Transcript Lyapunov coefficients - The University of Texas at Dallas

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Soft contact dynamics of an impacting bilinear
oscillator: numerical simulation and hints for
describing an impacted cantilever beam
Ugo Andreaus, Luca Placidi, Giuseppe Rega
Department of Structural and Geotechnical Engineering, “La Sapienza”, University of Rome
Research Workshop on
Bifurcation in Oscillators with Elastic and Impact Constraints
4 November – 6 November, 2009, Imperial College London, United Kingdom
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Introduction
Among the wide range of nonlinear dynamical systems, Piecewise
Smooth Systems (PSS) play an important role and can be
classified as continuous or discontinuous PSS.
An example of discontinuous PSS:
- The (hard) impact oscillator; see, e.g., Akashi (1958)
An example of continuous PSS:
- The (soft) bilinear oscillator; see, e.g., Shaw and Holmes (1983)
In this work, features of soft impact non-trivial dynamics of a
bilinear oscillator are analyzed via a return map approach.
Moreover, in order to grossly distinguish between a periodic and a
chaotic trajectory, we use Lyapunov coefficients, that measure the
average divergence of nearby trajectories, Ott (1993).
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For dynamical systems described by smooth differential equations
and for discrete maps, the calculation of Lyapunov coefficients is well
developed: Eckmann and Ruelle(1985), Müller (1995), Stefanski
(2000), Galvanetto (2000).
Discontinuity (or impact) maps are defined in such a way the event of a
certain discontinuity is given in terms of the foregoing discontinuity.
Discontinuity maps are used by de Souza and Caldas (2004), that apply the
standard method (see, e.g., Eckmann and Ruelle, 1985) for the calculation of
Lyapunov coefficients for discrete maps to the impact maps related to an
impact oscillator and to an impact pair system.
A novelty of the present work lies in the application of this technique to the
case of an impacting bilinear oscillator.
Finally, a reduced order model for describing single-mode dynamics of an
impacting cantilever beam is identified (on going).
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Index of the presentation
• Differential equations of the SDOF model
• The map approach
• Numerical simulations of the SDOF model
– Bifurcation analysis
– Lyapunov coefficients
• Hints for describing an impacting cantilever
beam
• Conclusions
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The Impacting Bilinear Oscillator
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Non dimensionalization
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Non dimensional equations
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Parameters for non-dimensionalization
Parameters for bifurcation-analysis
Parameters to be fixed on the basis of technical and experimental reasons
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Attenuation coefficient cs is chosen in such a way the oscillation amplitude is
attenuated at the rate of 10% every 10 cycles.
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The ratio between the velocity after and before the contact must be compatible with a
reasonable restitution coefficient
The ratio between the interval of time during which the mass is inside the obstacle and
the interval of time between successive impacts (flight-time) must be small
As a consequence,
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Numerical simulations
Return or discontinuity map
no grazing
Phase portrait
Discontinuity maps and phase portraits are evaluated as follows …
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The analytic solution inside the obstacle
as a function of the initial time and velocity:
The analytic solution outside the obstacle (in the system)
as a function of the initial time and velocity:
The initial time and velocity for the solution inside the
obstacle is the final time and velocity of the solution outside
the obstacle and viceversa.
The initial time and velocity can be evaluated only numerically !
How?
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A numerical evaluation of these equations in terms of t allows us to
Define the time series of impacts
t0 , t1 ,t 2 , t3 , t4 ,...
Define the velocity series at impacts
x0 , x1 , x2 , x3 , x4 ,...
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The evaluation of the functions Fo and Fs can be done only numerically!
Let us remark, however, that their
derivatives can be obtained in the
analytic form
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Discontinuity map or impact map
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Numerical simulations: sample motions
Period-1 orbit
Return map
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Phase portrait
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Period-1 orbit: Phase portrait
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Period-8 orbit
Return map
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Phase portrait
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Period-8 orbit: Phase portrait
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Chaotic case
Return map
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Phase portrait
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Chaotic case: Phase portrait
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Numerical simulations: Bifurcation analysis
Nondimensional time instants of contacts vs nondimensional external force frequency
Continuous bifurcation
Discontinuous bifurcation
Analysed by
Lyapunov coefficients
The point does not touch the obstacle
Enlarged in the following
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Nondimensional time instants of contacts vs nondimensional external force frequency
Enlargements
Discontinuous bifurcation
Continuous bifurcation
Chaotic phase range
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Intermittent chaos
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Portions of contact duration vs nondimensional external force frequency
Continuous bifurcation
Discontinuous bifurcation
The point does not touch the obstacle
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Nondimensional time instants of contacts vs nondimensional external force amplitude
Enlarged in the following
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Nondimensional time instants of contacts vs nondimensional external force amplitude
Enlargements
Continuous bifurcation
Discontinuous bifurcation
The point does not touch the obstacle
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Portions of contact duration vs nondimensional external force amplitude
Enlarged in the following
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Portions of contact duration vs nondimensional external force amplitude
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Enlargements
Continuous bifurcation Discontinuous bifurcation
The point does not touch the obstacle
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Lyapunov coefficients
To distinguish between periodic and chaotic orbits we evaluate the
two Lyapunov coefficients
h-th eigenvalue of the matrix
Where J is the Jacobian matrix of the impact map
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Discontinuity map or impact map
The evaluation of the functions F and G can be done only numerically!
However, their derivative can be obtained in the analytic form
Because of the chain rule derivative, the jacobian of
the discontinuity map is the following:
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The jacobian matrix must be evaluated on the time series of impacts, that can
be calculated only numerically.
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Lyapunov coefficients vs nondimensional external force frequency
Intermittent chaos
The point does not touch the obstacle
Enlarged in the following
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Lyapunov coefficients vs nondimensional external force frequency
Enlargements
Chaotic phase ranges
Periodic phase range
Due to intermittent chaos
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Lyapunov coefficients vs nondimensional external force amplitude
Enlarged in the following
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Lyapunov coefficients vs nondimensional external force amplitude
Enlargements
Chaotic phase range
The point does not touch the obstacle
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Relation between Lyapunov
coefficients and previous
bifurcation diagrams
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Relation between Lyapunov
coefficients and previous
bifurcation diagrams
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For a sinusoidally forced bilinear oscillator (with the discontinuity displaced by a gap
from the unstressed configuration of one spring) it is implemented a numerical iterative
method to derive the solution and discuss some relevant aspects.
The dynamics of the soft impact oscillator has been investigated based on the
discontinuity map obtained by slicing the three-dimensional phase space at the
discontinuity.
The definition of this transcendental map has been used to evaluate the Lyapunov
coefficients.
Phase portraits and return maps for three different response samples have been
shown: periodic, high-periodic and chaotic.
Besides classical bifurcation diagrams in terms of Lyapunov coefficients, more
characterizing diagrams have been given in terms of the time instants of contact when
the mass recovers the gap and impacts the obstacle, as well as of the percentage
portion of contact duration.
A very complex scenario has been highlighted, with a rich structured pattern that
includes continuous and discontinuous bifurcations, flip bifurcations and alternating
regular and chaotic regimes.
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Hints for describing an impacting cantilever beam
The considered bilinear oscillator can also represent a simple SDOF model for
effectively describing the impacting behavior of a cantilevered beam.
It can be useful
(i) to explore a wide range of problem parameters with low computational efforts,
(ii) to capture interesting features of the dynamic response, and
(iii) to get hints for further investigations of the beam via more complex tools, e.g.
1-D FEM models.
F0 Sinωt
EI

