1. science
2. mathematics
3. algebra

Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
Porto, Portugal, 30 June - 2 July 2014
A. Cunha, E. Caetano, P. Ribeiro, G. Müller (eds.)
ISSN: 2311-9020; ISBN: 978-972-752-165-4
A virtual experiment for the detection of specific features in the frequency response
of a coupled nonlinear and linear oscillator
Gianluca Gatti1, Stefano Marchesiello2, Michael J. Brennan3
Dipartimento di Ingegneria Meccanica, Energetica e Gestionale, Università della Calabria, 87036 Rende (CS), Italy
2
Dipartimento di Ingegneria Meccanica e Aerospaziale, Politecnico di Torino, 10129 Torino, Italy
3
Departamento de Engenharia Mecanica, Universidade Estadual Paulista, SP15385-000 Ilha Solteira, Brazil
email: [email protected], [email protected], [email protected]
1
ABSTRACT: This paper presents an investigation into some practical issues that may be present in a real experiment, when
trying to validate the theoretical frequency response curve of a two degree-of-freedom nonlinear system consisting of coupled
linear and nonlinear oscillators. Some specific features, such as detached resonance curves, have been theoretically predicted in
multi degree-of-freedom nonlinear oscillators, when subject to harmonic excitation, and the system parameters have been shown
to be fundamental in achieving such features. When based on a simplified model, approximate analytical expression for the
frequency response curves may be derived, which may be validated by the numerical solutions. In a real experiment, however,
the practical achievability of such features was previously shown to be greatly affected by small disturbances induced by gravity
and inertia, which led to some solutions becoming unstable which had been predicted to be stable. In this work a practical
system configuration is proposed where such effects are reduced so that the previous limitations are overcome. A virtual
experiment is carried out where a detailed multi-body model of the oscillator is assembled and the effects on the system
response are investigated.
KEY WORDS: Nonlinear oscillations; Detached resonance curves; Cubic stiffness; Parameter estimation.
1
INTRODUCTION
When a nonlinear oscillator is attached to a linear host
structure, which is excited by a harmonic force, multivaluedness of the steady-state response can occur [1]. In this
situation, a closed detached curve may appear within the main
continuous frequency response curve (FRC). The co-existence
of two types of the steady-state solutions, one of which is an
outer detached curve, was shown theoretically for a purely
cubic attachment [2], and experimental results confirmed
qualitatively the existence of this distinctive feature of the
response. The existence of an outer detached curve was also
identified in [3-6], where a purely nonlinear oscillator was
attached to a linear two degree-of-freedom system, illustrating
the changes in the FRCs when increasing the amplitude of
excitation.
Motivated by the dynamic testing of a nonlinear system, the
authors of this paper have recently shown [7,8] that if excited
by a harmonic force, a system consisting of a nonlinear
oscillator weakly coupled to a linear oscillator can have a
multivalued primary resonance response in which closed
detached resonance curves can appear also as inner detached
curves. These detached curves were found analytically and
their presence was confirmed numerically, both for the purely
cubic oscillator [7], and for the linear-plus-cubic oscillator [810] - the effects of the system parameters on the existence of
inner and outer detached curves were also investigated.
Recently, to investigate the effects related to the practical
implementation of an experimental rig used for validating the
theoretical predictions of such inner detached resonance
curves, the authors of this paper determined that gravity and
some of the physical parameters of the mechanical
components are critical in achieving a specific system
response [11, 12].
The aim of this paper is to investigate the reduction of
unwanted dynamic effects in a practical system configuration
so that a specific frequency response curve can be obtained. A
virtual experiment is carried out where a detailed multi-body
model of the system is assembled, and physical dimensions,
mass distribution and gravity are taken into consideration. As
in a real experiment, where some of the systems parameters
are usually unknown, parameter estimation is performed
based on the data obtained by the virtual experiment, so that
the performance of the nonlinear identification technique [1315] is also tested in case of disturbances.
2
2.1
SYSTEM DESCRIPTION AND MODELLING
Experimental rig and motivations
The practical system of interest in this work is depicted in
Figure 1(a) and (b). A suspended mass m, is attached to a
shaker via a support frame. The four springs between the
suspended mass and the support frame, which are made of
fishing line, are modelled as four elements of stiffness k and a
damper c. The initial tension in the wires was adjusted to have
a quasi-zero stiffness behavior [6]. When the suspended mass
vibrates in the horizontal direction, the springs stretch and this
generates a geometric nonlinearity.
Experimental tests (not reported in this paper) performed on
the real system depicted in Figure 1(a) did not confirm the
theoretical predictions, and this is why some further work on
this system has been recently performed [11, 12]. As shown in
that work, due to the low initial tension in the wires – which
was necessary to ensure a quasi-zero stiffness characteristic
and the consequent appearance of an inner detached resonance
curve [6] – and the physical dimensions of the suspended
mass, a rocking motion of this mass due to gravity was
observed. This was the main cause of the asymmetry, and the
1973
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
reason why some branches in the FRC were not observed in
practice.
A modification to the system, which is shown in the virtual
model shown in Figure 2, is proposed in this paper to
overcome the previous practical limitations. In this case, four
more springs are used in parallel to the previous set of springs
in order to limit the rocking motion, i.e. the “wobbling”,
caused by the physical length of the suspended mass and the
unbalance related to the offset between the center of gravity of
the mass and the point of intersection of the spring direction
lines.

