Applied Structural and Mechanical Vibrations 2009 Part 5 doc

where G
1
(s) and G
2
(s) are the transforms of g
1
(t) and g
2
(t), respectively. From
a table of transforms we get


so that the convolution integral is zero for t<t
1
, leaving only the first term in
eq (5.83) and

for t>t
1
. The inverse transformation of eq (5.83) finally yields


which, aside from the constant f
0
, are exactly eqs (5.10) and (5.43).
So far, we have not yet considered the possibility of obtaining directly the
final solution satisfying given initial conditions. This is one advantage of
solving linear differential equations with constant coefficients by the Laplace
transform method. By standard methods one finds a general solution
containing arbitrary constants, and further calculations for the values of the

constants are needed to solve a particular problem. The Laplace transforms
of derivatives given in Chapter 2 (eqs (2.39) and (2.40)) will now be used to
clarify this point.
Let us consider the general equation of motion for a damped SDOF system

(5.85)

with initial conditions and The Laplace transformation
of both sides gives

(5.86)

where, as customary, we are using lower-case letters for functions in the
time domain and capital letters for functions in the transformed domain.
Solving for X(s) and rearranging leads to

(5.87)

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The first term on the right-hand side (product of two functions of s)
transforms back to the convolution integral

(5.88a)

the second term transforms back to (see any list of Laplace transforms)

(5.88b)

and we have already considered the third term whose inverse transform is

(5.88c)

The sum of the three expressions (5.88a, b and c) finally gives the general
response


which is, as expected, eq (5.19) and shows explicitly how this method takes
directly into account the initial conditions in the calculation of the response
to external excitation.
5.4 Relationship between time-domain response and
frequency-domain response
From the discussion and the examples of preceding sections it appears that
both h(t), the impulse response function (IRF), and the frequency response
function (FRF) H(
ω
) (or the transfer function H(s)) completely define the
dynamic characteristics of a linear system. This fact suggests that we should
be able to derive one from the other and vice versa. The key connection
between the two domains is established by the convolution theorem and by
the Fourier (or Laplace) transform of the Dirac delta function. In Chapter 2
(eq (2.29a)) we determined that the Fourier transform of the convolution of
two functions g
1
(t) and g
2
(t)—provided that —is the
product of the two transformed functions. With our definition of the Fourier
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transform, as a formula this statement reads

and results, as we have seen, from an application of Fubini’s theorem.
Now, since we know from Section 5.2 that the time-domain response x(t)
of a linear system is given by the convolution (Duhamel’s integral) between
the forcing function f(t) and the system’s IRF h(t), i.e.


we can Fourier transform both sides of this equation to get the input-output
relationship in the frequency domain

(5.89)

Equation (5.89) justifies eq (5.76a) from a more rigorous mathematical point
of view. In fact the two equations (5.76a) and (5.89) are the same if we define

(5.90a)

and

(5.90b)
In this light, note that the functions F(
ω
) and X(
ω
) are the Fourier transforms
of the functions f(t) and x(t), respectively, but the FRF H(
ω
) (eq (5.90a))
differs slightly from the definition of the Fourier transform of the IRF h(t)
(there is no 1/(2π) multiplying factor). This is a consequence of our definition

of the Fourier transform (eqs (2.15) and (2.16)) and of the fact that the
fundamental input-output relationship for linear systems is almost always
found in the form given by eq (5.76a). However, this is only a minor
inconvenience since (Chapter 2) the position of the factor 1/(2π) is optional
so long as it appears in either the Fourier transform equation or the inverse
Fourier transform equation. Hence, the inverse transform equation
corresponding to eq (5.90a) is

(5.91)
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which, conforming to our definition of the inverse transform can be written
No such inconvenience arises in the case of Laplace
transforms because in this case, by virtue of the convolution theorem (eq
(2.43)), the Laplace transform of the Duhamel integral leads to

(5.92)

where we defined X(s) as the Laplace transform of the response x(t).
With the above general developments in mind, we may now recall that,
for a linear system, the function h(t) represents the system’s response to a
delta input, i.e. x(t)=h(t) when Consequently, since
(eq (2.74)), and eqs (5.89) and (5.92) give respectively

(5.93)

which tell us that, in the case of δ-excitation, the Fourier (Laplace) transform
of the system’s response is precisely its frequency response (transfer) function,
a circumstance which is also exploited in experimental practice.
With the definitions above, it is now not difficult to verify eq (5.76a)
(and hence eq (5.93)) for the case of, say, a viscously damped SDOF system.

