Transverse Vibrations of Strings Notes | EduRev

: Transverse Vibrations of Strings Notes | EduRev

 Page 1


Vibrations of Structures
Module I: Vibrations of Strings and Bars
Lesson 1: Transverse Vibrations of Strings
Contents:
1. Introduction
2. Mathematical Model
3. Examples
Keywords: Transverse vibrations, Taut string, Hanging chain, equation of
motion, non-homogeneous boundary condition
Page 2


Vibrations of Structures
Module I: Vibrations of Strings and Bars
Lesson 1: Transverse Vibrations of Strings
Contents:
1. Introduction
2. Mathematical Model
3. Examples
Keywords: Transverse vibrations, Taut string, Hanging chain, equation of
motion, non-homogeneous boundary condition
Transverse Vibrations of Strings
1 Introduction
A string is a one dimensional elastic continuum that does not transmit or
resist bending moment.
Elements that may be modeled as taut strings:
 Strings in stringed musical instruments such as sitar, guitar or violin
 Cables in a cable-stayed bridge or cable-car
 High tension wires
2 Mathematical Model
Assumptions in modeling:
 Motion is planar
 Slope of the string is small
 Longitudinal motion is negligible
 Tension does not change with displacement of the string
2
Page 3


Vibrations of Structures
Module I: Vibrations of Strings and Bars
Lesson 1: Transverse Vibrations of Strings
Contents:
1. Introduction
2. Mathematical Model
3. Examples
Keywords: Transverse vibrations, Taut string, Hanging chain, equation of
motion, non-homogeneous boundary condition
Transverse Vibrations of Strings
1 Introduction
A string is a one dimensional elastic continuum that does not transmit or
resist bending moment.
Elements that may be modeled as taut strings:
 Strings in stringed musical instruments such as sitar, guitar or violin
 Cables in a cable-stayed bridge or cable-car
 High tension wires
2 Mathematical Model
Assumptions in modeling:
 Motion is planar
 Slope of the string is small
 Longitudinal motion is negligible
 Tension does not change with displacement of the string
2
(a)
l
x
z
w(x; t)
; A; T
(b)
x x + dx
ds
T(x; t)
(x; t)
T(x + dx; t)
(x + dx; t)
p(x; t)ds
n(x; t)ds
Figure 1: (a) A taut string (b) An innitesimal string element
Consider a string of density , area of cross-section A(x), length l and under
a tension T(x; t) subject to distributed forces (force per unit length) p(x; t)
in the transverse direction, and n(x; t) in the axial direction . Let w(x; t) be
the transverse displacement eld variable.
Longitudinal dynamics:
Balance of forces in the longitudinal direction of the sting element in Fig. 1(b)
reads
T(x + dx; t) cos[(x + dx; t)] T(x; t) cos[(x; t)] + n(x; t)dx = 0
) [T(x; t)]
;x
=n(x; t) ; (1)
where []
;x
= @=@x. Here, we have assumed ds dx and cos  1.
3
Page 4


Vibrations of Structures
Module I: Vibrations of Strings and Bars
Lesson 1: Transverse Vibrations of Strings
Contents:
1. Introduction
2. Mathematical Model
3. Examples
Keywords: Transverse vibrations, Taut string, Hanging chain, equation of
motion, non-homogeneous boundary condition
Transverse Vibrations of Strings
1 Introduction
A string is a one dimensional elastic continuum that does not transmit or
resist bending moment.
Elements that may be modeled as taut strings:
 Strings in stringed musical instruments such as sitar, guitar or violin
 Cables in a cable-stayed bridge or cable-car
 High tension wires
2 Mathematical Model
Assumptions in modeling:
 Motion is planar
 Slope of the string is small
 Longitudinal motion is negligible
 Tension does not change with displacement of the string
2
(a)
l
x
z
w(x; t)
; A; T
(b)
x x + dx
ds
T(x; t)
(x; t)
T(x + dx; t)
(x + dx; t)
p(x; t)ds
n(x; t)ds
Figure 1: (a) A taut string (b) An innitesimal string element
Consider a string of density , area of cross-section A(x), length l and under
a tension T(x; t) subject to distributed forces (force per unit length) p(x; t)
in the transverse direction, and n(x; t) in the axial direction . Let w(x; t) be
the transverse displacement eld variable.
Longitudinal dynamics:
Balance of forces in the longitudinal direction of the sting element in Fig. 1(b)
reads
T(x + dx; t) cos[(x + dx; t)] T(x; t) cos[(x; t)] + n(x; t)dx = 0
) [T(x; t)]
;x
=n(x; t) ; (1)
where []
;x
= @=@x. Here, we have assumed ds dx and cos  1.
3
Transverse dynamics:
Newton's second law for the string element is given by
Adxw
;tt
= T(x + dx; t) sin[(x + dx; t)] T(x; t) sin[(x; t)] + p(x; t)dx
) A(x)w
;tt
 [T(x; t)w
;x
]
;x
= p(x; t) (2)
where sin  tan = w
;x
.
The linear partial dierential equation (2), along with (1), represents
the dynamics of a taut string. The complete description of the free vibra-
tion problem requires specication of two boundary conditions (w(0; t) and
w(l; t)), and two initial conditions (w(x; 0) and w
;t
(x; 0)).
3 Examples
Uniform taut string:
For a uniform taut unforced string (see Fig. 1(a)), A(x) = A and n(x; t) = 0
(implying T
;x
= 0). The equation of motion simplies to
w
;tt
 c
2
w
;xx
= 0 (3)
where c =
p
T=A is a constant having the dimension of speed. Since there
is no transverse displacement at the boundaries, the appropriate boundary
conditions are given by w(0; t) = 0 and w(l; t) = 0.
A string with a free/sliding boundary condition is shown in Fig. 2. At
the right boundary, the massless and frictionless roller cannot support any
4
Page 5


