1. UNIVERSITÀ DEGLI STUDI DI NAPOLI “FEDERICO II”
SCUOLA POLITECNICA E DELLE SCIENZE DI BASE
DIPARTIMENTO DI INGEGNERIA INDUSTRIALE
TESI DI LAUREA MAGISTRALE IN INGEGNERIA
AEROSPAZIALE
DESIGN AND CHARACTERIZATION OF A NONLINEAR
BEAM WITH CHANGEABLE NONLINEARITY
Relatore Candidato
Ch.mo Prof. Ing. Francesco Marulo Piero Rendina
Matr. M53/384
Correlatore
Dott. Andrea Cammarano
Anno Accademico 2015/2016

2. To those who always believed in me.

3. Contents
Abstract………………………………………………………………………..1
1. Introduction to nonlinear vibrations
1.1 Introduction to vibrations in flexible structures……………………………4
1.2 Causes of nonlinear vibrations……………………………………………..6
1.2.1 Material properties…………………………………………………...6
1.2.2 External forces, freeplay, impact and friction………………………...7
1.2.3 Geometric nonlinearity………………………………………………9
1.3 Linear and nonlinear vibrations modelled using sine waves………………10
1.4 Multiple degrees-of-freedom……………………………………………..16
2. Nonlinear vibration phenomena
2.1 Stata space analysis……………………………………………………….19
2.2 Equilibrium points………………………………………………………..22
2.2.1 Equilibrium points for a two-state linear harmonic oscillator……….24
2.2.2 Potential functions………………………………………………….29
2.3 Periodic and non-periodic oscillations: limit cycles………………………32
2.4 Bifurcations………………………………………………………………35
2.5 Bifurcations in forced nonlinear oscillations……………………………...46
3. Theory background and design of the experimental rig
3.1 Backbone curves………………………………………………………….50
3.2 Harmonic Balance………………………………………………………...52
3.3 Quick review of modal analysis theory…………………………………...54

4. 3.4 Beams…………………………………………………………………….56
3.5 Project and design of the experimental rig………………………………...60
4. Experimental rig: horizontal springs solution
4.1 Introduction……………………………………………………………....70
4.2 Mathematical model……………………………………………………...71
4.3 Backbone curve: analytical and experimental computation………………76
4.4 Conclusions………………………………………………………………79
5. Experimental rig: nonlinearity obtained through magnetic force
5.1 Introduction………………………………………………………………82
5.2 Magnetic model…………………………………………………………..83
5.3 State space analysis of the system………………………………………...86
5.4 Hardening case……………………………………………………………90
5.5 Bi-stable case……………………………………………………………..97
5.6 Frequency response……………………………………………………...101
5.6.1 Hardening case………………………………………………………104
5.6.2 Bi-stable case………………………………………………………..105
5.7 Conclusions……………………………………………………………..107

6. 1
Abstract
With many engineering system becoming increasingly lightweight and flexible,
the ability to model the behaviour resulting from nonlinear characteristics is
becoming increasingly important. A very important role in the study of
nonlinear systems is played by the analytical approach, together with an
experimental approach. This is because an analytical description of the problem
can offer a great insight into the mechanisms behind the dynamical behaviour,
and can provide excellent suggestions for procedures such as optimization.
There are many analytical techniques used to describe the responses of
unforced, undamped nonlinear systems, one of these is using the so-called
backbone curves. These curves can be used to understand the underlying
dynamics of the nonlinear systems subjected to forcing and damping: the
backbone curves can be used to extract information about the shape and the
location of the peaks in the frequency response. After a quick overview of the
nonlinear phenomena and their causes, this work focus on the project of a
system, which exhibit a nonlinear dynamic behaviour. On this system, some
free response tests and some frequency response tests will be conducted, in
order to experimentally characterize its dynamics. The backbone curves and the
frequency response will be obtained, and from these information on the type of
nonlinearity will be derived. When a system exhibits a nonlinear behaviour, the
response of the system to a periodic forcing with variable frequency changes
depending on the fact that the frequency increases, or the fact that it decreases.
This phenomenon is called hysteresis loop, where hysteresis in this contest
means that the system responds differently depending on the forcing frequency
increasing or decreasing. The idea developed in this work is to realize a linear
structure and to introduce the nonlinearity in the system. Two different ways to
introduce nonlinearity in the system have been thought. Only one of the two
approaches led to consistent results. The structure realized is a cantilever beam,
with a frame at the free end used to connect the features that introduce the

7. 2
nonlinearity in the system. In the first approach, the nonlinearity is introduced
by some horizontal springs, but the results are not satisfactory, because the
nonlinearity is very weak. In the second approach the nonlinearity is introduced
by an external force, a magnetic force, generated by some magnets placed on
the frame and on the beam. In this case, the nonlinearity is more consistent, and
the results are satisfying: the system exhibit a nonlinear hardening behaviour.
Furthermore, the distance between the magnets can be changed, and so the
intensity of the magnetic force, and therefore of the nonlinear contribution can
be increased or decreased. Even if the system considered in this work is a simple
cantilever beam, the phenomena experienced by the structure are the same
experienced by several engineering structures in their operative life. To provide
and example, nonlinear vibrations occur in presence of collapsing or imperfect
constraints. The approach used in this work is experimental. Because of the
nature of the system analyzed, we focused on the backbone curve related to the
first mode shape, because for this mode shape the nonlinear effect is more
consistent. The measurement system consists in a piezoelectric accelerometer,
connected to a data acquisition system, which communicates with a script
realized in Matlab®. The excitation system is composed by a signal generator,
which generates the excitation function, an amplifier, which amplifies the
output of the signal generator by increasing the current intensity, and a shaker,
which excites the beam. The results have been post-processed with Matlab®, in
order to obtain the backbone curves and the frequency response diagrams.
Before to introduce the nonlinearity, an experimental characterization of the
beam has been made, computing the first natural frequencies of the structure,
and the results have been compared with the results of a numerical simulation
with Matlab®, and with the results of a finite elements method simulation.

8. 3
CHAPTER 1
INTRODUCTION TO NONLINEAR VIBRATIONS
SUMMARY:
1.3 Introduction to vibrations in flexible structures
1.4 Causes of nonlinear vibrations
1.2.1 Material properties
1.2.2 External forces, freeplay, impact and friction
1.2.3 Geometric nonlinearity
1.3 Linear and nonlinear vibrations modelled using sine waves
1.4 Multiple degrees-of-freedom

9. 4
1.1 Introduction to vibrations in flexible structure
Because of the increasing of the demand for ever lighter structures which
preserve the same level of safety and which in some cases guarantee a wider
range of operation regions, the interest of the scientific community towards the
dynamics of nonlinear structures is growing faster. When the structure becomes
lighter, large deflections must be considered, and the theory commonly used to
study the dynamics of structures, based on the assumption of linear behaviour,
is no more applicable. As a consequence of this lightening these structures
usually are highly flexible and, because they are typically required to operate in
a dynamic environment, they can exhibit a nonlinear vibration behaviour related
to several factors, which naturally arise in flexible structural dynamics.
Vibrations occur in a wide range of structural and mechanical systems when the
system is shaken or suddenly disturbed by an external force. The typical form
of a vibration response is that one of a series of cyclic movements, named
oscillations, the most common of which are in the form of approximate sine
waves. For vibration to occur, the structure needs to have a restoring force,
which returns the structure towards its resting position when disturbed by an
external force, so that the structure behaves like a flexible body. Flexibility is a
common property of several structural elements like beams, rods, plates, shells,
membranes and so on, very commonly used in many fields of engineering.
Flexible structures also own two properties, which are very important for
vibrations, i.e. a mass distribution, which provides the inertia forces when the
structure vibrates, and some material damping, which has the effect of reducing
the magnitude of oscillations. The physical damping is usually very difficult to
model, but for most structural and mechanical systems is very small. So,
considering the forces involved in a vibration problem, the governing equations
of motion can be written in this form:
FI + FD + FR = FE (1.1)

10. 5
The terms in the equation above can be considered vectors corresponding to
forces applied to a discrete number of points on the structure. For linear
vibration problems the inertia forces are represented by FI=M𝒙̈ , the dissipation
forces are represented by FD=C𝒙̇, while the restoring forces are given by
FR=K𝒙. The matrices M, C and K are three N x N matrices (where N is the
number of degrees of freedom in the structural model) named respectively
Mass, Damping and Stiffness matrices. Substituting these expressions into
equation (1.1) gives
M𝒙̈ + C𝒙̇ + K𝒙 = FE (1.2)
which is the fundamental governing equation for linear vibrations of structural
and mechanical systems, where FE is the dynamic forcing vector. Among the
various types of external forces that can be applied, there is a continuous single
frequency sine wave, the harmonic forcing. The response of the system to this
kind of forcing can be divided into two parts: an initial transient response and,
after some length of time, the steady-state response. The length of time
necessary to reach the steady-state response depends on many factors, but if the
forcing is harmonic and the restoring force is linear-elastic, the steady-state
vibrations will be harmonic as well, with the same frequency as the forcing. If
the forcing is not harmonic but periodic, this means that it has a repeating
pattern as the harmonic one but it is not necessary limited to a single frequency
sine wave, and so the response will contain multiple components each with
different frequency. Now, nonlinear vibration problems are those that cannot be
modelled by eq. (1.2), for different reasons that will be discussed further.

11. 6
1.2 Causes of nonlinear vibrations
In this section some causes of non-linearity will be discussed. There are several
physical phenomena which lead to nonlinear vibration problems. We will focus
more on the geometric nonlinearity, and on the nonlinearity caused by external
forces, but a description of the other causes of non-linearity will be treated as
well.
1.2.1 Material properties
The constitutive relationships for any material are usually expressed as stress-
strain or force-displacement relationships, and are typically nonlinear. A linear
relationship exists only up to an elastic limit, named limit of proportionality. Up
to this limit we can say, for example for the stress-strain relationship, that 𝐄 =
𝛔/𝛆, where E is the elastic modulus (or Young modulus) of the material. In this
elastic region the stress can be considered proportional to the strain, and the
force proportional to the displacement (Hooke’s law). Beyond elastic limit
metals typically yield and there is a region of non-elastic behaviour, in which
the material exhibit a plastic behaviour: usually in this zone the unloading path
is not the same as the loading path, and in some cases unloading in the non-
elastic region leads to a switch back into linear behaviour but with some residual
strain. This behaviour is known as material hysteresis.
Fig. 1.1 Stress-strain relationship for an axially loaded aluminium rod. (Wagg-Neild).

