Structural Vibration - Science topic

To expand on the answer provided by Vyacheslav Ryaboy, when performing the increasing-frequency sine sweep test, you would pass through the first resonance frequency of the cantilever beam. This is what is referred to as "hitting" the first resonance frequency. If you continue to increase the frequency, you will eventually reach and pass through the second resonance frequency. If you continue to increase the frequency further, you will eventually reach and pass through the third resonance frequency.

Depending on the equipment at your disposal, you may wish to use an impact hammer to simultaneously excite a number of modes of vibration. The measurement system would likely have the ability to compute the transfer function between the response and the excitation force. The frequency response curve would generally show the first few natural frequencies of vibration of the cantilever beam, dependent somewhat on the bandwidth of the force impulse and also the point of application of the transient excitation force.

The links provided by Om Prakash Chhangani and Mohamed-Mourad Lafifi give the formulas for the first three natural frequencies of a cantilever beam, as well as a graphical representation of the associated mode shapes. There are some suggestions for how to go about experimental testing there too.

What test equipment do you have available to conduct the experiments with?

As Christian Willberg already pointed out, a 2 DOF system can only have 2 eigenvalues and consequently 2 mode shapes. These mode shapes describe all possible movement/displacement fields of the system. That is to say, each system response is a linear combination of those 2 modes. Everything else is not possible. Thus, if you have 3 modes shapes you must have a 3 DOF system.

In the modal decomposition method, the modes can be used to decouple the equations of motion (provided that we fulfill certain assumptions with respect to the material damping).

Therefore, you must restate your question and provide more details on your system, if you expect a meaningful answer. At this point, the question you would like to be clarified is not physical. Please go back to your problem and think a bit about what you actually want/need to know.

Kind regards,

Sascha

Hi Murana Awad

The reduction in stiffness will be high if the defect/crack is present in regions of high stress. In a cantilever beam, the maximum stress occurs at the fixed end. Therefore, if the defect/crack is near the free end you can expect a greater reduction in the natural frequency from that of an undamaged beam as compared to when the defect/crack is present elsewhere. You can find a discussion on the same in my research article. I have attached it below for your perusal.

Hope it helps.

Regards,

Jatin

Thank you so much for your reply, and have a nice day. Best regard.

Stick the accelerometer to a weight of known mass. Excite the weight with a known force using instrumented hammer or shaker. The acceleration and force will be related by F=ma. If you know the mass m, you can work out an appropriate calibration factor quite easily.

Dear Jatin,

Not always does a finer mesh result in a more exact solution. A mesh convergence study should always be performed to guarantee the descending trend of the error as the mesh size gets smaller.

Having this verified, yes, a finer mesh reduces the stiffness of the model. Because FE approximates the the PDE solution by forcing the element into specific modes of displacement which yields a stiffer element. But as the element size decreases, the FE solution converges to the analytical solution of PDE.

Eigenvalue can be physically interpreted as how stiff the structure is in the eigenvector direction. So it follows the same pattern as stiffness.

@ Rachid Ait Maalem Lahcen,

Thanks for your valuable suggestion. Are there any literature available for machine learning which is used in computational mechanics or structural vibration control?

The masses are given as none diagonal form therefore you can not sum up them. 153000 t is accurate.

Dear Yathish

Within SAP2000 or ETABS, a TMD may be modeled using a spring-mass system with damping. Guidelines for this subsystem are described in the following link:

Regards,

The effectiveness of tuned mass dampers (TMDs) in reducing the seismic response of civil structures is still a debated issue. The few studies regarding TMDs on inelastic structures indicate that they would perform well under moderate earthquake loading, when the structure remains linear or weakly nonlinear, while tending to fail under severe ground shaking, when the structure experiences strong nonlinearities. TMD seismic efficiency should be therefore rationally assessed by considering to which extent moderate and severe earthquakes respectively contribute to the expected cost of damages and losses over the lifespan of the structure.

Likely to be desired to use the tranducers at resonance to emit maximal amounts of energy.