Vibrations of Structures
Module I: Vibrations of Strings and Bars
Lesson 1: Transverse Vibrations of Strings
Contents:
1. Introduction
2. Mathematical Model
3. Examples
Keywords: Transverse vibrations, Taut string, Hanging chain, equation of
motion, non-homogeneous boundary condition
Transverse Vibrations of Strings
1 Introduction
A string is a one dimensional elastic continuum that does not transmit or
resist bending moment.
Elements that may be modeled as taut strings:
 Strings in stringed musical instruments such as sitar, guitar or violin
 Cables in a cable-stayed bridge or cable-car
 High tension wires
2 Mathematical Model
Assumptions in modeling:
 Motion is planar
 Slope of the string is small
 Longitudinal motion is negligible
 Tension does not change with displacement of the string
2
(a)
l
x
z
w(x; t)
; A; T
(b)
x x + dx
ds
T(x; t)
(x; t)
T(x + dx; t)
(x + dx; t)
p(x; t)ds
n(x; t)ds
Figure 1: (a) A taut string (b) An innitesimal string element
Consider a string of density , area of cross-section A(x), length l and under
a tension T(x; t) subject to distributed forces (force per unit length) p(x; t)
in the transverse direction, and n(x; t) in the axial direction . Let w(x; t) be
the transverse displacement eld variable.
Longitudinal dynamics:
Balance of forces in the longitudinal direction of the sting element in Fig. 1(b)
reads
T(x + dx; t) cos[(x + dx; t)] T(x; t) cos[(x; t)] + n(x; t)dx = 0
) [T(x; t)]
;x
=n(x; t) ; (1)
where []
;x
= @=@x. Here, we have assumed ds dx and cos  1.
3
Transverse dynamics:
Newton's second law for the string element is given by
Adxw
;tt
= T(x + dx; t) sin[(x + dx; t)] T(x; t) sin[(x; t)] + p(x; t)dx
) A(x)w
;tt
 [T(x; t)w
;x
]
;x
= p(x; t) (2)
where sin  tan = w
;x
.
The linear partial dierential equation (2), along with (1), represents
the dynamics of a taut string. The complete description of the free vibra-
tion problem requires specication of two boundary conditions (w(0; t) and
w(l; t)), and two initial conditions (w(x; 0) and w
;t
(x; 0)).
3 Examples
Uniform taut string:
For a uniform taut unforced string (see Fig. 1(a)), A(x) = A and n(x; t) = 0
(implying T
;x
= 0). The equation of motion simplies to
w
;tt
 c
2
w
;xx
= 0 (3)
where c =
p
T=A is a constant having the dimension of speed. Since there
is no transverse displacement at the boundaries, the appropriate boundary
conditions are given by w(0; t) = 0 and w(l; t) = 0.
A string with a free/sliding boundary condition is shown in Fig. 2. At
the right boundary, the massless and frictionless roller cannot support any
4
l
x
z; w
; A; T
w(0; t) = 0
Tw
;x
(l; t) = 0
Figure 2: Taut string with geometric and natural boundary conditions
transverse force (given by T sin (l; t)). Therefore, the boundary condition
at the roller is Tw
;x
(l; t) = 0 (taking sin  tan = w
;x
).
A condition on the geometry/kinematics at a boundary is know as a ge-
ometric/essential boundary condition (the left end boundary condition in
Fig. 2). A condition on the force at a boundary is know as a dynamic/natural
boundary condition (the right end boundary condition in Fig. 2).
The hyperbolic partial dierential equation (3) is known as the linear one-
dimensional wave equation, and c is known as the wave speed. In the case of
a taut string, c is the speed of transverse waves on the string. This implies
that a disturbance created at any point on the string propagates with a speed
c. It should be clear that the wave speed c is distinct from the transverse
material velocity (i.e., the velocity of the particles of the string) which is
given by w
;t
(x; t).
Uniform hanging string/chain:
For a hanging string (see Fig. 3), n(x; t) = Ag. From (1), the expression of
5
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