Forced Vibration

students, imagine pushing a playground swing at just the right time 🎢. If you push randomly, the swing still moves, but not very much. If you push in rhythm, the swing can go much higher. That idea is the heart of forced vibration in Solid Mechanics 2. In this lesson, you will learn what forced vibration means, why it matters in engineering, and how it is analyzed in simple vibration models.

Lesson Objectives

By the end of this lesson, students, you should be able to:

Explain the main ideas and terms used in forced vibration.
Describe how an external force causes a system to vibrate.
Connect forced vibration to single-degree-of-freedom vibration models.
Understand why resonance can produce large motion.
Use basic reasoning to interpret vibration behavior in real systems.

Forced vibration is one of the most important topics in vibration fundamentals because many machines, vehicles, buildings, and structures experience outside forces all the time. Examples include an unbalanced motor, a car driving over a rough road, a washing machine spinning, or a bridge responding to wind 🌉.

What Forced Vibration Means

A vibrating system is said to undergo forced vibration when an external input causes it to move. That input may be a periodic force, a repeating motion at the support, or another disturbance applied from outside the system.

For a single-degree-of-freedom system, the simplest model is a mass-spring-damper system. The equation of motion is often written as:

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

Here:

$m$ is mass,
$c$ is the damping coefficient,
$k$ is stiffness,
$x(t)$ is displacement,
$F(t)$ is the applied force.

This equation says that the motion is controlled by the balance between inertia, damping, stiffness, and the external force. If $F(t)=0$, the system is in free vibration. If $F(t)\neq 0$, the system is in forced vibration.

A key idea is that the response depends not only on the size of the force, but also on how the force changes with time ⏱️. A steady force can create a static displacement, while a time-varying force can create oscillation.

Why External Forces Create Oscillation

To understand forced vibration, students, think about a mass on a spring. If you pull the mass and release it, the spring pulls it back and the mass overshoots. That is free vibration. Now imagine someone keeps tapping the mass. Each tap adds energy into the system. If the tapping happens repeatedly, the motion can become large.

In general, the external force may be:

Harmonic: $F(t)=F_0\sin(\omega t)$ or $F(t)=F_0\cos(\omega t)$,
Periodic but not sinusoidal,
Random or transient.

In Solid Mechanics 2, harmonic forcing is especially important because it gives clear results and models many real situations. A rotating unbalanced mass, for example, creates a force that changes sinusoidally with time.

For a harmonic force, the system often reaches a steady-state response after the initial motion fades. The total response can be thought of as two parts:

a transient response, which depends on initial conditions and dies out when damping is present,
a steady-state response, which remains as long as the force continues.

This split is very useful in engineering because it shows that damping helps remove unwanted motion over time.

Single-Degree-of-Freedom Forced Vibration

A single-degree-of-freedom system has one independent coordinate needed to describe its motion, such as the vertical displacement of a mass moving on a spring. This is the standard starting point for forced vibration analysis.

For harmonic forcing, the system is often modeled as:

$$m\ddot{x}+c\dot{x}+kx=F_0\sin(\omega t)$$

where $F_0$ is the force amplitude and $\omega$ is the forcing frequency.

A very important quantity is the natural frequency of the system:

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

This is the frequency at which the system tends to vibrate on its own. Another important quantity is the damping ratio:

$$\zeta=\frac{c}{2\sqrt{km}}$$

These values help determine how the system responds when forced.

If the forcing frequency $\omega$ is far from $\omega_n$, the response is usually moderate. If $\omega$ gets close to $\omega_n$, the motion can become much larger. This strong growth near natural frequency is called resonance. Resonance is one of the most important ideas in vibration because it can create serious problems in structures and machines ⚠️.

Undamped Forced Vibration

In the undamped case, $c=0$, so the equation becomes:

$$m\ddot{x}+kx=F_0\sin(\omega t)$$

This model is simpler, and it shows why resonance is so important. When the forcing frequency is not equal to the natural frequency, the solution has a steady oscillation. But when $\omega=\omega_n$, the system can build up very large motion over time.

