My book defines natural vibrations as:
THE PERIODIC VIBRATIONS OF A BODY IN THE ABSENCE OF ANY EXTERNAL FORCE ON IT, ARE CALLED NATURAL VIBRATIONS.THE TIME PERIOD IS CALLED ITS NATURAL PERIOD AND THE FREQUENCY IS CALLED NATURAL FREQUENCY.
Let's say I clamp a elastic string at two ends and pluck the string in middle in a vacuum. The body would start vibrating with a fixed frequency.But when I apply more force the string should start vibrating with increased frequency. So my question is 1)which is the natural frequency of the string?
2) Are there more than one natural frequency for a body?
And further for resonance to occur the frequency of the external periodic force should be equal to that of the natural frequency of the body. So
3) Which natural frequency is being referred while discussing resonance?
Best Answer
The definitions of natural vibration, natural period and natural frequency in your book is correct. However, there is a small misunderstanding on your part. The example you give, that is, an elastic string clamped at the two ends needs a little more consideration. Let me try to explain.
If you attach a mass $m$ to a spring with a spring constant $k$, the natural frequency $\omega_0$ will be $$ \omega_0=\sqrt{\frac{k}{m}}$$ If you attach a mass to a pendulum of length $\ell$ under the gravitational acceleration of $g$, the natural frequency $\omega_0$ will be $$ \omega_0=\sqrt{\frac{g}{\ell}}$$ If you force such a system to vibrate with its natural frequency then the system resonates, that is, the amplitude increases rapidly. But these are "free" systems and an elastic string clamped at the two ends is not a free system but a bounded system. When you create a vibration on such a system you will create so called standing waves.
In such a bounded system there are infinitely many (natural) frequencies that the system can oscillate with, which are called the normal modes. This answers your first and the second questions, that is, yes, in this system there is more than one natural frequency.
How can one calculate these normal mode frequencies? Simple! Assuming that the string has length $L$ and linear mass density of $\mu$ and is under the tension of $T$ then the first normal mode frequency will be $$f_1=\frac{1}{2L}\sqrt{\frac{T}{\mu}}$$
and the higher frequencies can be calculated by
$$f_n=nf_1\ (n=1,2,3,...)$$
When you pluck such a string, you will most likely create a pattern that is not any of these normal modes. However, with time, due to inference, most of the other waves with different frequencies will die out and the string will oscillate with a combination of normal mode frequencies. This answers your last question.