Methods of Applied Mathematics - NYU Courantobuhler/Oliver_Buhler/Teaching_11_files/MAM...Methods of...
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Methods of Applied Mathematics Fall 2011 — Prof. Oliver B¨ uhler Homework set 4 Out: Tue 4. October 2011 Due back in two weeks: Tue 18. October 2011. 1. Multiple scales for weakly nonlinear pendulum Consider the following initial-value problem for Φ(t): d 2 Φ dt 2 +Φ= Φ 3 with Φ(0) = 0, Φ 0 (0) = 1. (1) Using a slow time scale τ = t and the asymptotic expansion Φ(t,τ,) ∼ Φ 0 (t, τ )+Φ 1 (t, τ ) find the leading-order solution valid for t = O(1). Hint: it’s useful to write the general solution of the linear harmonic oscillator equation in t as A(τ ) sin(t + θ(τ )) where A(τ ) is a slowly varying amplitude and θ(τ ) is a slowly varying phase shift. What is the connection with question 3 from the second homework sheet? 2. The children’s swing problem A child can amplify a swing’s motion by slightly varying the swing’s length periodically with a frequency that is a multiple of half the swing’s natural frequency. A mathematical model for the case where the forcing frequency is equal to the natural frequency is the initial-value problem for a linear pendulum such that d 2 φ dt 2 + (1 + cos t)φ =0, φ(0) = 0, φ 0 (0) = 1. (2) Notably, the slow time scale in this problem turns out to be τ = 2 t, not t. Pose the multi-scale expansion φ(t, )= φ(t,τ,) ∼ φ 0 (t, τ )+ φ 1 (t, τ )+ 2 φ 2 (t, τ ) (3) and find the evolution of φ 0 for times up to τ = O(1). Hint: you will find that φ 0 = A(τ ) cos t + B(τ ) sin t and subsequently find that A and B satisfy a two-by-two set of ODEs. Determine whether this ODE is stable or not. Bonus question: replace (2) by d 2 φ dt 2 + (1 + α 2 + cos t)φ =0, φ(0) = 0, φ 0 (0) = 1 (4) where the constant α models a slight detuning of the pendulum from unit frequency. Adapt the multiscale analysis to this case and find the range of α for which there is instability. 3. Kapitza’s horizontal pendulum Adapt the multi-scale computation given in class to the case of a nonlinear pendulum whose point of support is oscillating horizontally. The scaled governing equations are d 2 φ dt 2 + sin φ + β cos(t/) cos φ =0. (5) Find the effective potential energy V (φ 0 ) for the leading-order solution and plot it for different values of β . Where are the stable fixed points as a function of β ? 1
Transcript of Methods of Applied Mathematics - NYU Courantobuhler/Oliver_Buhler/Teaching_11_files/MAM...Methods of...
Methods of Applied MathematicsFall 2011 — Prof. Oliver Buhler
Homework set 4Out: Tue 4. October 2011
Due back in two weeks: Tue 18. October 2011.
1. Multiple scales for weakly nonlinear pendulum
Consider the following initial-value problem for Φ(t):
d2Φdt2
+ Φ = εΦ3 with Φ(0) = 0, Φ′(0) = 1. (1)
Using a slow time scale τ = εt and the asymptotic expansion Φ(t, τ, ε) ∼ Φ0(t, τ)+εΦ1(t, τ)find the leading-order solution valid for εt = O(1).
Hint: it’s useful to write the general solution of the linear harmonic oscillator equationin t as A(τ) sin(t + θ(τ)) where A(τ) is a slowly varying amplitude and θ(τ) is a slowlyvarying phase shift.
What is the connection with question 3 from the second homework sheet?
2. The children’s swing problem
A child can amplify a swing’s motion by slightly varying the swing’s length periodicallywith a frequency that is a multiple of half the swing’s natural frequency. A mathematicalmodel for the case where the forcing frequency is equal to the natural frequency is theinitial-value problem for a linear pendulum such that
d2φ
dt2+ (1 + ε cos t)φ = 0, φ(0) = 0, φ′(0) = 1. (2)
Notably, the slow time scale in this problem turns out to be τ = ε2t, not εt.
Pose the multi-scale expansion
φ(t, ε) = φ(t, τ, ε) ∼ φ0(t, τ) + εφ1(t, τ) + ε2φ2(t, τ) (3)
and find the evolution of φ0 for times up to τ = O(1).
Hint: you will find that φ0 = A(τ) cos t + B(τ) sin t and subsequently find that A and Bsatisfy a two-by-two set of ODEs. Determine whether this ODE is stable or not.
Bonus question: replace (2) by
d2φ
dt2+ (1 + αε2 + ε cos t)φ = 0, φ(0) = 0, φ′(0) = 1 (4)
where the constant α models a slight detuning of the pendulum from unit frequency. Adaptthe multiscale analysis to this case and find the range of α for which there is instability.
3. Kapitza’s horizontal pendulum
Adapt the multi-scale computation given in class to the case of a nonlinear pendulumwhose point of support is oscillating horizontally. The scaled governing equations are
d2φ
dt2+ sinφ+
β
εcos(t/ε) cosφ = 0. (5)
Find the effective potential energy V (φ0) for the leading-order solution and plot it fordifferent values of β. Where are the stable fixed points as a function of β?
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