Mechanical Vibrations: Single Degree of Freedom

Understanding the vibration of simple systems is the cornerstone of mechanical engineering, with direct applications ranging from designing earthquake-resistant buildings to isolating sensitive laboratory equipment from floor vibrations. By mastering the analysis of a single degree-of-freedom (SDOF) system—a model with mass, stiffness, and damping—you build the foundational skills to predict, control, and mitigate oscillatory motion.

Defining the System Components

An SDOF system is the simplest model for vibratory motion, implying that its position can be completely described by a single coordinate, such as the vertical displacement $x(t)$ of a mass. The model consists of three idealized elements working in parallel. First, the mass $m$ represents the system's inertia, resisting acceleration according to Newton's second law ($F=m\ddot{x}$). Second, the spring provides a restoring force proportional to displacement via its stiffness $k$; this force always acts to return the mass to its equilibrium position. Third, the viscous damper (or dashpot) provides a force proportional to velocity, modeled by the damping coefficient $c$. This force dissipates energy, typically as heat, and opposes the direction of motion. The interplay of these three elements—inertia ($m$), restoration ($k$), and energy dissipation ($c$)—defines all vibrational behavior of the system.

Free Vibration and Natural Frequency

When a system is displaced from equilibrium and released with no external force acting on it, it undergoes free vibration. If damping is absent ($c=0$), the motion is undamped free vibration, described by the simple harmonic oscillator equation: $m\ddot{x}+kx=0$. The solution is a pure sinusoid, $x(t)=A\cos ({\omega }_{n}t-\varphi )$, where $A$ is the amplitude and $\varphi$ is the phase angle. The rate of this oscillation is the system's natural frequency, denoted ${\omega }_n$ (in radians per second) and calculated from stiffness and mass alone: ${\omega }_n=\sqrt{k/m}$. Its cyclic counterpart in Hertz is $f_n={\omega }_n/2\pi$. This frequency is an intrinsic property of the system; a stiffer spring or a lighter mass results in a faster natural oscillation.

The Role of Damping

In real systems, energy is always dissipated, making damped free vibration the more practical model. The equation of motion becomes $m\ddot{x}+c\dot{x}+kx=0$. The character of the solution—and the system's behavior—depends entirely on the damping ratio $\zeta$, a dimensionless measure defined as $\zeta =c/(2\sqrt{mk})=c/(2m{\omega }_n)$. This ratio categorizes the system's response into three regimes:

• Underdamped ($\zeta <1$): The system oscillates with an exponentially decaying amplitude. This is the most common case for structural vibrations. The frequency of this damped oscillation is ${\omega }_d={\omega }_n\sqrt{1-{\zeta }^2}$.
• Critically damped ($\zeta =1$): The system returns to equilibrium in the minimum possible time without oscillating. This is often a design target for vehicle suspensions and door closers.
• Overdamped ($\zeta >1$): The system returns to equilibrium sluggishly without oscillation, as the damping force is so strong it prevents any overshoot.

Forced Vibration and Resonance

When a persistent external force $F(t)$ acts on the system, it undergoes forced vibration. A critically important case is harmonic excitation, $F(t)=F_0\cos (\omega t)$, where $F_0$ is the force amplitude and $\omega$ is the excitation frequency. The steady-state solution (after initial transients die out) is a harmonic oscillation at the driving frequency $\omega$, but with an amplitude $X$ that depends on the system properties. Analyzing the equation $m\ddot{x}+c\dot{x}+kx=F_0\cos (\omega t)$ yields a crucial relationship: the amplitude of vibration is magnified or reduced based on the frequency ratio $r=\omega /{\omega }_n$ and the damping ratio $\zeta$.

The magnification is most dramatic at resonance, which occurs when the excitation frequency $\omega$ is close to the system's undamped natural frequency ${\omega }_n$. For an undamped system at exact resonance ($\omega ={\omega }_n,\zeta =0$), the theoretical amplitude grows to infinity. In real, damped systems, resonance occurs at $\omega ={\omega }_n\sqrt{1-2{\zeta }^2}$, and the amplitude at this point is given by $X_{max}\approx F_0/(2k\zeta )$ for light damping. This explains why even a small periodic force, like an out-of-balance rotating part, can cause destructively large vibrations if its frequency matches a system's natural frequency. A plot of amplitude versus frequency ratio (a frequency response function) clearly shows a peak at resonance, with the peak's height and sharpness being inversely related to the damping ratio.

Vibration Transmissibility and Isolation

The concepts of forced vibration lead directly to two key engineering applications: transmissibility and isolation. Transmissibility $T_r$ is defined as the ratio of the force transmitted to the foundation (or the motion of a mounted equipment) to the applied disturbing force (or base motion). For a system with base excitation, the transmissibility ratio is derived from the same frequency response analysis and depends on $r$ and $\zeta$: $$T_r=\sqrt{\frac{1+(2\zeta r{)}^2}{(1-r^2{)}^2+(2\zeta r{)}^2}}$$

This equation informs vibration isolation strategy. To isolate a sensitive machine from a vibrating floor (or to isolate a vibrating machine from its foundation), you want to minimize transmissibility ($T_r<1$). The plot of $T_r$ reveals that isolation only occurs when the frequency ratio $r>\sqrt{2}$, regardless of damping. Therefore, a practical isolation design involves selecting spring mounts (which set ${\omega }_n$) so that the system's natural frequency is much lower than the disturbance frequency $\omega$. While damping reduces the resonant peak, it slightly degrades isolation performance in the $r>\sqrt{2}$ region—a classic engineering trade-off.

Common Pitfalls

1. Confusing Natural Frequency with Forcing Frequency: The natural frequency ${\omega }_n$ is a fixed property of the system (mass and stiffness). The forcing frequency $\omega$ is an independent input. Resonance is the condition where they are equal or very close.
2. Misapplying the Damped Natural Frequency Formula: Remember that the frequency of damped oscillation, ${\omega }_d={\omega }_n\sqrt{1-{\zeta }^2}$, is only valid for the underdamped ($\zeta <1$) free vibration case. It does not apply to forced vibration, where the steady-state response always occurs at the driving frequency $\omega$.
3. Overlooking the Isolation Region: A common mistake is thinking that adding more stiffness always reduces vibration. For isolation ($T_r<1$), you need a softer spring (lower $k$, and thus lower ${\omega }_n$) to achieve a high frequency ratio $r>\sqrt{2}$.
4. Assuming Resonance is Exactly at ${\omega }_n$: For damped forced vibration, the peak amplitude (resonance) actually occurs at $\omega ={\omega }_n\sqrt{1-2{\zeta }^2}$, which is slightly less than ${\omega }_n$. For lightly damped systems, the difference is negligible, but it's an important conceptual point.

Summary

• The single degree-of-freedom (SDOF) model with mass ($m$), stiffness ($k$), and damping ($c$) is the essential building block for analyzing mechanical vibrations.
• The system's natural frequency ${\omega }_n=\sqrt{k/m}$ dictates its inherent oscillation rate, while the damping ratio $\zeta$ determines whether free vibration is oscillatory ($\zeta <1$), critical ($\zeta =1$), or non-oscillatory ($\zeta >1$).
• Under harmonic forced vibration, the system's steady-state amplitude depends on the frequency ratio $r=\omega /{\omega }_n$ and $\zeta$. Resonance, leading to large amplitudes, occurs when $r\approx 1$.
• Transmissibility describes how much force or motion is transferred across the system. Effective vibration isolation requires designing the system so that $r>\sqrt{2}$, typically by using soft springs to achieve a low natural frequency relative to the disturbance frequency.