# Why and how are resonance frequencies of a system dependent upon the shape, mass and the way it is constrained?

I was reading a general overview on the frequencies before conducting a modal analysis for my structure using ANSYS, in hopes of obtaining the natural frequencies for my structure. I read in some article that if you change the geometrical shape of your structure, change its overall mass, and even change the locations of the supports on it while conducting the modal analysis, it is very probable you will see a different set of modal frequencies. My question is why and how? Moreover, I also want to ask that after getting the natural frequencies for my structure, does this mean that ALL the structure must be excited by an external frequency equal to natural frequency, to expect some dangerous and catistropic failure? Or even if I excite certain parts of my structure, for example some specific locations on my structure where it is connected to another part (which is essentially causing my structure to vibrate at these connecting locations) should also be dangerous?

• Have you researched natural frequency? Mar 28 at 19:17
• Yes ofcourse, thats what I am aiming to get for my structure. Actually, my structure is a composite whose thickness can be changed by increasing/decreasing the number of plies. I want to know how would a change in thickness change the natural frequencies of my structure. Mar 28 at 19:24
• So, if you did research the natural frequencies then you would have found the formulae... Mar 28 at 19:30
• So you might find this useful: brown.edu/Departments/Engineering/Courses/En4/Notes/… Mar 28 at 19:36

In order to answer that question you need to build up your knowledge in vibrational dynamics. This usually takes a whole semester in undergraduate physics at the latter stage of an engineering curriculum.

So you need to progress -at least- through the following concepts:

• The free response of the undamped Harmonic Oscillator with 1 dof (mass spring no external excitation)
• The free response of the damped Harmonic Oscillator with 1 dof (mass damper spring no external excitation)
• The forced harmonic response of the damped Harmonic Oscillator with 1 dof (mass damper spring with external sinusoidal excitation)
• The forced harmonic response of the damped Harmonic Oscillator with 1 dof (mass damper spring with external sinusoidal excitation)

Then you need to start thinking about mdof systems, and the matrix form of the problem (this is very helpful especially with the FE), and you need to do the following:

• Start with the simplest 2DOF (undamped no excitation) to see the form of the system.

The main difference from the SDOF systems, is the interactions between the position of the masses (and therefore the springs and dampers).

It is very crucial to understand at that there can be many different transformations for the same problem depending on the generalised coordinates you select. A simple example of what I mean is the following problem. In the above problem you can describe the motion equations with coordinates $$x_1$$ and $$x_2$$ (the spring displacements), or equivalenty by $$x,\theta$$ (the motion of the center of gravity and the rotation). The result should always be the same, however the individual coordinate responses will vary.

• Then you can proceed to Modal analysis

What is important to understand is that among the infinite number of generalised coordinates , where the responses are coupled, and additionally that there is at least on set of coordinates (sometimes called principal coordinates) where the transformed coordinate is decoupled (and are solved like the sdof - hence why you need to understand the behaviour of SDOF systems).

To obtain that set of coordinates, one way is to use the eigenvalues and eigenvectors of the mass normalised stiffness matrix. The cool thing is that you can rotate the initial problem, to obtain the principal coordinates, solve for the response of each decoupled system separately and then translate back to the original coordinates. The rotational transformation does not change the eigenfrequencies, in fact in the decoupled system you have a mode shape and a sinusoidal response. So each "mass" is pulsing proportionally to the other following one eigenfrequency. When you rotate back at the original masses/coordinate system, you get a contribution from both decoupled responses (hence the seemingly chaotic behavior).

• The final step is the Forced response of MDOF systems.

The same transformation that is applied to the Masses and springs can be applied to the force matrix (both directions). If you are planning to excite at a resonant frequency the mass you can apply a resonant force on the principal coordinate system and the transform that force matrix to the original matrix. When you do that you see that in the general case, you need to apply on all of the original coordinates a portion of the force with eigenfrequency $$\omega_i$$ in order to obtain the mode shape.

The transformation is essentially a rotation matrix, which can be applied to the excitation matrix. However this means that in order to excite the if you excite the nodes with the frequency at those eigenfrequencies,

• You do describe the steps one should take in order to understand some of the mathematics behind it, but those are not strictly necessary in order to have some intuitive understanding of it. Also, it is worth mentioning that doing a finite element analysis using matrices has its limitations, since the underlying physics is actually a partial differential equation (PDE). Only the first few lowest natural frequencies obtained from those matrices are usually accurate, because the matrices are obtained by discretizing space which gives an approximation of the underlying PDE. Mar 29 at 15:42
• I was writing my reply, probably at the same time as you were. I do confess, my approach was more of the undergraduate textbook, for discrete systems. The way I see it you took the continuum system which applies better to real structures, and you took a more hand's on approach. To be honest, the question was very vague, and there are different interpretations. I do feel that both posts have their own merit. Mar 29 at 16:09
• I agree that both our answers have merit. And I agree that the question itself is a bit vague, probably because of the lacking theoretical knowledge. This also why I tried to keep a lot of the underlying theory out of it. I myself find it useful to first gain some intuition and maybe spark some curiosity before diving deeper into underlying mathematical concepts. But hopefully the combination of our two answers might inspire some people reading them to learn more about it. Mar 29 at 17:00

Natural frequencies inside structures are standing waves that are reflected and propagated throughout the structure. The frequency of such wave depends on the shape of that wave and the speed at which sound travels through the material the structure is made of. The shape of possible waves in a structure I think can be well illustrated with Chladni figures. Adding supports constraints what kind of wave shapes can appear inside a structure, which you can also see in this video where putting a finger against the Chladni plate can be seen as a support. I have to note that in that video the natural frequencies of the plate are not actually altered, the temporary support initially limits which natural frequency/mode is excited.

Those standing waves do eventually die out if no energy is supplied to excite that natural frequency/mode. This is due to ways of dissipating energy, such as sound or damping behavior the material itself. But if energy at the "right" frequency is supplied to a structure at "right" locations, then the rate of energy dissipation might be too small, causing the energy of that natural frequency/mode to build up to dangerous levels until something might break. These "right" locations means that one isn't supplying the energy at nodes of that wave shape.

Adding mass to a structure can affect the way sound is reflected near that mass in the structure and thus affect wave shapes. This also closely related to impedance, which is demonstrated in this video.

When exciting a structure, not near any of the nodes of the natural frequencies, one does not have be exactly at any of the natural frequencies in order to get a large response of the structure. Though, the closer you are to one of the natural frequencies the larger the response. This can be illustrated with frequency response functions, sometimes also referred to as Bode plots, whose core concept I think is demonstrated well in this video.