What is the first, second etc mode? I cannot find online explanations. Is it the shape of vibration? Does a thing have more than one natural frequencies (first, second, etc) and it vibrates with different modes in these frequencies, named 1st, 2nd etc modes? thanks!
[Physics] What are the first, second etc modes of vibration
frequencyvibrations
Related Solutions
It's easy to see without doing any math, but just by looking at the picture.
Let's consider first the case of low $k_{12}$. In this case, $m_1$ and $m_2$ basically don't notice $k_{12}$ because it is so weak that it is drowned out by the other springs. So the low $k_{12}$ case basically gives the same value as the $k_{12}=0$ case for all three frequencies (you can check this).
As an exercise, let's think about what would happen if you removed $k_{20}$ and $k_{23}$ (i.e., set them equal to zero). Now $m_2$ can see $k_{12}$ because there are no other springs drowning out its effect, and you should get one frequency changing as $k_{12}$ goes to zero. It is only one because two of the frequencies will just be for the symmetric and antisymmetric modes of $m_1$ and $m_3$, which don't really care about $m_2$. The third one will be slower.
Now lets like about the high $k_{12}$ limit. Here $m_2$ sees only $k_{12}$, and since $k_{12}$ is so big, $m_1$ and $m_2$ are basically rigidly attached. Thus two modes will be the symmetric and antisymmetric modes of $m_3$ and $m_1+m_2$ (you can check this, you have to add $k_{10}$ and $k_{20}$ as well as $k_{13}$ and $k_{23}$ to get the effective spring constants), and the third mode will be a quick oscillation of $m_2$ relative to $m_1$. You can check this too, the frequency ought to be $\sqrt{k_{12}/\mu}$, where $\mu$ is the reduced mass for $m_1$ and $m_2$: $\frac{1}{\mu} = \frac{1}{m_1}+\frac{1}{m_2}$
Your understanding of resonance seems about right on a qualitative level. If one were to ignore losses like friction, drag, or the like, "driving" a system at its resonance frequency would indeed result in feeding it more and more energy which is stored in the form of a large amplitude of the oscillation. For a completely lossless system, the amplitude would grow to infinity. In reality, every system is lossy to a certain degree but for "high-Q"$^1$ systems (meaning the losses are small), the amplitude can grow so big that they rip the system apart. This is referred to as a resonance catastrophy and is what happened to the Tacoma Narrows Bridge.
Generally, the losses will be proportionally to the amplitude of the oscillation and also to their frequency. I.e. at resonance where the amplitudes are big, so are the losses. Another way to think about it is in terms of energy transfer. At resonance, it is easy for the object to pick up energy from the driver. The energy stored in the object grows at first, but it will eventually be balanced by the increasing losses. Instead, if the system is driven off resonance, it has trouble picking up energy from the driver, so the stored energy decreases (and so do the losses) until the losses balance the little energy intake that is still provided by the driver.
In summary, at equilibrium (meaning in the steady-state), the object will always dissipate as much energy as it picks up from the driver, otherwise its energy content (and thus the amplitude) would change. At resonance, it is easy for the object to do that, so there also the energy dissipation is greatest.
$^1$ Quality factor or $\mathit{Q}-\text{factor}$
Best Answer
Modes of vibration are particularly, though by no means exclusively, associated with musical instruments. It is the shape of vibration, and most musical instrument have more one mode of vibration, of they would be fairly limited in their musical range. Compare the sounds of a violin (with 4 to 7 strings) with a musical triangle, which only emits one note.
The first 3 modes of vibration of a guitar string.
For a more extreme example of the various vibration modes possible, here are some computer generated modes from a drumhead.
Images and Extracts from Modes of Vibration
When you pluck a stretched string, you always hear a sound with a definite musical pitch. By altering the length, tension or weight of the string, all familiar to musicians, you can alter this pitch. Strings and stretched drumheads are all suitable for producing a variety of vibrations, so they make musical instruments with a wide range of sounds possible. If instead you used a brick, or a frying pan, there is very little scope for musicical variety, as their vibration modes are limited.
Here is another example, but not musical, of modes of vibration
Galloping Gertie Movie Tacoma Narrows Bridge Collapse "Gallopin' Gertie" - YouTube