vibrating string

1 Introduction

D. Bernoulli (Op.Cit. in [1]) and Euler [2] were the first in studying, in more or less complete form, the transverse vibrations occurring in a string. The solution of the pertinent partial differential equation, in terms of arbitrary functions, is due to D’Alembert [3]. Such studies, whose objective was to explain the vibratory phenomena, helped in great way to the establishment of the general principles of mechanics. In the last century, were several the scientists who were dedicated to the researching about the vibratory phenomena that happens in an elastic string. We can mention a few examples. Carrier [4], who got a solution of this problem by the method of perturbations. Fermi, Pasta and Ulam [5] from Princeton, studied the equipartition of energy on a discretized string’s model. Also Zabusky [6], by using the method of characteristics in partial differential equations of hyperbolic type, arrived to an equation whose solution had been obtained by Riemann [7] when he was researching the wave’s propagation phenomena in an isentropic gas.

Aside from the historical importance that has been attributed to the studies of vibrating string, it is not less relevant the fact than it has been the starting point in the researching of the vibratory phenomena that happens in cables; for instance, the case about suspension bridges.

Here we will make an elementary approach on vibrating string, via integral equations. Next section is dedicated to the statement and hypotheses about the problem.

2 Restrictive hypotheses

Let us consider a flexible and elastic string with length $l$ which obeys Hooke’s law and it ends fixed at $x=0$ and $x=l$, under the following assumptions.

1. Weightless string.

2. $d\ll l$, where $d$ is the string’s diameter.

3. It does not take into account the elastic properties of string’s cross section.

4. Transverse vibrations, i.e. the abscissa of string’s points remaining unaltered.

5. Small deformations, i.e. $y(x,t)\ll l$, $0\le x\le l$, , where $y(x,t)$ is an arbitrary string’s deflection.

6. Tension on the string is uniform along it.

3 Green’s function

We assume Cartesian coordinates with origin $O$ and $y>0$ downwards. At $t=0^{-}$, the string rests on $x$-axis at its natural undeformed length $\overline{OA}=l$. At $x=\xi$, is downward applied to the string a vertical load of magnitude $P$, so that the initial static deformed configuration of the string will correspond to the triangle $OAC$, at $t=0$. A free diagram of node $C$ and Newton’s second law $\sum F_y=0$, leads to

$$T_0\frac{\delta }{\xi }+T_0\frac{\delta }{l-\xi }=P,\text{or}\mathit{\hspace{1em}\hspace{1em}}\delta =\frac{P\xi (l-\xi )}{T_{0}l},$$

being $T_0$ the string’s tension and $\delta$ the static deflection at $t=0$.
Denoting by $Y(x)$ the arbitrary deflection at $x$, and from the prior analysis, we obtain

$$Y(x)=PG(x,\xi ),$$

where

$$\{\begin{array}{cc}G(x,\xi )=\frac{x(l-\xi )}{T_{0}l},0\le x\le \xi ,\hfill & \\ G(x,\xi )=\frac{(l-x)\xi }{T_{0}l},\xi \le x\le l.\hfill & \end{array}$$ (1)

This is the Green’s function that we need. It is clear that $G(x,\xi )=G(\xi ,x)$. Let us suppose that on the string acts a distributed force per unit length $p(\xi )$, then the force element on $(\xi ,\xi +\delta \xi )$ it should be $p(\xi )\delta \xi$ and therefore, by the superposition principle, the string takes the form

$$Y(x)={\int }_0^{l}G(x,\xi )p(\xi )𝑑\xi .$$

4 Discussion of cases

Following to Petrovski [8], we will consider the following problems.

1. To find out the force’s density distribution $p(\xi )$, under whose action the string takes a given form $Y=Y(x)$. Thus, we arrive to the integral equation of the first kind

   $$Y(x)={\int }_0^{l}G(x,\xi )p(\xi )𝑑\xi$$ (2)

   respect to the unknown function $p(\xi )$.