Hi Majid Tavakolinia

I second Einar Strømmen & Pranoy Nair responses, also I would like to add a point that when we add a TMD we are introducing damping into the system. I am not really sure how you would define MPMR in such a system as modal analysis works only for undamped systems.

Saranya Bharathi I am wandering , which algorithm did you use ?

Hi Anders

Contact time, as indicated in your title, matters. That said, I tend to balance energy rather than impulse. See below for some ramblings.

As a rule of thumb, for an effective contact time, dt, the force spectrum is flat up to the frequency 1/dt and contains 90% of its signal energy below 2/dt.

So, yes, you do concentrate energy into a lower frequency range.

At the same time, bag mass, M, has the velocity, V, and the kinetic energy, M*V²/2, directly before impact. The kinetic energy is handled by the flexible buffer with stiffness K and the potential energy KX²/2. This yields, X = sqrt(M/K)*V.

We see that for a given impulse velocity, V, the higher the mass and the softer the spring, the larger the buffer compression stroke, X. As force is F = K*X, we find also that F = sqrt(K*M)*V and hence, the stiffer the spring and the higher the mass & velocity, the higher the maximum impulse force.

The stroke, velocity and contact time (simplistically) relate as X = V*dt/2 as dt is the time require to reach the stroke X and as 2/dt describes 90% of the total force.

We tend to find that the larger the stroke and the longer the contact time, the lower the contact force and the lower the frequency range that is excited.

The above assumes a perfectly linear spring. Buffer stops tend to be nonlinear which is why Force-Displacement curves tend to be provided for such components.

Now, to further compound matters - we need to take a look at shock response. Long story short, shock theory tells us that we get the highest dynamic response when the contact time, dt = T_j/2, where T_j is the mode j period time.

A soft mat tends to help, though you may be unfortunate and end up shaping the pulse for maximum dynamic response.

So, as per usual - it depends.... ;)

Hope this helps

Claes

Rui Yan, were you able to resolve this behaviour? I'm encountering a really similar problem!

Hello Daniel, FEM is fine to characterize isolated components, but your challenge relate to the boundary conditions ! Your very low frequency modes relate to bending + torsional motions of your girder flying in the air, while they will be constrained by the coupling with the ship hull which ultimately sits on water. The uncertainty on all these boundaries makes IMPOSSIBLE to trust a FEM model at >30Hz (= >1800rpm). In addition, assuming your engine is a Diesel, the most problematic frequency is not the RPM but the firing frequency (6 times higher if you have a 12cylinders engine) ! And if it is a propulsion engine, you have also the gearbox tones... You don't say either if your design uses rubber isolators. It is not impossible to use FEM but it takes years of learning, trials and errors to get the right way to model by FEM such complex structures. FYI a ship as a whole presents several "families" of natural frequencies: "beam modes" of the whole ship (flexure and torsion) at few Hz, local resonances (bulkheads and decks) at some 10 Hz and local resonances (machinery rafts and seatings) up to 100Hz - so there are always possible resonances and this is why silent ship design requires using rubber decoupling and structural damping devices to isolate the sources of vibration. Few shipyards master the whole predictive chain - just remember the Collins Class difficult infancy on this matter !

The best book you can borrow from your Library :https://www.dymocks.com.au/book/isogeometric-analysis-and-applications-2014-by-bert-juttler-and-bernd-simeon-9783319794709?utm_source=googleps&gclid=EAIaIQobChMI5_jdwvfc5gIVwRErCh3j8wIbEAYYBCABEgIUcvD_BwE

Number you change according t o your requirement

sample

TB,SDAMP,1,,1 TBFIELD,FREQ,30 TBDATA,1,0.02 TBFIELD,FREQ,70 TBDATA,1,0.018 TBFIELD,FREQ,120 TBDATA,1,0.016 TBFIELD,FREQ,200 TBDATA,1,0.014 TBFIELD,FREQ,240 TBDATA,1,0.012 TBFIELD,FREQ,308 TBDATA,1,0.01

The followings are good sources for ground vibration test (GVT):

http://www.garteur.org/Technical%20Reports/SM_AG-19_TP-115_OPEN.pdf

https://apps.dtic.mil/dtic/tr/fulltext/u2/a108262.pdf

I hope it is helpful for you.

