Undamped Systems

Undamped single-degree-of-freedom (SDOF) systems are linear mechanical systems with mass and stiffness but no energy dissipation (damping = 0). Their motion is governed by simple harmonic oscillation: free vibration is purely sinusoidal at the natural frequency, and forced harmonic excitation can produce large responses near resonance. In multiple-DOF undamped systems, modal analysis yields real-valued natural frequencies and mode shapes that are orthogonal with respect to the mass and stiffness matrices. Because no damping is present, total mechanical energy (kinetic + potential) is conserved in free vibration; in forced vibration at forcing frequency equal to a natural frequency the ideal linear undamped model predicts unbounded steady-state amplitudes (mathematical resonance).

Governing FormulaAssumptions: linear stiffness k (N/m), mass m (kg), no damping (c = 0), small displacements, SI units. Key equations for an undamped SDOF system: - Equation of motion (time domain): m x''(t) + k x(t) = F(t) - Natural (circular) frequency: ω_n = sqrt(k/m) (rad/s) - Natural frequency (Hz): f_n = ω_n / (2π) - Free vibration general solution: x(t) = C cos(ω_n t) + D sin(ω_n t) = A cos(ω_n t - φ) - Forced harmonic response to F(t)=F0 sin(ω t): steady-state amplitude X = F0 / (k |1 - r^2|), where r = ω/ω_n (undamped). Note: at r = 1 this expression becomes unbounded (mathematical resonance). - Base (support) harmonic excitation y(t)=Y sin(ω t) transmissibility (displacement ratio) for undamped SDOF: T = |X/Y| = r^2 / |1 - r^2|, where r = ω/ω_n. - Rotating unbalance harmonic forcing amplitude: F0 = m_e e ω^2 (N), where m_e (kg) is the eccentric mass, e (m) eccentricity, and ω (rad/s) rotational speed. - Total mechanical energy (free vibration): E = 1/2 m x'(t)^2 + 1/2 k x(t)^2 = constant.

Knowledge Check

10 Questions

1.Which differential equation correctly represents the equation of motion for a linear undamped SDOF system subjected to an external force F(t)?

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

3.If the undamped natural circular frequency is ω_n = 20 rad/s, what is the natural frequency f_n in Hz?

4.What is the general form of the free-vibration solution for an undamped SDOF system?

5.For an ideal undamped SDOF subjected to a harmonic force F(t)=F0 sin(ω t), what happens to the steady-state amplitude as ω approaches ω_n (exact resonance)?

6.Which expression gives the displacement transmissibility T (ratio of mass displacement amplitude to base displacement amplitude) for an undamped SDOF under harmonic base excitation at frequency ω (r = ω/ω_n)?

7.Compute the displacement transmissibility for an undamped SDOF with m = 1.0 kg, k = 100 N/m (so ω_n = 10 rad/s) when the base is harmonically excited at ω = 20 rad/s.

8.For an undamped linear multiple-DOF system with distinct natural frequencies, which statement about mode shapes is correct?

9.A rotating unbalance consists of an eccentric point mass m_e = 0.100 kg at eccentricity e = 0.005 m rotating at ω = 100 rad/s. What is the amplitude of the out-of-balance horizontal harmonic force F0 (N)?

10.In free vibration of an undamped SDOF system, which of the following quantities is conserved (constant in time)?