12. 7
The loading and unloading curves form a loop called hysteresis loop. When
cyclically loaded and unloaded, some materials can exhibit a progressive
reduction in the maximum point on the loading curve. This type of behaviour
can occur during the progressive failure of a material caused by the cyclic
loading and is called cyclic softening. Some materials can show opposite
behaviour, in this case called cyclic hardening. These behaviours are dependent
on the rate at which the loading is applied, and this rate dependence is very
significant for nonlinear vibration problems. The simplest rate-dependent
behaviour is where the force is proportional to the velocity, i.e.
𝑓𝑜𝑟𝑐𝑒 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 ∙ 𝑥̇, often used to model the viscous damping, as in the
equation (1.2). Sometimes some viscous behaviours are more accurately
modelled using nonlinear relationship, as in some acoustic damping
applications, where the viscous damping is described using a nonlinear
relationship like force=constant∙(𝑥̇)n
, where typical values of n are 2 ≤ 𝑛 ≤ 3.
Other factors, which have a significant effect on material behaviour, include
whether the material behaves in an isotropic and/or homogeneous manner, or if
some residual stresses, or damages, are present.
1.2.2 External forces, freeplay, impact and friction
Nonlinearity can be caused by external forces acting on a linear system, as in
the case of the interaction between aerodynamic forces and structures, known
as aeroelasticity, an important branch of structural vibration in aerospace and
civil engineering. An example of this can be provided by a single-degree-of-
freedom system invested by an airflow, as shown in Figure 1.2: as the airflow
invest the aerofoil with velocity U, the aerofoil pitches upward with positive
angle of attack, θ. The total aerodynamic moment is M(U, θ), function of the
velocity and of the angle of attack. The rotational spring kθ provides a restoring
moment which tends to reduce the angle of attack up to the initial resting
position, θ=0. As long as the angle of attack is small, the restoring force can be

13. 8
expressed by the linear relationship 𝐾𝜃 ∙ 𝜃. Nevertheless, because the
aerodynamic moment is a function of the velocity and of the induced angle of
attack, as M increases there comes a point where it overcomes the restoring
moment and the system becomes statically unstable, leading to large damaging
oscillations: this point of static instability is known as divergence, and its
dynamic equivalent is called flutter. This phenomenon is related to the fact that
the aerodynamic moment M actually is a nonlinear function of the angle of
attack, and flutter is a typical case of static equilibrium position becoming
unstable when the system is subjected to certain dynamic excitations.
Fig. 1.2 Single-degree-of-freedom aerofoil. (Wagg-Neild).
Nonlinear behaviour can be also caused by the interaction of structural
elements. The manufacture of any components takes into account some degree
of tolerance, particular when two or more components interact, and this leads to
a degree of freeplay, or backlash, as in the case of an aileron mounted on a wing.
In this case the aileron, subjected to an aerodynamic force, is allowed to vibrate
to and fro in the freeplay zone, and here the movement of the aileron cannot be
controlled: this leads to the particular “buzzing” sound. Another cause of
nonlinear behaviour resides in the presence of motion-limiting constraints,
which can cause repeated impacts. This type of behaviour is often called “vibro-
impact” motion, an example of which is shown in Figure 1.3, where a vertically
clamped cantilever beam is represented. The beam, sinusoidally forced, has a
motion-limiting constraint and as the beam vibrates, it has an impact every time
the beam hits the constraint. Although the beam has linear dynamics, the
nonlinear effect of the impact makes the problem nonlinear.

14. 9
Fig. 1.3 Cantilever beam subjected to a motion-limiting constraint (impact stop) near the tip, and
forced with a sinusoidal input. (Wagg-Neild).
When structural elements are already in contact, friction occurs along the
contact surface. When the elements move it triggers repeated stick and slip
cycles that can induce vibration in flexible structural elements: friction effects
are very significant in machines.
1.2.3 Geometric nonlinearity
Geometric nonlinearity occurs when the nonlinear effect derives entirely from
the geometry of the problem considered. The classic example of such
phenomenon is the pendulum, whose behaviour is linear until the angular
displacement is small, but becomes nonlinear when the angular displacement
get larger. Let’s consider the pendulum in Fig. 1.4, made of a bob with mass m
pivoted at point O and constrained to rotate in a circular plane by a rigid rod of
length l (assumed to be massless). Disturbing this system from its rest position,
gravity provides a restoring force mg𝑠𝑖𝑛 𝜃 tangent to the circular arc of motion.
Applying the equation (1.1) to this particular system, since the inertia force is
given by ml𝜃̈ (the acceleration along the arc is 𝑠̈ , but 𝑠 = 𝑙𝜃 , so 𝑠̈ = 𝑙𝜃̈), the
governing equation of the problem is: 𝒎𝒍𝜽̈ + 𝒎𝒈 𝒔𝒊𝒏 𝜽 = 𝟎 ,which becomes
𝜽̈ +
𝒈
𝒍
𝒔𝒊𝒏 𝜽 = 𝟎 (1.3)

15. 10
It’s usual to set
𝑔
𝑙
= 𝜔2
so that the frequency ω is a function of the pendulum
length, assuming gravity to be constant. Since sin 𝜃 ≅ 𝜃 −
𝜃3
3!
+
𝜃5
5!
,for small
oscillations it is possible to approximate sin 𝜃 ≅ 𝜃 , but as the oscillation
increases the nonlinear terms become more significant and this linearization is
no longer representative of the problem.
Fig. 1.4 Geometric nonlinear behaviour: simple pendulum. (Wagg-Neild).
Another common source of geometric nonlinearity is represented by large
deflections of beams, subjected to a load, which deflects into a deformed shape
with a certain curvature. The basic modelling assumption is that the bending
moment in the beam is proportional to the curvature. Assuming small
deflections the nonlinear terms arising from curvature can be neglected and a
linear equation of motion is obtained. This is true until the beam is very close
to being straight, but any deviation from this will lead to geometric
nonlinearities and errors in the modelling process if these nonlinearities are not
accounted for.
1.3 Linear and nonlinear vibrations modelled using sine waves
Before talking about nonlinear vibration’s modelling, it is important to talk
about the modelling of linear ones. For a linear system, if the forcing is
sinusoidal the response will be sinusoidal too, but not the same as the excitation.
Both the amplitude of the response sine waves and the phase will be different

16. 11
from the input signal. So, considering the linear single-degree-of-freedom
system
𝑚𝑥̈ +𝑐𝑥̇+𝑘𝑥 =FE (1.4)
where m, c and k are scalar parameters of the system and FE is the forcing, if
𝐹𝐸 = 𝐹0 𝑠𝑖𝑛(𝛺𝑡), the displacement response will be x=Xr 𝑠𝑖𝑛(𝛺𝑡 − 𝜙), where
F0 is the amplitude of the forcing input, Xr is the amplitude of the displacement
response, Ω is forcing frequency and ϕ is the phase shift between the two sine
waves, as shown in Figure 1.5 :
Fig. 1.5 Linear vibration of a single-degree-of-freedom system, forcing and response representation.
(Wagg-Neild).
In general, the solution to ordinary differential equations of this type is made of
two parts. The first one is the transient response, which corresponds to the
solution to the homogeneous equation (i.e. FE=0) and depends on the initial
conditions, while the second one is the steady state response (i.e. long term
response) corresponding to the particular solution depending on the forcing
function being applied to the system. The transient response dies quickly, so the
steady state response is usually the one of primary interest. A useful way to
represent the forcing sine wave is to use complex functions, namely:
𝐹𝐸 =
𝐹0
2𝑖
(𝑒 𝑖Ω𝑡
− 𝑒−𝑖Ω𝑡
) (1.5)
where F0 is real.

17. 12
The response sine wave is written as:
𝑥 =
𝑋
2𝑖
𝑒 𝑖Ω𝑡
−
𝑋̅
2𝑖
𝑒−𝑖Ω𝑡
where 𝑋 is a complex constant and 𝑋̅ is its complex conjugate. Using complex
functions, both amplitude and phase information can be included in the response
sine wave. The amplitude function, 𝑋𝑟 , is the modulus of 𝑋, while the phase Φ
is the argument of 𝑋. Substituting FE and x in the governing equation and
comparing the coefficients of the exponential terms we find:
( 𝑘 − 𝑚Ω2
+ 𝑖𝑐Ω) 𝑋 = 𝐹0 (1.6)
( 𝑘 − 𝑚Ω2
− 𝑖𝑐Ω) 𝑋̅ = 𝐹0 (1.7)
Considering that 𝜔 𝑛 = √
𝑘
𝑚
is the natural frequency and ζ =
c
2m𝜔 𝑛
is the damping
ratio, these relationships can be rewritten as:
(1 − (
Ω
𝜔 𝑛
)
2
+ 𝑖2ζ
Ω
𝜔 𝑛) 𝑘𝑋 = 𝐹0 (1.8)
(1 − (
Ω
𝜔 𝑛
)
2
− 𝑖2ζ
Ω
𝜔 𝑛) 𝑘𝑋̅ = 𝐹0 (1.9)
Manipulating these two equations (or equally the previous ones), and separating
the real and the imaginary part, the modulus Xr and the phase Φ can be obtained.
For both 𝑋 and 𝑋̅ the modulus Xr is given by:
𝑋𝑟 = (
𝐹0
𝑘
)
1
√(1−(
Ω
𝜔 𝑛
)
2
)
2
+4(ζ
Ω
𝜔 𝑛
)
2
(1.10)
As there are two complex vectors, there are two distinct values for the phase
delay, namely

18. 13
arg( 𝑋) = arctan (
−2ζ
Ω
𝜔 𝑛
1−(
Ω
𝜔 𝑛
)
2) = −𝛷 (1.11)
arg( 𝑋̅) = arctan (
2ζ
Ω
𝜔 𝑛
1−(
Ω
𝜔 𝑛
)
2) = 𝛷 (1.12)
Fig. 1.6 (a) Amplitude-frequency and (b) phase-frequency plots for a linear oscillator. (Wagg-Neild).
These functions define the response of the linear system as
𝑥 =
𝑋 𝑟
2𝑖
(𝑒 𝑖(Ω𝑡−𝛷)
− 𝑒−𝑖(Ω𝑡−𝛷)
) = 𝑋𝑟 sin(Ω𝑡 − 𝛷), (1.13)
which corresponds to two counter-rotating complex vectors in the complex
plane. These vectors always have equal and opposite imaginary parts, whit the
result that the sum of the two vectors is a sine wave in the real plane, as shown
in Figure 1.7.
Fig. 1.7 Linear response of a single-degree-of-freedom system showing how the two counter-rotating
response vectors form a sine wave. (Wagg-Neild).

19. 14
From equation (1.10) we can note that, when Ω=0, the response is 𝑋𝑟 =
𝐹0
𝑘
,
which corresponds to the static force displacement relationship 𝑥 𝑠 =
𝐹0
𝑘
, when
there is no dynamic excitation. When Ω = 𝜔 𝑛√1 − 2𝜁2 the response of the
linear system reaches the maximum value, i.e. the resonance peak. The concept
of resonance is very important both for linear and nonlinear systems, especially
for those systems which have a small amount of damping: the smaller the
damping ratio, the larger the maximum response amplitude. Furthermore,
lightly damped resonances usually lead to larger than desired displacements in
most structural systems. Let’s now consider a nonlinear system, whose
governing equation is:
𝒙̈ + 𝟐𝜻𝝎 𝒏 𝒙̇ + 𝝎 𝒏
𝟐
𝒙 + 𝜶𝒙 𝟑
= 𝑭 𝐬𝐢𝐧(Ω𝒕) (1.14)
This equation is usually known as Duffing oscillator, and is often used to model
nonlinear force-displacement behaviour such as hardening and softening,
shown in Fig. 1.8.
Fig. 1.8 Nonlinear force-displacement behaviour. (Wagg-Neild).
As seen in the linear case, the harmonic forcing can be represented using
complex functions, 𝐹 sin(Ω𝑡) =
𝐹
2𝑖
(𝑒 𝑖Ω𝑡
− 𝑒−𝑖Ω𝑡
), where 𝐹 =
𝐹0
𝑚⁄ is real.
Assuming the response is in the harmonic form shown in Eq. (1.13) and
substituting it in the governing equation the response is obtained. Because of

20. 15
the cubic term, for this nonlinear system with a single-frequency harmonic
forcing, a response can be expected at other frequencies than just the input
frequency. In this case, the effect of the cubic nonlinearity is to generate an
additional harmonic response at a frequency equal to 3Ω. In fact we have that:
𝑥3
= (
𝑋 𝑟
2𝑖
)
3
(𝑒 𝑖(𝛺𝑡−𝛷)
− 𝑒−𝑖(𝛺𝑡−𝛷)
)
3
=
𝑋 𝑟
3
4
(3 sin(𝛺𝑡 − 𝛷) − sin(3(𝛺𝑡 − 𝛷)))
This is caused by the cross-coupling terms which appear when the solution
assumed (Eq.(1.13)) is substituted in the governing equation, Eq.(1.14). This is
contrary to the original assumption that the response is a single-frequency sine
wave with frequency Ω.
Let’s consider now a simplified unforced and undamped system:
𝑥̈ + 𝜔 𝑛
2
𝑥 + 𝛼𝑥3
= 0 (1.15)
For this system any non-zero initial conditions, 𝑥(0) and 𝑥̇(0), will lead to
periodic oscillations. Now let’s assume a solution 𝑥 = 𝑋𝑟 sin( 𝜔𝑟 𝑡), where 𝜔𝑟
is the frequency of the response motion, and where no phase term has been
considered cause there is no forcing sine wave to compare with. Substituting
this solution in the governing equation and gathering the coefficients of the
harmonic terms, sin( 𝜔𝑟 𝑡) and sin(3𝜔𝑟 𝑡) gives :
[( 𝜔 𝑛
2
− 𝜔𝑟
2) 𝑋𝑟 +
3𝛼𝑋 𝑟
3
4
] sin(𝜔𝑟 𝑡) −
𝛼𝑋 𝑟
3
4
sin(3𝜔𝑟 𝑡) = 0 (1.16)
To simplify the problem let’s assume sin(3𝜔𝑟 𝑡) to be negligible. So the
approximate value of the response frequency is 𝝎 𝒓 ≈ 𝝎 𝒏√𝟏 +
𝟑𝜶𝑿 𝒓
𝟐
𝟒𝝎 𝒏
𝟐 . This result
implies that the frequency of the response, 𝜔𝑟, depends on the amplitude of the
response, 𝑋𝑟. This amplitude dependence is another fundamental difference
between nonlinear and linear vibration problems.

