Dampers

Dampers (viscous damping elements) provide force proportional to relative velocity and dissipate mechanical energy into heat. In single-degree-of-freedom (SDOF) linear systems the damper is represented by damping coefficient c (units N·s/m). Damping changes transient decay, shifts peak amplitudes under harmonic forcing, and defines the damping ratio (dimensionless) that classifies under-, critically-, and over-damped responses. For base-excited systems the displacement transmissibility quantifies vibration transmitted to the mass as a function of excitation frequency and damping. Common damping models include viscous dampers (force = c v) and Rayleigh (proportional) damping C = alpha·M + beta·K used in modal analysis.

Governing FormulaKey equations (SI units): - Viscous damper force: F_d = c v (N) - Equation of motion (forced SDOF): m x¨ + c x˙ + k x = F(t) - Natural frequency: omega_n = sqrt(k/m) (rad/s) - Critical damping: c_cr = 2 m omega_n = 2 sqrt(k m) (N·s/m) - Damping ratio: zeta = c / c_cr = c / (2 m omega_n) - Damped natural frequency: omega_d = omega_n sqrt(1 - zeta^2) (rad/s) for zeta < 1 - Steady-state harmonic response (force F0 sin(omega t)): amplitude X = (F0/m) / sqrt((omega_n^2 - omega^2)^2 + (2 zeta omega_n omega)^2) - Displacement transmissibility for base excitation (ratio of mass displacement amplitude X to base displacement amplitude Y): T(r) = sqrt((1 + (2 zeta r)^2) / ((1 - r^2)^2 + (2 zeta r)^2)), where r = omega/omega_n - Energy dissipated per cycle for harmonic motion x = X sin(omega t): E_diss,cycle = pi c omega X^2 (J) - Rayleigh (proportional) damping: C = alpha M + beta K (used in modal damping assignment).

Knowledge Check

10 Questions

1.Which expression correctly gives the instantaneous force in a linear viscous damper?

2.For an SDOF system with mass m and stiffness k, what is the critical damping coefficient c_cr (the boundary between under- and over-damped)?

3.A SDOF system has m = 10.0 kg, k = 20 000 N/m and a viscous damper c = 600 N·s/m. What is the damping ratio zeta (dimensionless)?

4.Using the same system as Q3 (m = 10.0 kg, k = 20 000 N/m, c = 600 N·s/m), what is the damped natural frequency omega_d (rad/s)?

5.For the system in Q3 (zeta ≈ 0.6708), compute the displacement transmissibility T for base excitation at frequency ratio r = omega/omega_n = 2.0. Use T(r) = sqrt((1+(2 zeta r)^2)/((1 - r^2)^2 + (2 zeta r)^2)).

6.A viscous damper c = 600 N·s/m is subjected to harmonic motion x(t) = X sin(omega t) with omega = 90 rad/s and amplitude X = 0.010 m. What is the energy dissipated by the damper per cycle (J)?

7.A rotating unbalance produces a harmonic force F(t) = F0 sin(omega t) with F0 = m_u e omega^2. For m = 5 kg, k = 20 000 N/m, unbalance m_u = 0.010 kg, e = 0.002 m, and damping ratio zeta = 0.05, what is the steady-state mass amplitude X when the spin speed equals the undamped natural frequency omega = omega_n? (Use X = (F0/m) / (2 zeta omega_n^2) at resonance.)

8.Which statement best describes Rayleigh (proportional) damping used in modal analysis?

9.Two linear viscous dampers with c1 = 100 N·s/m and c2 = 300 N·s/m are connected in series between a mass and ground. What is the effective damping coefficient c_eq (N·s/m)?

10.For an underdamped SDOF (zeta < 1/sqrt(2)), the frequency ratio r_peak at which the transmissibility (for harmonic base excitation) reaches its maximum is given by which expression?