In fact, in this case we know from the second of eqs (5.7a) that


so that, with the aid of a table of integrals from which we get


we can calculate the term


by noting that, in our specific case and
It is then left to the reader to show that the actual calculation leads to

(5.94)
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where, as usual, If we multiply this result by
we obtain explicitly the right-hand side of eq (5.76a) for our viscously damped
SDOF system. Then, the left hand-side can be obtained by virtue of eq (5.5)
and it can be determined immediately that, as expected,


In the light of these considerations we can write the response of a system
to an input f(t), whose Fourier and Laplace transforms are F(
ω
) and F(s), as

(5.95)

or, when more convenient, as the inverse Laplace transform of the product
H(s)F(s). Thus the following three equivalent definitions of the FRF H(
ω

)
can be given:

1. H(
ω
) is 2π times the Fourier transform of h(t).
2. For a sinusoidal input, i.e. is the coefficient of the
resulting sinusoidal response
3. Provided that in the frequency range of interest, H(
ω
) equals
the ratio where X(
ω
) is the Fourier transform of x(t).

Figure 5.8. is a frequently found schematic representation of the fact that
the dynamic characteristics of a ‘single-input single-output’ system are fully
defined by either h(t) or H(
ω
)
A final note of interest concerns two systems connected in cascade as shown
in Fig. 5.9. Denoting by h(t) and H(
ω
) the IRFs and FRFs of the combined
Fig. 5.8 Symbolic representation of a linear system.
Fig. 5.9 Systems in cascade.
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system, by and H
1
(

ω
) and H
2
(
ω
) the relevant functions of the
two subsystems we have

(5.96a)

and

(5.96b)
5.5 Distributed parameters: generalized SDOF systems
Up to the present we have always considered the simplest type of SDOF
system, i.e. a system where all the parameters of interest—mass, damping
and elasticity—are represented by discrete localized elements. This is, of
course, an idealized view. However, even when more complicated modelling
is required for the case under study, we have to remember that the key aspect
of SDOF systems is that only one generalized coordinate is sufficient to
describe their motion; if this characteristic is maintained it can be shown
that the equation of motion of our SDOF system, no matter how complex,
can always be written in the form

(5.97)

where z(t) is the single generalized coordinate and the symbols with asterisks
(not to be confused with complex conjugation) represent generalized physical
properties—the generalized parameters—of our system with respect to the
coordinate z(t). This latter statement means that a different choice of this

coordinate leads to different values of the generalized parameters. The possibility
of writing an equation such as (5.97) allows us to extend all the considerations
that we have made so far to a broader class of systems: assemblages of rigid
bodies with localized spring elements, systems with distributed mass, damping
and elasticity (bars, plates, etc.) or combinations thereof can be analysed in
this way once the relevant generalized parameters have been determined. Their
values can be obtained, in general, from energy principles such as Hamilton’s
or the principle of virtual displacements (Chapter 3) but general standardized
forms of these expressions can be given for practical use.
It is important to note that the SDOF behaviour of the system under
investigation may sometimes correspond very closely to the real situation
but, more often, is merely an assumption based on the consideration that
only a single vibration pattern (or deflected shape in case of continuous
systems) is developed. For example, a beam that deforms in flexure is, as a
matter of fact, a system with an infinite number of degrees of freedom, but
in certain circumstances a SDOF analysis can be good and accurate enough
for all practical purposes.
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For continuous systems in particular, the success of this procedure—which
is a particular case of the assumed modes method (Chapter 9)—depends on
the validity of the assumption above and on an appropriate choice of a
shape (or trial) function (x) which, in turn, depends on the physical
characteristics of the system and also on the type of loading. Ideally, the
selected shape function should satisfy all the boundary conditions of the
problem. At a minimum, it should satisfy the essential boundary conditions.
In this light, a few definitions are given here and then some examples will
clarify the considerations above.