21. 16
1.4 Multiple degrees-of-freedom
Most flexible structures exhibit multiple degrees-of-freedom, and modelling the
behaviour of such structures is a big challenge. Let’s start introducing a linear
multi-degree-of-freedom system, and consider the classic governing equation:
𝑀𝑥̈ + 𝐶𝑥̇ + 𝐾𝑥 = 𝐹𝐸
Generally the matrices M, K and C are not diagonal, that means the equations
are coupled. Anyway it is possible to simplify the system of equations using a
transformation that replace M, K and C with equivalent diagonal matrices. To
do this, a substitution has to be made, i.e. 𝒙 = 𝜳𝒒, which leads to :
𝑀𝛹𝑞̈ + 𝐶𝛹𝑞̇ + 𝐾𝛹𝑞 = 𝐹𝐸
Pre-multiplying by 𝜳 𝑻
gives
𝛹 𝑇
𝑀𝛹𝑞̈ + 𝛹 𝑇
𝐶𝛹𝑞̇ + 𝛹 𝑇
𝐾𝛹𝑞 = 𝛹 𝑇
𝐹𝐸
The matrix 𝛹 is called modal matrix and has to be chosen so that
𝛹 𝑇
𝑀𝛹, 𝛹 𝑇
𝐶𝛹 , 𝛹 𝑇
𝐾𝛹 are simultaneously diagonalized. Now, setting 𝐹𝐸 = 0
and assuming sinusoidal solutions, we can find the system eigenvalues and
eigenvectors. The N eigenvalues are related to the N natural frequencies of the
system, while the modal matrix 𝛹 is formed so that the columns contain the
eigenvectors. As eigenvectors are non-unique, 𝛹 is normalized such that
𝛹 𝑇
𝐶𝛹 = 𝐼, where I is the identity matrix. Then 𝜳 𝑻
𝑲𝜳 = [𝝎 𝒏𝒊
𝟐
] becomes a
diagonal matrix containing the squared natural frequencies. The C matrix is
diagonalized assuming that C is proportional to the mass and stiffness matrices:
this type of damping is called proportional damping, and leads to 𝜳 𝑻
𝑪𝜳 =
[𝟐𝜻𝒊 𝝎 𝒏𝒊], a diagonal matrix with 𝜁𝑖 as modal damping coefficient. This
diagonalization results in the building of a system of N equations in the form
𝒒𝒊̈ + 𝟐𝜻𝒊 𝝎 𝒏𝒊 𝒒̇ + 𝝎 𝒏𝒊
𝟐
= 𝜳 𝑻
𝑭 𝑬𝒊,

22. 17
that can be solved in the same way of the single degree case: this type of linear
system has the property of superposition, that means that each equation of the
system can be solved separately, and the results added together to get the total
response. This principle is a key part of the analysis of vibrating systems.
Superposition generally cannot be applied to nonlinear systems. However, there
are many aspects of linear vibration that can be used in the study of nonlinear
vibrations, like using linear approximations to the nonlinear system, in a small
local region of the system parameters. So, linearizing the system locally, we can
find eigenvalues to give information on the system dynamics.

23. 18
CHAPTER 2
NONLINEAR VIBRATION PHENOMENA
SUMMARY:
2.1 Stata space analysis
2.2 Equilibrium points
2.2.1 Equilibrium points for a two-state linear harmonic oscillator
2.2.2 Potential functions
2.3 Periodic and non-periodic oscillations: limit cycles
2.4 Bifurcations
2.5 Bifurcations in forced nonlinear oscillations

24. 19
2.1 State space analysis
The investigation of the behaviour of nonlinear systems usually pass through
approximate analysis and numerical simulations. This form of analysis is also
known as dynamical systems theory, and is based on the using of the state space
of a system. The state of a system is measured by the state vector 𝑥 =
{ 𝑥1, 𝑥2, … , 𝑥 𝑛} 𝑇
, where 𝑥𝑖 are the variables, like position and velocity, which
describe the state of the system. In a dynamical system the states evolve with
time, so any state is a function of the time, 𝑥𝑖(𝑡). For this reason the states are
referred to as dependent variables, while the time is an independent variable,
because it doesn’t depend on any other variable. The state vector has n states,
so it is possible to represent x in an n-dimensional Euclidean space, called state
space, or often phase space of the system. A plot representing the solutions in
a phase plane is called phase portrait. One of the most used and important phase
portrait for a system is the two-dimensional plot of displacement against
velocity. An example of this kind of problems can be provided by a
harmonically forced linear oscillator:
𝑥̈ + 2𝜁𝜔 𝑛 𝑥̇ + 𝜔 𝑛
2
𝑥 =
𝐹0
𝑚
cos(Ω𝑡) (2.1)
The state vector of this system is 𝑥 = { 𝑥1, 𝑥2} 𝑇
, where 𝑥1 = 𝑥 is the
displacement and 𝑥2 = 𝑥̇ is the velocity. It is possible to obtain a first order
system from this second order differential equation by using the definitions of
𝑥1 and 𝑥2 given above:
 𝑥1̇ = 𝑥2 = 𝑥̇
 𝑥2̇ = 𝑥̈
Using these positions it is possible to obtain a first order system:
 𝑥1̇ = 𝑥2
 𝑥2̇ = −2𝜁𝜔 𝑛 𝑥2 − 𝜔 𝑛
2
𝑥1 +
𝐹0
𝑚
cos(Ω𝑡)

25. 20
This system can be written in a matrix form:
 [
𝑥1̇
𝑥2̇
] = [
0 1
−𝜔 𝑛
2
−2𝜁𝜔 𝑛
] [
𝑥1
𝑥2
] + [
0
𝐹0
𝑚
cos(Ω𝑡)]
 𝒙̇ = 𝐴𝒙 + 𝑭( 𝑡) (2.2)
The Eq.(2.2) is in a state space form. The following pictures show the
displacement vs. time and the displacement vs. velocity (phase portrait)
plots, for the harmonic oscillator of Eq.(2.1), for different damping values:
Fig. 2.1 Time series and steady state periodic solutions for the harmonic oscillator, numerical results.
(Wagg-Neild).

26. 21
The phase portrait in Fig. 2.1 shows elliptical orbits named limit cycles, which
will be discussed later. If 𝜔 𝑛 = 1 these would have been circular orbits. The
size of each orbit depends on the balance of energy between the forcing input
and the amount of energy dissipated by damping. The bigger is the damping,
the smaller is the size of the ellipse, because more energy is dissipated by the
damper. The following figure shows a state space solution for a second-order
nonlinear system. There are three projections of this three-dimensional solution
curve, namely the displacement vs time plot, the velocity vs. time plot, and
finally the displacement vs. velocity plot, which is the mostly used phase
portrait in the analysis of second-order nonlinear systems.
Fig. 2.2 State space for a second-order nonlinear oscillator. (Wagg-Neild).
The solution curves shown in Fig. 2.2 are called either trajectories or orbits,
and the time evolution of these trajectories is known as the flow of the dynamical
system.

27. 22
2.2 Equilibrium points
An equilibrium point is a point for which f(x,t)=0. It is denoted as 𝑥∗
, so that
𝑓( 𝑥∗
, 𝑡) = 0, and it is also referred to as fixed point. Equilibrium points play an
important role in the analysis of nonlinear systems, because the dynamic
behaviour of a system near an equilibrium point can be locally analysed using
linear analysis. This is the reason why the study of a nonlinear system always
starts with the identification of the equilibrium (or fixed) points. The
linearization of a nonlinear system, close to an equilibrium point, starts from a
change of coordinates that makes the equilibrium point the origin of a new
reference system. First, a new coordinate vector is defined as 𝝃 = 𝒙( 𝒕) − 𝒙∗
, so
that 𝝃 = 𝟎 correspond to the equilibrium point of interest. Let’s assume that f is
not dependent on time, but only on the state variable x. Such a function is called
autonomous function. So, in this hypothesis
𝑑𝝃
𝑑𝑡
=
𝑑𝒙
𝑑𝑡
= 𝑓( 𝑥) = 𝑓( 𝑥∗
+ 𝜉) (2.3)
because
𝑑𝒙∗
𝑑𝑡
= 0, as 𝑥∗
is a constant. The next step is to get the Taylor series
expansion of 𝑓( 𝑥∗
+ 𝜉):
𝑓( 𝑥∗
+ 𝜉) = 𝐷 𝑥∗ 𝑓 𝜉 + 𝑂(‖ 𝜉‖2
) (2.4)
Where 𝐷 𝑥∗ 𝑓 is the Jacobian matrix evaluated at 𝑥∗
and 𝑂(‖ 𝜉‖2
) represents the
second order and higher terms, which are here ignored. This means that the
analysis is valid only in a small region close to the equilibrium point. The
Jacobian matrix contains the linear gradients of the nonlinear function f with
respect to the states variables 𝑥𝑖. So, 𝐷 𝑥∗ 𝑓 is obtained substituting the state
values at 𝑥∗
into the Jacobian expression. Due to the fact that we have assumed
that the nonlinear function is autonomous, this matrix is just a constant matrix,
so we define 𝑫 𝒙∗ 𝒇 = 𝑨, where 𝐴 is a 𝑛 × 𝑛 matrix of constant terms. Finally
we obtain:

28. 23
𝑑𝝃
𝑑𝑡
= 𝐴𝜉 (2.5)
The Eq. (2.5) represent a linear system, valid as an approximation of the
nonlinear system near to an equilibrium point, i.e. for 𝜉 small.
Let’s now consider a typical case, a system having two states, 𝑥 = { 𝑥1, 𝑥2} 𝑇
,
and let’s solve Eq. (2.5) close to an equilibrium point. To do this we have to
assume a solution of the form 𝝃( 𝒕) = 𝒄𝒆 𝝀𝒕
, where 𝑐 = { 𝑐1, 𝑐2} 𝑇
is a vector of
arbitrary constants. Substituting this solution into Eq. (2.5) gives:
𝒄𝝀𝒆 𝝀𝒕
= 𝑨𝒄𝒆 𝝀𝒕
, which leads to the linear eigenvalue problem ( 𝑨 − 𝝀𝑰) 𝒄 = 𝟎.
Ignoring the banal solution 𝒄 = 𝟎, which involves no dynamics, the solution to
the eigenvalue problem is obtained by solving |(𝑨 − 𝝀𝑰)| = 𝟎. For a two
dimensional system this expression can be written as
|
𝑎11 − 𝜆 𝑎12
𝑎21 𝑎22 − 𝜆
| = 0
which leads to the characteristic equation:
𝜆2
− ( 𝑎11 + 𝑎22) 𝜆 + ( 𝑎11 𝑎22 − 𝑎21 𝑎12) = 0 (2.6)
where ( 𝑎11 + 𝑎22) = 𝑡𝑟(𝐴) and ( 𝑎11 𝑎22 − 𝑎21 𝑎12) = det(𝐴) are respectively
the trace and the determinant of the matrix A. The Eq. (2.6) can be rewritten as
follow
𝜆2
− 𝑡𝑟(𝐴)𝜆 + det(𝐴) = 0
whose solutions are: 𝜆1,2 =
1
2
(𝑡𝑟(𝐴) ± √𝑡𝑟( 𝐴)2 − 4det(𝐴)
The terms under the square root represent the discriminant ∆, so we can rewrite
:
𝜆1,2 =
1
2
(𝑡𝑟(𝐴) ± ∆
1
2)

