Free Vibration

Free vibration is the motion of a mechanical system after it is given an initial disturbance (displacement and/or velocity) and then allowed to vibrate with no external forcing. For single-degree-of-freedom (SDOF) systems the response depends on mass, stiffness, and damping. Undamped free vibration is simple harmonic motion at the natural frequency. Damped free vibration decays exponentially with a damped natural frequency for underdamped systems. For multi-degree-of-freedom (MDOF) systems free vibration decomposes into modal motions (eigenvalues and eigenvectors), and modal orthogonality facilitates modal analysis. Assumptions in the questions below: linear behavior, small displacements, constant parameters, SI units.

Governing FormulaKey equations (SDOF): - Equation of motion (with external force F(t)): m x¨ + c x˙ + k x = F(t). For free vibration F(t)=0. - Undamped natural frequency: ωn = sqrt(k/m) [rad/s], fn = ωn/(2π) [Hz]. - Critical damping: cc = 2 m ωn. Damping ratio: ζ = c/(2 m ωn). - Damped natural frequency (underdamped, 0<ζ<1): ωd = ωn sqrt(1-ζ^2). - Undamped solution (initial conditions x(0)=x0, x˙(0)=v0): x(t) = A cos(ωn t - φ), A = sqrt(x0^2 + (v0/ωn)^2), tan φ = v0/(ωn x0). - Energy (undamped): total energy E = 1/2 k A^2 = 1/2 m (ωn A)^2; KE(t)=1/2 m x˙^2, PE(t)=1/2 k x^2. - MDOF eigenproblem (undamped): (K - ω^2 M) Φ = 0; mode orthogonality: 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 best defines free vibration of a mechanical system?

2.An undamped SDOF system has m = 2.00 kg and k = 2000 N/m. What is the natural frequency fn in Hz?

3.For an SDOF system with m = 1.00 kg, k = 400 N/m and viscous damping c = 4.00 N·s/m, what is the damped natural frequency ωd (rad/s)?

4.In undamped free vibration of an SDOF with amplitude A, where is the kinetic energy maximum and what is its value in terms of A, k? (Assume linear stiffness k.)

5.A mass m = 5.00 kg attached to a spring k = 12500 N/m is critically damped. What is the critical damping coefficient cc (N·s/m)?

6.For an undamped SDOF (m=1.00 kg, k=9.00 N/m) with initial conditions x(0)=0.100 m and v(0)=1.00 m/s, what is the oscillation amplitude A (m)?

7.A symmetric 2-DOF undamped system has equal masses m1=m2=1.00 kg and three identical springs such that the stiffness matrix yields eigenvalues proportional to {1,3}·(k/m). If k = 200 N/m, what is the lower natural frequency f1 in Hz?

8.For an underdamped SDOF with ζ=0.02 and ωn = 100 rad/s, how long (t in seconds) until the envelope amplitude decays to 5% of its initial value? Use exponential decay envelope exp(-ζ ωn t).

9.Which property describes modal orthogonality for an undamped MDOF system (K and M are symmetric positive-definite matrices)?

10.An undamped SDOF has m = 2.00 kg and k = 8.00 N/m. If released from x(0)=0 with initial velocity v(0)=1.00 m/s, what is the maximum displacement reached (m)?