•An essential (or geometric) boundary condition is a specified condition
placed on displacements or slopes on the boundary of a physical body

(e.g. at the clamped end of a cantilever bar both displacement and slope
must be zero).
•A natural (or force) boundary condition is a condition on bending
moment and shear (e.g. at the free end of a cantilever bar the bending
moment and the shear force must be zero).
•A comparison function is a function of the space coordinate(s) satisfying
all the boundary conditions—essential and natural—of the problem at
hand, plus appropriate conditions of continuity up to an appropriate
order.
•An admissible function is a function that satisfies the essential boundary
conditions and is continuous with its derivatives up to an appropriate
order. For a specific problem, the class of comparison functions is a
subset of the class of admissible functions.
•An assumed mode (or shape function) is a comparison or an admissible
admissible function used to approximate the deformation of a continuous
body.

Example 5.8. Let us consider the rigid bar of length L metres and mass m
kilograms shown in Fig. 5.10. The angle
θ
of rotation about the hinge, where
at static equilibrium, can be chosen as the generalized coordinate. The
vertical displacement z(t) of the tip of the bar can be another choice; for
small oscillations as shown in the figure.
Since the bar is considered rigid, the system has distributed mass (along
the length of the bar), localized stiffness, damping (the spring and the dashpot)
and is subjected to a localized force f(t).
For small oscillations about the equilibrium, we assume the shape function
The virtual displacement is then given by and
from the principle of virtual displacements it is not difficult to obtain

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Our method assumes that only one mode is developed during the motion
and represents the deflected shape u(x, t) of the beam as the product

(5.100)

where (x) is the chosen admissible function and z(t) is the unknown
generalized coordinate. The principle of virtual displacements considers all
the forces that do work and reads

(5.101)

where from the definition of potential energy V. It will be
shown in later chapters that the strain potential energy of a beam undergoing
a transverse deflection u(x, t) is given by

(5.102)

where we indicated for simplicity of notation It follows
from eq (5.102)

(5.103)

since
For the inertia forces we get

(5.104)

since and
Fig. 5.11 Schematized beam on elastic foundation.

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Similarly, for damping forces we get

(5.105)

and for the distributed spring and external forces

(5.106)
(5.107)

Addition of the various terms leads to

where all the integrals are taken between 0 and L. The virtual displacement
δ
z
is arbitrary and therefore can be cancelled out, leaving the equation of motion
of our system in the form of eq (5.97) where, after rearranging, the generalized
parameters are given by

(5.108a)
(5.108b)
(5.108c)
(5.108d)

It is now evident how the particular choice of the shape function affects
the generalized parameters of the system. Moreover, it is also clear that the
application of Hamilton’s principle leads the same expressions of eqs
(5.108a–d).
The most general case of the type shown above consists of a system which
is a combination of distributed and localized masses, springs, dampers and

external forces. Again, the displacement is assumed of the form
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where z(t) is the unknown generalized coordinate and the
following standardized expressions for the generalized parameters can be
given:

(5.109)

(5.110)

(5.111)

(5.112)

where the integrals take into account the distributed parameters of the system
under investigation (the symbols with a caret (^) are intended per unit length)
and the summations take into account the discrete elements. For example,
the contribution to the generalized damping of a localized dashpot of constant
c
1
at a distance x
1
from the origin of the axes is given by in the
summation of eq (5.110). With regard to the generalized mass, the second
summation accounts for the rotation effects of localized rigid-body masses:
I
0j
is the mass moment of inertia of the jth mass and the first derivative of
at the point x
j

represents the rotation at that point. Referring back to the
example of Fig. 5.10, it is not difficult to see that the generalized parameters
of eqs (5.99a–d) are particular cases of eqs (5.109)–(5.112) where the assumed
motion was and the connection to the general case is given by


