Vibration Isolation

Vibration isolation is the use of mounts, springs, dampers or combinations thereof to reduce the transmission of vibratory energy from a source (machine) to a support or environment. For a single-degree-of-freedom (SDOF) isolated mass, the key idea is to design the isolator so the system natural frequency is low compared with the dominant excitation frequency; above a certain frequency ratio the transmitted amplitude decays approximately as the inverse square of the frequency ratio. Damping reduces resonant amplification but can increase transmitted force in the isolation (high-frequency) range if excessive. For rotating unbalance, the excitation is a harmonic force proportional to e m_e ω^2 (eccentricity × unbalance mass × speed^2). Modal analysis extends these ideas to multiple degrees of freedom where independent modal responses are superposed.

Governing FormulaCommon equations (SI units): - SDOF natural frequency (rad/s): ω_n = sqrt(k/m). In Hz: f_n = ω_n/(2π) = (1/2π) sqrt(k/m). - Critical damping: c_c = 2 m ω_n. Damping ratio: ζ = c / c_c. - Base-excited transmissibility (steady-state) for displacement excitation: T( r , ζ ) = sqrt(1 + (2 ζ r)^2) / sqrt((1 - r^2)^2 + (2 ζ r)^2), where r = ω/ω_n. - Force from rotating unbalance: F_unbalance = m_e e ω^2 (amplitude of harmonic force). - Steady-state vibration amplitude for harmonic force F0 at ω: X = F0 / sqrt((k - m ω^2)^2 + (c ω)^2). - For r >> 1 and light damping, transmissibility ≈ 1 / r^2. Assumptions: linear time-invariant SDOF models unless otherwise stated, small displacements, viscous damping.

Knowledge Check

10 Questions

1.What is the primary objective of a vibration isolation system for an industrial machine?

2.Compute the natural frequency in hertz of an isolated mass m = 10.0 kg supported by a spring of stiffness k = 40,000 N/m. (Use f_n = (1/2π) sqrt(k/m))

3.For base excitation with frequency ratio r = ω/ω_n = 3 and damping ratio ζ = 0.05, what is the transmissibility T (use the standard transmissibility formula)?

4.A mass m = 5.0 kg is supported by a spring k = 8,000 N/m. What is the critical viscous damping coefficient c_c for this single-degree-of-freedom system? (c_c = 2 m ω_n)

5.A single-degree-of-freedom system has m = 2.0 kg, k = 50,000 N/m, and viscous damping corresponding to ζ = 0.02. A rotating unbalance with m_e e = 0.01 kg·m forces the mass at f = 20.0 Hz. Estimate the steady-state vibration amplitude (in mm). Use X = F0 / sqrt((k - m ω^2)^2 + (c ω)^2) with F0 = m_e e ω^2 and c = 2 ζ m ω_n.

6.For base excitation at resonance (r = 1) the transmissibility magnitude simplifies to T_res = 1/(2 ζ). What damping ratio ζ is required to limit the resonant transmissibility to T_res = 2?

7.Under low damping, what minimum frequency ratio r = ω/ω_n is required so that transmissibility T < 1 (i.e., the isolator actually isolates rather than amplifies) for harmonic base excitation?

8.Which statement about mode shapes (eigenvectors) of an undamped linear multiple-degree-of-freedom system is true?

9.A rotor has an unbalance m_e e = 0.005 kg·m and spins at 1800 rpm. What is the amplitude of the harmonic unbalance force F_unbalance = m_e e ω^2 (in newtons)?

10.A machine produces a dominant vibration at 60 Hz. For light damping and transmissibility requirement T < 0.1, what is the maximum allowable isolator natural frequency f_n (Hz) using the high-frequency approximation T ≈ 1/r^2?