29. 24
The discriminant ∆, the trace 𝑡𝑟(𝐴) and the determinant det(𝐴) are fundamental
in order to determine the nature of the eigenvalues and therefore the type of
equilibrium points. This will be shown later.
2.2.1 Equilibrium points for a two-state linear harmonic oscillator
Let’s now consider the linear harmonic oscillator, whose governing equation
is: 𝒎𝒙̈ + 𝒄𝒙̇ + 𝒌𝒙 = 𝑭(𝒕). The analysis of equilibrium points can be simplified
by first considering the unforced system, i.e. setting 𝐹( 𝑡) = 0. The space
representation of the system becomes
[
𝑥1̇
𝑥2̇
] = [
0 1
−
𝑘
𝑚
−
𝑐
𝑚
] [
𝑥1
𝑥2
]
If all the parameters are different from zero, the only equilibrium point is 𝑥1 =
𝑥2 = 0. This is physically explained by the fact that, for any initial condition
(i.e. initial displacement and velocity), an unforced but damped system
gradually loses energy until it reaches the ‘at rest’ position, where displacement
and velocity are zero. The ‘at rest’ position corresponds to the equilibrium point.
Since for all initial conditions the solution curves end up at the equilibrium
point, this equilibrium point is said to be attracting. A fixed point is said to be
attracting if all the trajectories that starts near the fixed point approach it as 𝑡 →
∞. That is 𝑙𝑖𝑚
𝑡→∞
𝑥( 𝑡) = 𝑥∗
. Such a fixed point actually attracts all trajectories in
the phase plane, so it can be called globally attracting. Previously we’ve seen
that the determinant and the trace of the matrix A determine the type of
equilibrium points. For the linear harmonic oscillator 𝑡𝑟( 𝐴) = −
𝑐
𝑚
and
det( 𝐴) =
𝑘
𝑚
, while ∆=
𝑐2−4𝑘𝑚
𝑚2
, and
𝜆1,2 = −
𝑐
2𝑚
±
1
2𝑚
√ 𝑐2 − 4𝑘𝑚

30. 25
As previously seen, the sign of ∆ determine the type of eigenvalues. Let’s
consider the case when ∆< 𝟎. In this case the eigenvalues are complex, and this
physically corresponds to underdamped vibrations. Rewriting the expression for
the eigenvalues in terms of undamped natural frequency, damping ratio and
damped natural frequency gives 𝜆1,2 = −𝜁𝜔 𝑛 ∓ 𝑖𝜔 𝑑, where 𝜔 𝑑 = 𝜔 𝑛√1 − 𝜁2
is the damped natural frequency. Substituting this into the assumed solution
𝜉( 𝑡) = 𝑐𝑒 𝜆𝑡
leads to this expression for the solution:
𝜉( 𝑡) = 𝑐𝑒−(𝜁𝜔 𝑛 𝑡)
sin(𝜔 𝑑)
This is a sine wave multiplied by an exponential envelope. Depending on
whether the damping is positive or negative two possible cases can occur. For
positive damping, 𝜁 > 0, the exponential envelope causes the sinusoidal
oscillation to decay, while for negative damping, 𝜁 < 0, the oscillations grow.
The idea of stability can be related to whether the oscillations grow or decay. If
they decay, such that solution curves of the governing equation of motion are
attracted to the equilibrium point, this is a stable behaviour. If the oscillations
grow, so that the solution curves are repelled from the equilibrium point, then
this is an unstable behaviour. These two behaviours can be shown in a phase
portrait as well, as shown in Fig. (2.3). In the case where 𝜁 > 0 the equilibrium
point is a stable spiral (or a sink), while in the case where 𝜁 < 0 the equilibrium
point is an unstable spiral (or a source).

31. 26
Fig. 2.3 Dynamics close to equilibrium point for (a),(b) positive damping, and (c),(d) negative
damping. (Wagg-Neild).
The importance of 𝑡𝑟( 𝐴), det( 𝐴) and ∆ on the type and stability of all
equilibrium points for the linear oscillator with two states, 𝑥 = { 𝑥1, 𝑥2} 𝑇
, can be
seen in Fig. (2.4). Remember that ∆= 𝑡𝑟( 𝐴)2
− 4det(𝐴).
Fig. 2.4 Two state linearized system, type and stability of the fixed points. (Wagg-Neild).

32. 27
Different types of equilibrium points are shown in Fig. 2.4:
 The centres are equilibrium points surrounded by closed orbits. These
orbits correspond to periodic motions, i.e. oscillations. The centres are
said to be neutrally stable, since nearby trajectories are neither attracted
to nor repelled from the fixed point. Analytically, centres occur if the
eigenvalues are complex conjugate with real part equal to zero (because
𝑡𝑟( 𝐴) = 0), i.e. when det( 𝐴) > 0 and ∆< 0. Centres occur very
commonly in frictionless mechanical systems, where energy is
conserved. In fact, the case when 𝑡𝑟( 𝐴) = 0 corresponds to zero
damping, and often is used to provide a simplified analysis of the
behaviour of systems with small damping.
 If the system is damped, in the same conditions (det( 𝐴) > 0, ∆< 0) the
fixed point is a spiral: as seen before for the linear harmonic oscillator,
the trajectory of the spiral is due to the fact that the system loses energy
on each cycle. Analytically this means that the eigenvalues are complex
conjugate, so we have oscillations which grow or decay depending on
the sign of the real part of the solution, i.e. depending on the sign of
𝑡𝑟(𝐴). In fact, as we can see from Fig. 2.4, when 𝑡𝑟( 𝐴) < 0 the fixed
point is a stable spiral, while when 𝑡𝑟( 𝐴) > 0 the fixed point is an
unstable spiral.
 If det( 𝐴) > 0 and ∆> 0 the fixed points are said to be nodes. In this
case the eigenvalues are real with the same sign. The case ∆> 0
corresponds to 𝜁 > 1, i.e. the overdamped case. The stability of the
nodes and spirals is determined by 𝑡𝑟(𝐴). When 𝑡𝑟( 𝐴) < 0 both
eigenvalues have negative real parts, so the fixed point is stable.
Unstable spirals and nodes have 𝑡𝑟( 𝐴) > 0. Neutrally stable centres live
on the borderline 𝑡𝑟( 𝐴) = 0, where the eigenvalues are purely
imaginary.

33. 28
 When 𝑑𝑒𝑡( 𝐴) < 0 the fixed points are said to be saddle points. In this
case there are two real eigenvalues, with opposite signs, which means
that the first eigensolution grows exponentially, while the second one
decays: this is the reason why such a fixed point is called saddle point.
 The case when ∆= 0, represented by the parabola in Fig. 2.4
corresponds to a critically damped system, 𝜁 = 1. In this case we have
two equal real eigenvalues, 𝜆1 = 𝜆2 = 𝜆, and two possibilities can
occur: either there are two independent eigenvectors corresponding to λ,
or there’s only one. In the first case, assuming 𝜆 ≠ 0, all trajectories are
straight lines to the origin, and the fixed point is a star node. The other
possibility is that there is only one eigenvector: in this case the fixed
point is a degenerate node. The degenerate node is on the borderline
between a spiral and a node. The trajectories try to wind around in a
spiral, but they don’t quite make it.
 If 𝑑𝑒𝑡( 𝐴) = 0, at least one of the eigenvalues is zero. Then the origin is
not an isolated fixed point. There is either a whole line of fixed points,
or a plane of fixed points, if 𝐴 = 0.
Saddle points, nodes and spirals are the major types of fixed points. They
occur in a large open region of the plane shown in Fig. 2.4. Centres,
degenerate nodes, non-isolated fixed points are borderline cases, which
occur along curves in the (𝑑𝑒𝑡( 𝐴), 𝑡𝑟( 𝐴)) plane.
The effect of small nonlinear terms
Linearizing the second order system close to an equilibrium point requires
neglecting the second order and higher terms in Eq. (2.4). The linearized
system obtained making this assumption provides a qualitatively correct
picture of the phase portrait close to the equilibrium point as long as the
equilibrium point for the linearized system is not one of the borderline cases
discussed above. In other words, if the linearized system predicts a saddle,

34. 29
node, or a spiral, then the fixed point really is a saddle, node, or a spiral for
the original nonlinear system. The borderline cases are much more delicate,
as they can be altered by small nonlinear terms. It can be shown that small
nonlinear terms can change a centre into a spiral and also change its
stability. Similarly, stars and degenerate nodes can be altered by small
nonlinearities, but unlike centres, their stability doesn’t change. This means
that, for example, a stable star may be changed into a stable spiral but not
into an unstable spiral. This is plausible, because stars and degenerate nodes
live squarely in the stable or unstable region, whereas centres live on the
separation line between stability and instability. In conclusion, fixed points
can be classified focusing on the concept of stability (and not focusing on
the geometry of trajectories) as follows:
 Repellers (also called sources): both eigenvalues have positive real
part.
 Attractors (also called sinks): both eigenvalues have negative real
part.
 Saddles: one eigenvalue is positive and one is negative.
There are also some marginal cases:
 Centres: both eigenvalues are pure imaginary.
 High-order and non-isolated fixed points: at least one eigenvalue is
zero.
Thus, from the point of view of stability, the marginal cases are those where
at least one eigenvalue satisfies 𝑅𝑒( 𝜆) = 0.
2.2.2 Potential functions
There is a direct connection between the system state space and the energy in
the system. Let’s consider the unforced, undamped, nonlinear oscillator:

35. 30
𝑚𝑥̈ + 𝑝( 𝑥) = 0
Considering that 𝑥̇ = 𝑣 =
𝑑𝑥
𝑑𝑡
we can write
𝑑𝑣
𝑑𝑡
𝑑𝑥 = 𝑣𝑑𝑣 and so
𝑚𝑣
𝑑𝑣
𝑑𝑥
+ 𝑝( 𝑥) = 0,
where 𝑝( 𝑥) is the stiffness function. If we consider the work done over a small
increment of distance 𝑑𝑥, as the mass moves from resting position to an
arbitrary 𝑥 value, integrating we obtain the total energy of the system.
∫ 𝑚𝑣
𝑑𝑣
𝑑𝑥
𝑑𝑥 + ∫ 𝑝( 𝑥) 𝑑𝑥 = 𝐸𝑡 →
1
2
𝑚𝑣2
+ 𝑉( 𝑥) = 𝐸𝑡
𝑥
0
𝑥
0
(2.7)
𝑉( 𝑥) = ∫ 𝑝( 𝑥) 𝑑𝑥
𝑥
0
is called potential function, and it has not to be confused
with the concept of potential energy. The potential function is the integral of the
stiffness function. The following figure shows the relationship between the
potential function and the phase portrait, relative to the case of the unforced,
undamped ‘escape equation’ 𝒙̈ + 𝒙 + 𝒙 𝟐
= 𝟎. The underlying concepts are
valid in general.
Fig. 2.5 Potential function and phase portrait for the undamped escape equation. (Wagg-Neild).

36. 31
This system has two fixed points: ( 𝑥1, 𝑥2) = (0,0), which is a centre, and
( 𝑥1, 𝑥2) = (−1,0) which is a saddle point. Considering the closed orbit A in
Fig. 2.5, the constant energy level for this orbit is shown up in the energy
diagram. As the size of the closed orbit increases, so does the energy level. At
the point where the closed orbit touches the saddle point, the maximum energy
level is reached. Beyond this point the system becomes unstable, and the
solutions become infinitely large. The closed orbit which separates the stable
area of solutions from the unstable area is called separatrix. The stable part of
the energy function containing the closed orbits is called potential well. Let’s
now consider a system with a cubic nonlinearity, like the undamped ‘Duffing
oscillator’ 𝒙̈ − 𝒌 𝟏 𝒙 + 𝒌 𝟑 𝒙 𝟑
= 𝟎, which in this case presents a negative linear
stiffness. This system has three fixed points: ( 𝑥1, 𝑥2) = (0,0), ( 𝑥1, 𝑥2) = (1,0)
and ( 𝑥1, 𝑥2) = (−1,0) which respectively are a saddle point and two centres, as
shown in Fig. 2.6.
Fig. 2.6 Potential function and phase portrait for the undamped Duffing oscillator. (Wagg-Neild).

