Damped Free Vibration

Introduction

In Solid Mechanics 2, vibration problems show up whenever a structure or machine moves back and forth around an equilibrium position. Think of a car suspension after hitting a bump, a ruler clamped on a desk and released, or a tall building swaying slightly in the wind 🌬️. In many real systems, the motion does not continue forever. Instead, the vibration gradually gets smaller because energy is lost to friction, internal material effects, or air resistance. That is called damped free vibration.

In this lesson, students, you will learn how damping changes the motion of a single-degree-of-freedom system, why the displacement decays with time, and how engineers describe the different damping cases. By the end, you should be able to connect damped free vibration to the wider topic of vibration fundamentals and use the standard equations used in Solid Mechanics 2.

Objectives

Explain the main ideas and terminology behind damped free vibration.
Apply Solid Mechanics 2 reasoning to damped vibration models.
Connect damped free vibration to vibration fundamentals.
Summarize how damping affects free motion.
Use examples to interpret damped vibration behavior.

What Damping Means in a Vibration System

A free vibration problem is one where the system moves after an initial disturbance and no external forcing acts on it. For a simple mass-spring-damper model, the system has mass $m$, damping coefficient $c$, and spring stiffness $k$. The motion is measured by displacement $x(t)$ from equilibrium.

Damping is a mechanism that removes mechanical energy from the system. In many engineering models, the damping force is taken as proportional to velocity, so the resisting force is $c\dot{x}(t)$. This means that when the object moves faster, the damping force becomes larger. This model is called viscous damping.

The equation of motion for damped free vibration in a single-degree-of-freedom system is

$$m\ddot{x}(t)+c\dot{x}(t)+kx(t)=0$$

This equation is the starting point for almost all basic vibration analysis in this topic. The three terms represent inertia, damping, and stiffness. Together they determine whether the motion dies out slowly or quickly, and whether the system oscillates while dying out.

A useful idea is that damping does not usually stop motion instantly. Instead, it reduces the total mechanical energy over time. As a result, the peaks of the motion become smaller and smaller 📉.

Key Parameters and Terminology

To analyze damped free vibration, engineers use a few standard quantities.

The undamped natural frequency is

$$\omega_n=\sqrt{\frac{k}{m}}$$

It is the frequency the system would have if there were no damping at all.

The critical damping coefficient is

$$c_c=2m\omega_n=2\sqrt{km}$$

This is a reference value used to compare the actual damping in the system.

The damping ratio is

$$\zeta=\frac{c}{c_c}$$

This dimensionless number is one of the most important ideas in vibration fundamentals. It tells us how strong the damping is relative to the amount needed for critical damping.

These parameters help classify the motion into three main cases:

Underdamped when $0<\zeta<1$
Critically damped when $\zeta=1$
Overdamped when $\zeta>1$

There is also the special case of undamped vibration when $\zeta=0$, but that is not damped free vibration. In real systems, some damping is almost always present.

Underdamped Free Vibration: The Most Common Case

Most engineering systems of interest are underdamped, meaning the system still oscillates but the amplitude decays with time. This is the case for a mass attached to a spring with some energy loss, such as a laboratory test rig or a bouncing vehicle suspension.

For $0<\zeta<1$, the motion is

$$x(t)=e^{-\zeta\omega_n t}\left(C_1\cos(\omega_d t)+C_2\sin(\omega_d t)\right)$$

where the damped natural frequency is

$$\omega_d=\omega_n\sqrt{1-\zeta^2}$$

This formula shows two important effects at the same time. First, the motion oscillates with frequency $\omega_d$. Second, the exponential factor $e^{-\zeta\omega_n t}$ causes the amplitude to decay. The larger the damping ratio $\zeta$, the faster the decay.

A useful real-world example is a car suspension after the car passes over a speed bump 🚗. The wheel and body may move up and down several times, but each bounce becomes smaller because shocks and internal resistance remove energy.

The constants $C_1$ and $C_2$ come from the initial conditions, such as initial displacement $x(0)$ and initial velocity $\dot{x}(0)$. Engineers use those conditions to calculate the exact motion.

Critical Damping and Overdamping

Not every damped system oscillates. If damping is large enough, oscillations may disappear entirely.

When $\zeta=1$, the system is critically damped. The motion returns to equilibrium as quickly as possible without oscillating. The general solution has the form

$$x(t)=(C_1+C_2t)e^{-\omega_n t}$$

This case is important in design when fast settling is desired, such as in some measuring instruments and door closers.

