Systems with Multiple Degrees of Freedom

Multi-degree-of-freedom (MDOF) vibration systems have multiple independent coordinates describing motion. Their dynamics are described by matrix equations of motion. Free undamped vibration leads to an eigenvalue problem whose solutions are natural frequencies and mode shapes. Mode shapes are orthogonal with respect to the mass and stiffness matrices, enabling decoupling via modal coordinates. Damping that is classical (proportional) preserves modal decoupling; non-proportional damping couples modes. Forced responses can be analyzed by modal decomposition: each mode behaves like an independent SDOF system with its modal mass, modal stiffness and modal damping characterized by projections of physical matrices and forcing onto mode shapes.

Governing FormulaEquations and key relations (SI units): 1) Matrix equation of motion: M x¨(t) + C x˙(t) + K x(t) = F(t) where M (kg), C (N·s/m), K (N/m), x (m), F (N). 2) Undamped eigenproblem (free vibration): (K - ω^2 M) φ = 0 -> det(K - ω^2 M) = 0 ω_n^2 = eigenvalues, φ_n = corresponding eigenvectors (mode shapes). 3) Modal orthogonality (distinct modes i ≠ j): φ_i^T M φ_j = 0, φ_i^T K φ_j = 0. 4) Modal properties for mode n: modal mass m_n = φ_n^T M φ_n modal stiffness k_n = φ_n^T K φ_n = m_n ω_n^2 modal participation factor Γ_n = (φ_n^T F) / m_n (for force vector F). 5) Single-mode (modal) steady-state amplitude under harmonic force at frequency ω: q_n_amp = Γ_n / (k_n sqrt((1 - r^2)^2 + (2 ζ_n r)^2) ), r = ω/ω_n at resonance (r = 1): q_n_amp = Γ_n / (2 ζ_n k_n) 6) Proportional (classical) damping assumption: C = α M + β K -> modes remain uncoupled in modal coordinates; modal damping ratios ζ_n known from c_n = φ_n^T C φ_n and ζ_n = c_n / (2 sqrt(k_n m_n)).

Knowledge Check

10 Questions

1.Which matrix equation correctly represents the general equations of motion for a linear MDOF mechanical system?

2.For undamped free vibration of an MDOF system the eigenvalue problem is:

3.Which statement about mode shapes for distinct (non-repeated) natural frequencies in an undamped MDOF system is correct?

4.Consider a 2-DOF translational system: m1 = 2.0 kg connected to ground by k1 = 2000 N/m and to m2 by k2 = 1000 N/m; m2 = 1.0 kg connected only to m1 by k2. For the undamped system, what are the natural frequencies (rad/s) ordered high then low?

5.For the same 2-DOF system (m1=2 kg, m2=1 kg, k1=2000 N/m, k2=1000 N/m), what are the (unnormalized) mode shapes corresponding to the low and high natural frequencies respectively?

6.Using φ = [1;2] for the low mode of the previous system, compute the modal mass m_n = φ^T M φ (in kg) using M = diag(2,1) kg.

7.If the damping matrix can be written as C = α M + β K (proportional damping), which statement about modal equations is true?

8.For the low mode φ = [1;2] (m_n = 6 kg) and a harmonic force vector F = [0; 1 N] applied to mass 2, what is the modal participation factor Γ_n = (φ^T F)/m_n?

9.How many natural frequencies (distinct ω_n) does a linear undamped mechanical system have relative to the number of generalized coordinates (DOF)?

10.Using the low mode data (φ=[1;2], m_n=6 kg, k_n = m_n ω_n^2 with ω_n = 22.361 rad/s) and assuming modal damping ratio ζ_n = 0.02, what is the steady-state amplitude of mass 2 when a unit harmonic force F(t)=1·sin(ω_n t) N is applied at mass 2 at resonance (ω = ω_n)? (Report result in mm.)