All the best.

Following the answers

You can use Inverse FFT to transform from freq. Domain to time domain as it is your case ..see the attachment

Artificial intelligence, Genetic Algorithms (and its variants like Genetic Programming and Gene Expression Programming) are very good tools to use in your research for optimization purpose. I would suggest you to understand the basic concepts first (you can find alot of material regarding this on internet). Also you must have some data in hand before taking a start (data can be yours or secondary data from already published literature). ANN and GEP are the most common techniques used in Civil Engineering and you can goto www.sciencedirect.com and give these keywords.

I have also attached some articles on GEP and ANN which are mostly related to structures.

Hope it helps.

Hi Farzad

I assume you are referring to FE of a bounded region?

There are two kinds of modes that can be computed, Real modes and Complex modes. The former is standing waves that conserve energy, i.e. with node position (=minimum response) that stand still. The latter uses damping information and describes energy transport across the system and, hence, has node positions that move.

As indicated by their names, real modes have real valued response while complex modes have complex values response.

Real modes can be computed without any damping information.

As indicated by Giuseppe Pennisi damping can be added afterwards with the tacit assumption that it is evenly distributed and light. Other common damping models are viscous modal damping and hysteretic material damping.

The former (viscous) is a weak approximation of sound radiation, the latter (hysteretic) a weak appoximation of internal material friction. The Raleigh damping model mentioned above is used for mathematical convenience and does not originate from any physical damping mechanism.

In real life, damping can be many things. Some ramblings of mine on this topic can be found here https://qringtech.com/2014/06/22/designed-damping-types-mechanisms-application-limitation/

Sincerely

Claes

see

Exponential definition of friction coefficient is the most convenient one.

... attached file.

Best regards

Ch. Oertel

dear. Fikrat Almahdi.

Do you have a PDF file of book: Modal analysis theory and testing ?

could you share to me ?

thank you so much

Thank you very much for this resource, Dr. Pany!

Dear Mohammad,

here is a link to a review paper that discusses temperature effects on vibration properties of civil structures:

this paper reviews existing literature on the variation of vibration properties of civil structures under changing temperature conditions, mostly bridges though.

As far as I know, depending on the intensity of temperature variation, little frequency changes are observed when the temperature varies in a normal manner. having this in mind, among ambient conditions, temperature holds a high hand, and it's influence on vibration properties is more than others.

you can check the references of the suggested article, or the papers that have cited this article to find more.

Wish you success in finding what you are looking for.

Thank you. I am trying to model the phenomena of forming a composites using a beam which is already prestressed and having an another beam bonded to it which is non prestressed. And then we perform the modal analysis. Is this problem setup feasible.

Preload has a huge effect on elastomeric materials. Both tangent and secant stifness change with static preload. Some materials soften, but most of the elastomers generally harden.

When using elastomer pads, usually, the pad producer provides a chart giving the dynamic stifness as a function of force applied (which is generally provided by the weight of the machinery).

Does it worth? Well it depends on what you need to do. Generally, preload on pads can be increased by adding mass between the machin pedestal and the pad itself using blocks of steel or iron. Sometimes it's useful, but you increase stifness together with mass.. hence the system gets tuned nearby the original frequency. In fact, this is usually done to limit the static deflection when loading the machine itself with process material, or to compensate for static misalignment.

To conclude, I wouldn't reccomend to preload mechanically (let's say, with a clamp) the elastomer pad, since creep is relevant for this material. So if you apply preload by imposing a certain displacement, after a while (it can be weeks) your preload force disappear. So with elastomer pads it is important to apply preload by means of constant forces.

Hi Claes,

I almost lost hope that somebody will reply to this question. I am surprised that none of the Aerodynamics researchers on RG commented on this topic which should be quite straightforward for them.

Nevertheless, thank you for the information. Helpful as always!