kb
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An example of phase portrait, using a nonlinear Finite Element tool
• F0=100N
• ω=66,82 Hz (resonance of free beam)
• kb=106 N/m
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Towards a reduced order SDOF model
F0 Sinωt
EI, 

kb
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A strategy for the identification of ks: the static analogy is used
The tip displacement of the beam
Equivalent to the displacement of the spring
3EI
ks  3
L
A strategy for the identification of m: the first mode dynamic identification
The beam first mode of vibration equivalent
to the mode of the mass-spring system
m
ks
I 
2
A strategy for the identification of k0: The resonance of the impacted
cantilever beam equivalent to the resonance of the bilinear oscillator...
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Resonance curves for the beam model
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Maximum and minimum displacement vs external force frequency
Each colour is for a given spring rigidity of the obstacle
The higher the spring rigidity of the obstacle, the higher the resonance frequency
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Resonance curves for the SDOF model
Maximum and minimum displacement vs external force frequency
Each colour is for a given spring rigidity of the obstacle
The higher the spring rigidity of the obstacle, the higher the resonance frequency
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Resonance frequencies vs spring rigidities of the obtacle in the two models
beam
sdof
The identification is done in terms of the same resonance:
The two spring rigidities related to the two models will be called
identified when the resonances of the two nonlinear models will be
equivalent.
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Identification of the SDOF spring rigidity
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First results for this kind of identification
Attachment velocity vs external force frequency (far from resonance!)
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Conclusions
Forced motion of a bilinear oscillator (SDOF) has been analysed by means
of return maps and Lyapunov coefficients, in order to highlight periodic and
chaotic regimes.
A problem of technical interest was considered: An elastic cantilever beam
(Euler model!) impacting a deformable obstacle
The SDOF bilinear oscillator has been identified in order to simulate the beam
behaviour at low computational cost.
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