signal changing with time: T T T
transient response
steady-state response
Analysed time data:
(steady-state response) A B C
Disadvantage:
The test time is longer than the test time using pseudo-random or random signal

55. Periodic Pulse Special case of pseudo random signal
Rectangular, Hanning, or Gaussian pulse with user definable D t repeated with a user definable interval, D T
D t
D T
Time
The line spectrum for a Rectangular pulse has a shaped envelope curve
sin x x
1/D t
Frequency
Leakage can be avoided using rectangular time weighting
Transient and exponential time weighting can be used to increase Signal/Noise ratio
Gating of reflections with transient time weighting
Effects of non-linearities are not averaged out
The signal has a high crest factor

56. Periodic Pulse Advantages Disadvantages Fast No leakage
(Only with rectangular weighting)
Gating of reflections
(Transient time weighting)
Excitation spectrum follows frequency span in baseband
Easy to implement
No linear approximation of non-linear system
High Crest Factor
High peak level might excite non-linearities
No Zoom
Special time weighting might be required to increase Signal/Noise Ratio . This can also introduce leakage
Periodic Pulse

57. Guidelines for Choice of Excitation Technique
For study of non-linearities:
Swept sine excitation
For slightly non-linear system:
Random excitation
For perfectly linear system:
Pseudo random excitation
For field measurements:
Summary
Here some very coarse guidelines concerning best choice of excitation technique in different situations are given.
Impact excitation
For high resolution field measurements:
Random impact excitation

58. Different Modal Analysis Techniques
f(t)
x(t) k c m
SDOF and MDOF Models
Different Modal Analysis Techniques
Exciting a Structure
Measuring Data Correctly
Modal Analysis Post Processing = +
+    +

59. Garbage In = Garbage Out!
A state-of-the Art Assessment of Mobility Measurement Techniques
– Result for the Mid Range Structure ( Hz) –
D.J. Ewins and J. Griffin
Feb. 1981
Transfer Mobility
Central Decade
Transfer Mobility
Expanded
Present state-of-the Art
A incredible amount of effort is put into the development of new sophisticated curve fitting algorithms and excitation techniques and perhaps too little attention has been placed on the frequency response function measurements themselves.
A standardized test object that comprised of a simple structure was sent around to some of the leading structural laboratories in Europe and the USA. The FRF curves returned have been overlaid in the figure above. The spread of data clearly seen speaks for itself. There is still a lot to learn.
Frequency
Frequency

60. Plan Your Test Before Hand!
Select Appropriate Excitation
Hammer, Shaker, or OMA?
Setup FFT Analyzer correctly
Frequency Range, Resolution, Averaging, Windowing
Remember: FFT Analyzer is a BLOCK ANALYZER!
Good Distribution of Measurement Points
Ensure enough points are measured to see all modes of interest
Beware of ’spatial aliasing’
Physical Setup
Accelerometer mounting is CRITICAL!
Uni-axial vs. Triaxial
Make sure DOF orientation is correct
Mount device under test...mounting will affect measurement!
Calibrate system

61. Where Should Excitation Be Applied?
Driving Point
Measurement j 1 2 i
i = j
In a test model a single point is described with maximum six degrees of freedom (DOF). Three translational and three rotational.
The notation of the FRF includes index with information of the DOF. Hij indicates the frequency response function measured at point i and excited at point j.
J=I means that the structure is excited in the same point as it is measured. This is called a driving point measurement. The opposite is called a transfer measurement.
The FRF model is then organized as a matrix where the columns contain information from the same excitation point and the rows for the same response point.
In the example it is seen that the vibration in DOF1 is a summation of the driving point FRF multiplied by the force in DOF 1 and the transfer FRF between DOF 1 and 2 and the force in DOF 2.
Transfer
Measurement j
i ¹ j i

62. Check of Driving Point Measurement
Im Hij
All peaks in
Mag Hij
An anti-resonance in Mag Hij must be found between every pair of resonances
In driving point measurements, all modes are in phase with the force. Hence, a characteristic of a driving point frequency response function is an anti resonance between each resonance, and a phase curve which varies between 0 and 180.
If due to practical reasons there is a physical distance between the excitation and the response point higher modes may show wrong driving point behavior.
Phase Hij
Phase fluctuations must be within 180

63. Driving Point (DP) Measurement
The quality of the DP-measurement is very important, as the DP-residues are used for the scaling of the Modal Model
DP- Considerations:
Residues for all modes must be estimated accurately from a single measurement
DP- Problems:
Highest modal coupling, as all modes are in phase
Highest residual effect from rigid body modes
m Aij
Log MagAij
Measured
In the previous section the importance of the correct driving point measurement is highlighted. The scaling of the test model is derived from one single measurement.
This sets the requirements for this measurement. Normally if the modal coupling is high the information of mode separation is carried by the phase information.
As all the modeshapes at the driving point are in phase the modal separation may be bad if only the driving point is considered. This also counts for the out of band modes such as the rigid body modes.  =
Re Aij
Without rigid
body modes  

64. Tests for Validity of Data: Coherence
Measures how much energy put in to the system caused the response
The closer to ‘1’ the more coherent
Less than 0.75 is bordering on poor coherence
Before doing the analysis it is important to validate the measurements. Even the best modal software will give wrong results if the measurements are wrong. “Garbage In - Garbage Out” (GIGO)
The Coherence function provides a means of assessing the degree of linearity between input and output signals. The interpretation of the Coherence function is that for each frequency it shows the degree of linear relationship between input and output.
Linearity can also be detected by measuring at different levels and observing the proportionality between input and output. This is most often tested using a sine wave.
Reciprocity is the phenomena that a response measured in DOF j and excited in DOF i gives the same result as a response measured in DOF i excited in DOF j.

