Dinamica dei Sistemi Meccanici– Prof. Stefano Bruni
Lesson 2
Dynamics of continuous systems – transversal vibration of
tensioned strings
1
Lumped parameter and distributed parameter vibrating systems
Lumped parameter systems (discrete systems) are vibrating systems that can be seen as formed by
rigid bodies (and / or point masses) interconnected by mass-less springs, dampers and by different
types of constraints (e.g. hinges, clamps etc.). In this way, the inertial properties of the system are
lumped in the rigid bodies / point masses, the elastic properties in the spring elements and the
dissipative effects in the damper elements. As an example, when studying the vibration of a car in
the vertical plane, one could establish a simple model by considering the carbody as a rigid body
connected to the ground through the front and rear suspensions, each one modelled as a
combination of mass-less springs and dampers. Lumped parameter systems have a finite number n
of degrees of freedom. Methods to write and solve a set of equations of motion for this kind of
systems is assumed here as a pre-requisite.
There are however other mechanical systems for which the separation into portions behaving
(approximately) as rigid bodies and other ones that may be reduced to lumped springs and dampers
is not feasible. For instance, if we consider the case of the shaft of a steam turbine + generator, we
recognise that this system may be modelled as a beam resting on several supports: if we want to
study the bending vibrations that may be produced by an unbalance in the shaft, we necessarily
need to define an approach taking into account the presence of inertial, elastic, viscous forces as
“distributed” inside the whole structure, and hence we need then to change our approach from the
study of a “discrete” system, to the study of a “continuous” system (or system with “distributed
parameters”, see Meirovitch, Fundamentals of Vibrations chapters 8 and 9).
This lesson and some of the following ones deal with the dynamics of continuous systems by
establishing appropriate mathematical approaches to take into account the “distributed parameter”
nature of a new category of mechanical systems.
Before entering into this new problem, we comment a little bit further on the need for an approach
to study the vibration of continuous systems. By the above comparison between the two examples
of a road vehicle and a steam turbine + generator shaft, we have introduced the need for discrete vs.
continuous models based on the physical properties of the system under study. However, we must
also recognise that the choice between o concentrated parameters vs. a distributed parameters model
also depends upon the frequency range where the system is to be studied. In other words, whether a
lumped parameter or distributer parameter model should be defined for a given physical system,
also depend on the scope of the analysis and particularly on the field of frequency covered
addressed. The following example aims at clarifying this concept.
The assumption of modelling the carbody of a road vehicle as a rigid body naturally sounds as
reasonable, but should be carefully verified with respect to the scopes of the mathematical model
being established. Indeed, any body in nature will intrinsically be deformable, and the decision to
use a rigid body model will always entail a simplification to be considered with care. Indeed, it is
possible to show that when the carbody is considered as a deformable body, modes of vibration
associated with deformable motions of the carbody (like e.g. bending and torsion of the structure)
are found in a frequency range above say 25÷30 Hz. Therefore, if the problem to be dealt with is
e.g. to study suspension road holding, a problem to which vibrations in the field below 10 Hz are
pertinent, the car body can be considered as a rigid body because the vibrations of the carbody in
the frequency range of interest are almost exclusively rigid motions produced by the elastic
deformation of the suspensions. However, if the problem to be studied is noise emission and
propagation in the interior of the car, which is a problem related to high frequency vibration of parts
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Dinamica dei Sistemi Meccanici– Prof. Stefano Bruni
of the carbody and interior furniture, a rigid body model would no longer be correct, and a more
detailed model representing in detail non-rigid deformations of the various parts inside the car
(doors, roof, covering panels etc.) would be needed.
2
Transversal vibrations of a tensioned string: assumptions and equation of
motion
A string is a one-dimensional continuum which, differently from a beam, is only able to transmit an
internal force directed as the axis of the string. This force is called ‘tension’. We consider the case
of a tensioned string of length l supported by two pins at its ends, as shown in Figure 2.1 below.
The transversal displacement of the string is defined by w, which is function of the spatial
coordinate x spanning the length of the string from the left end (x=0) to the right end (x=l) and of
time t.
w(x,t)
x
l
Figure 2.1: transversal vibration of a pinned-pinned tensioned string
We will treat the problem under the two following assumptions:
1) Small displacements. This means w(x,t) is considered as a small quantity, so that w2 can be
considered as a higher order infinitesimal quantity compared to w;
a. This also implies that any variation T of the tension in the string occurring due to
the transversal movements of the string is negligible when compared to the tention T
in the static undeformed configuration of the string. In other words, tension T is a
constant parameter of the problem;
2) the linear mass m of the string does not depend on the spatial coordinate x, so it is another
constant parameter of the problem.
3) damping effects are neglected;
4) No force is acting on the string (including gravitational forces), except at the two ends where
the pins are introducing a constraint force.