f s  8kz  1 




2
2
z  d 
d0
(1)
where d0  d is the length of the unstretched spring. Using
the Taylor-series expansion to the third order for small z, Eq.
(1) can be written as
f s  k1 z  k3 z 3
(2)
where k1  8k 1  d0 d  and k3  4k d0 d 3 .
The electro-dynamic shaker, which is used to excite the
system, can be modelled as a linear system consisting of a
parallel combination of a spring ks and a damper cs connected
to a mass ms, which is made up of the moving mass of the
shaker and the support frame.
Using the approximate expression for the spring restoring
force, the equations of motion of the system depicted in
Figure 2 are
ms xs  cs xs  ks xs  cz  k1 z  k3 z 3  f  t 
(a)
mxs  mz  cz  k1 z  k3 z 3  0
(3a,b)
where xs is the shaker displacement, x is the displacement of
the suspended mass and z  xs  x is the corresponding
relative displacement. If the shaker is driven with a constant
current at each frequency, the excitation can be modelled as a
harmonic force with constant amplitude, f  t   F cos t  ,
and Eqs. (3a,b) can be written in non-dimensional form as
ys 1     2 s ys  ys  cos      w
(b)
Figure 1. Preliminary experimental rig: photo (a) and virtual
model (b).
w  2 w  02 w   w3  ys
(4a,b)
where   d  d , in which   s t is non-dimensional
time; ys  xs x0 , y  x x0 and w  z x0 are the normalized
absolute displacements and normalized relative displacement,
respectively, in which x0  F ks ;    s is the nondimensional frequency in which s  ks ms and   m ms
is the mass ratio;   k3 x02  ks is the nonlinear coefficient;
 s  cs 2mss and   c 2ms are the damping ratios; and
the following non-dimensional parameter is also defined
0  1 s , where 1  k1 m.
Figure 2. Virtual model of the proposed system modification.
2.2
Amplitude-frequency equations
The relationship between the applied static force fs in Figure
2, and the resulting relative displacement z is given by
1974
Approximate solutions for the equations of motion given by
Eqs. (4a,b) are found by applying the harmonic balance
method. For this purpose, the normalized absolute
displacement of the shaker and the normalized relative
displacement between the shaker and the suspended mass,
ys  Ys cos    s 
given
respectively
by
and
w  W cos      , are substituted into Eqs. (4a,b) to give
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
aW 6  bW 4  cW 2  d  0
Ys2 
1W 4 g W 2h
e
(5a,b)
where
3
g
b   02   2    4 ,
2
e
2
h
c  02   2   4 2  2   4 ,
e
2
4 1
2
d   , e  1   1      4 s2  2 ,
(6a-g)
e
3
g    1   2 1     ,
2
4 2
h     2 1   2 1    02   2   8 s  2 .
a
9 2
 ,
16
Equations (5a,b) form a set of coupled algebraic equations in
which Eq. (5a) is cubic in W2 , and Eq. (5b) gives the
displacement amplitude of the shaker as a function of the
relative displacement amplitude between the suspended mass
and the shaker. This means that the system of equations Eqs.
(5a,b) may yield up to three steady-state solutions for W and
Ys, the type of which depends on the discriminant  of the
cubic polynomial on the left-hand side of Eq. (5a), which is
given by
  18abcd  27a2 d 2  4b3 d  b2c2  4c3 a.
correspond to those of the experimental rig shown in Figure
1(a). By then applying the approximate relation given in
Eq.(2), the linear and cubic stiffness coefficients are found to
be k1  28.235 N/m and k3  1.382 108 N/m3.
Table 1. Dynamic parameters of the physical system.
ms
[kg]
0.154
ks
[N/m]
1.88∙104
cs
[Ns/m]
9.86
m
[kg]
0.0052
k
[N/m]
4∙104
c
[Ns/m]
0.026
Table 2. Geometric parameters of the physical system.
d
[m]
0.034
d0
[m]
0.033997
Lm
[m]
0.025
Rm
[m]
0.004
LG
[m]
Lm/8
The Nonlinear Subspace Identification (NSI) method [14] is
used to identify the nonlinear coefficient k3 and the frequency
response functions (FRFs) of the underlying linear system (i.e.
of the linear part of the system equations of motion), from the
