mode shape and method for deriving/visualization 2

[Dmitry]

Hi All,
I'm doing analysis of modal test data. I found that mode shapes look different depending on method for deriving/visualization!

For particular frequency:
By First method - Amplitude ("Displays the modal vector in a derived real mode format.&quot - I see bending of the structure in vertical plain
By Second method - Quadrature ("Displays only the imaginary coefficient of the modal vector.&quot - I see similar bending of the structure in vertical plain

I think that if there is no apparent overlapping of the FRF pikes then it does not matter which method is used - resulting shape must be really close!

It is so weird. What is the matter?

Regards,
Dmitry

 

[Dmitry]

Sorry. By Second method - I see similar bending of the structure in HORIZONTAL plain

 

[GregLocock]

Hi Dmitry

Usually for a (confusingly called) Real mode the Imaginary component of the response is the mode shape. This is the quadrature method. This forces the response to be at +/- 180 degrees, as would be expected for a single degree of freedom system.

I'm a bit confused by your description of Amplitude method, it may be that it is animating the real and imaginary parts, ie allowing phases other than + or - 180 degrees.

This is often called a complex mode (in the literature) and is typical of complex structures and higher order modes generally.

Perhaps you could look at the mode shape list and compare it back to your Eigenvectors to see if this is correct.

Anyway, if I'm right then the Real part of the shape is bigger than the Imaginary part, so when you switch to Quadrature method you are only viewing a small part of the motion.

There are other more complex possibilities, depending on how you extracted the mode shape in the first place. If the data is fairly small you could send me the FRFs in ASCII and I'll tell you how many modes there are! Cheers

Greg Locock

 

[Dmitry]

Hi Greg,
I have access only for animated mode shapes. They are animated in STARReport system. Complete help on the topic is:

METHOD DESCRIPTION
AMPLITUDE Displays the modal vector in a derived real mode format.
AMPL/PHASE Displays the complete modal vector both real and imaginary coefficients.
COINCIDENT Displays only the real coefficient of the modal vector.
QUADRATURE Displays only the imaginary coefficient of the modal vector.

I think that first method handling amplitudes from Bode plot for each point of the structure or an curve-fitted data, while Quadrature method - amplitudes from imaginary part of signal spectrum.

Glad to hear from you!
Dmitry

 

[Dmitry]

I do have three spectrums. Each is the sum of signals for all the accelerometers in the specified direction. So they are not for a single point. They give only an idea about FRF.

Regards,
Dmitry

 

[MikeyP]

I am pretty sure that the AMPLITUDE visualisation works like this

1) Simple version (how I do it)
Consider the complex values of the eigenvector in the Argand plane. If it lies above the real axis then assign it a POSITIVE sign and give it an imaginary part equal to the magnitude of the eigenvector. If it lies below the real axis then assign it NEGATIVE sign and give it an imaginary part equal to the magnitude of the eigenvector. The eigenvector is then purely imaginary (ie a 'real' mode - confusing as Greg says!)

This can be summarised mathematically for one element, x, of the eigenvector as:
new_x = sgn(im(x))* |x| * i

example:
the eigevector {1+5i, -1+2i, 2-4i) would become:
{+|1+5i|i, +|-1+2i|i, -|2-4i|i}

Try plotting this on an Argand diagram and you will see what I mean.

2) more sophisticated approach
Again in the Argand plane. Rotate the whole eigenvector about the origin until there is minimum total deviation from the imaginary axis of all elements of the eigenvector (minimum 'phase scatter'). Now take the imaginary parts of the rotated eigenvector.

Of course, these normalisations are only of use if the mode is nearly 'real' to start off with. If your curve fitting has yielded significantly 'complex' modes and you want to compare them with normal modeshapes from an FE analysis for example, then you need to use a 'phase resonance' approach (also called 'force appropriation') which can (under favourable conditions) give you the undamped normal mode shape.

The reason that you see more of a difference with modes which are close in frequency than those which are well separated is because of non-proportional damping. The modes are coupled (even in modal space) by damping forces. This is the cause of the modal complexity. These coupled forces are greater when two modes are close in frequency.

You may also be getting errors if you are using a single degree-of-freedom curve fit such as a Nyquist circle fit. These methods will only work if the modes are well separated in frequency. A global multi degree-of-freedom curve fit (eg the LSCE/Polyreference family of methods) is much better at separating the modes.