and the generalized mass accounts for translational motion of the centre of
mass (at x=L/2) and the rotational motion around the centre of mass (mass
moment of inertia ).
Care must be taken in the calculation of the generalized stiffness when
‘destabilizing’ forces are acting, since they add a further contribution to eq
(5.111). Destabilizing forces may arise in different situations and may be of
various nature. Gravity, for example, can be such a force in the case of an
inverted pendulum, where a mass M is mounted on the tip of a light rigid
bar as in Fig. 5.12.
The reader is invited to determine that the effective stiffness of the system
can be written as k–k
G
where k
G
depends on the weight Mg. When
the system becomes unstable.
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we obtain

(5.113)

The virtual work done by P(x) is finally obtained as

(5.114)

The term of eq (5.114) must now be added to eq (5.101) to give a generalized
stiffness in the form where k* is as before and

(5.115)

In the case of a simple beam under the action of the axial compressive load
only—if P does not depend on x and can be taken out of the integral—we can
obtain the critical buckling load from the condition as

(5.116)

which is, obviously, relative to the assumed shape (x).
As the simple examples above show, and as the word itself implies,
destabilizing forces lead to stability problems. Stability is a broad subject in
its own right and extends outside our scope. Some of its basic aspects will be
considered when and if appropriate in the course of the book. For the moment,
it suffices to say that stability is in general connected to situations in which
the physical parameters (m, k or c or their generalized counterparts; one or
more of them) become negative. The motion is not well-behaved in these
cases and may diverge, i.e. increase without bounds, with or without
oscillating. An example of diverging motion, even if no destabilizing forces
are active, is the undamped oscillator excited at resonance.
One final word to point out that the assumed mode procedure can be
extended to more complicated elements. For example, if the element
undergoing flexure is two dimensional—i.e. a rectangular membrane or a
plate—we can assume the displacement of the centre as the generalized
coordinate z(t) and write
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where, again, (x, y) is a reasonable shape function consistent with the

boundary conditions. However, a good choice of the shape function becomes
more and more difficult as the number of dimensions increases and, as a
consequence, the method may lead to unreliable results.
5.5.1 Rayleigh (energy) method and the improved Rayleigh
method
Often, in practical situations, the quantity of main concern is the fundamental
frequency of vibration of a given structural or mechanical system. When the
system is complex, the exact determination of such a quantity may not be
an easy task, long computation time and difficult calculations being involved
in the process. The basis of a class of approximate methods to obtain the
needed result is the so-called Rayleigh’s method.
When an undamped elastic system vibrates at its fundamental frequency,
each part of the system executes simple harmonic motion about its equilibrium
position. The principle of conservation of energy applies for such a system
and during the motion two extreme situations occur:

• All the energy is in the form of potential strain energy at maximum
displacement.
• All the energy is in the form of kinetic energy when the system passes
through its equilibrium position (maximum velocity).

Conservation of total energy requires that the potential energy at maximum
displacement must equal the kinetic energy at maximum velocity, i.e.

(5.117)

The Rayleigh method calculates these maximum values, equates them and
solves for frequency, since this quantity always appears in the kinetic energy
term as a consequence of the simple harmonic motion of the system.
The undamped harmonic oscillator of Fig. 4.2 is the simplest example:

the motion of such a system can be written as (Chapter 4)


the potential and kinetic energies are then
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Their maximum values are


Equation (5.117) follows because, individually, the two terms above must
equal the total energy. Solving for
ω
n
we get the well-known result

(5.118)

Another example can be the beam of Fig. 5.10 in free vibration; we make
it conservative by considering c=0 and f(t)=0: no energy is removed from the
system and no energy is fed into it. We assume as before and a
harmonic motion of the generalized coordinate given by The
maximum values of the potential and kinetic energies are now
equating the two energies and solving for frequency gives

