🎓 Complete Mathematical Theory
The Governing Equation
∂⁴y/∂x⁴ + (ρA/EI)(∂²y/∂t²) = 0
This fourth-order partial differential equation describes transverse displacement y(x,t) of a beam subject to no external forces (free vibration).
Physical Interpretation: The left term represents how the bending moment changes (related to curvature). The right term represents how mass accelerates. These must balance for free vibration.
Boundary Conditions (Simply Supported)
At x = 0 and x = L:
y(0,t) = 0 and y(L,t) = 0 (no vertical displacement)
∂²y/∂x²(0,t) = 0 and ∂²y/∂x²(L,t) = 0 (no moment)
These constraints force the solution to be a sine function!
Separation of Variables Solution
Assume y(x,t) = Y(x)·T(t)
This splits the PDE into two ODEs:
Y''''(x) - β⁴Y(x) = 0
T''(t) + ω²T(t) = 0
where β⁴ = ρAω²/(EI)
Spatial Solution: The boundary conditions force Y(x) = sin(nπx/L) for n = 1, 2, 3, ...
Temporal Solution: T(t) = A·cos(ωt) + B·sin(ωt) — sinusoidal oscillation
Natural Frequencies (Eigenvalues)
From the constraint λ = nπ (where λ is the eigenvalue):
ω_n = (λ_n²/L²)√(EI/ρA)
f_n = ω_n/(2π) = (λ_n²/2πL²)√(EI/ρA)
λ_1 = π ≈ 3.14 → First mode constant ≈ 9.87
λ_2 = 2π ≈ 6.28 → Second mode constant ≈ 39.48
λ_3 = 3π ≈ 9.42 → Third mode constant ≈ 88.83
Why These Specific Constants? They come from solving sin(λ) = 0 with the boundary conditions. The first non-trivial solution is λ ≈ π (not exactly, but 9.87 comes from numerical solution).
Modal Superposition Principle
Complete solution: y(x,t) = Σ[A_n·sin(nπx/L)·cos(ω_n·t + φ_n)]
The motion is a superposition of all possible modes. Higher modes require more energy to excite, so they usually have smaller amplitudes (A_n decreases with n).
Energy Perspective: Each mode stores energy independently. Total energy = kinetic + potential. As the beam oscillates, energy constantly exchanges between these two forms. Natural frequency represents maximum efficiency of this energy exchange.
Moment of Inertia Derivation
For rectangular cross-section (width b, height h):
I = ∫∫ y² dA = ∫₀^h ∫₀^b y² dx dy
I = (1/12)·b·h³
The h³ term explains why height is so powerful!
Design Implications
Resonance Avoidance: If external forcing frequency matches natural frequency, response amplitude grows without bound. Engineers design structures to have natural frequencies far from excitation sources.
Practical Damping: Real beams have damping (energy dissipation through material deformation, air resistance, etc.). This prevents infinite amplification and causes oscillations to decay over time.
Multi-Degree-of-Freedom Systems: Real structures have infinite DOF. We extract the first few modes (usually 3-10) and ignore higher ones because they carry little energy and are rarely excited.
The Complete Picture: Beam vibration is one of the foundational problems in structural mechanics. It appears everywhere: buildings, bridges, aircraft wings, machine tools, turbines. Understanding this one system unlocks understanding of all linear vibration problems.