In order to write the equation of motion for the transversal vibration of the string, we consider a
piece of string having infinitesimal length dx, enclosed between the generic section x and the
generic incremented section x+dx (see figure 2.2) and we write an equation by taking the projection
along the vertical direction of:
r
r
F = mdxa
This equation reads:
T sin( ( x , t )) + T sin( ( x + dx , t )) mdx
and, because
2
w
t
2
=0
is a small angle:
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(2.1)
Dinamica dei Sistemi Meccanici– Prof. Stefano Bruni
T
(x+dx)
2
(x)
mdx
T
w
t2
Figure 2.2: Forces acting on a piece of string having infinitesimal length dx.
(x , t ) + T (x + dx , t )
T
mdx
expanding the value of angle
( x ,t ) + T ( x ,t ) + T
T
2
w
t
2
=0
in the incremented position x+dx we get:
2
x
(2.2)
dx mdx
w
t2
=0
(2.3)
the first two terms at left hand side in the equation above cancel each other, and in the remaining
terms the length dx of the element can be simplified obtaining:
T
=m
x
2
w
t
2
(2.4)
finally, we recall that for small displacements w, the inclination
string is:
w
x
=
of the tangent to the deformed
(2.5)
so that replacing this result in the previous equation we get the following second-order, linear
partial derivative differential equation:
T
2
w
x
2
=m
2
w
t
2
(2.6)
representing the equation of motion for the transversal vibration of the tensioned string under the
assumptions declared above. We note that while for a discrete (n-dof) system the set of independent
coordinates was represented by n functions of time (x1(t), x2(t), …, xn(t)), for a distributed parameter
system as the one treated in this lesson we had to take as independent kinematical parameter one
single function w(x,t), depending upon both the spatial coordinate x and the time coordinate t. If we
consider that the displacement function w is defined for infinite values of coordinate x, spanning the
whole length of the string, we recognise that introducing function w(x,t) as the independent
coordinate for the string corresponds to introducing an infinite number of time dependent
coordinates (namely, one for each of the possible infinite values of coordinate x). Correspondingly,
instead than a set of n total derivative differential equations, we get one single partial derivative
differential equation. In some sense, the passage from a discrete to a continuous system may then be
regarded as the limit case where the number n of degrees of freedom of the discrete system tends to
infinity.
Before closing this section, we observe that eq. (2.6) may be rewritten in the form:
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Dinamica dei Sistemi Meccanici– Prof. Stefano Bruni
2
w
t
2
=c
2
2
w
(2.7)
x2
with:
c2 =
T
m
;
c=
T
m
(2.8)
Equation (2.8) represents a more general form of (2.7). Indeed, as will be shown in a next lesson, by
introducing different expressions for parameter c, equation (2.7) represents a variety of vibration
problems for distributed parameter systems including, besides the transversal vibrations of a
tensioned string studied here, the axial vibration of a bar and the torsional vibration of a rod.
Parameter c takes the meaning (as will be shown in the next section) of the speed of propagation of
a perturbation inside the distributed parameter medium.
3
Propagative solution
In this section we start discussing the solution of eq. (2.6) / (2.7). It may be shown that the general
solution for equation (2.7) takes the form:
w( x ,t ) = f1 ( x ct ) + f 2 ( x ct )
(3.1)
where f1 and f2 are two arbitrary functions, each one being twice derivable. We easily recognise that
expression (3.1) is solution of equation (2.7) since (by denoting as f’’ the second derivative of
functions f1 and f2):
2
w(x ,t )
x
2
= f1'' ( x ct ) + f 2'' ( x ct ) ;
2
w( x ,t )
t
2
= c 2 f1'' ( x ct ) + c 2 f 2'' ( x ct )
In order to provide a physical meaning for solution (3.1) to problem (2.7) we define:
s1 = x ct
;
s 2 = x + ct
then, we concentrate initially on the first term of solution (3.1), as if it was f2=0. We may observe in
this case that f1(s1) represents a waveform (with arbitrary shape) defined in terms of the auxiliary
coordinate s1. If we consider time t=0, then the auxiliary coordinate s1 coincides with the space
coordinate x, so that we may identify function f1 with the deformation of the string at the initial time
t=0 (under the above assumption f2=0). If we then consider a subsequent time t>0, the relationship
between s1 and x becomes x= s1+ct, so that the same value of coordinate s1 representing the initial
string deformation in a given position s1 is now obtained for t>0 in a different location that is
displaced by a quantity ct in the positive direction of the x axis, which means rightwards according
to the conventions previously selected. Thus, f1(x-ct) represents a waveform of deformation that is
propagating rightwards inside the string, while keeping the same waveform. This situation is
depicted in the upper part of figure 3.1.
