Damped Systems

Damped systems are dynamical systems in which energy is dissipated (commonly via viscous damping) so that free vibrations decay in time. Key behaviors depend on the damping ratio ζ: underdamped (0 < ζ < 1) exhibit oscillatory decay, critically damped (ζ = 1) return to equilibrium fastest without oscillation, and overdamped (ζ > 1) return non-oscillatorily but slower. Forced responses show resonance near the undamped natural frequency ω_n with peak amplitudes reduced by damping. For linear viscous damping, equations of motion remain linear and modal superposition applies if damping is proportional (classical).

Governing FormulaAssume linear viscous damping, single-degree-of-freedom (SDOF): - Equation of motion: m x¨ + c x˙ + k x = F(t) - Undamped natural frequency: ω_n = sqrt(k / m) [rad/s] - Critical damping: c_c = 2 sqrt(m k) [N·s/m] - Damping ratio: ζ = c / (2 sqrt(m k)) (dimensionless) - Damped natural frequency (underdamped): ω_d = ω_n sqrt(1 - ζ^2) [rad/s] - Free response (underdamped): x(t) = A exp(-ζ ω_n t) cos(ω_d t + φ) - Logarithmic decrement δ = ln(x(t_n)/x(t_{n+1})) = 2π ζ / sqrt(1 - ζ^2) - Displacement transmissibility for base excitation: T(r,ζ) = sqrt((1 + (2 ζ r)^2) / ((1 - r^2)^2 + (2 ζ r)^2)), where r = ω/ω_n.

Knowledge Check

10 Questions

1.For a single-degree-of-freedom linear system with mass m and stiffness k and viscous damping coefficient c, which expression gives the damping ratio ζ (dimensionless)? Assume m>0, k>0.

2.Given an SDOF oscillator with mass m = 2.0 kg and stiffness k = 800 N/m, what is the undamped natural frequency ω_n (rad/s)?

3.For the same system as above (m = 2.0 kg, k = 800 N/m) with damping ratio ζ = 0.10, what is the damped natural frequency ω_d (rad/s)?

4.Which expression correctly describes the envelope of the underdamped free-vibration displacement x(t) for an SDOF system (initial amplitude A)?

5.Compute the critical damping coefficient c_c for a system with m = 2.0 kg and k = 800 N/m.

6.A system has m = 5.0 kg, k = 1250 N/m and viscous damping coefficient c = 50 N·s/m. What is the damping ratio ζ and the damping classification (underdamped/critical/overdamped)?

7.For base excitation of an SDOF system, displacement transmissibility at excitation frequency equal to ω_n (i.e., r=1) is approximately T = 1/(2 ζ) for small ζ. If ζ = 0.05, what is the approximate displacement transmissibility at resonance?

8.A measured logarithmic decrement between successive peaks is δ = 0.200. Estimate the damping ratio ζ (use the exact relation).

9.A small rotating unbalance has mass m_e = 0.10 kg located at an eccentricity e = 5.0 mm on a rotor spinning at f = 5.00 Hz. What is the amplitude of the centrifugal excitation force (peak) in newtons? (Use ω = 2π f.)

10.Which statement about modal behavior and damping is correct for a multi-degree-of-freedom linear system with classical (proportional) damping?