Regards,

Mahmoud

Please refer to these articles:

Dear Abdelrahman Abdelaleem, that warning may be indicative of any modeling error. By default, shift is a very low value. If there are modes below this value, it means there are mode shapes with very low stiffness and, likely, some connectivity problem.

There is a discuss forum on this topic here: http://www.sepakistan.com/topic/1748-ritz-analysis-problem/

I would suggest running some easy tests to check the connectivity. You can also try to use merging

options of nodes. There can be nodes that apparently are connected but they are only close in space and no truly connected.

Hope it may be of any help. Best.

Hi,

Visualizing software frequently has an option to automatically determine point group (for example I use Chemcraft). On the other hand results of the calculation assuming C1 symmetry should more or less match results of calculation assuming real symmetry of your molecule (but maybe on the larger calculation time).

Hello Pranav Lad

  It is better to answer your question from the assessment.of free vibration of a sdof system governed by the ode

         MU_tt  + CU_t + KU =0                                                                                        The solution to this equation is   U(t) = A*exp(-ω_dt)Sin(ω_dt +α)                                   where  ω_d = ω_n*sqrt(1-D²)   and ω_n = circular natural frequency and D is the damping ratio = C/C_c.   The decay factor is exp(-ω_dt).   The higher the frequency the faster the decay and the more the damping. Thus, the decay rate will vary with frequency.  As is well known the viscous damping coefficient for a sdof system  is defined by                             c = 2DMω_n                                                                This idea can be extended to strings and beams. 

 I hope this answer will be useful to you.  

There are various ways to formulate stiffness, but we can skip that and go straight to the explanation.

Lets define up to be in an axial direction. We apply a torsional moment to all radial cross sections, for example as an upwards force acting on the outer lower corner, balanced by a pressure on the upper edge. This leads to a small and equal rotation of all these cross sections.

The upmost part of the cross sections move inwards. The lower parts move outwards.

Now lets slice a tangential, curved plane, cut at any radius  R you like.  All parts are 2piR horizontally. The upper part used to be on a larger radius. The upper part of our ring has just become shorter than it was. The lower part has just become longer than it was. The centre though has stayed on radius R and is just as long as it were. Its a little sheared though.

Sounds familiar? This is the same strain deformation as in bending.

I would not call this torsion, my own mother language has more precise terms. Perhaps someone with better language knowledge could come up with the right term? Wringing?

I also agree with Claes.

Moreover, there is also the option of isolate the entire building using viscodampers in the link between the building and its foundation. We have applied this technology in a building in Barcelona for the isolation against underground traffic and the results are satisfactory.

Tuned mass dampers is something in which we are working on in our research center. But, since they are only interesting when the vibration signal is close to be tonal (very well defined peaks in the vibration frequency spectrum), I think that they aren't a good general solution for your problem.

Bests,

 Thank you for your answer and publication you have recommended, I just wonder which kind of model she will use in pulsatile flow research for crossflow. 

Hi

This is a classic problem & a nuisance. 

Try these steps. 

The FFT is cyclic so, doing 2x FFTs in a row only returns your original time signal - try it. Note that you need long time records for the approach to work at least so so. 

The above can be done in Matlab.  

A better way, if you can repeat your test, is to use mechanical filters to remove the high frequency content that upsets the sensor. Take a look here. 

/Claes

I think can explain the phrase "frequency range based on the loads that are relevant in your application" with another example:

Assume a three storied building with an electricity generator (gen-set) at the top. We know the RPM and the force generated (from the manufacturer's spec). We can analyse the FE model for obtaining the eigen frequencies. Then the most important eigen modes are those having frequencies within the gen-set's operating frequency plus or minus 20%. So consider all modes from lowest to the gen-set's operating frequency plus 20%.

