Operator Split Techniques Applied to Nonlinear Dynamical Systems
M. A. SAVI
Department of Mechanical and Materials Engineering Instituto Militar de Engenharia
22.290.270 - Rio de Janeiro - RJ BRAZIL
P. M. C. L. PACHECO
Department of Mechanical Engineering CEFET/RJ
20.271.110 - Rio de Janeiro - RJ BRAZIL
A. M. B. BRAGA
Department of Mechanical Engineering Pontifícia Universidade Católica do Rio de Janeiro
22.453.900 - Rio de Janeiro - RJ BRAZIL
Abstract: The present contribution reports on the operator split technique applied to nonlinear dynamical systems. A brief discussion of the method is presented. This technique permits to evaluate the response of nonlinear dynamical systems using a combination of classical algorithms. A collection of results is presented for two different one-degree of freedom systems: Shape Memory and Elasto-plastic oscillators. Numerical simulations of the free and forced response are considered.
Key-Words: Operator Split Technique, Nonlinear dynamics, Chaos, Shape Memory, Elasto-plasticity.
1 Introduction
The operator split technique has been used on the solution of different nonlinear problems and its basic idea is to promote a partition of state space in sub-spaces which may be solved separately. The solution of each sub-space permits to perform a sequence of updatings where the results of the previous part is used as an input on the next one.
The main advantage of this procedure is the use of a combination of classical algorithms to evaluate the response of each part of the system.
The operator split technique has been applied on the simulation of different physical systems. As examples one could mention the elasto-plastic behavior [1,2], shape memory and pseudoelastic effects [3] and nonlinear dynamical systems [4,5].
This contribution reports on a collection of results regarding the operator split technique when
applied to nonlinear dynamical systems. Two different one-degree of freedom oscillators are considered: Shape Memory and Elasto-plastic. The results presented show that this technique is useful to deal with this kind of problems.
2 Operator Split Technique
In order to present the operator split technique, one can envision a dynamical system described by the following equations of motion,
Rn
x x f
x= ( ), ∈ (1)
The technique considers a partition of the Rn space in sub-spaces which may be solved separately. Initially, one separates the system in N parts, which may be written as,
) , , ,
(
1i 2i mii
x i
x x
x = (i = 1, 2, …, N) (2)
where xi∈Rmi. After this partition, the equations of motion can be rewritten as,
) , ( t
x fxi= i (i = 1, 2, …, N) (3) Suppose that for each part, it is possible to define a differencing scheme for updating the variable x from time-step n to time-step n+1. This scheme is defined by a classical integration procedure and may be associated with some specific algorithm used to treat nonlinearities of the problem. Hence,
) , (
1 11 + +
+ = n n
i i
n F x t
x (i = 1, 2, …, N) (4)
where F represents the nonlinear function f associated with the differencing scheme applied to the vector field x.
The operator split technique conceives a sequence of updatings, as follows,
=
=
=
=
+ + +
+ + +
+ +
+ + +
+ +
+ +
+ +
+
) , , , , , (
) , , , , , (
) , , , , , (
) , , , , , (
1 1 3
1 2
1 1
1 1
1 3
1 2
1 1
1 3 3
1
1 3
2 1 1
1 2 2
1
1 3
2 1
1 1 1
1
n N n n
n n N N n
n N n n
n n n
n N n n n n n
n N n n n n n
t x x
x x F x
t x x
x x F x
t x x x x F x
t x x x x F x
(5)
It should be pointed out that this sequence permits solving each sub-space separately by considering the variables of the other parts as known parameters. The solution of the previous part is used as an input on the next one. An iterative procedure may be used to assure the convergence of the process.
3 Shape Memory Oscillator
The operator split technique can be used to analyze the response of a shape memory oscillator.
The shape memory effect is associated with thermoelastic martensitic transformations.
Plastically deformed objects made of shape memory alloys (SMA) may recover their original form after going through a proper heat treatment.
Because of such a remarkable property, SMA have found a number of applications in engineering [6].
