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Nonlinear Mechanical Systems
A linear spring has reactive force as kx and nonlinear springs may have an added component of force as x^3. In the presence of an external force having periodic behavior, a model of mechanical system becomes
kx x A wt dt
dx C dt
d x m (^) 2 3 cos
2
Where, x(t) is the displacement.
It is a nonlinear second order ODE with a non-homogeneous term.
We can simulate it using one of explicit techniques after converting it into two first order differential equations.
Let us consider for simplicity , m = C = = w = 1 and k = -1, then the set of first order equations (also called forced Duffing Equations ) is
v(t) dt
dx( t) , (8.10a)
v(t) x(t) x (t) Acost dt
dv( t) ^3 , (8.10b)
Such a system has critical points at (-1, 0) and (1, 0) respectively.
Let us simulate the response of such a system to different applied forces having period of 2 w = 1 in the above model).
Figure 8.4 shows this behavior for different values of force amplitude, A (0.6 through 0.8) for same initial conditions x(0) = 0 & v(0) = 0. It show that mass has periodic oscillations and period doubling for force amplitude lower than A = 0.8. When external force amplitude is increased to the value of 0.8, the period in oscillations is lost (see xt-diagrams, Fig. 8.4a).
It becomes more visible when we compare phase diagrams for A = 0.8 and A = 0.75. There is not a single attractor but a chaotic structure. The system never returns to a state or set of states but forms a complex structure in phase space. Such a behavior is not seen in linear mechanical systems and is peculiar to nonlinear systems only. You may try this for different parameters (m, k, b, C and A etc.,) and measure the period for various cases. It will be very hard to see any period when the system becomes chaotic.
Try to plot the xn on x-axis and xn+1 on y-axis for first 100 time steps for integration in each case. Its called Lorenz plot. For a chaotic system, it will change when initial conditions are slightly modified.
Figure 8.4 Simulation of forced doffing equation with force amplitude A = 0.6 (a), 0.7 (b), 0.75 (c), and 0.8 (d). Left side has phase diagrams and the right side shows the x(t) versus time for each case. Chaos occurs for A = 0.8.
Nonlinear Electrical Systems
One of the famous studies of nonlinear oscillator circuits in early development of radio transmission was done by Balthasar van der Pol (in 1920s) when he used RLC series circuit and replaced resistance R by an active element (a vacuum tube). Let us consider the model here. The voltage drop across such an element is a function, f(I) , of current I. This leads to the following balance equation:
C
f(I) Q dt
LdI. (8.12)
Differentiating it, we get
2 C
I(t) dt
dI(t) dt
df(I) dt
d I(t) L. (8.13)
Van der Pol assumed that voltage drop across the tube can be represented as a nonlinear function of current with two constants ( a & b actually measurable in experiments):
f (I) bI^3 (t) aI(t). (8.14)
Therefore the model equation transforms into the following form:
2 C
I(t) dt
dI(t) ( bI (t) a) dt
d I(t) L. (8.15)
If we substitute dx/dt=y and say I = px , then
2 C
x dt
( bp x a)dx dt
Ld x. (8.16)
With new constants, p a/( 3 b) and a C/L , a standard form of van der
Pol’s equation emerges:
dt
dx y , (8.17a)
(x^2 (t) 1 )y(t) x(t) 0 dt
dy(t)
. (8.17b)
Nonlinear Electrical Systems (continued)
- Results of simulation for positive value of the parameter (Eqs. 8.17a & 8.17b) in phase space and xt-space are shown as Fig. 8.6a and 8.6b respectively.
- For = 1.5 the solutions are periodic as seen in x(t) & y(t) versus time curves (Fig. 8.6 b).
- The model becomes very stiff as is allowed to have large values.
- It has been used to test the robustness of ODE solvers for large . The origin (0, 0) is the spiral point for all 0 < < 2. Solution always converges to same region after oscillations. Try it for a range of values from 0 to 1000 and observe effect on periods.
Figure 8. 6 (a) The phase portrait of van der Pol’s equation for nonlinear electrical systems ( = 1.5). (b) The x(t) and y(t) versus time for = 1.5.