For the undamped case, the response amplitude is strongly affected by the ratio $\frac{\omega}{\omega_n}$. When this ratio is close to $1$, the system is near resonance. In a perfect mathematical model without damping, the amplitude at exact resonance can grow without bound. Real systems always have some damping, so actual motion does not become infinite, but it can still become dangerously large.

A real-life example is pushing a swing. Small pushes at the right rhythm cause larger and larger motion. Another example is a bridge that responds to repeating footsteps or wind gusts. Engineers must design structures so that expected forcing frequencies do not align too closely with the natural frequencies.

Damped Forced Vibration

Most real systems have damping, so the more realistic model is:

$$m\ddot{x}+c\dot{x}+kx=F_0\sin(\omega t)$$

Damping removes energy from the system, usually by friction, material effects, or fluid resistance. Because of damping, the response at resonance is limited instead of becoming unbounded.

For a damped system, the steady-state displacement can be written in the form:

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

where $X$ is the response amplitude and $\phi$ is the phase lag between force and motion. The amplitude depends on the forcing frequency, stiffness, mass, damping, and force level.

As the forcing frequency changes, the response changes too. Near resonance, the amplitude increases, but the amount of increase depends on damping. More damping means a lower peak response and a wider, flatter resonance curve. Less damping means a sharper and higher peak 📈.

The phase angle $\phi$ is also important. At low frequencies, the motion is nearly in phase with the force. Near resonance, the phase shifts significantly. At high frequencies, the mass tends to lag behind the force. This phase behavior helps engineers understand how a system reacts over a range of input frequencies.

Real-World Examples and Engineering Meaning

Forced vibration appears in many ordinary and industrial systems.

A spinning washing machine may vibrate if the clothes are unevenly distributed. That imbalance creates a time-varying force. If the spin speed gets close to a natural frequency of the machine frame, the vibration can become strong.

In cars, the suspension system is designed to reduce forced vibration from road irregularities. The tire and suspension act like a mass-spring-damper system. Good design lowers the vibration transmitted to passengers and protects the vehicle components.

In rotating machinery, small imbalances can create harmonic forcing. If not corrected, the resulting vibration may damage bearings, reduce accuracy, or cause failure.

In buildings and bridges, wind, machinery, traffic, or earthquakes can act as external forces. Engineers study forced vibration so that structures remain safe, comfortable, and functional.

These examples show why forced vibration is not just a theoretical topic. It is a practical tool for predicting motion and preventing damage.

How Forced Vibration Fits in Vibration Fundamentals

Forced vibration is closely connected to the other parts of vibration fundamentals.

In undamped free vibration, the system moves on its own after an initial disturbance.
In damped free vibration, the motion dies out because energy is lost.
In forced vibration, the motion continues because an external input keeps supplying energy.

So forced vibration extends the earlier ideas by adding a real outside source. It combines the system’s natural properties with the frequency and strength of the forcing. This is why the same structure may behave very differently depending on how it is excited.

Understanding forced vibration helps you see that the response is not only about the system itself. It is also about the interaction between the system and the outside world.

Conclusion

students, forced vibration describes motion caused by an external force or input. In Solid Mechanics 2, it is commonly studied using the single-degree-of-freedom mass-spring-damper model:

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

For harmonic forcing, the response depends on the forcing frequency, natural frequency, and damping ratio. Resonance can greatly increase motion, but damping limits the response in real systems. Forced vibration is essential for understanding the behavior of machines, vehicles, and structures in everyday life. It completes the picture of vibration fundamentals by showing how systems respond when the world around them keeps pushing them.

Study Notes

Forced vibration happens when an external force or input causes a system to oscillate.
The standard single-degree-of-freedom model is $m\ddot{x}+c\dot{x}+kx=F(t)$.
For harmonic forcing, common forms are $F(t)=F_0\sin(\omega t)$ and $F(t)=F_0\cos(\omega t)$.
The natural frequency is $\omega_n=\sqrt{\frac{k}{m}}$.
The damping ratio is $\zeta=\frac{c}{2\sqrt{km}}$.
The response often has a transient part and a steady-state part.
Resonance occurs when the forcing frequency is close to the natural frequency.
Damping reduces the peak response and prevents unbounded motion in real systems.
Forced vibration is important in vehicles, machinery, buildings, and bridges.
It connects directly to free vibration by showing how external energy changes the system response.