Therefore, consider a one-degree of freedom oscillator where the nonlinear restoring force is provided by a shape memory element and the damping is assumed to be linear. By considering a
constitutive equation proposed by Fremond [7], it is possible to present the following equations of motion, discussed by Savi and Braga [4],
2 1 x x =
) ( ) ( 4 3 1
2
2 x x x x g t
x =−ξ − −α − + 3 1
[
1 (θ 1) λ1 λ3]
η − − + −
= x L
x (6)
4 1
[
1 ( 1) 2 3]
λ λ η − − θ− + −
= x L
x
where α, ξ and L are constants, θ is associated with temperature and λ1, λ2 and λ3 are Lagrange multipliers. The function g(t) is a forcing term where t is the dimensionless time. The variable x1
represents the displacement and x2 the velocity.
The volumetric fraction of a martensitic variant is represented by x3 = β1, referred as positive martensite (M+), while x4 = β2 is the volumetric fraction of other variant referred as negative martensite (M-). The system is also subjected to the following Kuhn-Tucker conditions [8].
3 0
1x =
λ , λ2x4=0, λ3
(
x3+x4−1)
=0, and≥0
λi (i=1,2,3). (7)
For this particular system, the problem is split into two parts:
) , ( 1 2
1 x x
x = and x2=(x3,x4) (8) The solution of this system considers fourth order Runge-Kutta to solve part (1). Part (2) is split into two sub-parts using Euler's method in the first sub-part of (2), combined with an orthogonal projection in the second [4]. It should be pointed out that after all variables at the instant tn+1 are calculated, the variables associated with part (1), (x1, x2), have been evaluated using the values of x3
and x4 at the instant tn. Hence, it is necessary to return back to part (1) to recalculate the actual state for the pair (x1, x2) using values of (x3, x4) at the instant tn+1. This procedure must be repeated until the values converges for two consecutive iterations.
To illustrate the behavior of the shape memory oscillator, some of the results obtained by Savi and Braga [4] are here presented. Good convergence rates are obtained with step size ∆t = 2π/100. This is possible, since the orthogonal projections tends to correct small errors which otherwise would
propagate in the solution. In all calculations α = 1, L = 10, η = 0.1 are considered.
Free vibrations of the oscillator are governed by equations (6) with g(t) = 0. In addition to the initial displacement and velocity, one also has to prescribe the internal state of the system represented by x3(0) and x4(0). These values are determined by the proportions of the three different crystalline structures (austenite and two variants of martensite) initially present in the SMA element.
The unforced response of the undamped system (ξ = 0) at different temperatures is illustrated in the phase plane orbits shown in Fig.1. Three distinct behaviors can be identified. At θ < 1, when martensite is stable, the phase plane portrait exhibits two centers. When (1 < θ < 1+α/L), the number of stable fixed points is three. This is the temperature range where the shape memory effect is manifested. As the temperature is raised above this range, the element starts to exhibit a pseudoelastic behavior, and the nonlinear system will oscillate periodically around the only fixed point at x1= x2=0.
(a) (b)
(c)
Figure 1. Orbits in the phase plane. (a) θ = 0.69;
(b) θ = 1.06; (c) θ = 1.39.
Figs.2-3 show in detail the response of the undamped system at a temperature where the martensitic phase is stable (θ < 1). The SMA element is completely martensitic at this temperature range. If the only type of martensite present is that whose volumetric fraction is measured by x3, i.e. (x3, x4) = (1,0), the unloaded configuration of the SMA element will correspond to x1 = α. The other possibility is (x3, x4) = (0,1),
which is the initial internal state prescribed in the simulation presented in Figs.2-3. In this case, the stable equilibrium configuration of the unstretched element will be such that x1 = -α. Fig.2 shows an orbit in the phase plane. Friction loss associated with phase changes is the only dissipative mechanism considered here. For the initial values of x1 and x2 prescribed in the simulation, the system reaches a periodic oscillatory motion around the center (-α,0) after a transient. The response in the plane of internal variables is shown in Fig.3. The alloy initially contains only negative martensite. While steady state is not reached, the crystalline structure alternates between the two kinds of martensite. Finally, the alloy returns to the internal state associated with negative martensite.
