Dynamics: Forced Vibration Response

Understanding forced vibration is crucial for engineers designing everything from skyscrapers and bridges to automotive suspensions and precision machinery. It explains how structures and mechanical systems behave when subjected to oscillatory forces, allowing you to predict amplitudes, avoid catastrophic resonances, and ensure operational safety and longevity. This analysis focuses on the steady-state response—the persistent vibration that remains after initial transients die out—of damped systems to harmonic excitation.

Harmonic Forcing and the System Model

We begin by modeling a standard single-degree-of-freedom damped system. It consists of a mass $m$, a linear spring with stiffness $k$, and a viscous damper with damping coefficient $c$. When this system is subjected to an external harmonic forcing function, the equation of motion becomes: $$m\ddot{x}+c\dot{x}+kx=F_0\cos (\omega t)$$ Here, $F_0$ is the force amplitude and $\omega$ is the forcing frequency (in rad/s). The natural frequency of the undamped system is ${\omega }_n=\sqrt{k/m}$, and the damping ratio is defined as $\zeta =c/(2\sqrt{km})$. The harmonic force is sinusoidal, representing common real-world excitations like unbalanced rotating parts or periodic wind loads.

Deriving the Particular Solution for Forced Response

The complete solution to the differential equation has two parts: the complementary solution (transient response) and the particular solution for forced response (steady-state response). For a linear system with harmonic input, the steady-state response will also be harmonic at the same frequency $\omega$, but with a possible phase shift. Therefore, we assume a particular solution of the form: $$x_p(t)=X\cos (\omega t-\varphi )$$ where $X$ is the amplitude of the steady-state response and $\varphi$ is the phase angle by which the response lags behind the forcing function. To find $X$ and $\varphi$, we substitute this assumed solution back into the equation of motion. This leads to a system of algebraic equations after equating coefficients of $\cos (\omega t)$ and $\sin (\omega t)$.

Amplitude and Phase as Functions of Frequency Ratio

Solving the algebraic equations yields explicit formulas for the amplitude and phase. It is most insightful to express these in terms of the frequency ratio $r=\omega /{\omega }_n$. The steady-state amplitude is: $$X=\frac{F_0/k}{\sqrt{(1-r^2{)}^2+(2\zeta r{)}^2}}$$ The phase angle is: $$\varphi ={\tan }^{-1}\left(\frac{2\zeta r}{1-r^2}\right)\text{for}0\le \varphi \le \pi$$ These equations show that both $X$ and $\varphi$ depend critically on $r$ and $\zeta$. The term $F_0/k$ represents the static deflection if the force were applied slowly. The denominator's square root modifies this static deflection based on dynamics. For example, at very low frequency ($r\ll 1$), the amplitude approaches $F_0/k$ and $\varphi \approx 0$, meaning the response follows the force in phase. At very high frequency ($r\gg 1$), the amplitude becomes very small and $\varphi \approx 180^{\circ }$, meaning the response is essentially out of phase with the force.

Magnification Factor and Frequency Response Curves

Engineers often use the magnification factor (or dynamic magnification factor) $M$ to normalize the response amplitude. It is defined as the ratio of the dynamic amplitude to the static deflection: $$M=\frac{X}{F_0/k}=\frac{1}{\sqrt{(1-r^2{)}^2+(2\zeta r{)}^2}}$$ Plotting $M$ and $\varphi$ against $r$ for various $\zeta$ values generates frequency response curves. These curves are fundamental design tools. The magnification factor curve peaks near $r=1$ (resonance) for light damping. The peak amplitude occurs at $r=\sqrt{1-2{\zeta }^2}$ and has a value of $M_{\text{max}}=1/(2\zeta \sqrt{1-{\zeta }^2})$. For small $\zeta$, this is approximately $1/(2\zeta )$. The phase curve shows a gradual shift from 0 to 180°, passing through 90° at $r=1$ regardless of damping. These plots allow you to quickly assess how a system will amplify or attenuate input forces across a frequency spectrum.

Beat Phenomenon from Near-Resonant Forcing

A special condition occurs when the forcing frequency is very close to, but not exactly equal to, the natural frequency ($\omega \approx {\omega }_n$). In lightly damped systems, this leads to the beat phenomenon. Beats are characterized by a slow, periodic variation in the amplitude of vibration. The total response in this near-resonant case can be viewed as the sum of two oscillations at closely spaced frequencies. The beat frequency is given by $|\omega -{\omega }_n|$, and the amplitude modulates at half this difference. For example, if a system with ${\omega }_n=100$ rad/s is forced at $\omega =102$ rad/s, you would observe amplitude oscillations at a frequency of 1 rad/s. This phenomenon is important for detecting small frequency mismatches in tuning and for understanding temporary large amplitudes before the steady-state is fully established.

Common Pitfalls

1. Ignoring the Phase Angle in System Alignment: Students often focus solely on amplitude, but the phase angle $\varphi$ is critical for understanding timing. In multi-component systems, incorrect phase relationships can lead to destructive interference or added stresses. Correction: Always calculate and consider both $X$ and $\varphi$. For instance, in a rotating machine, the phase lag determines where imbalance forces act relative to the displacement.
2. Confusing Transient and Steady-State Response: It's easy to assume the full solution is just the particular solution. However, the complete response includes transient terms that decay over time. On exams, you might be asked for the total response at a specific early time. Correction: Remember $x(t)=x_{\text{transient}}(t)+x_p(t)$. The transient part depends on initial conditions and decays as $e^{-\zeta {\omega }_{n}t}$.
3. Misapplying the Resonance Condition: A common mistake is stating that maximum amplitude always occurs exactly at $r=1$. This is only true for undamped systems. With damping, the peak frequency shifts slightly lower, as given by $r_{\text{peak}}=\sqrt{1-2{\zeta }^2}$. Correction: Use the correct formula for damped resonance frequency, especially when calculating critical speeds for machinery.
4. Overlooking Units in Frequency Ratio: The frequency ratio $r$ is dimensionless, but it requires $\omega$ and ${\omega }_n$ in the same units (e.g., both in rad/s or both in Hz). Mixing units (like rad/s and Hz) will lead to incorrect calculations. Correction: Consistently convert all frequencies to rad/s before computing $r$, recalling that $\omega =2\pi f$.

Summary

• The steady-state response of a damped system to a harmonic force $F_0\cos (\omega t)$ is itself harmonic, with amplitude $X$ and phase lag $\varphi$ given by functions of the frequency ratio $r=\omega /{\omega }_n$ and damping ratio $\zeta$.
• The magnification factor $M$ normalizes the dynamic amplitude against static deflection, and its plot versus $r$—the frequency response curve—shows how systems amplify inputs near resonance, with damping controlling the peak sharpness.
• The phase angle $\varphi$ transitions from 0° to 180° as $r$ increases, always passing through 90° at $r=1$, which is a key identifier of resonance in phase plots.
• Under near-resonant forcing ($\omega \approx {\omega }_n$) with light damping, the beat phenomenon occurs, producing a slowly oscillating amplitude envelope at a frequency equal to the difference between the forcing and natural frequencies.
• Mastery of these concepts enables you to predict vibration levels, design systems to avoid excessive resonance amplitudes, and interpret experimental frequency sweep data correctly.