When $\zeta>1$, the system is overdamped. The displacement still returns to equilibrium without oscillation, but more slowly than in the critically damped case. The motion can be written as a sum of two decaying exponentials. The exact form depends on the system parameters, but the key idea is that the response is non-oscillatory and sluggish.

These cases matter because they show that damping is not simply “good” or “bad.” Too little damping may cause long-lasting oscillations, while too much damping may make a system slow to respond.

How Damping Changes Energy and Response

Damping changes the system by converting mechanical energy into heat or internal losses. In a spring-mass system, energy continually shifts between kinetic energy and strain energy. Without damping, the total energy stays constant. With damping, the total mechanical energy decreases over time.

That is why the response envelope shrinks. If you imagine marking the tops of successive peaks, they form a decaying curve. This is one of the clearest signs of damped free vibration.

Another helpful quantity is the logarithmic decrement, which compares the size of two successive peaks. If $x_1$ and $x_2$ are consecutive maximum displacements, then

$$\delta=\ln\left(\frac{x_1}{x_2}\right)$$

For underdamped systems, this can be extended over multiple cycles to estimate the damping ratio. In practice, engineers use it in experiments when they record a vibration signal and want to identify damping from measured data.

For example, if a machine frame vibrates after being hit and the first peak is twice the second peak, then

$$\delta=\ln(2)$$

This indicates measurable damping. Such calculations are common in testing and diagnostics.

Solving a Basic Damped Free Vibration Problem

When solving a damped free vibration problem, the usual steps are:

Write the equation of motion
Calculate $\omega_n$ and $\zeta$
Classify the damping case
Write the correct solution form
Use initial conditions to find constants

Suppose a system has $m$, $c$, and $k$. First compute

$$\omega_n=\sqrt{\frac{k}{m}}$$

Then compute

$$\zeta=\frac{c}{2m\omega_n}$$

If $0<\zeta<1$, use the underdamped form with

$$\omega_d=\omega_n\sqrt{1-\zeta^2}$$

Then apply initial displacement $x(0)$ and initial velocity $\dot{x}(0)$ to find the constants.

This procedure is a standard Solid Mechanics 2 method because it combines physical understanding with mathematical modeling. It also links directly to single-degree-of-freedom vibration analysis, which is the foundation of more advanced topics.

Why Damped Free Vibration Matters in Engineering

Damped free vibration is not just a classroom idea. It is used in many real systems:

Buildings and bridges must avoid excessive oscillation after wind or earthquakes 🏗️
Vehicle suspensions must settle quickly for safety and comfort
Machine tools must reduce vibration for accuracy
Measuring devices need controlled motion to produce reliable readings

In each case, engineers care about the damping ratio because it affects how fast the vibration dies out and whether oscillations are present. A well-designed system balances stability, speed of response, and comfort.

Damped free vibration is also a bridge to more advanced vibration topics. Once students understands this lesson, it becomes easier to study forced vibration, resonance, vibration isolation, and multi-degree-of-freedom systems.

Conclusion

Damped free vibration describes the natural motion of a system when no external force acts, but energy is lost because of damping. The core model is

$$m\ddot{x}(t)+c\dot{x}(t)+kx(t)=0$$

The main parameters are $\omega_n$, $c_c$, and $\zeta$. These quantities determine whether the motion is underdamped, critically damped, or overdamped. In the most common case, the response oscillates while its amplitude decays exponentially. This behavior explains why real systems eventually settle down after being disturbed.

Understanding damped free vibration gives students a strong foundation for the rest of vibration fundamentals and for many engineering applications where motion control matters.

Study Notes

Damped free vibration is motion after an initial disturbance with no external forcing, but with energy loss.
The standard equation is $m\ddot{x}(t)+c\dot{x}(t)+kx(t)=0$.
The undamped natural frequency is $\omega_n=\sqrt{\frac{k}{m}}$.
The critical damping coefficient is $c_c=2m\omega_n=2\sqrt{km}$.
The damping ratio is $\zeta=\frac{c}{c_c}$.
If $0<\zeta<1$, the system is underdamped and oscillates with decaying amplitude.
The damped natural frequency is $\omega_d=\omega_n\sqrt{1-\zeta^2}$.
If $\zeta=1$, the system is critically damped and returns fastest without oscillation.
If $\zeta>1$, the system is overdamped and returns without oscillation more slowly.
Damping removes mechanical energy, usually converting it to heat or internal losses.
The logarithmic decrement is $\delta=\ln\left(\frac{x_1}{x_2}\right)$ for successive peaks.
Damped free vibration is a core topic in vibration fundamentals and helps explain real machine and structural behavior.