37. 32
Systems like this one, with a negative linear stiffness, can be used to model an
interesting class of systems which have bi-stability. This means that this systems
have two stable configurations (like the two equilibrium points at ±1 in Fig.
2.6) separated by an unstable configuration (the saddle). An example of this
kind of problems is given by buckled beams. The form of the potential function
𝑉(𝑥) shown in Fig. 2.6 is called double potential well. Energy levels for two
different closed orbits are shown as well: orbit A is inside the potential well
around the equilibrium point ( 𝑥1, 𝑥2) = (1,0), while orbit B has much higher
energy level and enclose both the potential well. In this case the separatrix
marks the boundary between the orbits confined to the potential wells around
each of the centre equilibrium points and the orbits which enclose both.
2.3 Periodic and non-periodic oscillations: limit cycles
One of the clearest differences between linear and nonlinear systems is that the
second ones can show multiple solutions. This means that a nonlinear system,
like the Duffing oscillator seen above, can have more than one stable
equilibrium point. The Duffing oscillator of the previous example actually has
two stable equilibrium points, one at 𝑥1 = −1 and the other at 𝑥1 = 1. If
damping was added to the system, these equilibrium points would become
attracting spirals, as shown in the phase portrait in Fig. 2.7.
Fig. 2.7 Damped Duffing oscillator: phase portrait. (Wagg-Neild).

38. 33
This means that for a particular set of initial conditions 𝑥1( 𝑡0), 𝑥2(𝑡0), there are
two potential finishing points. Starting points which are close to each other can
anyway finish at different equilibrium points. The regions containing initial
conditions values that are attracted to an equilibrium point are called basins of
attraction. The basins of attraction can define eventually the steady-state
behaviour of the oscillator or, recording the time to reach a steady-state, they
can provide information about the transient behaviour of the oscillator.
Fig. 2.8 Example of two close starting points finishing at different equilibrium points. (Wagg-Neild).
For unforced, undamped systems, steady-state periodic orbits can be observed
which have amplitudes that are dependent on the initial conditions. However,
real structures and engineering systems are both forced and damped, and for
these systems a common steady-state response is in the form of a periodic orbit,
whose amplitude depends on the energy balance in the system. These types of
periodic orbits are called limit cycles. A limit cycle is an isolated close
trajectory. Isolated means that nearby trajectories are not closed; they spiral
either toward or away from the limit cycle. If nearby trajectories approach the
limit cycle, it is said to be stable or attracting, otherwise it is said to be unstable.
Stable limit cycles are used to model systems that exhibit self-sustained
oscillations, i.e. systems which oscillate even in absence of an external periodic

39. 34
forcing. An example of such phenomena can be the flutter of a wing or
dangerous self-excited oscillations in bridges. In each case, there is a standard
oscillation of some preferred period, waveform and amplitude, and if the system
is perturbed slightly, it always returns to the standard cycle.
Fig. 2.9 Stable and unstable limit cycles. (Wagg-Neild).
Limit cycles are inherently nonlinear phenomena; they can’t occur in linear
systems. Linear systems can have, of course, closed orbits, but they won’t be
isolated. This means that if 𝑥(𝑡) is a periodic solution for the linear system,
𝒄𝑥(𝑡) will be a solution too, for any constant, 𝑐 ≠ 0. So there will be a family
of closed orbits depending on the parameter c, and consequently the amplitude
of a linear oscillation is set entirely by its initial conditions.
Fig. 2.10 Closed orbits for a linear system. (Wagg-Neild).
In contrast, limit cycle oscillations are determined by the structure of the system
itself, not only by initial conditions. In addition to limit cycles, some other types
of behaviour can be encountered. For example, a closed orbit that takes two
forcing periods to repeat the motion is called period-two orbit. Non-periodic
responses can also occur, including quasi-periodic motion.

40. 35
Quasi-periodic motion occurs when the response is composed by two or more
signals with frequencies which are not integer multiples of each other. This type
of motion can occur in vibration problems with multiple frequencies in the
response. Linear natural frequencies are typically non-integer multiples, but
they are spaced such that vibration modes at other frequencies are less
significant to the response. However, closely spaced modes can result in a multi-
frequency response which can appear similar to a quasi-periodic response for a
single degree-of-freedom system.
2.4 Bifurcations
In a nonlinear system a range of complex dynamic responses can occur over a
relatively short parameter range and, as a parameter is varied, key changes take
place between different dynamic responses. In this broader context we find the
concept of bifurcation. In particular, as one or more parameters are varied, fixed
points can be created or destroyed, or their stability can change. This could
happen for closed orbits as well. The parameter values at which these
phenomena occur are called bifurcation points. In one-dimensional systems
bifurcations usually involve fixed points, which, as said above, can be either
created or destroyed, or can change their stability. In two dimensional systems
this behaviour involves the closed orbits as well. This means that oscillations
can be turned on or off, in several ways. So, in this case we can say that if the
topological structure of the phase portrait changes, a bifurcation has occurred.
To investigate the steady-state behaviour of a particular system, one or more of
the system parameters can be varied. In vibration analysis, the amplitude and
the frequency of the external forcing terms are often used to characterize the
steady-state system response, but other system parameters can also be used. For
the linear (or linearized) systems, a common way to investigate the stability of
the system is to plot the system eigenvalues in the complex plane. Then, if the
real parts of the eigenvalues are in the left-hand plane the system is stable,

41. 36
otherwise if 𝑅𝑒(𝜆) are in the right-hand plane the system is unstable. This is
because of the exponential form of the solution 𝝃( 𝒕) = 𝒄𝒆 𝝀𝒕
. For nonlinear
systems, we consider each equilibrium point individually. If, for a particular
equilibrium point, the eigenvalues of the Jacobian matrix 𝑫 𝒙∗ 𝒇 are in the left-
hand plane, then the equilibrium point is said to be locally stable. If a system
parameter is varied, the position of the eigenvalues of the linearized system will
vary, as shown in Fig. 2.10.
Fig. 2.11 Linear (or linearized) system: eigenvalues varying position as a parameter changes. (Wagg-
Neild).
As the system parameter varies, the real part of one or more of the eigenvalues
will become zero, and here is where the bifurcation occurs, meaning a
significant change in the behaviour of the system. As shown in Fig. 2.10, the
eigenvalues could either be a pair of complex conjugate becoming real as a
parameter is varied, or could be a pair of real eigenvalues becoming complex
conjugates. In terms of vibrations, complex eigenvalues represent underdamped
vibrations, while real eigenvalues represent overdamped vibrations. In terms of
bifurcations, real eigenvalues crossing the imaginary axis are related to static
bifurcations such as buckling of structures, while complex eigenvalues crossing
the imaginary axis are related to dynamic bifurcations such as the sudden
appearance of oscillations like flutter. The bifurcations of fixed points have the
same characteristics in one, two, or more dimensions. So, to have a qualitative
understanding of this type of bifurcations we will refer to one-dimensional
systems. Before describing different types of bifurcations of fixed points in one-
dimensional systems, let’s briefly describe how to define the stability of a fixed

42. 37
point in these systems. An equilibrium point is said to be stable if all sufficiently
small disturbances away from it damp out in time, while it is said to be unstable
if the disturbances grow in time. To define if an equilibrium point is stable or
unstable we have to plot the first order equation and sketch the vector field,
which is to the right when 𝑥̇ > 0, and to the left when 𝑥̇ < 0. Then, if the local
flow is toward the equilibrium point, that equilibrium point is a stable one,
otherwise it is an unstable equilibrium point. To clarify, let’s consider the one-
dimensional system represented by the equation 𝑥̇ = 𝑥2
− 1, and let’s plot the
phase portrait for this system.
Fig. 2.12 Definition of the stability of fixed points for one-dimensional system. (Strogatz).
As said above, the vector field is to the right when 𝑥̇ > 0, and to the left when
𝑥̇ < 0. So this system has a stable fixed point (plain dot), and an unstable fixed
point (empty dot).
Saddle-Node Bifurcation
The saddle-node bifurcation is the basic mechanism by which fixed points are
created and destroyed. As a parameter is varied, two fixed points move toward
each other, collide and mutually annihilate. The prototypical example of a
saddle-node bifurcation is given by the first order system
𝒙̇ = 𝒓 + 𝒙 𝟐
(2.8)
where r is a parameter, which can be positive, negative or zero.

43. 38
Fig. 2.13 Example of saddle-node bifurcation. (Strogatz).
Eq. (2.8) is the equation of a parabola, and if 𝑟 < 0 there are two different fixed
points (±√ 𝑟), one stable and one unstable, which coalesce in one single half-
stable fixed point when 𝑟 = 0. When 𝑟 > 0 there are no fixed points, so in this
example a bifurcation occurred at 𝑟 = 0. A common way to depict such a
bifurcation is to plot 𝑥 𝑣𝑠. 𝑟, where r is the parameter which can vary, as shown
in Fig. 2.13.
Fig. 2.14 Saddle-node (or fold) bifurcation: bifurcation diagram. (Strogatz).
To distinguish between stable and unstable fixed points, it is used a solid line
for stable points and a broken line for unstable ones. Such a plot is named
bifurcation diagram. Because the bifurcation diagram of the saddle-node
bifurcation shows a fold in it, this kind of bifurcation is often called fold
bifurcation. The term saddle-node is used in two-dimensional case, when
saddles and nodes can collide and annihilate.

44. 39
Transcritical Bifurcation
There are certain situations where a fixed point must exist for all values of a
parameter and can never be destroyed. Such a fixed point, however, may change
its stability as a parameter is varied. The transcritical bifurcation is the standard
mechanism for such changes in stability. The prototypical example (also
referred to as normal form) for a transcritical bifurcation is:
𝒙̇ = 𝒓𝒙 − 𝒙 𝟐
(2.9)
Fig 2.15 Example of transcritical bifurcation. (Strogatz).
Note that there is a fixed point at 𝑥∗
= 0 for all values of r. For 𝑟 < 0 there are
two fixed points, an unstable one in 𝑥∗
= 𝑟 and a stable one in 𝑥∗
= 0. As r
increases, the unstable fixed point approaches the origin, and coalesces with it
when 𝑟 = 0. Finally, when 𝑟 > 0, the origin has become unstable, and now
𝑥∗
= 𝑟 is a stable fixed point. Between the two fixed points, a change of stability
has occurred. Let’s note the crucial difference between the fold bifurcation and
the transicritical bifurcation: in the transcritical case, the two fixed points don’t
disappear after the bifurcation, they just switch their stability. Figure 2.15 shows
the bifurcation diagram for the transcritical bifurcation. The parameter r is
regarded as the independent variable.

