MTH2032 Chap.5 Forced Oscillations, BVPs and Reduction of Order
Forced Oscillations, BVPs and Reduction of Order
Week 5 adds an external force to the spring and then changes the kind of problem being solved. Undamped forcing produces beats away from the natural frequency and resonance at it; damping removes the special case and leaves a decaying transient plus a steady state.
Boundary value problems fix conditions at both ends of an interval, may have no solution or many, and lead to the eigenvalue problem at the heart of separation of variables. Reduction of order closes Part I. Each part ends with a calculation you will meet again in the separation-of-variables chapters.
What this chapter covers
- 01
Undamped forcing and beats
- 02
Resonance and the factor t
- 03
Damped steady state and transient
- 04
Dirichlet, Neumann and mixed BVPs
- 05
The eigenvalue problem X″ = KX
- 06
Reduction of order and missing variables
Worked example · free
Dirichlet eigenvalues on an interval of length 3
- 1If K is positive, X = A cosh(√K x) + B sinh(√K x); X(0) = 0 gives A = 0 and X(3) = 0 gives B = 0.
- 1If K = 0, X = A + Bx; both conditions give A = B = 0.
- 1If K is negative, X = A cos(√(−K) x) + B sin(√(−K) x); X(0) = 0 gives A = 0.
- 1X(3) = B sin(3√(−K)) = 0 with B non-zero needs 3√(−K) = nπ.
- 1So K_n = −(nπ/3)² and X_n = sin(nπx/3) for n = 1, 2, 3 and so on.
Key terms
- Beats
- The slowly varying amplitude produced when a system is driven near, but not at, its natural frequency.
- Transient Solution
- The homogeneous part of a damped forced response, which decays to zero over time.
- Boundary Value Problem
- A differential equation on a fixed interval with conditions imposed at both ends.
- Reduction of Order
- A method that uses one known solution, or a missing variable, to lower the order of an ODE by one.
- Steady-State Amplitude
- The size of the long-run oscillation of a damped, driven system, equal to the square root of A² + B².
Forced Oscillations, BVPs and Reduction of Order FAQ
Why does resonance not occur when there is damping?
With damping the characteristic roots have a non-zero real part, so neither cosine nor sine at the driving frequency solves the homogeneous equation. The ordinary trial works without the factor t, and the response stays bounded. The notes make this point when they treat damped forcing as a single case.
Can a boundary value problem have no solution?
Yes. Unlike an initial value problem, existence and uniqueness depend on the data. For u″ + u = 0 with u(0) = 0, every candidate is a multiple of sin x, so a nonzero value at π cannot be met, while at π/2 a unique multiple works. Always test whether the constants can be fixed.
Which eigenvalue case should I check first?
Check all three, because markers expect to see each one. For Dirichlet and Neumann conditions a positive K gives only the zero solution; K = 0 gives a constant eigenfunction under Neumann conditions but nothing under Dirichlet ones; and K < 0 gives the sine or cosine eigenfunctions with K_n = −(nπ/L)².
What is the difference between transient and steady-state behaviour?
In a damped, driven system the homogeneous part of the solution decays because every characteristic root has a negative real part; that is the transient. The particular solution oscillates at the driving frequency forever; that is the steady state. After a long time only the steady state is visible, whatever the initial conditions were.
Exam move
Treat this chapter as the bridge to Part II. First, make the forced-oscillation results your own: derive the non-resonant particular solution and the resonant one with the factor t, and for a damped example solve the two-by-two system for the steady-state coefficients rather than plugging into the printed formula.
Sketch, at least once, the bounded beat and the linearly growing resonant response so that you can describe them in words. Second, practise boundary value problems by fitting constants on short intervals and deciding whether there are zero, one or infinitely many solutions.
Third, and most important for the final, drill the eigenvalue problem for Dirichlet, Neumann and both mixed combinations until you can produce the eigenvalues and eigenfunctions in under five minutes, with the three cases written separately. Finally, practise reduction of order with a known first solution, and one problem of each missing-variable type, because the substitution y′ = z(y) is easy to forget under time pressure.
For each boundary value problem you solve, record whether it had no solution, one solution or infinitely many, and which feature of the data caused that outcome.
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