(5.119)

where the generalized stiffness and mass are the same as in eqs (5.99a) and
(5.99c).
It is clear at this point that Rayleigh’s method is strictly related to the
assumed modes method (which is used to obtain the equation of motion),
and the generality of the symbolic relation —often referred to

as the Rayleigh quotient—can be appreciated.
Further generalization will be given in later chapters when appropriate.
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As a last example of SDOF system we consider a simple cantilever beam
(i.e. a beam that is clamped at one end and free at the other) that undergoes
flexural vibrations without energy loss during its motion. We assume the x-
axis in the horizontal direction.
The assumption characterizes the SDOF behaviour of
this system and, again, characterizes the harmonic time
dependence of this motion. The maximum potential and kinetic energies are

(5.120)

(5.121)

respectively, where EI is the flexural rigidity, is the mass per unit length.
Equating and solving for the frequency gives

(5.122)
At this point it is interesting to test the effect of different choices for (x)
on the calculated frequency. Since the exact deformation shape can only be
obtained by solving the equation of motion (but in this case the value of the
fundamental frequency would be determined also) and therefore it is not
known, we will try three trial functions

(5.123)
(5.124)
(5.125)

All of these functions give zero displacement at the clamped end (x=0)

and unit displacement at the free end (x=L) of the cantilever, the rationale
being as follows:
1
is the simplest function one can think of in the given
situation,
2
is a reasonable sinusoidal function and
3
is the static deflection
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curve of a cantilever beam under uniform load. In all of the cases above we
can calculate the Rayleigh quotient and obtain an approximate value for the
fundamental frequency of our system.
After some calculation that the reader is invited to try, we get the following
results:

(5.126)
(5.127)
(5.128)

respectively, where the superscript in parentheses refers to the chosen trial
function. It will be determined in Chapter 8 that the exact fundamental
frequency of the system we are considering is

(5.129)

The first consideration is that all of the trial functions produce a result
that overestimates the exact value; this is a fundamental characteristic of the
Rayleigh quotient and will be proven rigorously on a mathematical basis.
On physical grounds, one can observe that additional constraints must be

applied to the system if it is forced to vibrate in a shape that is different from
its natural one; these constraints add stiffness to the system and hence an
increase in frequency. Obviously, if the exact shape function (the lowest order
eigenfunction) is used for (x), the result is eq (5.129).
In addition, we note that the degree of approximation is rather crude
(27% high) for the first function, but satisfactory for the other two.
Qualitatively, we can say that the more the trial function resembles the true
deflection shape, the more accurate the result will be. A closer examination
requires the analysis of the boundary conditions. For the cantilever beam
the following boundary conditions must be satisfied:

1. zero displacement and slope at the clamped end (x=0), i.e.
(5.130)
which we recognize as essential boundary conditions;
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2. zero bending moment and shear force at the free end (x=L), i.e.

(5.131)

which we recognize as natural boundary conditions.

All the trial functions satisfy the conditions of eqs (5.130) but only
3
(x)
satisfies all four;
1
(x) does not satisfy the first of eqs (5.131) and
2
(x) does
not satisfy the second of eqs (5.131).

In general, the deflection produced by a static load is a good candidate
for Ψ(x) because it automatically satisfies all the necessary boundary
conditions and simplifies the calculation of the potential strain energy that
can be obtained as the work done by the static load to produce the desired
deflection. Only the function (x) appears in this calculation and not its
second derivative.
A common assumption is to choose the deflection shape that results from
the application of the gravity load due to the mass of the structure. In this
case, the direction of gravity must be chosen to match the probable
deformation shape: in the analysis of the free vibrations of a vertical cantilever
for example, the direction of gravity must be horizontal if we are interested
in lateral motions of the structure. Obviously, this does not correspond to
anything real, it is just a useful expedient.
There are two are the reasons that justify the assumption above:

1. It is not necessary to spend much time in the choice of an assumed
shape because any reasonable function compatible with the essential
boundary conditions leads to acceptable results. It will be shown in
Chapter 9 that the error on the calculated frequency is of the order of
ε
2
, if
ε
is the error of the assumed shape with respect to the exact one.
2. The displacements in free vibration result, as a matter of fact, from the
application of inertia forces and these forces depend, in their turn, on
the mass distribution in our system.