45. 40
Fig. 2.16 Transicritical bifurcation: bifurcation diagram. (Strogatz).
Pitchfork Bifurcation
This bifurcation is common in physical problems that have a symmetry, for
example like a spatial symmetry between left and right. In such cases, fixed
points tend to appear and disappear in symmetrical pairs. To provide an
example, let’s consider the problem of the buckling of a beam. If the axial load
is small, the beam is stable in its initial position (let’s suppose it to be vertical),
so in this case there is a stable fixed point corresponding to zero deflection. If
the buckling threshold is exceeded, the beam can buckle to either the left or the
right, so the vertical position is gone unstable and two new symmetrical fixed
points have been born. Two types of pitchfork bifurcation are possible.
 Supercritical pitchfork bifurcation
The normal form of the supercritical pitchfork bifurcation is
𝒙̇ = 𝒓𝒙 − 𝒙 𝟑
(2.10)

46. 41
Fig. 2.17 Example of supercritical pitchfork bifurcation. (Strogatz).
When 𝑟 < 0 the only fixed point is 𝑥∗
= 0, and it is a stable one, as we can see
in figure 2.16. When r approaches 0 the origin is still stable, but weaker, since
the linearization vanishes and the solutions decay slower. Finally when 𝑟 > 0
the origin has become unstable, and two new stable fixed points appear on either
side of the origin, symmetrically located at 𝑥∗
= ±√ 𝑟. Figure 2.17 shows the
bifurcation diagram for this bifurcation. It is evident the reason why it is called
“pitchfork”:
Fig. 2.18 Supercritical pitchfork bifurcation diagram. (Strogatz).
 Subcritical pitchfork bifurcation
In the supercritical case the cubic term is stabilizing, acting like a restoring force
that pulls 𝑥(𝑡) back toward 𝑥 = 0. If instead the cubic term were destabilizing,
as in
𝒙̇ = 𝒓𝒙 + 𝒙 𝟑
(2.11)

47. 42
then we would have a subcritical pitchfork bifurcation. Figure 2.18 shows the
bifurcation diagram for the subcritical case.
Fig. 2.19 Subcritical pitchfork bifurcation diagram. (Strogatz).
The nonzero fixed points are now unstable, and exist only below the bifurcation
(𝑟 < 0), which motivates the term “subcritical”. Furthermore the origin is stable
for 𝑟 < 0 and unstable for 𝑟 > 0 as in the supercritical case, but now the
instability is not opposed by the cubic term, actually it is helped by it. We have
to say that in real physical system such an explosive instability is usual opposed
by the stabilizing influence of higher-order terms.
The concepts discussed above, relative to bifurcations of one-dimensional
systems, can be extended to the two-dimensional or higher dimensional cases
as well. In all of the examples above, the bifurcation occur when at least one of
the eigenvalues of the problem equals zero, i.e. when det( 𝐴) = 0. Because of
this, the saddle-node, transcritical and pitchfork bifurcations are also named
zero-eigenvalue bifurcations. Such bifurcations always involve the collision of
two or more fixed points. The next kind of bifurcation that will be considered
has no counterpart in one-dimensional systems, and provides a way for a fixed
point to lose stability without colliding with any other fixed points.

48. 43
Hopf Bifurcations
If we consider a two-dimensional system, which have a stable fixed point, the
possible ways this point can lose its stability as a parameter varies depend on
the eigenvalues of the Jacobian. If the fixed point is stable, then the eigenvalues
𝜆1, 𝜆2 must both lie in the left half-plane 𝑅𝑒( 𝜆) < 0.
Fig. 2.20 Eigenvalues related to a fixed point of a two-dimensional system. (Strogatz).
As we can see in figure 2.19, either the eigenvalues are both real and negative,
or they are complex conjugates. To destabilize the fixed point we need one or
both of the eigenvalues to cross into the right-half plane as a parameter varies.
The previous analysed cases involved a real eigenvalue passing through 𝜆 = 0
(saddle-node, transcritical, and pitchfork bifurcations). Now we consider a
scenario in which two complex conjugate eigenvalues simultaneously cross the
imaginary axis into the right half-plane.
 Supercritical Hopf bifurcation
Let’s consider a physical system in which small disturbances exponentially
decay until the system settles down to equilibrium (i.e. exponentially
damped oscillations). Now let’s suppose that the decay rate depends on a
control parameter 𝜇. If the decay becomes slower and slower and finally
changes to growth at a critical value 𝜇 = 𝜇 𝑐, the equilibrium state will lose
stability. The resulting motion is, in many cases, a small-amplitude
oscillation about the former steady state. In this case we say that the system

49. 44
is undergone a supercritical Hopf bifurcation. In terms of phase space, a
supercritical Hopf bifurcation occurs when a stable spiral changes into an
unstable spiral surrounded by a small, nearly elliptical limit cycle. Hopf
bifurcations occur in phase spaces of any dimensions 𝑛 ≥ 2.
Fig. 2.21 Oscillations changing from decaying to growing as the parameter 𝜇 reaches the value 𝜇 𝑐:
Supercritical Hopf Bifurcation. (Strogatz).
To provide an example of the supercritical Hopf bifurcation, let’s consider the
system:
𝑟̇ = 𝜇𝑟 − 𝑟3
𝜃̇ = 𝜔 + 𝑏𝑟2
In this system there are three parameters: μ controls the stability of the fixed
point at the origin, ω gives the frequency of infinitesimal oscillations, and b
determines the dependence of frequency on amplitude for larger amplitude
oscillations. When 𝜇 < 0 the origin 𝑟 = 0 is a stable spiral, whose sense of
rotation depends on the sign of ω. For 𝜇 = 0 the origin is still a weaker stable
spiral. For 𝜇 > 0 there is an unstable spiral at the origin and a stable circular
limit cycle at 𝑟 = √ 𝜇.
Fig. 2.22 Equilibrium point changing stability as the parameter μ passes from negative to positive
sign. (Strogatz).

50. 45
 Subcritical Hopf bifurcation
The subcritical case is more dramatic and potentially dangerous in many
engineering applications. In fact, after the bifurcation, the trajectories must jump
to a distant attractor, which may be a fixed point, or another limit cycle, or also
infinity. Let’s consider the system
𝑟̇ = 𝜇𝑟 + 𝑟3
− 𝑟5
𝜃̇ = 𝜔 + 𝑏𝑟2
The important difference from the supercritical case is that now the cubic term
is destabilizing, so it helps to drive trajectories away from the origin. As we can
see in Figure 2.23, for 𝜇 < 0 there are two attractors, a stable limit cycle and a
stable fixed point at the origin. Between them there is an unstable cycle, shown
as a dashed curve.
Fig. 2.23 Example of subcritical Hopf bifurcation. (Strogatz).
As μ increases, the unstable cycle tends to tighten around the fixed point, until
it engulfs the origin as μ approaches to 0, rendering the origin unstable. A
subcritical Hopf bifurcation occurs at 𝜇 = 0. In fact, for 𝜇 > 0 the stable limit
cycle is suddenly the only attractor and solutions that used to remain near the
origin are now forced to grow into large-amplitude oscillations.

51. 46
2.5 Bifurcations in forced nonlinear oscillations
When a damped nonlinear system is forced, one of the most likely steady-state
responses is for a limit cycle (also called periodic orbit) type of behaviour.
When the system has more degrees of freedom, limit cycles exist for each
degree of freedom (or mode of vibration). In the linear single and multi-degree
of freedom systems, varying the forcing frequency leads to changes in limit
cycle amplitude, but no changes in the structure of the limit cycles occurs. For
nonlinear systems, the limit cycle structure can change and the points at which
this happens are bifurcation points. Bifurcations of limit cycles lead to structural
changes in the resonance behaviour of nonlinear oscillators. One of the most
common examples of this phenomena is given by the Duffing oscillator, as we
can see in Figure 2.24. There are different approaches to obtain a bifurcation
diagram. One of them is the brute-force approach, which will be briefly
discussed now. The first step is to compute a time series of the system, allowing
a large enough number of forcing periods to decay so that steady-state behaviour
has been reached. Then plot the amplitude of one of the system states (usually
the displacement), before incrementing the parameter by a small amount and
then repeating. It is important not to reset the initial conditions after each
parameter increment. This means that we have to keep the last 𝑥 𝑛 and t values
from the steady state and use them as the initial conditions after the parameter
has been incremented. Once the maximum parameter value is reached, the
process should be repeated for decreasing parameter values through the full
range, back to the starting value. This will allow any regions of hysteresis to be
captured. There is a range of sophisticated associated numerical techniques as
well, which make it possible to start from a fixed point and then continue the
path of the fixed point in the state space, as a parameter is varied. Where brute
force will normally capture only stable steady state solutions, continuation

52. 47
methods can be used to capture both stable and unstable branches, as shown in
Figure 2.24.
Fig. 2.24 Nonlinear resonance with hysteresis (Duffing oscillator). (Wagg-Neild).
The parameter varied is the ratio of the forcing frequency to the linear natural
frequency of the oscillator, 𝜇 = 𝛺/𝜔 𝑛. The measurement taken, for each
frequency value, is the maximum displacement per forcing period. The resulting
bifurcation curve is then the curve defining the resonance amplitude of the
oscillator. It is similar to the linear case, but the resonance peak is bent to the
right, and contains two fold bifurcations, on either side of a region of hysteresis.
In this contest, hysteresis means that a different behaviour is obtained for
increasing or decreasing Ω. As Ω increases, a stable solution path gradually
increases in amplitude until it reaches fold A. Here, the stable path joins an
unstable path of solutions. If Ω is increased beyond the bifurcation point, there
is a jump to the lower stable branch. When decreasing Ω from above the
resonance, the stable path continues until fold B, where there is a jump up to the
upper stable branch. The region between fold A and fold B is the region of
hysteresis. This type of resonance is associated with hardening nonlinearity,

53. 48
meaning a spring which becomes stiffer as it displaces further. The opposite
case is a softening spring, meaning a spring which becomes less stiff as it
displaces further. This leads to a resonance peak that bends to left, as shown in
Figure 2.25.
Fig. 2.25 Nonlinear bifurcation diagram: softening spring. (Wagg-Neild).
Hardening and softening cases are both described by the Duffing equation,
which will be a centre point of this work:
𝑀𝑥̈ + 𝐶𝑥̇ + 𝐾𝑥 + 𝐾 𝑁𝐿 𝑥3
= 𝐹(𝑡)
In this equation, 𝐾 and 𝐾 𝑁𝐿 are respectively the linear and nonlinear stiffness.
Unlike the linear case, the elastic coefficients do not only influence the position
of the resonant peak, but also the shape of the entire frequency response. If we
consider 𝑘 > 0, two cases are possible:
 If 𝑘 𝑁𝐿 > 0 the system experiences a hardening case.
 If 𝑘 𝑁𝐿 < 0 the system experiences a softening case.

54. 49
CHAPTER 3
THEORY BACKGROUND AND DESIGN OF THE
EXPERIMENTAL RIG
SUMMARY
3.1 Backbone curves
3.2 Harmonic Balance
3.3 Quick review of modal analysis theory
3.4 Beams
3.5 Project and design of the experimental rig

55. 50
3.1 Backbone curves
In Chap.1, section 1.3, the response of a linear second-order oscillator has been
derived, and an expression for the amplitude of the response (Eq. 1.10) and for
the phase difference between the forcing sine wave and the response sine wave
(1.12) have been obtained as well. Let’s report here these results:
𝑋𝑟 = (
𝐹0
𝑘
)
1
√(1 − (
Ω
𝜔 𝑛
)
2
)
2
+ 4 (ζ
Ω
𝜔 𝑛
)
2
𝛷 = arctan (
2ζ
Ω
𝜔 𝑛
1 − (
Ω
𝜔 𝑛
)
2)
The resonance behaviour for the linear oscillator is defined by this two
equations. For lightly damped systems, resonance peaks are fundamental to
model the vibration response, because they represent significant amplifications
of the input forcing signal. For this reason it is important to have a model for
the frequency-amplitude behaviour of the system. While for the linear systems
this behaviour can be easily modelled, for nonlinear systems resonant behaviour
is much more complex, and the natural frequency is often a function of
oscillation amplitude. An example of a resonance curve for a nonlinear system
is shown in Figure 3.1(a). In this graph it is shown an important tool in the
understanding of nonlinear resonance curves: the backbone curve. A backbone
curve defines the natural frequency as a function of the amplitude of the
response of the undamped and unforced system. For a linear oscillator the
backbone curve is a vertical line at 𝛺 = 𝜔 𝑛, where 𝜔 𝑛 = √𝑘 𝑚⁄ is the natural
frequency, in the frequency-amplitude diagram. So, finding an approximate
backbone curve for a nonlinear oscillator gives an indication of the distortion
from linear resonance. In addition, as the backbone curve represents the system

