Undamped Free Vibration

Introduction

students, imagine pushing a playground swing and then letting it go 🚲. If we ignore air resistance and friction, the swing keeps moving back and forth forever. That idealized motion is the heart of undamped free vibration. In Solid Mechanics 2, this topic helps us understand how real structures like beams, machine parts, bridges, and car components can move when disturbed.

In this lesson, you will learn to:

Explain the main ideas and vocabulary of undamped free vibration.
Work with the standard single-degree-of-freedom model.
Solve the motion of a mass-spring system after an initial disturbance.
Connect this lesson to the larger study of vibration fundamentals.

This topic matters because engineers often begin with the simplest vibration model before studying more realistic cases with damping and forcing. Understanding the ideal case makes the real case much easier to analyze later ✅.

What Undamped Free Vibration Means

A vibration is repeated motion about an equilibrium position. In free vibration, the system moves after being disturbed, but there is no external driving force acting on it during the motion. In undamped vibration, we also assume no energy is lost to friction, air resistance, or internal material effects.

That means the system keeps exchanging energy between two forms:

Kinetic energy when the mass is moving fastest
Potential energy when the spring is stretched or compressed the most

A common single-degree-of-freedom model is a mass attached to a spring. If the mass is pulled away from equilibrium and released, it oscillates back and forth. The position is usually measured by a displacement variable such as $x(t)$.

Key terms you should know:

Equilibrium position: the resting position where forces balance
Displacement: the distance from equilibrium, written as $x(t)$
Amplitude: the maximum value of $|x(t)|$
Period: the time for one complete cycle, written as $T$
Frequency: the number of cycles per second, written as $f$
Natural frequency: the frequency at which the system vibrates on its own

For a simple mass-spring system, the restoring force from the spring is given by Hooke’s law:

$$F=-kx$$

where $k$ is the spring stiffness. The minus sign means the force acts opposite to the displacement, pulling the mass back toward equilibrium.

Building the Single-Degree-of-Freedom Model

A single-degree-of-freedom system needs only one coordinate to describe its motion. For the classic undamped free vibration model, that coordinate is $x(t)$.

Consider a mass $m$ connected to a spring with stiffness $k$. When the mass moves, Newton’s second law gives:

$$m\ddot{x}=-kx$$

Rearranging produces the standard differential equation:

$$m\ddot{x}+kx=0$$

This equation describes the balance between inertia and restoring force.

To make the equation easier to interpret, divide by $m$:

$$\ddot{x}+\frac{k}{m}x=0$$

We define the natural circular frequency as

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

so the equation becomes

$$\ddot{x}+\omega_n^2 x=0$$

This is the governing equation for undamped free vibration of a mass-spring system. It tells us that acceleration is always directed toward equilibrium, which creates oscillation 🔁.

Solving the Motion

The solution to

$$\ddot{x}+\omega_n^2 x=0$$

is a sinusoidal function:

$$x(t)=C_1\cos(\omega_n t)+C_2\sin(\omega_n t)$$

This can also be written in amplitude-phase form:

$$x(t)=X\cos(\omega_n t-\phi)$$

where $X$ is the amplitude and $\phi$ is the phase angle.

The constants are found from the initial conditions. If the mass starts at displacement $x(0)=x_0$ and velocity $\dot{x}(0)=v_0$, then:

$$x(0)=C_1=x_0$$

and

$$\dot{x}(t)=-C_1\omega_n\sin(\omega_n t)+C_2\omega_n\cos(\omega_n t)$$

$$\dot{x}(0)=C_2\omega_n=v_0$$

which gives

$$C_2=\frac{v_0}{\omega_n}$$

Therefore the complete response is

$$x(t)=x_0\cos(\omega_n t)+\frac{v_0}{\omega_n}\sin(\omega_n t)$$

This equation shows that the future motion depends on how the system was released. A small initial push can create a large oscillation if the system is very flexible or light.

Example

Suppose a $2\,\text{kg}$ mass is attached to a spring with stiffness $8\,\text{N/m}$. Then

$$\omega_n=\sqrt{\frac{8}{2}}=2\,\text{rad/s}$$

If the mass is pulled to $x_0=0.1\,\text{m}$ and released from rest, then $v_0=0$ and

$$x(t)=0.1\cos(2t)$$

This means the mass oscillates between $0.1\,\text{m}$ and $-0.1\,\text{m}$ forever in the ideal undamped model.

