vibrating string

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

.

1

1

For a solution via partial differential equations, see pahio: solving the

wave equation

due to D. Bernoulli (

http://planetmath.org/SolvingTheWaveEquationByDBernoulli

).

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

≪

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

)

≪

l

,

0

≤

x

≤

l

,

0

≤

t

<

∞

, 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

O

⁢

A

¯

=

l

. At

x

=

ξ

, 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

O

⁢

A

⁢

C

, at

t

=

0

. A free diagram of node

C

and Newton’s second law

∑

F

y

=

0

, leads to

2

2

From

hypothesis

5., it is clear
that

∠

⁢

A

⁢

O

⁢

C

=

θ

≪

1

, and

∠

⁢

C

⁢

A

⁢

O

=

α

≪

1

. Thus,

sin

θ

≈

tan

θ

=

δ

ξ

,

sin

α

≈

tan

α

=

δ

l

-

ξ

⋅

T

0

⁢

δ

ξ

+

T

0

⁢

δ

l

-

ξ

=

P

,

or

δ

=

P

⁢

ξ

⁢

(

l

-

ξ

)

T

0

⁢

l

,

being

T

0

the string’s tension and

δ

the static deflection at

t

=

0

.

Denoting by

Y

⁢

(

x

)

the arbitrary deflection at

x

, and from the

prior

analysis

, we obtain

Y

⁢

(

x

)

=

P

⁢

G

⁢

(

x

,

ξ

)

,

where

{

G

⁢

(

x

,

ξ

)

=

x

⁢

(

l

-

ξ

)

T

0

⁢

l

,

0

≤

x

≤

ξ

,

G

⁢

(

x

,

ξ

)

=

(

l

-

x

)

⁢

ξ

T

0

⁢

l

,

ξ

≤

x

≤

l

.

(1)

This is the Green’s function that we need. It is clear that

G

⁢

(

x

,

ξ

)

=

G

⁢

(

ξ

,

x

)

. Let us suppose that on the string acts a distributed force per unit length

p

⁢

(

ξ

)

, then the force

element

on

(

ξ

,

ξ

+

δ

⁢

ξ

)

it should be

p

⁢

(

ξ

)

⁢

δ

⁢

ξ

and therefore, by the

superposition principle

, the string takes the form

Y

⁢

(

x

)

=

∫

0

l

G

⁢

(

x

,

ξ

)

⁢

p

⁢

(

ξ

)

⁢

𝑑

ξ

.

4

Discussion of cases

Following to Petrovski

[

8

]

, we will consider the following problems.

1.

To find out the force’s density

distribution

p

⁢

(

ξ

)

, 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

)

=

∫

0

l

G

⁢

(

x

,

ξ

)

⁢

p

⁢

(

ξ

)

⁢

𝑑

ξ

(2)

respect to the unknown function

p

⁢

(

ξ

)

.

2.

Let us suppose that on the string acts an exciter force with density, at

x

=

ξ

and instant

t

, given by

p

⁢

(

ξ

,

t

)

:=

p

⁢

(

ξ

)

⁢

sin

⁡

ω

⁢

t

, being

ω

>

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

⁡

ω

⁢

t

.

Denoting by

ρ

⁢

(

ξ

)

the mass linear density at

x

=

ξ

, from the

D

’Alambert principle

3

3

i.e.

∑

F

y

⁢

d

⁢

ξ

+

{

-

ρ

⁢

(

ξ

)

⁢

∂

2

⁡

y

∂

⁡

t

2

⁢

(

ξ

,

t

)

⁢

d

⁢

ξ

}

≡

0

,

a

dynamical equilibrium

where the second term is the involved

inertia force

. It is very important realize that in order to represents the solution

y

⁢

(

x

,

t

)

through an integral equation, it is necessary to explicit the exciter force

p

⁢

(

ξ

)

⁢

sin

⁡

ω

⁢

t

. Thus,

∑

F

y

=

∑

F

y

′

+

p

⁢

(

ξ

)

⁢

sin

⁡

ω

⁢

t

. One also realizes that

∂

2

⁡

y

/

∂

⁡

t

2

may be obtained from the assumption above expressed for

x

=

ξ

. Hence the solution becomes

y

⁢

(

x

,

t

)

=

Y

⁢

(

x

)

⁢

sin

⁡

ω

⁢

t

=

∫

0

l

G

⁢

(

x

,

ξ

)

⁢

{

-

∑

F

y

′

}

⁢

𝑑

ξ

.