Hope I haven't missed anything important connected with the procedure.

regards,

Rahul Leslie

Moreover, you can simulate the ambient vibration caused by wind in your numerical models. For this goal, you need to introduce with aerodynamic forces such as self-excited forces and buffeting forces in wind-induced ambient vibration simulation. Although these forces are usually dominated in bridges. The following reference may be useful:

He, X., Moaveni, B., Conte, J.P., Elgamal, A., Masri, S.F., (2008). "Modal Identification Study of Vincent Thomas Bridge Using Simulated Wind‐Induced Ambient Vibration Data." Computer‐Aided Civil and Infrastructure Engineering 23(5): 373-388.

Hi Alireza,

I suggest you to look for solutions in the field of multivariate harmonic analysis. As a starting point, check for the following references:

Hoping it will be helpfull

Very Good tutorials for Finite Element Simulations with ANSYS Workbench 17

You can try the general-purpose multiscale constitutive modeling which can take a block of material and homogenization it to a beam. The code can be freely launched in the cloud at https://cdmhub.org/resources/scstandard

Dear Dieu,

If the load case type is based on mode shapes, specify the pushover

direction, the modes to be used, a scale factor for each mode,

and a base shear direction for each mode.

Use the Add, Insert, Replace and Delete buttons in

the usual way to build up a modal load list. Only horizontal

translational masses are considered for push-over loads. Vertical

translational masses are ignored, and rotational masses do not

affect push-over loads.

If you choose loads based on mode shapes, and if you use more than

one mode shape, you must specify a scale factor for each mode. Only

the relative values of the scale factors are important, since the concern

is the load distribution over the building height, not the load magnitude.

Consider first the case of a symmetrical 3D structure with mode shapes

that are clearly oriented along the H1 and H2 directions. The following

procedure is suggested.

(1) Get the mode shapes and periods. This usually means that you

must run an analysis to calculate the periods before you can define

the push-over load case.

(2) Examine the mode shapes and determine, for each of the H1 and

H2 directions, which is the "first" mode and which is the "second"

mode. For example, modes 1 and 4 might be the first and second

H1 modes, and modes 2 and 5 might be the first and second H2

modes. You may also consider the "third" modes. Ignore torsional

modes.

(3) Choose a design response spectrum. This is a separate task, and

you can not do it in PERFORM-3D. Only the shape of the

spectrum is important, not the intensity.

(4) For each mode, get the mode period and hence get the spectral

acceleration from the response spectrum. This is the scale factor

for the mode.

(5) Decide on a push-over direction, say H2. Specify the angle to this

direction (= 90 degrees for H2).

Chapter 13. Static Push-Over Load Cases

PERFORM-3D User Guide 13-9

(6) Specify the mode number for the "first" mode in the push-over

direction (= 2) and the scale factor for this mode from Step (4).

Choose "+D" for the base shear direction. Since the structure in

this example is symmetrical, the results will be the same for +D

and –D push-over.

(7) Specify the mode number for the "second" mode in the push-over

direction (= 5) and the scale factor for this mode from Step (4). If

you want the base shears for modes 1 and 2 to be in the same

direction choose "+D" for the direction, else choose "-D".

(8) If you wish, repeat Step(7) for the "third" mode.

This defines the load distribution and direction for this load case.

You may ask why it is necessary to specify the angle to the push-over

direction. For the symmetrical structure in this example, PERFORM-

3D could figure out the direction, since both mode 1 and mode 2 have

base shears only along H2. Also, since the structure is symmetrical it is

necessary to consider push-over only in the +H2 direction. In general,

however, it is not that simple.

actually, it is dynamic stiffness. Kd=F(w)/U(w)，w  denotes frequency

Your results seem plausible. The system oscillates in all 3 frequencies. Due to damping, the oscillations in the eigenfrequencies will vanish quickly, leaving only the steady-state response of constant amplitude in the forcing frequency. Without damping, you should always see a superposition of all 3 frequencies.

Eigen frequencies and modes are dependent on the stiffness and mass interaction between nodes. As you know, for a given mesh and no. of eigenvalues requested during extraction, you've only certain no of eigenvalues accurate (this is due to eigensolving technique).

Mass distribution is easy, as many codes utilize lumped mass system.