By the same reasoning, also the second part of solution (3.1), f2(x+ct), may be interpreted as the
propagation of a second perturbation with steady waveform inside the string, with the only
difference that in this case, due to the different sign in front of the ct term, this second term
represents a propagative term moving leftwards instead than rightwards.
Thus, we may conclude from this analysis that when an initial condition is assigned at the initial
time t=0, two waves are initialised in the string, having opposite directions but the same absolute
value of the speed defined by eq. (2.8). Thanks to this physical interpretation, the solution (3.1) of
equation (2.7) is called the “propagative solution” for the problem ostudied.
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Dinamica dei Sistemi Meccanici– Prof. Stefano Bruni
w(x,t)
ct
t=0
f1(s1=x)
t>0
f1(s1=x-ct)
V=c
x
w(x,t)
t>0
f2(s2=x+ct)
-ct
t=0
f2(s2=x)
V=-c
x
Figure 3.1: Interpretation of solution (3.1) as the propagation of two perturbations in the string
4
Stationary solution
As discussed in the previous section, the propagative solution (3.1) is very useful to deal with
problems concerning wave propagation in infinite media: this may be the case, e.g., for sound
propagation into a supposedly infinite mono-dimensional medium or for the vibration generated by
a pantograph sliding on a railway catenary. However, in the case of a system having finite length,
the two waves generated at t=0 will reach the ends of the string and will be reflected, so that
positive and negative interference will take place between the travelling and the reflected waves. By
decomposing the travelling waves into a Fourier series and considering properly the boundary
conditions at the extremities of the string, it may be demonstrated that this positive and negative
interference will produce stationary waveforms of vibration, which means a vibration that may be
described as a waveform fixed in space whose amplitude is modulated by a time-dependent
coefficient. This kind of so-called “stationary vibration” may be expressed in the form:
w( x ,t ) = ( x ) ( t )
(4.1)
where (x) is a function of space alone describing the waveform of the stationary vibration, while
(t) represents a time dependent amplitude coefficient being applied on the spatial waveform (x).
Figure 4.1 shows the physical meaning of the stationary solution, assuming e.g. that (t) oscillates
between ±1.
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Dinamica dei Sistemi Meccanici– Prof. Stefano Bruni
w(x,t)
t=t1, (t1) =+1
t=t2, (t2) =+0.5
x
t=t3, (t3) = -0.5
t=t4, (t4) = -1
Figure 4.1: Interpretation of the stationary solution (4.1) for the vibration of the string
The actual expressions for the two functions (x) and (t) could be obtained from the propagative
solution (3.1) by using a Fourier series expression in conjunction with a proper representation of the
boundary conditions at the ends of the string. However, we prefer to use here a simpler approach,
called “separation of variables”, which consists in replacing expression (4.1) into equation (2.7).
To this end, we observe that:
2
(x )
x
(t )
=
2
''
2
( x) (t )
;
(x )
t
2
(t )
= ( x) &&(t )
where '' ( x) stands for the second derivative of
relationships in eq. (2.7) we get:
( x ) &&( t ) = c 2
''
with respect to x. By introducing the above
( x) (t )
then, we separate at first side of the above equation all terms depending upon the time coordinate t,
from the terms depending upon the space coordinate x that are placed at the right hand side. Since
the expression at left hand side is time dependent while the one at the right hand side is space
dependent, the only way to satisfy the equality sign is to assume that both terms are equal to the
same constant. For reasons that will soon be clear, this constant must be negative valued, so that the
constant expression – 2 is used to make sure of the negative sign:
&&(t )
(t )
= c2
''
( x)
=
( x)
2
(4.2)
We now use the above equation to derive one differential equation for the time dependent function
(t) and one separate differential equation for the space dependent function (x):
&&(t ) + 2 (t ) = 0
2
( x) = 0
c
the first equation (in time) has the well known form of the equation describing the free vibration of
an undamped 1-dof system, and thus the solution takes the form of a harmonic function of time, that
may be represented as a cosine function with amplitude C and phase .
''
( x) +
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Dinamica dei Sistemi Meccanici– Prof. Stefano Bruni
( t ) = C cos( t +
)
(4.3)
From the above result the reason why we required in equation (4.2) the constant – 2 to be negative
valued should now be clear, since in case of positive sign, we would have obtained a differential
equation for the time dependent solution (t) having the form:
&&(t )
2
(t ) = 0
that would lead to a diverging solution (t) whereas the system under study is conservative and
indeed inherently stable.