Michael

 

[MikeyP]

Dmitry,

You cannot get modeshape information from the summed responses. You need to keep the data from each transducer separate.

Michael

 

[Dmitry]

Thanks Michael,
Indeed I want to identify modes and compare them with normal mode shapes from an FE analysis, but I am trying to figure out if all curve fitting is already done or not.
Judging from yours and Greg postings:
* In case of AMPLITUDE - I am working with mode shape, eigenvector or "modal vector in real mode format"
* In case of AMPL/PHASE method - with ODS (Operating Deflection Shape) that is "modal vector". This is why I see some delay in movements of different points of the structure, while using other methods all points seems to move simultaneously.
* In case of COINCIDENT - they animate relative phase shift between Input and Output??
* In case of QUADRATURE - they animate ODS, which is taken equal to Im part of a FRF at resonant frequency for each point of the structure.

As it happens I do not know what is represented by Im and Re parts of eigenvector. What sense does it make to animate them for real 3D structure?

Regards,
Dmitry

 

[MikeyP]

Dmitry, perhaps by incorrectly calling the complex modeshapes generated by the curve fitter "eigenvectors" I have confused you.

The true eigenvector is always made up of real values in a linear system and is the undamped normal mode shape for that mode.

The curve fitting will generally give you "complex" modeshapes (not strictly ODSs). If you use the AMPLITUDE/PHASE visualisation then you are looking at the "complex mode" which the curve fitter has generated.

A purely real value in the fitted modeshape is one where the response displacement or acceleration (NOT velocity) is in phase or 180 degrees out of phase with the input force (coincident). An imaginary value is one where the response is + or - 90 degrees out of phase with the input (quadrature). Of course your software may use a different phase reference, but mine uses these conventions.

In a single degree of freedom mechanical system the point where the response and force are 90 degrees out of phase occurs at the natural frequency. An ideal modal analysis with ideal data performed on this SDOF system would give a mode shape which has purely imaginary values. You could check this by looking at the coincident and quadrature visualisations side by side. The coincident response would be zero and the quadrature response would look the same as the amplitude/phase visualisation. The true normal modeshape (the eigenvector) will be the same as the imaginary (quadrature) part.

When you have a multi-degree of freedom system and less than ideal FRF data things are not so simple. Non-proportional damping and modes being close in frequency confuse the curve fitter and it does it's best by giving you a complex mode shape. If you use these complex mode shapes along with the natural frequencies and complex damping ratios (the ratios themselves are not complex but they are the damping ratios of a complex mode) to systhesise an FRF this should still agree with the measured FRF. However these complex modeshapes are of no use if you are trying to validate an FE model. You need the normal mode shapes.

Most of the time the curve fitter does a good job if you have modes which are well separated in frequency. When you look at the amplitude/phase visualisation for these modes everything moves pretty much in phase. If you then look at the coincident and quadrature visualisations side by side then the quadrature response should be large compared with the coincident response. In this instance, you can use the AMPLITUDE visualisation to ignore any coincident response. This will give you a real valued modeshape which you can compare with the eigenvector produced by your FE model.

Where the modes are close in frequency, there may be a large coincident reponse. You can also see this on the amplitude/phase visualisation. The nodal points and nodal lines where there is no motion will appear to move around during the animated cycle. In this case, if you ignore the coincident information, you will introdude significant errors into the mode shape.

It is in these situations where a force appropriation analysis is required.

I hope this makes things a bit clearer.

Michael

 

[Dmitry]

Michael,
I really appreciate you help. Now the issue is clear.
Since at least some of my modes call for force appropriation analysis, can you explain the base principle of it and how new is the method?

Thanks,
Dmitry

 

[MikeyP]

Dmitry,

Force appropriation aims to excite a mode at it's undamped natural frequency and in its undamped normal mode shape.

The main thing is that there must be AT LEAST TWO exciters. For 'soft tuning' (explained below) you could use one exciter (hammer, shaker etc) and measure 2 or more sets of FRFs with the exciter in different positions. For 'hard tuning' you need two or more exciters on the structure at the same time and these exciters must be capable of producing a sine wave at a specified amplitude and phase (ie a hammer won't do it).