This latter consideration, together with the serious disadvantage that the
method does not allow us to estimate e if the exact (x) is not known, leads

to the improved Rayleigh method, whose line of reasoning is as follows.
Suppose that the true deflection (x) is the same as the deflection produced
by an external load f(x). Then, deflection is produced by a
load zf(x) and the potential strain energy is the work done by this force to
give the displacement z , i.e.
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If, as before, we get


Equating to the maximum kinetic energy of eq (5.121) gives

(5.132)

which states that the load of eq (5.132), where we recognize inertia forces,
produces the exact vibration shape. Equation (5.132) is true if (x) is the
true shape. Our assumed shape, which we call now
0
(x), is probably
different from the true one and hence the load will produce a
shape different from
0
(x), let us call it This function cannot be
calculated because of the unknown factor, but intuition suggests that it
is likely to be a better approximation than
0
for the true deflected shape.
Nevertheless, the function (not to be confused with the
function of eqs (5.123) which was a particular
0
for the cantilever problem)

can be obtained from

(5.133)

and we can write the maximum potential energy as

(5.134)

Equating to the maximum kinetic energy gives

(5.135)

At this point it seems reasonable to proceed further and use
1
also for the
kinetic energy, so that

(5.136)
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Equating to E
p,max
of eq (5.134) gives now

(5.137)

Further iteration—that is, the use of
1
to obtain an even better
approximate function
2

and use the latter to calculate the frequency—is
generally not worth it.
We still do not have an estimate for the error
ε
, but indirectly we can
have an idea by looking at the difference between the frequencies obtained
from eq (5.122) and eq (5.135) or (5.137). If this difference is large, the
function
0
is not a very good approximation for the true deflected shape
and
ε
is large as well; if it is small,
ε
is small as well and
0
is a good choice.
Now, going back to the cantilever problem, we show an application of
the improved Rayleigh method. We start from the function —
which produced the result of eq (5.126)—and use the inertia forces to
calculate a better deflected shape. We know from beam theory that

where
1
is the shape that results from the application of the forces on the
right-hand side. By integrating four times and calculating the constants of
integration from the boundary conditions of eqs (5.130) and (5.131) we get
(5.138)

which can be used in eq. (5.135) to obtain

(5.139)

or in eq (5.137) to get

(5.140)
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These values are much better estimates of the exact frequency (given by
eq (5.129)) and the large difference between the frequencies obtained from
eq (5.126) and (5.139)—26.7% with respect to the lower value—indirectly
suggests that the assumed
0
was not a good approximation for the true
deflected shape and hence the relative error on the frequency must have
been large as well.
5.6 Summary and comments
Chapter 5 continues the discussion on SDOF systems. When the excitation
is not a simple sinusoidal function, the response of the system can be obtained
by means of various techniques, which obviously apply to harmonic excitation
as well. The main distinction is between time-domain and frequency-domain
techniques.
If the functions involved are analysed in the time domain, a fundamental
concept is the impulse response function h(t), whose convolution with the
forcing exciting function provides the time response of our SDOF system.
This particular form of convolution is known as Duhamel integral, which,
in turn, can be visualized as a sum of the input excitation ‘weighted’ by an
appropriately shifted form of the impulse response h(t). As far as dynamic
aspects are concerned, the function h(t) is an inherent property of the system
and characterizes it completely.
In this light, the response to the frequently encountered situation of