The second equation (in space) defining the solution (x) has actually the same form as the
previous one, except for the fact that the independent variable is now x (space) instead than t (time)
2
and for the fact that constant 2 is replaced by ( c ) . It is then easy to see that the corresponding
solution will be in the form of a harmonic function of space with the ratio c in the place of the
constant . However, for the space dependent part of the solution (x) we prefer (for reasons that
will be soon clear) the use of a solution in the form based on the linear combination of a cosine and
a sine function (without phases) instead than the one based on a single cosine function with generic
phase:
( x) = A cos
c
x + B sin
c
(4.4)
x
it is left to the reader to verify that the c ratio has dimensions length at power -1, so that the
argument of the sine and cosine functions in the above solution are pure numbers.
Now, if we put together the partial solutions (4.3) and (4.4) and we embed constant C from eq. (4.3)
into constants A and B of eq. (4.4) we get:
u ( x, t ) = ( x) (t ) = A cos
c
x + B sin
c
x
cos( t +
)
(4.5)
Expression (4.5) represents the general solution for the stationary free vibration of a tensioned
string, regardless the boundary conditions acting at the boundaries of the system. In order to
introduce the concept of natural frequencies and associated modes of vibration for the problem
under study, we have then to take into account the boundary conditions introduced by the pins
placed at the two ends of the string. These read:
w( x ,t ) x =0 = 0 for any time t
w( x ,t ) x =l = 0 for any time t
if we replace the first of these two boundary conditions in solution (4.5) we get:
A=0
(4.6)
where the time dependent expression has been simplified since the equality must hold for any time
t. If we now replace in (4.5) the second boundary condition, taking into consideration that A=0 we
get:
B sin
c
l =0
l = 0 . The
c
first of the two solutions would imply A=0, B=0 and then no motion at all for the system (what is
called the “trivial” solution). In order to get non trivial solutions, we must then require:
that may be satisfied in two alternative ways, either by setting B=0, or by setting sin
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Dinamica dei Sistemi Meccanici– Prof. Stefano Bruni
sin
c
l =0
c
l=k
k = 1,2 ,...
(4.7)
from the above relationship, we conclude that the free vibration of the pinned-pinned string is
possible if the angular frequency parameter assumes any of the values defined by formula:
k
=
k
k
c=
l
l
T
m
k = 1,2 ,...
(4.8)
where k takes the meaning of the k-th natural frequency of the pinned-pinned tensioned string. We
observe that in (4.8) there is no upper limit posed on the value of k, which means that formula (4.8)
provides an infinite sequence of natural frequencies for the continuous system, which is in line with
our previous observation that a continuous system may be seen as the limit case of a n-dof system
where nJ . We will come back soon on this point.
In order to define the solution of motion associated with the k-th natural frequency of the system,
we now replace (4.6) and (4.7) into the general expression of stationary vibration (4.5), and we get:
wk (x ,t ) = Bk sin
k
x cos(
l
kt
+
k
)
(4.9)
which represents the k-th component of the stationary vibration of the string, associated with the
natural angular frequency k. The general motion of the system will be a combination of all the
possible components with expression (4.9), and will thus result into:
w( x ,t ) =
k =1
wk ( x , t ) =
k =1
Bk sin
k
x cos(
l
kt
+
k
)
(4.10)
where the amplitude parameters Bk and the phase parameters k have to be determined by
considering a particular set of initial conditions for the system.
The space dependent part of the k-th vibration component (4.9), that is:
k
(x ) = Bk sin
k
x
l
(4.11)
represents the shape of the vibration, and thus defines the k-th mode of vibration, associated with
the natural angular frequency k. As for the discrete systems, the mode of vibration is defined with
an arbitrary amplitude Bk1, which may be conventionally set to 1, so that the modes of vibration
become sine functions with wavelengths that are entire sub-multiple of a base wavelength equal to
the double of the string span. Figure 4.2 shows the shape of the first modes of vibration of the
transversal vibration for the pinned-pinned string: in the first mode of vibration all the sections are
vibrating in phase, with the largest amplitude of vibration occurring at centre-span, in the second
mode of vibration, the left and right halves of the string are vibrating in counter-phase, with the
maximum amplitude of displacement occurring at ¼ and ¾ of the span length, and with a nodal
section at centre-span. We recall that a nodal point is a point that does not move in a specified mode
of vibration of the system, despite not being subject to a constraint. The third and fourth modes of
vibration introduce a larger number of nodes and portions of the string moving in counter-phase
each other, with all even modes (n.2, n.4 etc.) having a node in the central section. We note by the
way that for increasing frequency, the associated modes of vibration become increasingly
“winding”: this is a quite general rule that applies not only to the problem considered here, but in
general to the vibration of discrete and continuous systems.
1
Actually, this function only defines the shape of the vibration component, not the amplitude that will depend upon the
particular initial conditions imposed to the system
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Dinamica dei Sistemi Meccanici– Prof. Stefano Bruni
Figure 4.2: Shape of the first four modes of vibration for the vibration of a pinned-pinned string
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