BASIC MODELING (Let's put it this way) 4

[Dmitry]

Hi All

There is a long beam with known mass (m), moment of inertia (I), material properties (E - Young's modulus) and dumping decrement (d). The task is to make spring/mass model which amplitude (in the middle of span) and frequency will be the same as for the beam first bending mode.
The model described as: M * x" + C * x' + K * x = 0.
How model parameters are related to parameters of the real system?
I think in this case:
M = m;
K = 1/ (E*I);
C = d * (2*pi*f0) 2*m;
Where f0 – undumped frequency.

Let me know, please, if I'm mistaking.

With best wishes,
D. Semenov

 

[thierrybelge]

I think this depends on the way the Beam is held.

If the beam is clamped at one side and free at the other side, you know that deformation at the end of the beam is given by:
x=F*l^3/(3*E*I)
And you know that F=K*x

So I would say that the K is given by:
K=(3*E*I)/l^3 with l: length of the beam

 

[Dmitry]

Hi thierrybelge!
As I think, this way we do can describe system behavior under external load. In this case beam deformation in the point of interest is dependent from end conditions (as you said), as well as, from external force type (for example distributed or point applied).
But what about the case of free decaying oscillations? If system is linear then characteristics of her free vibration can't depend from the type of excitation!

Regards,
D.Semenov

 

[brad]

Dmitry,
I'm a bit puzzled by the statement " . . .characteristics of her free vibration can't depend from the type of excitation"

They will depend on the initial conditions of the system, which is dependent on loads and where they are applied. The free oscillation must have initial displacements or velocities to introduce energy to dissipate. The amount of energy is very much dependent on load (excitation) type.

Also,if you are trying to model the system as a single spring-mass system, then your spring constants will be dependent on the loading assumptions, as thierrybelge noted.

Regards,
Brad

 

[ANTOINE]

Dmitry,

If you aretalking about free response, the modal characteristics, i.e. natural frequency, deformed shape and associated modal damping depends only on the dimensions, material properties and boundary conditions of the beam in the case of a linear system.

Maybe a simple way to handle this, is to calculate the natural frequency of the first mode using appropriate formula depending on the boundary conditions (Blevins), use the stiffness given by thierrybelge (if it is relevant of your BC) and get the "modal" mass from the knowledge of the stiffness and frequency,and use the damping coefficient from the material properties...

 

[Dmitry]

Bradh,
I’m after the modal parameters (characteristics). My goal is predict modal parameters of real system by the most simple model. In fact this task is quite similar to devising a one-degree of freedom curve fitter for particular pike of experimental mobility function.
The difference.
In case of curve fitter, parameters of a mathematical model adjusted to minimize error between computed and experimental mobility function.
In my case. I want make use of dimensions, material properties and boundary conditions knowledge in other to estimate M and K (for particular mode of simply supported beam). As I understand C can be estimated analytically only in very special cases. Once these parameters are estimated, mobility function can be computed and compared with exact solution or results of an experiment.
As I have noticed, reading the BASIC MODELING tread, you and Greg has offered solutions for this task, although the task was in “rotational statement”. May be because of unfamiliar statement I failed to understand you and Greg as well.

P.S. I'm sorry for such long delay. My Internet limit was overdrawn in the middle of the month.

Best regards,
D. Semenov

 

[Dmitry]

ANTOINE,
I wold like to use “… appropriate formula…”, but Formula is solution of certain equation (as usual) and equation implies existence of a set of assumptions on account of system properties. If I do not like any one of these assumptions, then usage of the Formula becomes questionable.
As it seems now, more flexible way is to describe real system by equation (ODE or system of ODE) and make solution by an iterative solver. In this case you can easily modify the model, use various dumping models and excitation force types without any change of solution process and results representation.
The main advantage of such approach is that, as result will be received a signal (for example acceleration vs. time). Properties of this signal can be compared with experiment result directly.

Best regards,
D. Semenov

 

[GregLocock]

Hi Dmitry,

make sure that you don't re-invent the wheel!

There are many curve fitting approaches out there, which model an MDOF frequency response curve as a collection of SDOF systems. LMS in Belgium supply one.

There is also an older one used by SDRC which was a TIME SERIES based MDOF estimation technique. This retransformed the FRF into an impulse response and then somehow successively knocked SDOF contributions out of the waveform. I used to know how this worked. There may be technical papers on it.

I also wrote a frequency based MDOF optimiser to run on a GenRad computer, as an alternative to the SDRC routine. One of the harder decisions with real life data is to decide what the error function should be. Cheers

Greg Locock

 

[Dmitry]

Hi Greg!
You are a bit late, I have already developed one and probably a month earlier I received curve fitter (CF) evaluation copy from Vibrant Technology, Inc.

http://www.vibetech.com

But CF development and usage is interesting but quite separate question that deserves independent tread. Would you mind send some Internet shortcuts as “Helpful Tip”?

Regards,
D. Semenov