system response to a random force excitation. The physical
parameters of the underlying linear system are then extracted
from those FRFs in a classical way, as discussed below.
According to the proposed method, it is assumed that
nonlinearities are localized and that an underlying linear
regime of vibration exists. In this case, the equations of
motion may be written in the following general form
p
Mz  t   Cz  t   Kz  t    s j Lnj v j  t   f  t 
(7)
j 1
If   0 then there are three distinct real roots, which
correspond to three steady-state solutions (two stable, one
unstable); if   0 there is one real root and a pair of
complex conjugate roots, which corresponds to a single
steady-state solution, and if   0 then at least two roots
coincide so there are at least two coincident solutions, which
occur at the jump-up and jump-down frequencies [16,17].
These are the frequencies when there is a sudden
discontinuous change of the amplitude of the response when
the frequency is varied very slowly.
To determine the FRCs of Ys and W as functions of , Eqs.
(5a,b) are solved for particular values of the system
parameters, and the stability of these steady-state solutions is
calculated by applying Floquet theory as described in [18].
Unstable solutions are shown as dashed lines in the FRCs
shown in this paper.
3
PARAMETER IDENTIFICATION
In order to verify the analytical expressions for the FRCs
obtained theoretically in the previous section and to validate
the approximation used in Eq. (2), a multi-body model of the
system is implemented and shown in Figure 2. Each wire is
modelled as a linear spring and the suspended mass is
modelled as a cylindrical body with a specified length, Lm, and
radius, Rm. Its center of mass is defined relative to the
intersection point of the wires by the offset LG, while the wires
are pinned at the outer surface of the cylindrical body. The
dynamic and geometric parameters are listed in Table 1 and
Table 2, respectively, and the numerical values approximately
where M, C and K are the mass, damping and stiffness
matrices respectively, z(t) is the generalized displacement
vector, f(t) is the generalized force vector, and the nonlinear
term is expressed as a summation of p nonlinear components,
which depends on a scalar nonlinear coefficient sj, a location
vector Lnj related to the location of the nonlinear element in
the system equation of motion (whose entries may assume the
values 1, -1 or 0), and a scalar nonlinear function vj(t), which
specifies the class of the nonlinearity (e.g. friction, clearance,
quadratic damping, cubic stiffness, etc.). The summation of
the nonlinear terms in Eq. (7) may be interpreted as an
internal feedback force vector and may be shifted to the righthand side of the equation, so that it may be conveniently
turned into a discrete r-sampled state space model [14] as
x r 1  Ax r  Bu r
y r  Cx r  Du r
(8)
where the output yr contains the experimental measurements
(displacements, accelerations etc.), the input ur includes both
the applied forces and internal nonlinear feedback forces, the
state vector is xr, whose dimension defines the order of the
model, and A, B, C and D are the state space matrices, which
depend on the system parameters.
The nonlinear identification procedure is based on the
computation of system parameters (sj and those included in
M, C, K), once the state space matrices have been estimated
1975
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
H12   1   s1 H11  H 21 
H
Hs1L n1  s1  11

   
H 21 H 22   1 s1 H 21  H 22 
by a subspace method in the time domain [19], which involves
a Singular Value Decomposition.
An extended FRF matrix HE() of the system, which also
includes nonlinear terms, may be then calculated as [14]
and the extended FRF matrix becomes

