Understanding Single Degree-of-Freedom Systems and Harmonic Excitation

A single degree-of-freedom (SDOF) system is the simplest model used to analyze the dynamic behavior of structures and mechanical components. It represents a system whose motion can be described by a single coordinate. When such a system is subjected to a harmonic force, which is a repetitive force that varies sinusoidally with time, it experiences forced vibrations.

Many real-world systems, like bridges, buildings, or machine parts, can be simplified into SDOF models for initial analysis. The presence of viscous damping, which is a type of resistance proportional to velocity, is crucial in preventing infinitely large amplitudes at resonance.

Phase Angle in Vibrating Systems

When a system is excited by a harmonic force, its response (e.g., displacement) will also be harmonic, but it may not be "in phase" with the exciting force. The difference in timing between the peak of the exciting force and the peak of the displacement is called the phase angle or phase lag.

For a viscously damped single degree-of-freedom system subjected to a harmonic force \(F(t) = F_0 \sin(\omega t)\), the steady-state displacement response \(x(t)\) can be expressed as:

\(x(t) = X \sin(\omega t - \phi)\)

where \(X\) is the amplitude of the displacement, \(\omega\) is the excitation frequency, and \(\phi\) is the phase angle (or phase lag) of the displacement with respect to the exciting force. The phase angle \(\phi\) indicates by how much the displacement "lags behind" the exciting force.

The expression for the phase angle \(\phi\) is given by:

\(\tan\phi = \frac{2\zeta r}{1 - r^2}\)

Here:

• \(\zeta\) (zeta) is the damping ratio, a dimensionless measure of the amount of damping in the system.
• \(r = \frac{\omega}{\omega_n}\) is the frequency ratio, where \(\omega\) is the excitation frequency and \(\omega_n = \sqrt{\frac{k}{m}}\) is the natural frequency of the undamped system (\(k\) is stiffness, \(m\) is mass).

Resonance and Its Effect on Phase Angle

Resonance occurs when the excitation frequency (\(\omega\)) is equal or very close to the natural frequency (\(\omega_n\)) of the system. In other words, at resonance, the frequency ratio \(r = \frac{\omega}{\omega_n} = 1\).

At resonance, the amplitude of vibration tends to become very large, especially in systems with low damping. Let's analyze the phase angle at this critical condition for a single degree-of-freedom system with viscous damping.

Substituting \(r = 1\) into the phase angle formula:

\(\tan\phi = \frac{2\zeta (1)}{1 - (1)^2}\)

\(\tan\phi = \frac{2\zeta}{1 - 1}\)

\(\tan\phi = \frac{2\zeta}{0}\)

For any non-zero damping ratio \(\zeta\) (i.e., for a viscously damped system, \(\zeta > 0\)), the numerator \(2\zeta\) will be a positive value. When the denominator is zero, \(\tan\phi\) approaches infinity. This mathematical condition implies that the phase angle \(\phi\) is \(90^{\circ}\) (or \(\frac{\pi}{2}\) radians).

This means that at resonance, the displacement of the system lags the exciting force by exactly \(90^{\circ}\). The displacement response is in quadrature with the exciting force.

Therefore, for a single degree-of-freedom system with viscous damping excited by a harmonic force, at resonance, the phase angle of the displacement with respect to the exciting force is \(90^{\circ}\).