Stiffness distribution will be tricky as depending upon type of elements, each node will have certain no of stiffness components. So theoretically it's possible to retain in rigid body system with springs, but practically it'll be quite difficult. Just look at stiffness matrix of a simple spring. Each node interacts with the other node (off diagonal terms in stiffness matrix). Similarly in a beam element, each dof will interact with other dof of the same node + the node at the other end. Simple springs generate uncoupled stiffness matrix. If you know a way to account of inter dof interaction, then yes you can do what you want, otherwise no :)

Dear Sir,

I have been through this lately in my institution. I hope you'll find this list helpful:

1- Vibration analyzer (may be B&K PULSE system if the budget allows) or NI-CDAQ or CRIO for data acquisition together with Labview software and NI Sound and Vibration assistant which is a very powerful tool;

2- Vibration exciter with its power amplifier and signal generator or exciter control;

3- Accelerometers (different sizes to suit different applications); a triaxial accelerometer will be an asset; magnets and wax for mounting;

4- Hammer;

5- Force transducer;

6- Tachometer;

7- Proximity transducers;8- Conditioning amplifier.

Good luck with your upgrade.

This is really no different than a nonlinear rotor dynamics problem, and Guyan reduction is extremely effective in removing higher order frequencies that push the time step to extremely small values.

Hello Masoud,

Check the attached paper. I wish that can help you.

Best regards

Yes, there are several non-linear identification methods. I've dealt with the problem for a 2DoF system in the paper that I attach to the answer. You can think also about using an Unscented Kalman Filter, but for a 6DoF system (assuming you consider it having shear-type behaviour) the state-space formulation is quite tedious. For UKF I suggest to refer to the work of E. Chatzi and A. Smyth.

It seems trivial, but it is not. You can proceed measuring the global response with an accelerometer but you have a very large error near the resonances of the structure supporting the machine. That's why in modal testing you typically use a load cell where the shaker is in contact with the tested specimen: in order to clearly measure the real force. You can check on Ewins book "Modal Testing", it describes in detail the problem of measuring the real force between to moving object.

Hi

What Prof. Khulief writes is all true.

A small boat floats on top of the wave while a large boat drives through the wave, i.e. size versus wavelength greatly matters.

However, there are a few things more that goof things up for us. These have been known in the field of acoustics from experiments since the late 1920's and its theory was developed by M Schröder in the 1950's.

There is something called called the acoustic limit, or the Schröder limit.

Here it goes.

The asymptotic solutions to implies the modal count for a room implies that the number of modes, N, grow with frequency, f, as f³ for a volume. At high frequency, i.e. at a frequency where the wavelength is short in comparison with the dimensions of the volume, there will be a high number of modes as counted per Hertz.

The modal density, n, is the average mode-to-mode separation. 

The half power bandwidth, df_j,  of a mode, j, is dfj = eta/f_j , where eta is the damping loss factor.

Now, at high frequency, the half power bandwith will grow with frequency while the modal density will shrink.

Schröder showed that when three modes fall inside the half power bandwidth of a single mode, then a FRF will have random phase and one stadard deviation for the FRF amplitude is 5 dB to 6 dB, i.e. with about 90% confidence, the FRF amplitude variation is in the ballpark +/- 10 dB.

Incidentally, this is the criterion also for diffuse field conditions provided eta is moderate.

The kicker is that when passing through resonance, the FRF phase jumps 180 degrees. Add a tiny bit of uncertainty to the resonance frequency and start summing modes and you will readily see what happens. 

The FRF dispersion has been demonstrated through carefully made tests on production sets of nominally identical cars by various people in the 1990's when FRF measurement became easier to do. It had been shown for other product types long before this.

The mode separation, modal bandwidth relation is called Modal Overlap, where MO = n*eta/f_j . MO = 3 at the Schröder limits.

Reliable damping estimation uses some sort of modal algorithm for identification. Such algorithms tend to work up to MO <= 0.3. Sure, you can apply them at higher MO but you get unique results every time, i.e. tweak your assumptions just a tiny bit (e.g. the selected frequency range or vary algorithm parameters) and the mode set you identify will be a new set and so on.