HE    D  C Ie
1t
A

1
B
where t is the discrete-time step and I is the identity matrix.
The matrix HE() obtained as a result of Equation (9) is
then equated to its equivalent form [14] as
HE    H   H   s1Ln1
s H  H 21 
H
H E     11 1 11
.
H 21 s1 H 21  H 22 
(9)
H   s p Lnp 
which is solved, at each frequency, for the nonlinear
coefficients sj and for the matrix H(), which is the FRF
matrix of the underlying linear system.
In particular, the nonlinear coefficients identified from
HE() are complex-valued and frequency-dependent, but an
appropriate selection of the nonlinear functional forms vj(t)
should make the imaginary parts much smaller than the
corresponding real parts. The frequency dependence of the
coefficients should also remain small in a successful
identification.
In order to apply the NSI method to the actual two degreeof-freedom mechanical system, Eqs. (3a,b) are rewritten in
terms of the absolute displacements of the two masses and
then put in a matrix form, yielding
 ms 0   xs  c  cs c   xs 
   ...
 0 m   x    c
c   x 

  
 k  k k1   xs 
...  1 s
   ...
k1   x 
 k1
 f  t 
1
3
...k3    xs  x   


1
 
 0 
From this matrix the underlying linear system FRFs H11,H 21
and H 22 are extracted, together with the nonlinear coefficient
k3  s1 . In the first step, the first column is extracted, then the
nonlinear coefficient is obtained and finally H 22 is estimated.
It should be noted that errors on both the first column and the
nonlinear coefficient strongly affect the H 22 estimate. Also
note that the latter function can be obtained only because the
equations of motion are coupled via the nonlinearity – in the
classical (linear) modal analysis it is impossible to get an
estimate of the direct receptance of a degree of freedom
without applying any force directly to it.
Time histories were then generated by running a multi-body
simulation where gravity was set to 9.81 m/s2 and the system
was excited by a Gaussian random force (1.115 Nrms)
sampled at 2000 Hz for a total time of 100 seconds. The
estimated nonlinear coefficient is shown in Figure 3, where it
is noted that it has a mean value of 1.380·108 N/m3, which is
very close to the nominal value, the frequency dependence is
small and the imaginary part (not shown here) is one order of
magnitude smaller. The NSI estimated natural frequencies ni
and modal damping ratios ni for the two modes (i=1,2) of the
underlying linear system are summarized in Table 3.
where it is noted that the number of nonlinear terms is p=1,
the scalar coefficient is s1=k3, the location vector is
Ln1  1 1 and the scalar nonlinear function is v1(t)=(xs–
T
x)3. The direct point receptance and cross point receptance of
the underlying linear system are given as
H11   
Xs
F
and
H 21   
X
F
where X s and X are the complex displacement amplitudes
of the shaker and of the suspended mass, respectively.
The extended FRF matrix is a square 2×2 matrix, because
two displacements are measured (outputs) and two inputs are
considered in the model (the physical force and the internal
nonlinear feedback force). Taking into account the reciprocity
relationship H12  H 21 , the second column of H E   is
transformed into
Figure 3. NSI estimate of the cubic coefficient (real part).
Table 3. NSI estimated modal parameters.
Mode i
1
2
ni
ni 
[Hz]
11.7186 0.0339
55.7434 0.0917
A direct calculation of the FRFs yields
H11 
k1  ic  i 2 m
k1k s  i k s c  k1cs   i 2 k1ms  k1m  k s m  cc s   ...
...i 3 c  cs m  cm s   i 4 mms
1976
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
H 21 
k1  ic
k1k s  i k s c  k1c s   i 2 k1ms  k1m  k s m  cc s   ...
...i 3 c  c s m  cm s   i 4 mm s
k1  k s  i c  c s   i 2 ms
H 22 
k1k s  i k s c  k1c s   i 2 k1ms  k1m  k s m  cc s   ...
...i 3 c  c s m  cm s   i 4 mm s
which appear as ratios of polynomials. The coefficients of
these polynomials are computed by applying the Rational
Fraction Polynomial (RFP) method [20] to the FRF estimates
(reported in Figure 4 by solid blue lines), and then physical
parameters are obtained by solving a nonlinear parameter
estimation problem [12]. In Figure 4, it is noted that the FRFs
estimated by NSI (blue solid lines) are indistinguishably
overlaid to the FRFs computed by applying RFP (red dashed
lines).
All the estimated physical parameters of the system are
reported in Table 4, where it is noted that they are in excellent
agreement with the nominal pre-set parameters reported in
Table 1 and the corresponding value for k1 – the max
percentage error is less than 0.06%.
Furthermore, the clear evidence of a jump in the singular
value plot of Figure 5 confirms the appropriateness and
effectiveness of the identification process.
(b)
(c)
Figure 4. FRFs H11 (a), H21 (b) and H22 (c) of the underlying
linear system estimated by NSI and fitted by RFP.
Table 4. Identified parameters of the physical system.
ms
[kg]
0.1540
ks
[N/m]
1.8862·104
cs
[Ns/m]
9.8642
m
[kg]
0.0052
k1
[N/m]
28.2212
c
[Ns/m]
0.0260
Figure 5. Singular value plot for the NSI method, based on
multi-body simulations.
(a)
4
HARMONIC RESPONSE
The numerical values estimated for the physical system
parameters are then used to compute the corresponding nondimensional parameters as used in Eq. (4a,b). They are
reported in Table 5, and are adopted to plot the FRCs in
Figure 6 by using the amplitude-frequency relations given in
Eqs. (5a,b).
1977
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
Table 5. Non-dimensional system parameters.