2. Let us suppose that on the string acts an exciter force with density, at $x=\xi$ and instant $t$, given by $p(\xi ,t):=p(\xi )\sin \omega t$, being $\omega >0$ the angular frequency of the exciter force. Under its action, the string is putting in motion. Further, we will assume that the string’s transverse oscillations are periodic, described by the equation

   $$y(x,t)=Y(x)\sin \omega t.$$

   Denoting by $\rho (\xi )$ the mass linear density at $x=\xi$, from the D’Alambert principle applied to the string’s segment $(\xi ,\xi +\delta \xi )$, at instant $t$, we have

   $$y(x,t)=Y(x)\sin \omega t={\int }_0^{l}G(x,\xi )\{p(\xi )\sin \omega t+{\omega }^2\rho (\xi )Y(\xi )\sin \omega t\}𝑑\xi .$$

   Simplifying by $\sin \omega t$ and defining

   $$f(x):={\int }_0^{l}G(x,\xi )p(\xi )𝑑\xi ,K(x,\xi ):=\rho (\xi )G(x,\xi ),\lambda :={\omega }^2,$$

   we get

   $$Y(x)=\lambda {\int }_0^{l}K(x,\xi )Y(\xi )𝑑\xi +f(x).$$ (3)

Supposing known the function $p(\xi )$ and, therefore $f(x)$, we arrive on this manner to a Fredholm’s integral equation of the second kind for determination of the function $Y(x)$. Note that, by virtue of the definition of $f(x)$, we have $f(0)=f(l)=0$.
Wether the density $\rho (\xi )$ is constant and $f(x)$ is regular , it is not too hard solving this integral equation. In fact, from (1) and the definition of the symmetric kernel $K(x,\xi )$, results

$$Y(x)={\omega }^2\rho {\int }_0^x\frac{(l-x)\xi }{T_{0}l}Y(\xi )𝑑\xi +{\omega }^2\rho {\int }_x^l\frac{x(l-\xi )}{T_{0}l}Y(\xi )𝑑\xi +f(x)$$

or

$$Y(x)=\frac{{\omega }^{2}c}{l}(l-x){\int }_0^x\xi Y(\xi )𝑑\xi +\frac{{\omega }^{2}cx}{l}{\int }_x^l(l-\xi )Y(\xi )𝑑\xi +f(x),$$

Where $c=\rho /T_0$. By twice differentiation respect to $x$, we obtain (by application of Liebnitz’s rule)

$$Y^{\prime \prime }(x)=-{\omega }^{2}cY(x)+f^{\prime \prime }(x).$$ (4)

On the other hand, it can be shown that any solution of the differential equation (4), which vanishes for $x=0$ and $x=l$, it is also solution of the integral equation (3). To see this, we multiply the equation

$$Y^{\prime \prime }(\xi )=-{\omega }^{2}cY(\xi )+f^{\prime \prime }(\xi )$$

by $-T_{0}G(x,\xi )$ and integrating respect to $\xi$ from $\xi =0$ to $\xi =l$. Then (3) is obtained since, integrating by parts, it is easily shown that

$${\int }_0^l{T}_{0}G(x,\xi )U^{\prime \prime }(\xi )𝑑\xi =-U(x),$$

where $U(x)$ is any regular function, such that $U(0)=U(l)=0$. From the theory on ordinary differential equations, the general solution of (4) has the form

$$Y(x)=C_1\sin \mu x+C_2\cos \mu x+\frac{1}{\mu }{\int }_0^x{f}^{\prime \prime }(\xi )\sin \mu (x-\xi )𝑑\xi ,$$

Where $\mu =\omega \sqrt{c}$ and $C_1,C_2$ are arbitrary constants. From (1) and (3) is deduced that $Y(0)=Y(l)=0$. By making use of these conditions, and whenever $\sin \mu l\ne 0$, we have

$$Y(x)=-\frac{1}{\mu }\frac{\sin \mu x}{\sin \mu l}{\int }_0^l{f}^{\prime \prime }(\xi )\sin \mu (l-\xi )𝑑\xi +\frac{1}{\mu }{\int }_0^x{f}^{\prime \prime }(\xi )\sin \mu (x-\xi )𝑑\xi .$$ (5)