We are trying to find the vector of forces (one force for each exciter) which will a) excite the mode in question, and b) suppress the response of the other modes close to it. The vector of forces must achieve the 'phase resonance condition'. That is, the forces must be monophase (either in phase or 180 degrees out of phase). Also the responses must be monophase and in quadrature with the applied forces.

There are different ways of finding this 'appropriated force vector' but in my opinion the best method is the Modified Mode Indicator Function (MMIF). Briefly the MMIF is minimisation problem (The maths is actually pretty straightforward). It minimises the real part of the response with respect to the total response. The eigenvalues and eigenvectors of this cost function tell you a) where the modes are (from the eigenvalues) and b) what the appropriated force vectors are for those modes (from the eigenvectors). Just consider it as a black box. You put FRF data in one end and you get eigenvalues and eigenvectors out the other end. A little bit of interpretation of this output data by the user is required.

So now we have our appropriated force vector for the mode we are interested in. We have 2 options:

'soft tuning'. We simply multiply the matrix of FRF values at the undamped natural frequency by the appropriated force vector. This gives a response vector which, with ideal data, will be the undamped normal mode shape.

'hard tuning'. We apply the appropriated force vector to the structure in the form of a sine wave at the undamped natural frequency. We then measure the response which, with ideal data , will be the undamped normal mode shape.

The practicalities of hard tuning are not as simple as they may seem. You know the appropriated force vector. But what you need know is "What voltages do I apply to my shakers to produce that force vector?". This can be done, but it is a lot less straigtforward than soft tuning.

So why bother with hard tuning at all? Well, the estimate of the undamped natural frequency obtained from the MMIF eigenvalues is only as accurate as the frequency resolution of the original FRF. Small errors in the natural frequency can lead to large errors in the quality of the tuned mode. With hard tuning we can nudge the frequency up and down by tiny amounts until we get the best result.

A crucial element in the force appropriation analysis is the exciter positions. The exciters must be able to excite and isolate the mode in question. In theory, if you have 2 close modes then you can perfectly tune them with 2 exciters if they are in the right place, for 3 close modes you need 3 exciters. In practice, careful choice of exciter positions combined with good quality FRF data is enough to get pretty decent results.

This method is now well established particularly in the aerospace industy. Other sectors are now starting to use it as well. Many people only test with single exciters and so force appropriation is not an option. Some high end modal analysis software includes the ability to perform hard tuning. Soft tuning can be done easily with just a few lines of MATLAB code.

Michael

 

[GregLocock]

Neat. How well does this work with non-linear structures? When we used to do whole vehicle modals we were very keen on MIMO when it first became available, but in practice our reciprocity was so lousy that we went back to SIMO.

Cheers

Greg Locock

 

[MikeyP]

Dmitry

If you have hardening stiffness non-linearities (the most usual kind) then doing all your mesurements at a low excitation level will give you a good approximation of the underlying linear system. Of course keeping the displacements low may be difficult when you are deliberately exciting the structure at resonance. Using SISO instead of MIMO will not make the non-linearity go away! you will still have lousey reciprocity, you are just not measuring it!

There are methods for doing force appropriation on non-linear systems which can identify the non-linear stiffness and damping characteristics. Unfortunately, the only place in the world they exist at the moment is on my PC! In fact my current research could be neatly summed up as "Identification of Non-Linear Multi Degree of Freedom Systems Using Force Appropriation"

We are looking at two approaches.
1) Force Appropriation of Non-Linear Systems (FANS)
This again trys to isolate an individual mode, but this time the force vector will include 2nd, 3rd, 4th, 5th, harmonic terms. These are calculated using an optimisation process.

2) Resonant Decay of Non-Linear Systems (RDNS)
This works of the premise that most systems will have a mixture of linear uncoupled modes, linear modes coupled by non-proportional damping, non-linear uncoupled modes, non-linear modes coupled by non-linear terms.

You can use the usual modal analysis methods of curve fitting and force appropriation to identify the linear modes. For the non-linear modes, you excite them at a high excitation level with their linear appropriated force vector and see which other modes are also excited. You then use a curve fit on the response to the appropriated vector of all the active modes which includes non-linear terms. This gives you a modal model of the non-linear system which is essentially the undelying linear model with a few polynomial non-linear terms tacked on the end.

This works rather well with computer simulations. We are only just beginning to assess the method experimentally.

The method does rely on good quality estimates of the linear normal mode shapes and these are found by doing an initial modal analysis at a low excitation level.