Minus sign, above placed, it is conventional as the string’s tension balance (because the tension applied at the ends of string’s free diagram) is assumed upwards and, obviously,

y

⁢

(

x

,

t

)

is changing periodically of sign.

applied to the string’s

segment

(

ξ

,

ξ

+

δ

⁢

ξ

)

, at instant

t

, we have

y

⁢

(

x

,

t

)

=

Y

⁢

(

x

)

⁢

sin

⁡

ω

⁢

t

=

∫

0

l

G

⁢

(

x

,

ξ

)

⁢

{

p

⁢

(

ξ

)

⁢

sin

⁡

ω

⁢

t

+

ω

2

⁢

ρ

⁢

(

ξ

)

⁢

Y

⁢

(

ξ

)

⁢

sin

⁡

ω

⁢

t

}

⁢

𝑑

ξ

.

Simplifying by

sin

⁡

ω

⁢

t

and defining

f

⁢

(

x

)

:=

∫

0

l

G

⁢

(

x

,

ξ

)

⁢

p

⁢

(

ξ

)

⁢

𝑑

ξ

,

K

⁢

(

x

,

ξ

)

:=

ρ

⁢

(

ξ

)

⁢

G

⁢

(

x

,

ξ

)

,

λ

:=

ω

2

,

we get

Y

⁢

(

x

)

=

λ

⁢

∫

0

l

K

⁢

(

x

,

ξ

)

⁢

Y

⁢

(

ξ

)

⁢

𝑑

ξ

+

f

⁢

(

x

)

.

(3)

Supposing known the function

p

⁢

(

ξ

)

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

ρ

⁢

(

ξ

)

is

constant

and

f

⁢

(

x

)

is

regular

4

4

i.e.

f

⁢

(

x

)

∈

𝒞

2

⁢

[

0

,

l

]

.

, it is not too hard solving this integral equation. In fact, from (1) and the definition of the

symmetric

kernel

K

⁢

(

x

,

ξ

)

, results

Y

⁢

(

x

)

=

ω

2

⁢

ρ

⁢

∫

0

x

(

l

-

x

)

⁢

ξ

T

0

⁢

l

⁢

Y

⁢

(

ξ

)

⁢

𝑑

ξ

+

ω

2

⁢

ρ

⁢

∫

x

l

x

⁢

(

l

-

ξ

)

T

0

⁢

l

⁢

Y

⁢

(

ξ

)

⁢

𝑑

ξ

+

f

⁢

(

x

)

or

Y

⁢

(

x

)

=

ω

2

⁢

c

l

⁢

(

l

-

x

)

⁢

∫

0

x

ξ

⁢

Y

⁢

(

ξ

)

⁢

𝑑

ξ

+

ω

2

⁢

c

⁢

x

l

⁢

∫

x

l

(

l

-

ξ

)

⁢

Y

⁢

(

ξ

)

⁢

𝑑

ξ

+

f

⁢

(

x

)

,

Where

c

=

ρ

/

T

0

. By twice

differentiation

respect to

x

, we obtain (by application of Liebnitz’s rule)

Y

′′

⁢

(

x

)

=

-

ω

2

⁢

c

⁢

Y

⁢

(

x

)

+

f

′′

⁢

(

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

′′

⁢

(

ξ

)

=

-

ω

2

⁢

c

⁢

Y

⁢

(

ξ

)

+

f

′′

⁢

(

ξ

)

by

-

T

0

⁢

G

⁢

(

x

,

ξ

)

and integrating respect to

ξ

from

ξ

=

0

to

ξ

=

l

. Then (3) is obtained since, integrating by parts, it is easily shown that

∫

0

l

T

0

⁢

G

⁢

(

x

,

ξ

)

⁢

U

′′

⁢

(

ξ

)

⁢

𝑑

ξ

=

-

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

⁡

μ

⁢

x

+

C

2

⁢

cos

⁡

μ

⁢

x

+

1

μ

⁢

∫

0

x

f

′′

⁢

(

ξ

)

⁢

sin

⁡

μ

⁢

(

x

-

ξ

)

⁢

𝑑

ξ

,

Where

μ

=

ω

⁢

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

⁡

μ

⁢

l

≠

0

, we have

Y

⁢

(

x

)

=

-

1

μ

⁢

sin

⁡

μ

⁢

x

sin

⁡

μ

⁢

l

⁢

∫

0

l

f

′′

⁢

(

ξ

)

⁢

sin

⁡

μ

⁢

(

l

-

ξ

)

⁢

𝑑

ξ

+

1

μ

⁢

∫

0

x

f

′′

⁢

(

ξ

)

⁢

sin

⁡

μ

⁢

(

x

-

ξ

)

⁢

𝑑

ξ

.