Frequency, Period, and Physical Meaning

The natural circular frequency $\omega_n$ is measured in radians per second. Engineers also use ordinary frequency $f$, measured in hertz, where

$$f=\frac{\omega_n}{2\pi}$$

The period is the time for one cycle:

$$T=\frac{2\pi}{\omega_n}$$

For the example above, with $\omega_n=2\,\text{rad/s}$,

$$T=\frac{2\pi}{2}=\pi\,\text{s}$$

and

$$f=\frac{2}{2\pi}=\frac{1}{\pi}\,\text{Hz}$$

These values tell us how quickly the system repeats its motion.

The meaning of natural frequency is very important in engineering. Every structure has one or more natural frequencies. If a load pushes the structure repeatedly near one of these frequencies, the motion can become large. That idea is part of why vibration analysis matters in bridges, engines, rotating machines, and buildings 🏗️.

Energy View of Undamped Free Vibration

A helpful way to understand undamped vibration is through energy. In an ideal system, total mechanical energy stays constant.

The kinetic energy is

$$T=\frac{1}{2}m\dot{x}^2$$

and the spring potential energy is

$$V=\frac{1}{2}kx^2$$

So the total energy is

$$E=T+V=\frac{1}{2}m\dot{x}^2+\frac{1}{2}kx^2$$

Because there is no damping, $E$ does not decrease with time.

At maximum displacement, the velocity is zero, so kinetic energy is zero and spring potential energy is largest. At equilibrium, displacement is zero, so spring potential energy is zero and kinetic energy is largest. The energy keeps moving back and forth between these two forms.

This is why the amplitude remains constant in the ideal model. In real systems, damping causes energy loss, so the amplitude eventually decreases. That contrast is a key bridge from undamped free vibration to damped free vibration later in the course.

How This Fits into Vibration Fundamentals

Undamped free vibration is one of the first models studied in vibration fundamentals because it is simple but powerful. It teaches several core ideas:

Motion can be described with a differential equation.
Restoring force leads to oscillation.
Natural frequency depends on system properties like $m$ and $k$.
Initial conditions determine the exact response.
Energy conservation explains constant amplitude in the ideal case.

This topic also prepares you for more advanced vibration problems. Once you understand the undamped model, you can add damping, external forcing, and more degrees of freedom.

For example, if a machine part vibrates after being hit, the first approximation may be an undamped free vibration model. If a car suspension is analyzed, damping must be included because the system is designed to reduce motion. If a bridge is studied under traffic or wind, forcing and damping both matter. The undamped case is the clean starting point that helps organize those more complex situations.

Conclusion

students, undamped free vibration describes ideal oscillatory motion that continues without energy loss after an initial disturbance. For the basic mass-spring system, the equation of motion is

$$m\ddot{x}+kx=0$$

with natural circular frequency

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

The response is sinusoidal, the amplitude stays constant, and the total mechanical energy remains unchanged. Even though real systems usually have damping, this ideal model is essential because it reveals the structure of vibration behavior and introduces the main language of the subject.

Mastering undamped free vibration gives you a strong foundation for the rest of vibration fundamentals 🌟.

Study Notes

Vibration is repeated motion about equilibrium.
Free vibration means no external driving force acts during motion.
Undamped means no energy is lost to friction, air resistance, or internal dissipation.
The classic single-degree-of-freedom model is a mass $m$ attached to a spring with stiffness $k$.
The restoring force is given by $F=-kx$.
The governing equation is $m\ddot{x}+kx=0$.
The natural circular frequency is $\omega_n=\sqrt{\frac{k}{m}}$.
The displacement response is $x(t)=C_1\cos(\omega_n t)+C_2\sin(\omega_n t)$.
With initial conditions $x(0)=x_0$ and $\dot{x}(0)=v_0$, the response is $x(t)=x_0\cos(\omega_n t)+\frac{v_0}{\omega_n}\sin(\omega_n t)$.
The period is $T=\frac{2\pi}{\omega_n}$ and the frequency is $f=\frac{\omega_n}{2\pi}$.
Total mechanical energy is $E=\frac{1}{2}m\dot{x}^2+\frac{1}{2}kx^2$ and remains constant in the ideal undamped case.
This lesson is the starting point for understanding damped vibration and forced vibration later in Vibration Fundamentals.