We have been touting this method around the conference circuit for the last few months with some very favourable responses. I have just put in an application for some goverment funding on this very subject so let's hope the response there is favourable too otherwise I won't have a job by September!

Watch this space

Michael

 

[Dmitry]

Michael,
I will vote to give you the job if somebody is to ask me.
Can you tell how you are modeling output of nonlinear system: in FE code or in MatLab thru the system of equations?
How is error distributed between modal parameters in case when SISO applied to non-linear system? I think that
ž max is for damping
ž then mode shape is affected
ž min is for natural frequency
Dmitry

 

[MikeyP]

Dmitry, At the moment the simulations are being done in Matlab on a 9 degree of freedom lumped parameter model with cubic stiffness elements added to some of the masses. The next stage is to try it on a real lumped parameter system in the lab. Then a continuous FE type model will be simulated followed by experiments on a real continuous structure. Most non-linear identification algorithms have only been simulated on very small systems (1,2 or 3 DOF) so 9 DOF is already a huge advance.

The errors in modal parameter estimation when the system is non-linear depend on the type of excitation used in the modal test. When random excitation is used, the peaks broaden (leading to errors in damping estimation) and shift slightly in frequency (generally upward for a hardening non-linearity). If stepped sine excitation is used the shift in frequency is generally greater than for random excitation. For severe non-linearity, bifurcations or 'jumps' may appear in the FRF. The FRF will be different depending on whether you are stepping up or down in frequency. I suspect than damping and frequency will be most affected and these will make the curve fitter generate mode shapes which are also in error. All three parameters will interact to produce errors.

Michael

 

[Dmitry]

Michael,
Is it already defined how non-linearity will be modeled in FE?
I cannot agree on you estimation of the error distribution. Let us conceder impact testing (SISO or SIMO).
If resonance pike is identified, then for a mode at 50-200 Hz error in frequency cannot be more then 5%.
Mode shape can be distorted but it is still suitable for visual comparison with FE results. As usual there are other modes of similar type and looking for a pattern (how modes should be placed in spectrum) often helps.
And the largest error should be attributed to damping simply because it is difficult to estimate correctly the number of resonant pikes.

How lucky you are to do such work! It is providing a unique experience.
Dmitry

 

[MikeyP]

Dmitry, geometric non-linearity is usually included in an FE model by the software. In ABAQUS for example you just have to set the NLGEOM keyword. This will only have an effect in a time domain dynamic analysis. A simple eigenvalue analysis will just give you the natural frequencies and modeshapes of the underlying linear system.

You should understand that the concept of a normal mode is not strictly valid for a non-linear system. How would you define a normal mode? The maximum value of the peak? - Even a linear structure can have the peak at a different frequency to the normal mode, so that won't work. How can we define a normal mode shape of a non-linear structure if we can't even define the natural frequency? As I said before, we can "linearise" the non-linear structure by exciting it at a low level. This will work for most types of non-linearity (except friction). The algorithms I work with use this approach. The underlying linear model is identified and then extra terms are included to account for the non-linearities.

The effects of using a linear curve fit on non-linear response data will depend on a number of things. For example the type of non-linearity (stiffness, damping, friction, clearance etc.), the severity of the non-linearity, and the level at which you excite the non-linear structure.

A good illustration is to try it out with a simple FE model. Define a 1-dimensional beam which is clamped at both ends. 8 or 10 elements will be enough to get the first 3 modes. This will have a hardening cubic stiffness non-linearity. The stiffness curve will follow an x^3 shape. The higher the displacement, the stiffer the beam gets. You can test this with a simple static test. Apply a distributed static load to the beam and calculate the displacement. The relationship will not be proportional.

If we excite this with a sine wave of increasing frequency (a stepped sine test) and calculate the frf, the position of the peak of the first mode will shift upwards with increasing force level. I have done this with a plate with clamped boundary conditions and the peak can shift a lot more than 5%! If we then use a random excitation, the frf will look totally different to the stepped sine frf.

All this non-linear dynamic FE analysis takes a long time to run. We have developed methods to significantly reduce these computations by using our identified modal model with non-linear terms. A 256 element model of a clamped plate takes about 10 hours on a super-computer to collect 1 second of data at a smple frequency of 10kHz. With the modal model, this takes 2 seconds on a PC! The agreement between the two models is excellent, even for very high force levels.

Michael