56. 51
without damping and forcing, calculating an approximate curve is considerably
simpler than calculating the resonance curve for the full system. As the physical
systems usually considered are lightly damped (as well as weakly nonlinear),
the backbone curve is normally a reasonable approximation to the system
response.
Fig. 3.1 Frequency response for a nonlinear oscillator, representing: (a) the backbone curve for the
undamped and unforced system, and the resonance peak for the forced and damped system, and (b)
the phase lag for the forced and damped system. (Wagg-Neild).
However, like in the linear case, the extent of the nonlinear resonance curve
depends mainly on the magnitude of the damping in the system. Note that the
backbone curve defines how the undamped natural frequency increases with
amplitude. So, to define a backbone curve, an approximate relationship between
natural frequency and response amplitude is required. However, the backbone

57. 52
curve itself is not enough to define the resonance peak. Anyway, the value at
which the resonance curve crosses the backbone curve can be taken as an
approximation to the peak value of amplitude. To locate this point, let’s first
consider what happens in a linear resonance peak. In this case, as the resonance
peak crosses the backbone curve, there is a rapid change in phase, from values
close to zero to values close to 𝜋, with all cases passing through 𝛷 = 𝜋/2 , as
shown in Figure 1.6 (b). Nonlinear oscillators show a similar behaviour. The
phase at resonance can be approximated as 𝛷 = 𝜋/2 and can be used to
approximate the point at which the resonance curve crosses the backbone curve,
as shown in Figure 3.1. As we can see in this figure, there is a jump in the phase
plot close to resonance, corresponding to jumps in amplitude of the hysteresis
region. For hardening peak, like the one in Figure 3.1, it is increasing Ω that
gives a phase jump close to the maximum amplitude. For softening, decreasing
Ω would give equivalent response. So, in conclusion, to define a backbone curve
two things are required:
 A frequency-amplitude relationship.
 An estimate of the peak value.
There are different techniques which can be used to estimate the frequency-
amplitude relationship. In the next paragraph one of these techniques will be
briefly described: the harmonic balance.
3.2 Harmonic Balance
In the first chapter we have briefly seen that, for a given nonlinear system,
governed by the equation 𝒙̈ + 𝟐𝜻𝝎 𝒏 𝒙̇ + 𝝎 𝒏 𝒙 𝟐
+ 𝜶𝒙 𝟑
= 𝑭 𝐬𝐢𝐧(Ω𝒕), with a
single frequency harmonic forcing, supposing a response in the harmonic form
𝒙 = 𝑿 𝒓 𝐬𝐢𝐧(Ω𝒕 − 𝜱), and substituting it into the governing equation we obtain
a multi-frequencies response. This means we will have a response at other
frequencies than just the input frequency, and this behaviour is due to the

58. 53
nonlinear cubic term. Furthermore, we have seen that the frequency response is
a function of the response amplitude and its approximate expression is:
𝜔𝑟 ≈ 𝜔 𝑛√1 +
3𝛼𝑋𝑟
2
4𝜔 𝑛
2
The process of assuming a harmonic (sine and cosine waves) solution is the first
part of the harmonic balance technique. After substituting the assumed solution
into the governing equation, the second step is to balance the coefficients of the
harmonic terms. Harmonic balancing is a technique which allows the
calculation of the approximate steady-state system response. While for a linear
oscillator an exact balance can be obtained, for a nonlinear system finding a
solution to the system response is more difficult. This is because nonlinear
systems can have a multi-frequencies response. So, in order to fully balance the
harmonic terms, the trial solution for x must include a summation of all the
relevant harmonics and subharmonics. A general example of a trial solution for
a nonlinear system, with a primary response at frequency 𝜔𝑟 would have this
form:
𝑥 = 𝑎0 + ∑ 𝑎 𝑛
∞
𝑛=1
cos( 𝑛𝜔𝑟 𝑡) + 𝑏 𝑛 sin( 𝑛𝜔𝑟 𝑡) 𝑛 = 1,2,3 … ,
where 𝑎 𝑛 and 𝑏 𝑛 are coefficients. In summary, the harmonic balance approach
is first to apply a trial solution with a reduced number of terms (often limited to
just terms at frequency 𝜔𝑟) and then to balance just the terms present in the trial
solution, ignoring any higher frequency terms generated by the nonlinearity. By
applying this method, an approximated backbone curve expression, i.e. a
frequency-amplitude relationship, is found. As already observed, this curve
alone is not enough to understand the resonance behaviour of the damped and
forced system, as the peak amplitude of the response is dominated by the
damping. The response amplitude of the full (damped and forced) system can
be estimated by assuming two things. The first assumption is that the response

59. 54
is dominated by a component at the forcing frequency (this assumption is
similar to that one made when using harmonic balance including only the first
harmonic term). The second assumption is that, as in the linear case, the
resonance peak occurs when the phase lag of the response relative to the forcing
is 𝜋 2⁄ . Generally, the harmonic balance method is considered to be
inconsistent, as higher frequency terms are not balanced using a single
frequency substitution. However, in order to obtain a quick and approximate
idea of the system behaviour, it is an useful technique. Furthermore, this
technique is useful because it can be applied to systems with “large”
nonlinearities, unlike other techniques which requires nonlinear terms to be
“small” in order for the approximation to be valid. The last thing to note about
this method is that it provides a calculation of the steady-state response, without
providing information about the transient behaviour.
3.3 Quick review of modal analysis theory
Linear vibration theory defines a specific set of modes of vibration for the
system under consideration. This modes physically relates to a particular
geometric configuration. In the linear case, the superposition principle is
applied, and consequently the response of the system is given by the sum of the
responses from each mode. So, modal analysis provides a calculation of the
response of the system by considering its vibration modes. An important step to
do this is the modal decomposition, which is the process of transforming the
system from a physical to a modal representation. This is particularly useful in
linear systems because each mode has an associated resonance, and
understanding where resonances could occur in a structure is a key part of
analysing vibration problems. The underlying concept of modal analysis for
linear vibration is that the system response can be represented as the sum of
contributions from a series of mode shapes. It is assumed that each mode shape
is related to a specific physical configuration, function of spatial coordinates

60. 55
within the structure, and not function of the time. The contribution of a
particular mode shape to the total response is represented by the modal
amplitude, which is a function of the time but not of the spatial coordinates. For
example, the modal representation of the displacement response of a continuous
system is:
𝑤( 𝑥, 𝑡) = ∑ 𝛷𝑖( 𝑥) 𝑞𝑖(𝑡)∞
𝑖=1 (3.1)
where 𝛷𝑖 is the 𝑖 𝑡ℎ
mode shape and 𝑞𝑖 is its modal coordinate, which represent
the contribution of that mode to the overall response. In linear systems, each
vibration mode has an associated resonance frequency (or natural frequency)
which occurs at a clearly defined resonance peak. For each frequency value
there is a single amplitude value in the resonance peak: it is a single value
function, which increases monotonically up to the resonant frequency, and then
monotonically decreases after the resonant frequency. At, or near, the point of
resonance, the motion of the linear system is dominated by the vibration mode
corresponding to that particular resonance frequency. In multi-degree-of-
freedom linear systems this means that the state response reduces to a series of
single-degree-of-freedom harmonic oscillators for each of the modes. In
nonlinear systems, the shape of the resonance peaks are often amplitude-
dependent or distorted due to nonlinear effects such as those associated with
hardening or softening springs.

61. 56
3.4 Beams
In order to introduce the main object of this work, a brief discussion about
beams has to be made. In particular, we will focus on the small-deflection beam
theory, governed by the Euler-Bernoulli equation. As a starting point, let’s
consider the classical small-deflection (i.e. linear) approach to the analysis of a
cantilever beam. The beam is assumed to be homogeneous and isotropic such
that, for a constant cross-sectional area, the distributed mass and elasticity of
the beam are constant along its length. It is also assumed to be slender, so that
bending deformation dominates shear deformation, which can be neglected.
Any such continuum requires an infinite number of coordinates in order to
specify the position of every particle in the beam as it vibrates. So, in this sense
the beam possesses an infinite number of degrees of freedom and its governing
equation of motion is a partial differential equation: the Euler-Bernoulli
equation. It is important for the purpose of this work to derive this equation and
to discuss it, because it will be a fundamental starting point for the design phase
of the experiment. When a load acts on a beam, it will deflect into a deformed
shape with a certain curvature. To derive a mathematical model of the beam
behaviour, a series of assumptions must be made about the physical behaviour
of the beam. By considering statics, the basic modelling assumption for beam
bending is that the bending moment, M, at any point in the beam, is proportional
to the curvature. This relationship is represented in the form:
𝑀 = 𝐸𝐼
1
𝑅
= 𝐸𝐼
𝑑𝛹
𝑑𝑠
(3.2)
where 𝐸(𝑁/𝑚2
) is the Young’s modulus, 𝐼(𝑚4
) is the moment of inertia and
𝑅(𝑚) is the radius of curvature. The slope of the beam is defined as 𝛹 and hence
the curvature is defined by
𝑑𝛹
𝑑𝑠
, where 𝑠(𝑚) is the length along the beam.

62. 57
Fig. 3.2 Element of a vertical beam with a forcing function applied along its length. (Wagg-Neild).
The term EI represents the flexural rigidity of the beam and it represents the
resistance offered by a structure while undergoing bending. Further assumptions
are that E and I are constant along the length of the beam, which physically
means that the beam has a uniform cross-sectional area, uniform material
Young’s modulus E and uniform mass distribution. Now the loading on a small
element of beam will be considered. It is assumed that the beam deflection in z
direction, 𝑤(𝑥, 𝑡), is small and the deflection in x direction is negligible, and
finally it is assumed that the rotary inertia of the element can be ignored.
Considering the element of beam in figure 3.1, if the force per unit length at the
position x is F, the force per unit length at the position x+Δx is F+ΔF, and the
same thing counts for the bending moment M and for the shear force V. So,
considering the force equilibrium in the z direction, and of the moment
equilibrium about point O:
 → 𝑧 − ( 𝑉 + 𝛥𝑉) + 𝑉 + (𝐹 +
𝛥𝐹
2
) 𝛥𝑥 = 𝜌𝐴𝛥𝑠
𝜕2 𝑤
𝜕𝑡2
(3.3)
 ↻ 𝑂 ( 𝑉 + 𝛥𝑉)
𝛥𝑥
2
+ 𝑉
𝛥𝑥
2
+ 𝑀 − ( 𝑀 + 𝛥𝑀) = 0 (3.4)
In these equations 𝜌 is the density, A is the cross sectional area. As the
deflections are assumed to be small, it is assumed that 𝛥𝑠 ≅ 𝛥𝑥. Using this

63. 58
approximation and taking the limit as 𝛥𝑥 → 0, neglecting small terms, equations
(3.3) and (3.4) becomes:
 −
𝜕𝑉
𝜕𝑥
+ 𝐹( 𝑥, 𝑡) = 𝜌𝐴
𝜕2 𝑤
𝜕𝑡2
(3.5)
 𝑉 =
𝜕𝑀
𝜕𝑥
(3.6)
Now, substituting Eq.(3.5) into Eq.(3.4) gives
𝜕2
𝑀
𝜕𝑥2
+ 𝜌𝐴
𝜕2
𝑤
𝜕𝑡2
= 𝐹( 𝑥, 𝑡) (3.7)
Finally, the moment relationship of Eq. 3.1 can be approximated based on the
small deflection assumptions by writing 𝛹 ≅ 𝛥𝑤/𝛥𝑥 and noting that 𝛥𝑠 ≅ 𝛥𝑥
𝑀 = 𝐸𝐼
𝜕𝛹
𝜕𝑠
→ 𝑀 = 𝐸𝐼
𝜕2
𝑤
𝜕𝑥2
Using this relationship to eliminate M from Eq. (3.6) gives:
𝐸𝐼
𝜕4
𝑤
𝜕𝑥4
+ 𝜌𝐴
𝜕2
𝑤
𝜕𝑡2
= 𝐹( 𝑥, 𝑡) (3.8)
which is the linear undamped Euler-Bernoulli equation for beam vibration with
a time-varying distributed load applied along its length, and it can be applied to
model the transverse (z direction) vibration of the beam for small displacements.
Free vibration
In the absence of a transverse load, the equation (3.8) becomes a free vibration
equation. This equation can be solved using a decomposition of the
displacement into the sum of harmonic vibrations of the form:
𝑤( 𝑥, 𝑡) = 𝑅𝑒[ 𝑤̂( 𝑥) 𝑒−𝑖𝜔𝑡] (3.9)
where ω is the frequency of vibration.

