Forced Vibration

Forced vibration studies the steady-state and transient response of dynamical systems subjected to external time-dependent excitations. For engineering practice we commonly model single-degree-of-freedom (SDOF) systems with linear stiffness and viscous damping. Key behaviors include resonance (large amplitudes near natural frequencies), phase lag between input and response, transmissibility for base-excited systems, and force generation from rotating unbalance. Assumptions used below: linear time-invariant SDOF models, viscous damping, and harmonic steady-state forcing unless stated otherwise.

Governing FormulaEquations and useful relations (SI units): - Equation of motion (force excitation): m x'' + c x' + k x = F0 sin(ω t) - Natural frequency: ω_n = sqrt(k / m) [rad/s] - Critical damping: c_cr = 2 sqrt(k m) [N·s/m] - Damping ratio: ζ = c / c_cr = c / (2 sqrt(k m)) - Frequency ratio: r = ω / ω_n - Steady-state displacement amplitude (force excitation): X(ω) = (F0 / k) / sqrt((1 - r^2)^2 + (2 ζ r)^2) - Phase angle between force and displacement: φ = arctan((2 ζ r) / (1 - r^2)) (radians) - Rotating unbalance force amplitude: F0 = m_e e ω^2 (m_e = unbalanced mass, e = eccentricity) - Transmissibility for base (support) displacement Y(t)=Y sin(ω t): X / Y = sqrt(1 + (2 ζ r)^2) / sqrt((1 - r^2)^2 + (2 ζ r)^2) Notes: k = stiffness [N/m], m = mass [kg], c = viscous damping [N·s/m], ω = forcing circular frequency [rad/s], F0 = force amplitude [N].

Knowledge Check

10 Questions

1.Which equation correctly represents the linear single-degree-of-freedom (SDOF) equation of motion for a mass m, viscous damper c, stiffness k, subjected to a time-harmonic external force F(t)=F0 sin(ω t)?

2.Given m = 2.00 kg and k = 2000 N/m, what is the undamped natural circular frequency ω_n (rad/s)?

3.For the same system (m = 2.00 kg, k = 2000 N/m) with viscous damping c = 6.3246 N·s/m, what is the damping ratio ζ?

4.For the SDOF system m = 2.00 kg, k = 2000 N/m, c = 6.3246 N·s/m (ζ = 0.05), driven by F(t)=10.0 sin(30.0 t) N (ω = 30.0 rad/s), what is the steady-state displacement amplitude X (m)?

5.Using the same parameters (m = 2.00 kg, k = 2000 N/m, c = 6.3246 N·s/m) and forcing amplitude F0 = 10.0 N, what is the approximate steady-state amplitude at exact resonance (ω = ω_n)?

6.A rotor has a small unbalanced mass m_e = 0.010 kg located at eccentricity e = 0.005 m and spins at ω = 100.0 rad/s. What is the amplitude of the rotating unbalance force F0 (N)?

7.For base (support) harmonic excitation Y(t)=Y sin(ω t) on the same SDOF (m=2.00 kg, k=2000 N/m, ζ=0.05), if ω = 2 ω_n (r = 2) what is the transmissibility T = X/Y (dimensionless)?

8.What is the phase angle φ (in degrees) between the applied harmonic force F(t)=F0 sin(ω t) and the steady-state displacement x(t) for r = 0.95 and ζ = 0.05?

9.Regarding mode dominance in forced response of a multi-degree-of-freedom (MDOF) structure, which statement is correct?

10.Compute the critical damping c_cr and then the damping ratio ζ for m = 2.00 kg and k = 2000 N/m. (Report c_cr in N·s/m and ζ for c = 6.3246 N·s/m.)