loadings of short duration that may release a considerable amount of energy
can be considered. One generally speaks of transient or shock loading,
depending on a comparison between the time duration of the input load and
the system’s period of oscillation The ratio between these two latter
quantities is the natural abscissa axis for the representation of shock spectra,
where the maximum response of the system is plotted on the ordinate axis
without regard to the entire time history of the event. Shock spectra are
obtained considering an undamped SDOF system as a standard reference
and are widely used for design and comparison purposes in order to assess
the potential disruptive effects of various forms of shock.
Another class of loadings is given by periodic (i.e. with a repetitive pattern
in time) functions. Fourier’s theorem states that a general periodic (and well-
behaved) signal is the superposition of an infinite number of simple sinusoidal
functions with frequencies that are all integral multiples of a value
ω
0
. It
follows that its mathematical form is a convergent Fourier series of such
functions and, owing to the principle of superposition, the response of a
linear system is a similar Fourier series as well. Amplitudes and phases are
modified between input and output (excitation and response), but it is not so
for the frequency content and even if, rigorously, an infinite number of terms
appear in the mathematical representation of the above series, a finite and
limited number of terms often suffices for all practical purposes.
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The generalization of Fourier series to nonperiodic signals leads to the
Fourier and Laplace transformation integrals, which constitute the basis of
the frequency-domain approach. Besides the fact that they often allow a
simplification of the mathematics required to solve specific problems, their
importance cannot be overstated and the fundamental concepts of frequency

response function H(
ω
) and transfer function H(s) of a linear system follow
directly from their application.
These latter functions play a crucial role in almost every aspect of linear
vibration analysis. They completely characterize a linear system in the
frequency domain and—as h(t) in the time domain—they are inherent
properties of the system under study. Given these similarities, logic dictates
that it must be possible to obtain H(
ω
)—or H(s)—from h(t) and vice versa.
The connection is the Fourier (or Laplace) transform: h(t) and H(
ω
) are a
Fourier transform pair and, likewise, h(t) and H(s) are a Laplace transform
pair.
Unfortunately, both the time-domain and the frequency-domain
approaches often lead to integral expressions which cannot be evaluated
analytically and, therefore, recourse must then be made to computer
calculations on ‘sampled’ versions of the original signals. This sampling
process is not harmless and its effects will be considered in Part II of the
book which deals with electronic instrumentation. It is, however, important
to point out right away that some care must be exercised in these cases if we
do not want to run into undesirable consequences.
The last part of the chapter shows how, in some circumstances, an SDOF
analysis can be extended to a wide class of more complex systems when
some basic assumptions on the system’s behaviour can be made or when the
needed results can be accepted with a reasonable degree of approximation.
The concept of generalized parameters is introduced in order to obtain a
SDOF equation of motion or to calculate an approximate value of the

fundamental frequency by means of the Rayleigh energy method. The
assumption that only one vibration pattern (or ‘shape’) is developed during
the motion is particularly useful in an approximate examination of continuous
systems, where the static deflection under an appropriate load—often their
own weight—is a good choice for the assumed vibration shape in most cases.
The assumed shape then satisfies automatically the essential and natural
boundary conditions of the problem. Nevertheless, simpler shapes satisfying
only the essential boundary conditions can be chosen if great accuracy is not
needed. Needless to say, the exact value of frequency is obtained if one has
the luck (or the physical insight) to choose the correct deformed shape.
The improved Rayleigh method provides more accuracy in the calculation
of the fundamental frequency—which is always overestimated when the
assumed shape is not correct—and allows an indirect qualitative evaluation
of the error with respect to the (unknown) exact value by looking at the
improvement of the first one or two iterations involved in the process. Further
iterations are, in general, not needed.
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References
1. Cooley, J.W. and Tukey, J.W., An algorithm for the machine calculation of complex
Fourier series, Mathematics of Computation, 19, 297–301, 1965.
2. Harris, C.M. (ed.), Shock and Vibration Handbook, 3rd edn, McGraw-Hill,
New York, 1988.
3. Jacobsen, L.S. and Ayre, R.S., Engineering Vibrations, McGraw-Hill, New York,
1958.
4. Erdélyi, A., Magnus, W., Oberhettingher, F. and Tricomi, F.G., Tables of Integral
Transforms, 2 vols, McGraw-Hill, New York, 1953.
5. Thomson, W.T., Laplace Transformation, 2nd edn, Prentice Hall, Englewood
Cliffs, NJ, 1960.
6. Graff, K.F., Wave Motion in Elastic Solids, Dover, New York, 1991.
7. Inman, D.J., Engineering Vibrations, Prentice Hall, Englewood Cliffs, NJ, 1994.

