Modal Analysis

Modal analysis decomposes the dynamic response of a linear multi-degree-of-freedom (MDOF) system into a set of independent modal contributions (natural frequencies, mode shapes, and modal damping). For undamped or proportionally damped linear systems, the equations of motion can be diagonalized by the eigenvectors (modes) of the generalized eigenvalue problem, enabling modal superposition to obtain time- or frequency-domain responses. Modal quantities (modal mass, modal stiffness, modal damping, modal participation factors) quantify how each mode contributes to a given forcing or base excitation. Assumptions used in the quiz: linear elasticity, small deformations, time-invariant mass (M) and stiffness (K) matrices, and symmetric positive-definite M and K matrices. SI units used throughout (kg, N/m, s, rad/s, Hz, m).

Governing FormulaKey equations and relations used in modal analysis: 1) Equation of motion (time domain): M x¨ + C x˙ + K x = f(t) 2) Undamped eigenvalue problem: (K - ω_r^2 M) φ_r = 0, where ω_r is the r-th natural circular frequency (rad/s) and φ_r its mode shape. 3) Natural frequency (single DOF): ω_n = sqrt(k/m) (rad/s), f_n = ω_n/(2π) (Hz). 4) Modal orthogonality (for symmetric M and K): φ_i^T M φ_j = 0 and φ_i^T K φ_j = 0 for i ≠ j. 5) Modal mass and stiffness for mode r: m_r = φ_r^T M φ_r, k_r = φ_r^T K φ_r; modal natural frequency ω_r = sqrt(k_r/m_r). 6) Proportional (Rayleigh) damping: C = α M + β K ⇒ modal damping is diagonal and modal equations decouple. 7) Modal participation factor (force at DOFs described by vector r): Γ_r = (φ_r^T M r) / (φ_r^T M φ_r). If modes are mass-normalized (φ_r^T M φ_r = 1), then Γ_r = φ_r^T M r = φ_r^T r (for M=I). 8) Frequency-response function (modal expansion, single-input single-output H_ij(ω)): H_ij(ω) = Σ_r (φ_i,r φ_j,r) / (ω_r^2 - ω^2 + 2 i ζ_r ω_r ω), where ζ_r is modal damping ratio for mode r (mass-normalized modes assumed). Real/imaginary parts give magnitude/phase. 9) Peak (resonant) frequency of a lightly damped SDOF: ω_peak ≈ ω_n sqrt(1 - 2 ζ^2) for ζ < 1/√2. 10) Modal superposition (time-harmonic steady-state): x(t) = Re{ Σ_r φ_r q_r(ω) e^{iωt} }, where modal coordinate q_r(ω) = (φ_r^T f) / (k_r - ω^2 m_r + i ω c_r) or equivalently using modal parameters and modal damping.

Knowledge Check

10 Questions

1.What is the primary objective of modal analysis for linear structural dynamics?

2.For a single-degree-of-freedom (SDOF) undamped system with mass m = 2.0 kg and stiffness k = 800 N/m, what is the natural frequency f_n in Hz?

3.Consider a 2-DOF undamped system with M = diag(1,1) kg and stiffness matrix K = [[1500,-500],[-500,500]] N/m. What are the two natural frequencies (in Hz), rounded to three significant figures?

4.For symmetric positive-definite mass and stiffness matrices, the eigenvectors (modes) satisfy which orthogonality relations for i ≠ j?

5.Why is it common to mass-normalize mode shapes so that φ_r^T M φ_r = 1?

6.Which form of damping matrix guarantees that the modal equations of motion decouple (i.e., C is diagonal in the modal basis)?

7.Using the 2-DOF mass-normal form from question 3, take the two (unnormalized) mode shapes φ1 = [-2.4142; 1] and φ2 = [0.4142; 1]. For a unit force applied at DOF1 (force vector r = [1; 0]) and M = I, compute the modal participation factors Γ1 and Γ2 using Γ_r = (φ_r^T M r)/(φ_r^T M φ_r). Which pair (Γ1, Γ2) is correct (rounded to four decimal places)?

8.For a lightly damped single-degree-of-freedom system with damping ratio ζ < 1/√2, the peak (resonant) frequency ω_peak of the steady-state response to harmonic forcing is approximately which expression in terms of ω_n and ζ?

9.When truncating higher-frequency modes in a modal reduction, which of the following most directly determines whether truncation will produce a significant error in predicted response to a given forcing?

10.Which modal expansion correctly gives the single-input single-output frequency-response function H_ij(ω) (assuming mass-normalized modes and modal damping ζ_r)?