s

 

Xs [mm]
0.034 0.091 0.007 0.211 0.006
-1
10
10
20
30
40
50
Frequency [Hz]
60
70
80
60
70
80
60
70
80
(a)
0
X [mm]
10
-1
10
-2
10
10
20
30
40
50
Frequency [Hz]
(b)
0
Z [mm]
10
-1
10
10
20
30
40
50
Frequency [Hz]
From Figure 6 it is noted how a closed detached curve is
present, which lies inside the main continuous FRC of the
relative displacement amplitude and of the absolute
displacement amplitude of the suspended mass. This feature
seems to appear only for certain particular values of the
system parameters [7, 8], and in particular for a quasi-zerostiffness oscillator [6], which is the case under investigation. It
should be noted that the parameters used in the actual case are
such that the wires in the attachments are only slightly
tightened. For the configuration reported in Figure 1, this
causes the dynamics of the system be much more affected by
the undesired wobbling of the suspended mass due to its
inertia effects and its weight distribution [12], but in the actual
arrangement of Figure 2, unwanted dynamics are greatly
reduced.
Simulations are then used to determine the response of the
system to harmonic excitation, and to compare it to that
predicted by the analytical solutions of the FRCs given by
Eqs. (5a,b). To this purpose, gravity is added to the multibody simulation and a harmonic force of constant amplitude
equal to F=3.18N is applied to the shaker, where the initial
conditions of the system (in terms of the absolute
displacements and velocities of the two masses) are all set to
zero, as in a practical experimental condition. Simulations are
run for 3s and time histories of the displacements of the two
masses are recorded. Fourier coefficients, corresponding to
the excitation frequency, are then extracted from the recorded
time histories, and plotted as circles in the FRCs shown in
Figure 6.
It was observed that, for the case where the system is
excited from rest, after the initial transient a specific
amplitude of oscillation is achieved, corresponding to the
lower stable branch in the detached curve of the FRCs. When
multi-valuedness in the FRC is present at a specified
frequency, the higher branch, corresponding to the highest
amplitude for the response, is attained due to a perturbation of
the system, which is practically applied as a relatively light
impulsive force to the suspended mass. This would be
equivalent to “touching” the suspended mass when vibrating
during an experimental test, thus physically forcing a jump to
occur.
It was observed that, for the actual system configuration, at
all frequencies tested and for the parameters used, the
response of the system in terms of mass displacements was
predominantly harmonic at the frequency of excitation. The
amplitudes of the higher and lower harmonics were found to
be negligible – their magnitude given as a percentage of the
amplitude of the response at the excitation frequency was less
than 5%. The results reported in Figure 6 clearly show the
agreement between the analytical solution obtained for the
simplified 2-DOF model and the numerical solution obtained
from the detailed multi-body system. The results thus confirm
the effectiveness of the proposed experimental setup shown in
Figure 2, to validate the approximate analytical model given
in Section 2.
(c)
Figure 6. FRCs of the system. Shaker (a), suspended mass (b),
and relative (c) displacement. Stable solution (−), unstable
solution (- -), multi-body solution (o).
1978
5
CONCLUSIONS
In this paper a mechanical system has been proposed for the
validation of the dynamic behavior of a two degree-of-
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
freedom nonlinear oscillator, when subject to harmonic
excitation. Practical issues relating to an experimental
implementation have been discussed and a configuration has
been proposed which overcomes the limitations of previous
solutions. In particular, when gravity was added to the model,
it was observed that the weight of the suspended mass and its
mass distribution play an important role, generating a
wobbling motion, which prevents some of the specific
features in the FRC to be identified. The proposed system
configuration, whose effectiveness was verified using a virtual
experiment in a multi-body environment, appears to be
suitable for the implementation of a real experimental rig to
validate the analytical model described in the paper. It was
shown that the response of the virtual multi-body model had
detached resonance curves in the frequency response curve.
Furthermore, the proposed parameter identification method,
based on random input excitation, was shown to effectively
estimate the system parameters when gravity was included in
the simulations and related dynamics are present.
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