65. Reasons for Low Coherence
Difficult measurements:
Noise in measured output signal
Noise in measured input signal
Other inputs not correlated with measured input signal
Bad measurements:
Leakage
Time varying systems
Non-linearities of system
DOF-jitter
Propagation time not compensated for

66. Tests for Validity of Data: Linearity
X1 = H·F1
X2 = H·F2
X1+X2 = H·(F1 + F2)
a·X1 = H·(a· F1)
Linearity Þ
H()
F()
X()
More force going in to the system will equate to more response coming out
Since FRF is a ratio the magnitude should be the same regardless of input force
Before doing the analysis it is important to validate the measurements. Even the best modal software will give wrong results if the measurements are wrong. “Garbage In - Garbage Out” (GIGO)
The Coherence function provides a means of assessing the degree of linearity between input and output signals. The interpretation of the Coherence function is that for each frequency it shows the degree of linear relationship between input and output.
Linearity can also be detected by measuring at different levels and observing the proportionality between input and output. This is most often tested using a sine wave.
Reciprocity is the phenomena that a response measured in DOF j and excited in DOF i gives the same result as a response measured in DOF i excited in DOF j.

67. Tips and Tricks for Best Results
Verify measurement chain integrity prior to test:
Transducer calibration
Mass Ratio calibration
Verify suitability of input and output transducers:
Operating ranges (frequency, dynamic range, phase response)
Mass loading of accelerometers
Accelerometer mounting
Sensitivity to environmental effects
Stability
Verify suitability of test set-up:
Transducer positioning and alignment
Pre-test: rattling, boundary conditions, rigid body modes, signal-to-noise ratio, linear approximation, excitation signal, repeated roots, Maxwell reciprocity, force measurement, exciter-input transducer-stinger-structure connection
Quality FRF measurements are the foundation of experimental modal analysis!

68. Different Modal Analysis Techniques
f(t)
x(t) k c m
SDOF and MDOF Models
Different Modal Analysis Techniques
Exciting a Structure
Measuring Data Correctly
Modal Analysis Post Processing = +
+    +

69. From Testing to Analysis
Frequency
Measured
FRF
Curve Fitting
(Pattern Recognition)
Frequency
Modal Analysis

70. From Testing to Analysis
Modal Analysis
Frequency
f(t)
x(t) k c m
f(t)
x(t) k c m
f(t)
x(t) k c m
SDOF Models

71. Mode Characterizations
All Modes Can Be Characterized By:
Resonant Frequency
Modal Damping
Mode Shape

72. Modal Analysis – Step by Step Process
1. Visually Inspect Data
Look for obvious modes in FRF
Inspect ALL FRFs…sometimes modes will show up in one FRF but not another (nodes)
Use Imaginary part and coherence for verification
Sum magnitudes of all measurements for clues
2. Select Curve Fitter
Lightly coupled modes: SDOF techniques
Heavily coupled modes: MDOF techniques
Stable measurements: Global technique
Unstable measurements: Local technique
MIMO measurement: Poly reference techniques
3. Analysis
Use more than 1 curve fitter to see if they agree
Pay attention to Residue calculations
Do mode shapes make sense?

73. Modal Analysis – Inspect Data
1. Visually Inspect Data
Look for obvious modes in FRF
Inspect ALL FRFs…sometimes modes will show up in one FRF but not another (nodes)
Use Imaginary part and coherence for verification
Sum magnitudes of all measurements for clues
2. Select Curve Fitter
Lightly coupled modes: SDOF techniques
Heavily coupled modes: MDOF techniques
Stable measurements: Global technique
Unstable measurements: Local technique
MIMO measurement: Poly reference techniques
3. Analysis
Use more than 1 curve fitter to see if they agree
Pay attention to Residue calculations
Do mode shapes make sense?

74. Modal Analysis – Curve Fitting
1. Visually Inspect Data
Look for obvious modes in FRF
Inspect ALL FRFs…sometimes modes will show up in one FRF but not another (nodes)
Use Imaginary part and coherence for verification
Sum magnitudes of all measurements for clues
2. Select Curve Fitter
Lightly coupled modes: SDOF techniques
Heavily coupled modes: MDOF techniques
Stable measurements: Global technique
Unstable measurements: Local technique
MIMO measurement: Poly reference techniques
3. Analysis
Use more than 1 curve fitter to see if they agree
Pay attention to Residue calculations
Do mode shapes make sense?

