Introduction to Vibration

Introduction to Vibration: Vibration is oscillatory motion about an equilibrium position. In engineering this is often modeled as single-degree-of-freedom (SDOF) or multi-degree-of-freedom (MDOF) mechanical systems. Key behaviors: free vibration (no external forcing), forced vibration (external time-varying forcing), undamped vs damped (energy dissipative mechanisms). Natural frequencies and mode shapes characterize free vibration of linear systems; damping ratio determines amplitude decay and bandwidth in forced response. Resonance occurs when forcing frequency approaches a system natural frequency, producing large amplitudes for low damping. Rotating unbalance produces a sinusoidal forcing proportional to ω^2. Useful approximations: for lightly damped systems, modal superposition and orthogonality simplify MDOF analysis.

Governing FormulaKey formulas (SI units): - Equation of motion (SDOF, translational): m x'' + c x' + k x = F(t). - Undamped natural frequency (rad/s): ω_n = sqrt(k/m). Frequency (Hz): f_n = ω_n/(2π). - Damping ratio: ζ = c/(2 m ω_n) = c/(2 sqrt(m k)). - Damped natural frequency: ω_d = ω_n sqrt(1-ζ^2). - Steady-state amplitude for harmonic forcing F0 sin(ωt) (magnitude): X = (F0/m)/sqrt((ω_n^2-ω^2)^2 + (2 ζ ω_n ω)^2). - Rotating unbalance forcing amplitude: F0 = m_e e ω^2 (m_e: unbalanced mass, e: eccentricity, ω: rotational rad/s). - Resonance condition (undamped): ω = ω_n (forcing frequency equals natural frequency). - Modal orthogonality (undamped, symmetric systems): φ_i^T M φ_j = 0 and φ_i^T K φ_j = 0 for i ≠ j.

Knowledge Check

10 Questions

1.Which statement correctly describes undamped free vibration of a single-degree-of-freedom linear system?

2.A mass m = 2.00 kg is attached to a linear spring with stiffness k = 800 N/m. What is the natural frequency in Hz (use f_n = ω_n/(2π))?

3.For a SDOF system with m = 1.0 kg, k = 1000 N/m and viscous damping c = 10.0 Ns/m, what is the damping ratio ζ = c/(2 sqrt(m k))?

4.Which ordinary differential equation correctly represents a forced, damped SDOF translational system under harmonic forcing F(t) = F0 sin(ω t)?

5.For an undamped SDOF system under harmonic forcing, resonance (unbounded ideal amplitude) occurs when which condition is met?

6.A rotor has an unbalanced mass m_e = 0.050 kg located at eccentricity e = 0.0020 m and spins at 3000 rpm. What is the amplitude of the resulting sinusoidal force F0 = m_e e ω^2? (Use ω = 2π·(rpm/60).)

7.Which property correctly describes mode shapes φ_i of an undamped, symmetric MDOF system (mass matrix M, stiffness matrix K)?

8.The quality factor Q of a lightly damped oscillator is defined by Q = 1/(2 ζ). For ζ = 0.02, what is Q?

9.A damped SDOF system has ζ = 0.05 and ω_n = 20.0 rad/s. How long does it take for free vibration amplitude envelope to decay to one-half of its initial value? (Envelope ~ e^{-ζ ω_n t}.)

10.For a viscously damped SDOF under harmonic forcing, the frequency ratio r = ω/ω_n at which the steady-state amplitude reaches its maximum (for ζ < 1/√2) is given by which expression?