I enclose a presentation on this matter that was made by two of my former colleagues, B Gärdhagen and J. Plunt, which demonstrates the above and shows why people working with acoustics do things like sum energy (a squared quantity) and work with power based concepts.

At high frequency, it turns out that FRFs disperse while energy/power converges.

To summarize, for high observability, you must for small cracks go up to high frequency - where you may struggle with the Schröder limit. For large cracks, you may use FRFs at low frequency.  Note that the modal density function varies with object form, the modal count is very much different between volumes, plates and beams. Therefore, you may have more success for certain object shapes.

I do not mean to depress you but the above is old, hard won, knowledge that repeatedly has been proven.

Hope this helps

Claes

Could you please describe your problem in detail? Your load I understand: F1+0.5*(F2-F1)*sin(w*t+pi/2), but what parameter you want to vary? What is the goal of your analysis?

Thanks for your answer William!

I'm using Static (General), Frequency and finally Modal Dynamic steps in succession. In the Modal Dynamic step, while the string is vibrating, tension is released by returning the boundary displacements to the zero point. 

Hi

Your question is a class room question, but a good class room question that is worth delving on.

The simplest way to undrestand this is to look at the swing, i.e. an item most of us have an in-depth study from when we were kids,

The swing is a pendulum, i.e. its resonance is w = sqrt(g/L), where g is 9.82 m/s2 where I live and L is the pendulum length. This provides the same resonance frequency irrespective of passenger weight.

If you push the swing, you can do so at any freqyency. For simpicity, let us assume that you move the swing slowly at large stroke at a frequency much lower than w. This is not easy when the passenger is heavy as, in essence, you then must lift the passnger. The same applies if you try to push the swing much faster than w, only in this case, it is the passnger inertia that resist fast oscillation.

In both cases, as soon as you stop the forced excitation, the swing will use whatever kinetic and potential is avialble and distribute it into its most favorable setting, which is a split where all kinetic energy can shift to potential energy and vice versa, which can be achieved only at its natural frequency, w.

The reason why it is easier to push the swing at w is that it then stores energy when excited at resonance. Free vibration implies vibration at natural frequency.

If we take a mass spring system. Potential energy is 1/2*K*X^2, where K is spring stiffness and X is displacement. Kinetic energy is 1/2*M*V^2. For harmonic vibration X = A*e^jwt, we find that V = jw*A*e^jwt = jw*X.

Inputting and summing kinetic energy, we find that

1/2*K*X^2 + 1/2*M*(jw)^2 = 0

solving whis equation produces the natural frequency w = sqrt(K/M), which just is another way of stating the energy summation listed above . 

Multidof really is the same thing with information added as to where you drive and how this matches the mode shape.

However, to better understand things, take a look here

For modal summation, I enclose part of a course I hold now & then on structure vibration which intended for an audience without prior knowledge on vibration. This excerpt treats a lumped 2dof system and explains most of what you are asking for.

Have fun

Claes

The most common methods are subspace iteration and Lanzcos method. You will find further details in the books by Louis Komzsik What every engineer should know about computational techniques of finite element analysis and Klaus-Jürgen Bathe's Finite Element Procedures.

I thin by measuring the absolute distance between to markers, you can calculate delta_X, and ........

Best compensation for B component impact on A component would be forces applied by B on A on each node. Maybe you could try to analyze component B by itself to get reactions in nodes which are common for component A and B, and then that reactions apply on nodes on component A as loads. It would not be exact the same nodal displacement of A, but you could try.

Hi Nati,

you might be interested in a paper by Krenk and Hogsberg

"Tuned mass absorber on a flexible structure", Journal of Sound and Vibration, Vol. 333 (2014) pp. 1577--1595.

Best regards,

                              Nils

Hi Mahdi,

I am recently studying on the same subject with you. It is hard to visualize and understand all system. Sometimes I have problem with that. Can you please check out  word file that attached in the post? I hope this help you.

Best regards,

Osman

Hello, thanks for your interest.