Figure 2. Orbit in the phase plane at θ = 0.69.
Figure 3. Time evolution of the internal variables.
In the forthcoming analysis, one discusses the dynamics of the shape memory oscillator when it is subjected to a periodic excitation g(t) = δ sin(Ωt).
The spectrum of prospective responses is much richer now. In particular, one are interested on uncovering the possibility of chaotic behavior by the non autonomous system.
Fig.4 shows an orbit in the phase plane for θ = 0.95, ξ = 0.1, Ω = 1, and δ = 0.45. A Poincaré section of this orbit is also presented in Fig.4 [9].
For this set of parameters, one observes that the response is periodic, corresponding to a period-3 subharmonic oscillation. A Poincaré section of the response in the plane of internal variables is plotted in Fig.5 and the internal variables also change with the same periodicity.
Figure 4. Period-3 motion in the phase plane.
Figure 5. Poincaré section in the plane of internal variables for a period-3 motion.
A chaotic response is found when the damping coefficient ξ is increased to 0.2 while all other parameters are kept the same as those used in the simulation presented in Fig.4 and 5. This type of behavior is identified by the strange attractor [9] in the Poincaré section of the phase plane shown in Fig.6. The Poincaré section of the plane of internal variables (Fig.6) also exhibits a fractal structure.
Comparing the plots in Fig.6 with those in Figs.4 and 5, it is interesting to observe that only by changing the damping coefficient, even when the excitation is not altered, the character of the system response can be drastically modified.
Figure 6. Poincaré sections of a chaotic motion.
4 Elasto-Plastic Oscillator
In order to consider another example of operator split technique application, a nonlinear dynamics of an elasto-plastic oscillator with both kinematic and isotropic hardening is studied. Elasto-plasticity theory describes the well known behavior of bodies that present an elastic response until a limit, defined by the yield surface, is reached. After this
limit, the body presents a plastic response which is associated with irreversible strains. The hardening effect represents the way of how plastic strains modify the yield surface.
Consider a one-degree of freedom oscillator where the nonlinear restoring force is provided by an elasto-plastic element and the damping is assumed to be linear. Therefore, it is possible to present the following equations of motion, as discussed by Savi and Pacheco [5],
2
1 x
x =
) ( ) ( 1 3
2 0 2
2 x x x g t
x =−ξ −ω − + ) sign( 5
3 P x
x =λ − (9)
λ
4 = x
) sign( 5
5 H P x
x =λ −
where ξ, ω0 and H are constants, λ is the Lagrange multiplier and g(t) is a forcing term. The variable x1 = x represents the displacement and x2 = y the velocity. x3 = xp represents the plastic displacement, x4 = α is a variable associated with isotropic hardening, while x5 = β is associated with kinematic hardening. sign(x) = x/|x| and P is the force on the elasto-plastic element, defined as follows,
) (x1 x3 K
P= − (10)
K is a constant. The yield surface is defined by the following expression,
) (
) , ,
(P x4 x5 P x5 P Gx4
h = − − y + (11)
where Py and G are constants. The system is also subjected to the following Kuhn-Tucker conditions [8].
≥0
λ , λh=0
=0
λh if h=0 (12)
The numerical solution of this system considers a partition of the space of variables into two parts.
One part is the phase plane and includes the variables (x1, x2) and the other is the internal variables space and includes (x3, x4, x5),
) , ( 1 2
1 x x
x = and x2=
(
x3,
x4,
x5)
(13) With the proposed split, it is possible to use any integration scheme to solve part (1) and the return mapping algorithm [10], which consists in an elastic predictor step associated with a plasticcorrector, to solve part (2). An iterative process takes place until the convergence is achieved.
To illustrate the behavior of the elasto-plastic oscillator some of the results obtained by Savi and Pacheco [5] are now discussed. In all simulations, one considers K = 54.9MN/m, G = 345.6kN/m, H
= 78.5kN/m and Py = 31.4kN. Good convergence is obtained for time steps, ∆t, lower than 2π/(200ω0).
