(a) Pendulum

(b) Swing(c) Stability diagram for the Mathieu equation

Home assignment N 2- in ODE and Mathematical modeling MMG511/TMV161. Spring 2014.

Modeling of Kapitza pendulum.

Each modeling assignment consists of a theoretical part that requires some mathematical reasoning and an implementation partincluding writing a simple Matlab code solving an ODE, graphical output, analysis of numerical solutions and some conclusions.You are welcome to contact the teacher in his office L2034 or by e-mail to ask questions. If a larger group of students needs helpwith Matlab programming an additional lecture can be given

You are encouraged to work in small groups of 2-3 people, but each of you must write an own report (handwritten or in TEX) including: 1) analytical work, 2) theoretical argumentation, 3) numerical results with graphical output and theirinterpretation after the plan below, and names of all group members.

Send Matlab codes with clear comments by e-mail. Time for the work is 3,5 weeks including the vacations week: good tobe ready on the 8-th of May, because one more assignment will follow. Grades for your reports on home assignments will contribute30% to the final marks for the course.

Introduction.

In the second home assignment we consider two equations related to pendulum with vertically oscillating pivot (named oftenKapitza pendulum after Russian physicist Nobel laureate Pyotr Kapitza). Read some literature about it: Wikipedia, and pp. 66-67in the course book.

A free pendulum is described by the equation

l··θ = −g sin θ, (1)

(see at the picture) and has two stationary states: downward and upward: θ = 0,·θ = 0 and θ = π,

·θ = 0.

The downward stationary state of the free pendulum is stable (pendulum stays close to it if we deviate it slightly). The upwardstationary state of the free pendulum is unstable (pendulum falls down it if we deviate it slightly from exactly upward position).The pendulum with vertically oscillating pivot is described by the equation with ξ(t) added for the coordinate of the oscillatingpivot:

l··θ = −

[g +

··ξ(t)

]sin θ, (2)

and has a remarkable property. If the frequency of the pivot movement ξ(t) is high enough the pendulum can show stable oscillations

around the upper position θ = π,·θ = 0.

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Questions.

1. The first goal of the assignment is to find numerically frequency ω of the harmonic oscillations ξ(t) = A cos(ωt) such that the

pendulum will stay upward (π/2 < θ < π3/2) if it starts close to the upper position θ = π,·θ = 0. Keep the amplitude A smaller

than 5% of the the length of the pendulum. For this particular question choose the length of the pendulum 50 cm.2. The second goal of the assignment is to investigate if the downward orientation of the pendulum starting close to θ = 0,

·θ = 0 can be destabilized by choosing certain frequency ω. In this case we also keep the amplitude A smaller than 5% of the lengthof the pendulum. For this question you can choose the length of the pendulum 3 m to make it similar to a swing.

3. The third goal of the assignment is to apply the Floquet theory and analysis of characteristic multipliers to the linearizedmodel of the Kapitza pendulum. It is the Mathieu equation introduced on pp. 67-68 in the course book. You must investigatestability of solutions to the two Mathieu equations corresponding to the linearization of the pendulum equation around the upwardθ = π and downward θ = 0 positions. They will differ by signs in front of the terms with gravitational force. The output of thisinvestigation will be a diagram as above showing approximate stability and instability regions in the plane of two parameters relatedto the amplitude and the frequency ω of the pivot. Use your knowledge of Floquet theory to explain what happens with solutionsto the Mathieu equations for parameters from the stable and from the unstable part of the diagram. Support your theoreticalconclusions by 4 typical numerical examples for each of the equations.

You can also try to answer the following additional questions (not obligatory)*) At which particular points do stability zones meet at the axis ε = 0 ?**) Which of solutions you that have got for the original non-linear model (2) agree (upward and downward orientation stable or

unstable) with the stability diagram for the linearized model - Mathieu equation. Does the Mathieu equation give a fair descriptionof the Kapitza pendulum?

Working plan

p1. Rewrite the equation in the non-dimensional form: in particular change the time scale t 7−→ t′ so that the excitationfrequency under cos(ωt) will become unit: ωt 7−→ t′.

p2. Transform the non-dimensional equation for pendulum to a system of two first order equations for θ(t),·θ(t).

p3. Write a program in Matlab that solves this system of equations and illustrates solutions by several parametrized integral

curves (θ(t),·θ(t)) in the phase plane and lets choose different initial data.

p4. Compute several (might be many!) solutions starting close to the fixed points θ = 0,·θ = 0 and θ = π,

·θ = 0, with several

different values of parameters (especially frequency) and illustrate these solutions by curves in the phase plane. Answer Question1 and Question 2 using your numerical results.

p5. Linearize the original equation (2) around the fixed points θ = 0 and θ = π in the similar way as in the book. You willget two similar Mathieu equations that describe stability of the solutions around these points with different signs in the term with”gravitational force”.

··y + (±a ∓ 2ε cos t) y = 0.

The sign in front of the oscillating term does not matter in fact.

Investigate stability of the corresponding linear systems of equations for various values of the parameters aand ε related to the amplitude and frequency of the excitation. Use the variant of Floquet theory for the Mathieuequation. Answer the Question 3.

It will include the following steps:p6. Learn theory of the Mathieu equation in the book p.p. 67-68. Observe that we have two cases with different signs in fron

of a in the equation above.p.7 Write a Matlab program to create a stability diagram as in the picture above .Choose an array (rather large) of points (a, ε) in the parameter plane and make a loop in your program so that for each pair of

parameters you do the following:i) Compute numerically the monodromy matrix for the system of equations. Use Remark 3.5.1 in the book for computing the

monodromy matrix: Φ−1(t0)Φ(t0 + T ) = eTR . The notion monodromy matrix not used in the book.ii) Calculate characteristic multipliers (eigenvalues of the monodromy matrix eTR) and make conclusions about the stability of

the solutions for the particular pair of parameters. Add a marker into the point (a, ε) in the plain of parameters marking if thecorresponding solution is bounded or unbounded. These calculations is clever to organize as a loop over the chosen array of pointsso that at the end of the loop you get a stability diagram like above.

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