I have tried different solutions but I still do not have reached the correct one. Here is what I have done:

1. Dynamic analysis considering, instead of a point mass with gravity an equivalent concentrated force (but not considering the gravity applied to the beam)

2. Dynamic analysis considering, instead of a point mass with gravity an equivalent concentrated force (considering the gravity applied to the beam)

3. Dynamic analysis considering a point mass with gravity and the gravity applied to the beam

No one of this model works as I aspect. As I have seen in previous works the ABAQUS model should be exactly as the exact solution (see attached image :BLUE line).

What do you think could be the problem?

I take this opportunity to wish you merry Christmas.

Best regards

Giacomo

ANSYS APDL uses FEM platform. Through FEM analysis we get conjugate pairs of natural frequency. like if we take example of a beam we can get 23.5Hz for first mode and 23.3 hz for second mode, these two are mode shapes of a beam but however one may be bending and another may be torsional. Similarly we can get bending modes in different planes. To understand this you will have to study the theory behind the modal analysis using FEM. "DYNAMICS OF ROTATING MACHINES" book provides an easy and good understanding about conjugate pairs.

In experiment like yours the mode which you get depends on the placement of sensor and direction in which it is measuring the signal. So, through software take that mode which gives the displacement in the same plane as you are getting from experiment.

I hope I am able to explain it.

*MODAL DAMPING

Dear Hauke,

in addition to the references mentioned by Huangchao Yu, you may also want to have a look at these publications, if you are interested in wavelengths much larger than the tube radius:

Solution for thick-walled elastic pipes with infinitely high Young's modulus (two modes, one shear-dominated, one fluid-dominated):

Bernab ́e, Y. (2009a), “Oscillating flow of a compressible fluid through deformable pipes and pipe networks: Wave propagation phenomena”, in S. Vinciguerra and Y. Bernab ́e, eds, Rock Physics and Natural Hazards, Pageoph Topical Volumes, Birkhäuser Basel, 969–994

Solution for thin-walled elastic pipes (solid acts like an elastic shell, dissipation is captured phenomenologically):

Gautier, F., Gilbert, J., Dalmont, J.-P. and Vila, R. P. (2007), “Wave propagation in a

fluid filled rubber tube: Theoretical and experimental results for Korteweg’s wave”, Acta Acustica united with Acustica 93, 333–344.

Literature Review from 1975 with figures of dispersion relations:

Tijdeman, H. (1975), “On the propagation of sound waves in cylindrical tubes”, J. Sound Vib. 39(1), 1–33.

The limit-case for a rigid solid (fluid wave only) can be found in works of Biot and Womersley:

Biot, M. A. (1956b), “Theory of propagation of elastic waves in a fluid-saturated porous solid. II. Higher-frequency range”, J. Acoust. Soc. Am. 28, 179–191.

Womersley, J. R. (1955), “XXIV. Oscillatory motion of a viscous liquid in a thin-walled elastic tube-I: The linear approximation for long waves”, Philos. Mag. Series 7 46, 199–221.

Analytical solutions are difficult to derive for non-trivial cases as the differential equations easily get very complex and require numerical methods, e.g:

Bansevicius, R. and Kargaudas, V. (2005a), “Wave propagation in elastic cylindrical tubeswith viscous and heat-conducting fluid”, Ultragarsas 3(56), 7–10.

Bansevicius, R. and Kargaudas, V. (2005b), “Wave propagation in elastic cylindrical tubes with viscous and heat-conducting fluid”, Ultragarsas 3(56), 11–16.

Best regards

Hi

Here are good classic references for plates and shells

Leissa - Vibration of plates: http://ntrs.nasa.gov/search.jsp?R=19700009156

Leissa - Vibration of shells: http://ntrs.nasa.gov/search.jsp?R=19730018197

I believe that you find what you are looking for in the Leissa books. 

Sincerely

Claes

It seems you are mixed up between free vibration and forced vibration. Read Chapters 4 and 5 of Tedesco's book "Structural Dynamics, Theory and Applications" and they will guide you with your research.