64. 59
For each value of the frequency we can solve the equation:
𝐸𝐼
𝑑4
𝑤̂
𝑑𝑥4
− µ𝜔2
𝑤̂ = 0
Where 𝜌𝐴 = µ (i.e. mass per unit length). The general solution of the equation
is:
𝑤̂ = 𝐴1 cosh( 𝛽𝑥) + 𝐴2 sinh( 𝛽𝑥) + 𝐴3 cos( 𝛽𝑥) + 𝐴4 sin( 𝛽𝑥) (3.10)
with 𝛽 = (
𝜇𝜔2
𝐸𝐼
)
1
4
and the constants 𝐴1, 𝐴2, 𝐴3, 𝐴4 to be determined according to
the boundary conditions and are unique. The solution for the displacement is
not unique and depends on the frequency. These solutions are typically written
as
𝑤̂ 𝑛 = 𝐴1 cosh( 𝛽 𝑛 𝑥) + 𝐴2 sinh( 𝛽 𝑛 𝑥) + 𝐴3 cos( 𝛽 𝑛 𝑥) + 𝐴4 sin( 𝛽 𝑛 𝑥) (3.11)
with 𝛽 𝑛 = (
𝜇𝜔 𝑛
2
𝐸𝐼
)
1
4
. The quantities 𝜔 𝑛 are the natural frequencies of the beam.
Each of the displacement solutions is called a mode and the shape of the
displacement curve is called the mode shape. The importance of this paragraph
will be clear later, when using these relationships the natural frequencies of a
slender beam will be computed. A script in MATLAB® will be used for this
computation. The results will be then compared first with the results given by a
finite elements analysis made using the software ABAQUS and finally with the
experimental results on the real beam.

65. 60
3.5 Project and design of the experimental rig
Introduction
The purpose of this work is to build up a rig which exhibit a nonlinear hardening
behaviour. The structure that will be studied is a cantilever slender beam. In
order to reach this purpose a design work of the experiment has been made, as
will be discussed in this section. The first steps of this phase of the project are
the design of the clamping system, to which the beam will be fixed, and the
design of a frame, that will be used as a connection between the beam and the
features that will be used to generate the nonlinearity in the system. As it will
be discussed in the following sections, two different ways have been thought to
obtain a nonlinear hardening behaviour. The first one, which will be discussed
next, involves the use of two horizontal springs connected, through the frame,
to the free end of the beam. This one is a of nonlinearity deriving from
geometrical causes. The second way instead involves the use of some magnets,
and so in this case the nonlinearity derives from an external force acting on the
system. This last one will be the way that will produce the results sought by this
work.
Choice of the beam and design of the clamping system
The first step of the design project involves the choice of the beam. It needs to
be very slender, and for this purpose we chose a stainless steel ruler, whose
dimensions and mechanical characteristics are reported in the following table.
Material Young
Modulus
E
Density Length Width Thickness Mass
Stainless
steel
grade
4310
200 Gpa 7900
𝐾𝑔 𝑚3⁄
0.33 m 0.03 m 0.00105 m 0.0821Kg
Table 1 Mechanical and geometrical properties of the beam.

66. 61
Fig.3.3 Stainless steel beam designed using the software SolidWorks®.
The mass of the beam has been calculated by knowing the volume of the beam
and its density. Regarding the clamping system, first it has been designed with
the CAD software SolidWorks®, then it has been realized in iron by the
mechanical workshop of the University of Glasgow. It is constituted by a square
plan base, with two wings with holes, in order to fix the clamping system to a
table, and a square cover. The top of the base has four threaded holes that allow
to fix the cover to the base through four screws. The beam will be fixed between
these two components. In the following pictures are reported the drawings of
the two parts which compose the clamping system, and the clamping system
realized by the mechanical workshop. The dimensions have been expressed in
millimetres.

67. 62
Fig. 3.4 Drawings of the base of the clamping system, three views.
Fig. 3.5 Drawings of the cover of the clamping system, three views.

68. 63
Fig. 3.6 CAD design of the clamping system realized with SolidWorks®.
Fig. 3.7 Clamping system.

69. 64
After this, a characterization of the beam has been done, both experimentally
and numerically, in order to compute its natural frequencies. Given that the
beam is very slender and has a homogeneous cross sectional area along its
length, the Euler-Bernoulli beam theory can be used in order to derive the
natural frequencies of the beam. According to this theory, in the case of a free
vibration analysis, the deflection of the beam is given by Eq.(3.11):
𝑤̂ 𝑛 = 𝐴1 cosh( 𝛽 𝑛 𝑥) + 𝐴2 sinh( 𝛽 𝑛 𝑥) + 𝐴3 cos( 𝛽 𝑛 𝑥) + 𝐴4 sin( 𝛽 𝑛 𝑥)
Where 𝛽 𝑛 = (
𝜇𝜔 𝑛
2
𝐸𝐼
)
1
4
and 𝜔 𝑛 are the natural frequencies of the beam. Now,
considering the cantilever beam, applying the boundary conditions and
substituting Eq.(3.11) what we obtain is a function where the only unknown
term is 𝛽 𝑛.
Fig.3.8 Scheme of the cantilever beam
 𝑤(0) = 0
 𝑤′(0) = 0
 𝑤′
′( 𝐿) = 0
 𝐸𝐼𝑤′′′′( 𝐿) = 0
By substituting Eq.(3.11) in place of 𝑤 we obtain:
 𝐴 = −𝐶
 𝐵 = −𝐷
 𝐴 cos( 𝛽 𝑛 𝐿) + 𝐵 sin( 𝛽 𝑛 𝐿) − 𝐶 cosh( 𝛽 𝑛 𝐿) − 𝐷 sinh( 𝛽 𝑛 𝐿) = 0
 −𝐴 sin( 𝛽 𝑛 𝐿) + 𝐵 cos( 𝛽 𝑛 𝐿) − 𝐶 sinh( 𝛽 𝑛 𝐿) − 𝐷 cosh( 𝛽 𝑛 𝐿) = 0
Rearranging these equations we obtain:
0 L

70. 65
1 + cosh( 𝛽 𝑛 𝐿) ∙ cos( 𝛽 𝑛 𝐿) = 0
By using the MATLAB® function “fsolve” it is possible to find the zeros of this
function, that is the values of 𝛽 𝑛 which satisfy the equation 𝐹(𝛽 𝑛) = 0. In this
way, giving to fsolve two values in the neighbourhood of which found the zeros,
I have found the first two values of 𝛽 𝑛:
 𝛽1 = 1.875
 𝛽2 = 4.694
From these two values we obtain:
 𝜔1 = 7.84 𝐻𝑧
 𝜔2 = 49.1 𝐻𝑧
Now we are going to compare these two values with the same values computed
with a finite elements method, using the software Abaqus. Using Abaqus, a
finite elements model of the beam has been realized. The boundary conditions
of a clamping system at one end of the beam have been applied and then a free
response modal analysis has been accomplished. In Figure 3.5 are represented
the first two mode shapes of the cantilever beam:
Fig. 3.9 Finite elements representation of the first two mode shapes – Abaqus.

71. 66
Finally an experimental free response analysis of the beam has been conducted,
and the results have been compared with the ones deriving from MATLAB®
and Abaqus. In order to experimentally compute the natural frequencies of the
beam I used an ICP (Integrated Circuit Piezoelectric Sensor) accelerometer,
connected to the beam and to a data acquisition system. An ICP sensor is made
of piezoelectric material and converts mechanical deformations into electric
signals. The data acquisition system interfaces with a MATLAB® script, which
records the time response of the system, i.e. provides the Amplitude-vs-Time
plot, where the amplitude of the oscillation is expressed in volt, because the
output of the accelerometer is a voltage. By knowing the sensitivity of the
accelerometer (1.087 𝑚𝑉/ 𝑚 𝑠2⁄ ) it is then possible to pass from a voltage to
an acceleration. In order to experimentally characterize the beam, we have to
provide an initial displacement, different from zero, and then let the beam
vibrate until it reaches the rest position. With the data acquisition system we
obtain the so called “decay curve”: the amplitude of the oscillation decays with
the time, because of the presence of the damping (cf. paragraph 2.2.1). In Figure
10 is reported the decay curve of the beam, experimentally computed:
Fig. 3.10 Free response of the beam, decay curve. Experimental data.

72. 67
In order to obtain the natural frequencies of the beam, we have to switch from
the time domain to the frequency domain, using a Fourier transform. Fourier
analysis converts a signal from its original domain (often time or space) to a
representation in the frequency domain, breaking a waveform into an alternate
representation, characterized by a sum of sinusoidal functions. The frequency
response of the beam has been computed in this way, and the results are shown
in Figure 3.11:
Fig. 3.11 Frequency response of the beam.
In Figure 3.11 are visible two resonance peaks, related to the first two natural
frequencies of the beam. The results obtained are the following:
 𝜔1 = 7.1 𝐻𝑧
 𝜔2 = 44.3 𝐻𝑧
The following table summarize the results deriving from the experimental test
and from the numerical simulations.
𝜔1 𝜔2
MATLAB® 7.84 Hz 49.11 Hz
Abaqus 7.88 Hz 49.40 Hz
Experimental Test 7.13 Hz 44.25 Hz
Table 2 Beam characterization, comparison of the results

73. 68
Comparing the results we note that the results coming from the modelled beam,
both with MATLAB® and with Abaqus, are nearly the same. The experimental
results are slightly different, but this very little difference is reasonable and it’s
due to some factors, like some approximations made in the model. To provide
an example, an important factor which can affect the results is the thickness of
the beam, more exactly the difference between the nominal thickness (used in
the models) and the real one, which can be slightly different because of
fabrication tolerances. At this point, we have the clamping system and the beam,
so the last structural component which will complete our experimental rig is the
frame, which will be used as connection structure between the beam and the
features which will provide nonlinearity to the system. This structure is a simple
squared iron frame, and it has been realized by the mechanical workshop of the
University of Glasgow. In order to continue with the development of the
experiment, in the following section the first project attempt will be described,
and the reasons of its failure will be discussed. After that, the second and
definitive solution will be discussed.

74. 69
CHAPTER 4
EXPERIMENTAL RIG: HORIZONTAL SPRINGS
SOLUTION
SUMMARY
4.1 Introduction
4.2 Mathematical model
4.3 Backbone curve: analytical and experimental computation
4.4 Conclusions

75. 70
4.1 Introduction
As we said in the first chapter, several types of nonlinear behaviour can occur,
depending on the particular case under exam. Nonlinearity, in fact, can derive
from material properties of the system, or from its geometrical characteristics
for example, as well as from some other causes. This work aims to project and
realize a structure that exhibits a nonlinear behaviour, and the nonlinearity
sought is obtained modifying some geometrical features of the structure. In this
chapter, we will focus on a particular solution, which involves the use of two
springs in order to obtain a nonlinear behaviour. The idea is to connect two
springs to the beam, horizontally, through the squared frame placed at the free
end of the beam. The nonlinearity will derive from the restoring force of the
horizontal springs, more exactly from its component in the direction of the
deflection, as it will be better discussed next. In order to connect the springs to
the beam, a small pin has been welded to the free end of the beam, and two
hooks have been installed on the frame. Furthermore, two more hooks have been
installed on the frame, in order to connect two vertical springs to the free end of
the beam. These vertical springs can be used to tune the linear stiffness of the
system, by adding the effect of the stiffness of the springs to the effect of the
bending stiffness of the beam. In the next paragraphs, a mathematical model of
the system will be obtained, and the reasons why this solution does not allow us
to appreciate a consistent nonlinear behaviour will be explained. The following
picture represents the whole experimental system, composed by the clamping
system, the beam, the frame and the springs, both the horizontal and the vertical
ones.