(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

⁡

μ

⁢

l

≠

0

, that the function

f

⁢

(

x

)

be

continuous

, so that the condition about existence and, with more reason, about that

f

′′

⁢

(

x

)

be continuous, is superfluous. On the contrary, condition

sin

⁡

μ

⁢

l

≠

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

⁡

μ

⁢

l

=

0

, then

μ

=

k

⁢

π

l

,

ω

=

k

⁢

π

l

⁢

c

,

λ

=

k

2

⁢

π

2

l

2

⁢

c

,

(6)

where

k

∈

ℤ

. The values of

λ

, given in

(

6

)

3

for

k

∈

ℤ

+

, are the eigenvalues of integral equation (3), and the corresponding values of

ω

in

(

6

)

2

, are the

eigenfrequencies

of string’s oscillations. From (5) and

f

⁢

(

x

)

being regular, as

sin

⁡

μ

⁢

l

=

0

one realizes that the integral equation (3) has solution only if

∫

0

l

f

′′

⁢

(

ξ

)

⁢

sin

⁡

μ

⁢

(

l

-

ξ

)

⁢

𝑑

ξ

=

0

.

(7)

Integrating by parts and by making use of the fact that

sin

⁡

μ

⁢

(

l

-

ξ

)

=

0

and

f

⁢

(

ξ

)

=

0

for

ξ

=

0

and

ξ

=

l

, condition (7) becomes

5

5

From (8) one sees that indeed, condition for

f

⁢

(

x

)

, may be weakened to

f

⁢

(

x

)

∈

𝒫

⁢

𝒞

⁢

[

0

,

l

]

.

∫

0

l

f

⁢

(

ξ

)

⁢

sin

⁡

μ

⁢

(

l

-

ξ

)

⁢

𝑑

ξ

=

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

μ

for which

sin

⁡

μ

⁢

l

=

0

. In particular, condition (8) is satisfied if

f

⁢

(

x

)

≡

0

. Then, the integral equation (3) and the differential equation (4) become

homogeneous

. Consequently, all the solutions of the

homogeneous differential equation

Y

′′

⁢

(

ξ

)

+

ω

2

⁢

c

⁢

Y

⁢

(

ξ

)

=

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

)

=

λ

⁢

∫

0

l

K

⁢

(

x

,

ξ

)

⁢

Y

⁢

(

ξ

)

⁢

𝑑

ξ

are given by

Y

⁢

(

x

)

↦

Y

k

⁢

(

x

)

=

C

⁢

sin

⁡

μ

k

⁢

x

,

(9)

Where

C

is an arbitrary constant and

μ

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

⁡

μ

k

⁢

x

⁢

sin

⁡

ω

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

∈

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

ω

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

λ

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

]

.

References

1

C. Truesdell,

Essays in the History of Mechanics

, pp. 108-110, Springer-Verlag, Berlin-Heidelberg-New York, 1968.

2

L. Euler,

Mechanica sive motus scientis analytice exposita

, 2 tomes=

Opera Omnia II

, 1 and 2, Petropoli, 1736.

3

J. L.

D

’Alembert,

Sur la Corde Vibrante

, Memoires de l’Académic des Sciences, 216ff., Berlin, 1747.

4

G. F. Carrier,

On the non-linear

vibration problem

of the elastic String

, Quart. Appl. Math.,

3

¯

, 157, 1945.

5

E. Fermi, J. R. Pasta, S. Ulam,

Studies of Non-linear Problems I

, Los Alamos, Report

N

o

1940, May 1955. (Unpublished)

6

N. J. Zabusky,

Exact Solution for the Vibrations of a Nonlinear Continuous Model String

, J. Math., Phys.,

3

¯

, 1028, 1962.

7

G. F. B. Riemann,

Uber die fortpflanzung ebener Luftwellen von endlicher Schwingungsweite

, Abhandl., Konigl.Ges.Wiss. Göttingen,

8

¯

, 43, 1860.

8

I. G. Petrovski,

Lecciones de Teoría de las Ecuaciones Integrales

, 2da. ed., Trad. de la 3era. ed. Rusa, Edit. MIR, Moscú, 1976.

9

P. Fernández,

Solución Exacta para las Vibraciones transversales en un modelo de Cable No-Hookeano

, Trabajo de Ascenso a Prof. Agregado, UCV, Caracas, 1985.

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

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