Free vibrations of elasto-plastic oscillator are considered by letting g(t) vanish in equation (9). A system with no external dissipation (ξ = 0) is examined. Results from simulations are presented in the form of phase portraits, where each orbit is associated with different initial conditions. Fig.7 shows the phase portrait in the phase plane (Fig.7a) and in the internal variables plane (Fig.7b). Initial conditions in the elastic domain cause linear responses. Outside the elastic domain, initial conditions cause a variation of the equilibrium point which is associated with plastic displacements. The phase portrait in internal variables plane occupies a restrict region of the plane as a consequence of complementary conditions (11) (Fig.7b).
Fig.8 shows a particular orbit associated with initial conditions which are outside the elastic domain. The motion occurs around equilibrium point which position on x-axis is defined by the plastic displacement, x3. Fig.8a shows the phase plane while Fig.8b shows the force-displacement curve.
(a) (b)
Figure 7. Phase portrait of the oscillator with kinematic hardening. (a) Phase plane; (b) Internal
variables plane.
-4E-4 0 4E-4 8E-4 1.2E -3
x (m ) -5
0 5
y (m/s)
(a)
-4E -4 0 4E -4x (m ) 8E -4 1.2 E -3
-40 0 40
P (KN)
(b)
Figure 8. Response of the oscillator with kinematic hardening for a particular initial condition.
(a) Phase plane; (b) Force-displacement curve.
A forced vibration of the oscillator when it is subjected to a periodic excitation g(t) = δ sin(Ωt), is now in order. Since isotropic hardening tends to expand the yield surface and the loads are prescribed, it is expected that the oscillator with kinematic/isotropic hardening presents an elastic response in steady state. Orbits in phase plane tends to modify its form and position until steady state is reached when an elliptical shape is assumed. Fig.9 shows different orbits for the 10th and 100th cycles. When steady state is reached, plastic displacements assume constant values while displacement and velocity present periodic response. It must be observed the initial superharmonic response which is caused by the cyclic response of plastic displacements.
-4 E -4 0 4E -4 8E -4 1.2E -3 1.6E -3
x (m ) -2
0 2
y (m/s)
Figure 9. Phase plane orbits of the oscillator with kinematic/isotropic hardening and no external dissipation for the 10th () and 100th (- - - -)
loading cycles.
Now, one considers only kinematic hardening which imposes that yield surface modifies its position. Periodic excitation tends to cause a steady state response where plastic variables have a stabilized cycle. In transient response, the translation of yield surface causes a non-symmetric response for tensile and compressive behavior. In steady state, a symmetric situation occurs and system variables present periodic responses. Fig.10 shows steady state response for different driving force amplitudes (δ =41.4Kms−2and
4 2
.
51 −
= Kms
δ ) where the superharmonic
response is associated with odd multiples of forcing frequency [5].
-4E -3 -2E -3 0 2E -3 4E -3
x (m ) -4
0 4
y (m/s)
(a)
-8E -3 -4E -3 0 4E -3 8E -3
x (m ) -10
0 10
y (m/s)
(a)
Figure 10. Orbits of steady state response of the oscillator with kinematic hardening.
(a) δ =41.4Kms−2; (b)δ =51.4Kms−2
(b)
5 Conclusions
The operator split technique applied to nonlinear dynamical systems has been explored in this paper.
In order to present some examples of the technique applications, one considers the dynamic response of one-degree of freedom oscillators where the nonlinear restoring force is provided either by a shape memory or by an elasto-plastic element. This technique permits the evaluation of the response of nonlinear dynamical systems using a combination of classical algorithms. Numerical simulations of the free and forced response of the oscillators are considered. The results show the potentiality of the operator split technique when dealing with nonlinear dynamical systems.
6 Acknowledgements
The authors would like to acknowledge the support of the Brazilian Research Council (CNPq) and the Research Foundation of Rio de Janeiro (FAPERJ).
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