In this case, (3) possesses a unique solution for any function $f(x)$, whenever it be regular and $f(0)=f(l)=0$.
It can be shown that for the existence of the solution of integral equation (3) is sufficient, if $\sin \mu l\ne 0$, that the function $f(x)$ be continuous, so that the condition about existence and, with more reason, about that $f^{\prime \prime }(x)$ be continuous, is superfluous. On the contrary, condition $\sin \mu l\ne 0$ is absolutely essential in order to that the integral equation has solution for any function $f(x)$; moreover, for all function $f(x)$ differentiable any number of times.
In the case that $\sin \mu l=0$, then

$$\mu =\frac{k\pi }{l},\omega =\frac{k\pi }{l\sqrt{c}},\lambda =\frac{k^2{\pi }^2}{l^{2}c},$$ (6)

where $k\in \mathbb{Z}$. The values of $\lambda$, given in ${(6)}_3$ for $k\in {\mathbb{Z}}^{+}$, are the eigenvalues of integral equation (3), and the corresponding values of $\omega$ in ${(6)}_2$, are the eigenfrequencies of string’s oscillations. From (5) and $f(x)$ being regular, as $\sin \mu l=0$ one realizes that the integral equation (3) has solution only if

$${\int }_0^l{f}^{\prime \prime }(\xi )\sin \mu (l-\xi )𝑑\xi =0.$$ (7)

Integrating by parts and by making use of the fact that $\sin \mu (l-\xi )=0$ and $f(\xi )=0$ for $\xi =0$ and $\xi =l$, condition (7) becomes

$${\int }_0^{l}f(\xi )\sin \mu (l-\xi )𝑑\xi =0.$$ (8)

Reciprocally, it is easy to show that condition (8) is also sufficient for the existence of a solution of integral equation (3) for any $\mu$ for which $\sin \mu l=0$. In particular, condition (8) is satisfied if $f(x)\equiv 0$. Then, the integral equation (3) and the differential equation (4) become homogeneous. Consequently, all the solutions of the homogeneous differential equation

$$Y^{\prime \prime }(\xi )+{\omega }^{2}cY(\xi )=0$$

vanish for $x=0$ and $x=l$ and therefore, all the solutions of the homogeneous integral equation (indeed a Fredholm’s integral of the first kind)

$$Y(x)=\lambda {\int }_0^{l}K(x,\xi )Y(\xi )𝑑\xi$$

are given by

$$Y(x)\mapsto Y_k(x)=C\sin {\mu }_{k}x,$$ (9)

Where $C$ is an arbitrary constant and ${\mu }_k$ is one of the numbers ${(6)}_1$. Equation (9) give us the amplitude, at the point $x$, of the string’s free vibrations, that is

$$y_k(x,t)=C\sin {\mu }_{k}x\sin {\omega }_{k}t,$$ (10)

which takes place without the action of external exciter forces. As it has been seen so far, these oscillations do not take place with any frequency, but solely of one of the given in ${(6)}_2$ for $k\in {\mathbb{Z}}^{+}$.
As it is shown in (5), if condition (7) (or equivalently (8)) does not satisfied, then the amplitude $Y(x)$ of string’s periodic oscillations, at the point $x$, will increase indefinitely as the exciter’s force frequency $\omega$ yields to one of the string’s eigenfrequencies (or string’s natural frequencies). At limit, as these frequencies coincide, begins the resonance. Thus, in general, there are not periodic oscillations on the string. Accordingly, in general, there is not solution for the non-homogeneous integral equation (3), as $\lambda$ is one of the eigenvalues of this equation.
A more elaborated discussion (which takes into account string’s cross section properties), not for Hookean elastic but Maxwellian viscoelastic string’s model, is given in [9].

Title	vibrating string
Canonical name	VibratingString
Date of creation	2013-03-22 17:25:19
Last modified on	2013-03-22 17:25:19
Owner	perucho (2192)
Last modified by	perucho (2192)
Numerical id	11
Author	perucho (2192)
Entry type	Topic
Classification	msc 45B05