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6 Multiple-degree-of-freedom
systems
6.1 Introduction
In one way or another, almost any vibrating system could be modelled and
represented as an SDOF system. In general, the quality of this assumption
depends on the type of system and excitation, on the accuracy required and
on the scope of the investigation. In some circumstances the assumption is
correct; in some other cases it may lead to a description of the system’s
dynamic behaviour within an acceptable degree of accuracy but, in many
other cases, the assumption is just an extreme oversimplification leading to
inaccurate results which have almost nothing to do with the real situation.
Since, a priori, the true behaviour of a real system is in general not known,
the assessment of the validity of the results obtained from a SDOF analysis
may not be an easy task. Therefore, in order to obtain a meaningful
description for a wide class of systems, more complex representations are
needed from the outset.
When one coordinate is not sufficient to characterize the motion of a
given system, one speaks rightfully of two-, three-,…n- or, in general, multiple-
degree-of-freedom (MDOF) systems; where the number refers to the
independent coordinates necessary to describe completely the vibration
phenomenon.
The SDOF model enables us to explain—without particular mathematical
difficulties—many fundamental concepts such as free and forced vibrations,
natural frequency and resonance. Broadly speaking, all of these concepts
can be extended to MDOF models. However, some important differences
will appear; in anticipation we can say that the natural vibration of an MDOF
system may occur at a number of different frequencies. Each one of them
corresponds to a particular pattern (or ‘shape’ to give a pictorial view) of
the system’s motion and these different configurations, known as natural or

normal modes of vibration, play a crucial role in almost every aspect of
further analysis.
As discussed in Chapter 3, a set of n simultaneous ordinary differential
equations of motion—one for each degree of freedom—must now be obtained
in order to mathematically describe our system and a proper choice of the
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generalized coordinates is one of the key points of the whole process. In
fact, the theory of finite-dimensional vector spaces and the closely related
subject of linear matrix algebra are the most convenient mathematical tools
required to deal with MDOF systems: it is well-known that the choice of a
basis in a vector space determines the particular matrix representation of
operators and vectors and may considerably simplify the problem at hand.
6.2 A simple undamped 2-DOF system: free vibration
As a starting point, let us consider the simple system of Fig. 6.1. It consists
of two masses M and m connected by a spring k
2
, with the mass M suspended
from a fixed point by a spring k
1
. We assume that this system is constrained
to move only in the vertical direction; in this case the coordinates x
1
and x
2
that specify the position of the two masses at any instant are sufficient to
describe the motion completely and we have a 2-DOF system.
If we take the position of static equilibrium as a reference, the terms
representing the weights of the masses cancel out with the spring tensions at
equilibrium, exactly as stated in Section 4.2.1; as the masses move at x
1

and
x
2
, respectively, the tension in the lower spring is The (coupled)
equations of motion are
(6.1)


Fig. 6.1 Simple 2-DOF system.
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as can also be determined by using Lagrange’s equations. Let us consider
now if such a system can vibrate so that all the masses move with the same
frequency and, if this is possible, at how many different frequencies
this vibration can occur. Let us suppose further that, for example, M=3m,
k
2
=k and k
1
=4k. Equations (6.1) become
(6.2)
By analogy with the behaviour of SDOF systems, we look for harmonic
solutions. Using for example the sinusoidal notation we write
(6.3)
where A, B and
ω
are constants.
Substituting eqs. (6.3) in eqs. (6.2) leads to
(6.4)
We note in passing that the same phase angle is used because, were it not so,
only the trivial case A=0 and B=0 (i.e. no motion at all) would be a solution;

the case in which the phase angles differ by
π
is equivalent to a change of the
sign of B, which is, as yet, still arbitrary.
A nontrivial solution of eqs (6.4) exists if the determinant
vanishes. For reasons that will become clear in the following sections, the
problem of determining the values of frequency for which eqs (6.4) admit
nontrivial solutions is called the eigenvalue (or characteristic value) problem.
Equating to zero the determinant above leads to the frequency equation
(6.5)
with solutions
(6.6)
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