75. How Does Curve Fitting Work?
Curve Fitting is the process of estimating the Modal Parameters from the measurements 2s
Find the resonant frequency
Frequency where small excitation causes a large response
Find the damping
What is the Q of the peak?
Find the residue
Essentially the ‘area under the curve’
H
R/s w wd
-180
Phase
Frequency

76. Residues are Directly Related to Mode Shapes!
Residues express the strength of a mode for each measured FRF
Therefore they are related to mode shape at each measured point!
Second
Mode
First
Mode
Amplitude
Distance
Third
Mode
Beam
Frequency
Acceleration
The expression and figure shows how a MDOF system is decomposed in SDOF FRF models.
The modal parameters are known as the Natural Frequency, the Damping and the Mode shape. The Residue expresses the strength of the mode.
The Residue is a mathematical concept carrying the scaling of the FRF and relating to the Modeshape.
The Residue is expressed as the area beneath the FRF of a SDOF system. This means that the Residue is somehow related to the maximum peak of the FRF.
The figure shows the second mode of a cantilever beam with the excitation at i and response measured at point j.
Note that the resonance curve of all the DOFs shows the same frequency and damping (global parameters) but the residue depends of the DOF (local parameter).
Force
Force
Force
Force
Force
Force
Force
Force
Force
Force
Force
Force

77. SDOF vs. MDOF Curve Fitters
Use SDOF methods on LIGHTLY COUPLED modes
Use MDOF methods on HEAVILY COUPLED modes
You can combine SDOF and MDOF techniques!
SDOF
(light coupling)
H
MDOF
(heavy coupling) 

78. Local vs. Global Curve Fitting
Local means that resonances, damping, and residues are calculated for each FRF first…then combined for curve fitting
Global means that resonances, damping, and residues are calculated across all FRFs
H 

79. Modal Analysis – Analyse Results
1. Visually Inspect Data
Look for obvious modes in FRF
Inspect ALL FRFs…sometimes modes will show up in one FRF but not another (nodes)
Use Imaginary part and coherence for verification
Sum magnitudes of all measurements for clues
2. Select Curve Fitter
Lightly coupled modes: SDOF techniques
Heavily coupled modes: MDOF techniques
Stable measurements: Global technique
Unstable measurements: Local technique
MIMO measurement: Poly reference techniques
3. Analysis
Use more than one curve fitter to see if they agree
Pay attention to Residue calculations
Do mode shapes make sense?

80. Which Curve Fitter Should Be Used?
Frequency Response Function
Hij ()
Real
Imaginary
Nyquist
Log Magnitude
Another factor influencing how modes interact is the degree of modal coupling. With light modal coupling, a number of simplifications can be made, for instance, the assumption that the local value of the frequency response function is dominated by a single mode. Light coupling allows modal analysis by quadrature picking, whereby the value of the imaginary part of the frequency response function the residue, or strength of each mode at each degree of freedom. Another implication of light coupling is that single degree of freedom curve fitting techniques can be used.

81. Which Curve Fitter Should Be Used?
Frequency Response Function
Hij ()
Real
Imaginary
Nyquist
Log Magnitude
Heavy coupling means that the modes are highly interactive, and the situation becomes more complicated. Heavy coupling means that multiple degree of freedom curve fitting techniques should be used.

82. Modal Analysis and Beyond
Experimental
Modal Analysis
Dynamic Model
based on
Modal Parameters
Structural
Modification F
Response
Simulation
Hardware Modification
Resonance Specification
Simulate Real
World Response

83. Conclusion
All Physical Structures can be characterized by the simple SDOF model
Frequency Response Functions are the best way to measure resonances
There are three modal techniques available today: Hammer Testing, Shaker Testing, and Operational Modal
Planning and proper setup before you test can save time and effort…and ensure accuracy while minimizing erroneous results
There are many curve fitting techniques available, try to use the one that best fits your application
Sound Pressure Level = 10log (P/P0)2
The range of human hearing is from 0 dB (2 × Pa = Threshold of hearing) to 130 dB (Threshold of pain)

84. Literature for Further Reading
Structural Testing Part 1: Mechanical Mobility Measurements
Brüel & Kjær Primer
Structural Testing Part 2: Modal Analysis and Simulation
Modal Testing: Theory, Practice, and Application, 2nd Edition by D.J. Ewin
Research Studies Press Ltd.
Dual Channel FFT Analysis (Part 1)
Brüel & Kjær Technical Review # 1 – 1984
Brüel & Kjær Technical Review # 2 – 1984

85. Appendix: Damping Parameters
3 dB bandwidth ,
Loss factor
Damping ratio
Decay constant
Quality factor

86. Appendix: Damping Parameters
, where the Decay constant is given by e-st
The Envelope is given by magnitude of analytic h(t):
h(t)
Decay constant
Time constant :
Damping ratio
Loss factor
Quality
Time
The time constant, t, is determined by the time it takes for the amplitude to decay a factor of e = 2,72… or
10 log (e2) = 8.7 dB