KAHIBARO
Table of Contents
External Cause of Oscillation
A driving force is an external force that acts on an oscillating system again and again, usually in a periodic way. Its role is to supply energy to the system from outside. Without this repeated input, real oscillators gradually lose energy because of damping. With a driving force, the motion can be maintained, increased, or controlled.
A simple idea is a child on a swing. If the swing is left alone, friction and air resistance slowly reduce its motion. If someone gives regular pushes, the swing keeps moving. Those pushes are the driving force.
Periodic Driving
In many important cases, the driving force changes with time in a regular pattern. A common mathematical form is
$$
F(t) = F_0 \cos(\omega t)
$$
where $F_0$ is the maximum value of the force and $\omega$ is the driving angular frequency. The force may also be written with sine instead of cosine. The choice does not change the physics, it only changes the starting time reference.
This kind of force does not act just once. It acts continuously and repeatedly, so the oscillator is being fed energy over time.
A driving force is an external time-dependent force, often periodic, that keeps an oscillator moving by supplying energy.
A common model is
$$
F(t) = F_0 \cos(\omega t)
$$
where $F_0$ is the driving amplitude and $\omega$ is the driving angular frequency.
What the Driving Force Does
The driving force tries to make the system move according to its own rhythm. The oscillator itself also has its own natural tendency to oscillate. The actual motion depends on the competition between the external forcing and the properties of the system.
If the applied force is weak, the response may be small. If the force is stronger, the amplitude of the oscillation can become larger. If energy losses are present, the driving force may exactly balance those losses, producing a steady repeated motion instead of a motion that dies away.
In this steady state, the oscillator moves at the driving frequency, not necessarily at its natural frequency.
Examples in Physics
A driving force appears in many physical situations. A loudspeaker cone is driven by an alternating electrical signal. A building can be driven by periodic ground motion during an earthquake. A child’s swing is driven by repeated pushes. An electrical circuit with an alternating voltage is also an example of a driven oscillating system, although the details belong to later topics.
The important common feature is that some outside agent provides repeated forcing.
Driving Frequency and Response
The driving force has a frequency, often written as $f$ or angular frequency $\omega$. If this frequency is changed, the response of the system changes. Some driving frequencies produce only small motion, while others produce much larger motion.
This dependence on driving frequency is one of the most important features of forced oscillations. It leads directly to resonance, which is treated in its own chapter. Here, the key point is simply that the driving force does not just add energy randomly. Its timing matters.
Phase Relation
The driving force and the motion are not always perfectly synchronized. The oscillator may respond with a delay. This delay is described by a phase difference.
If the displacement is written as
$$
x(t) = A \cos(\omega t - \phi)
$$
then $\phi$ is the phase difference between the driving force and the resulting motion. The value of $\phi$ depends on the system and on the driving frequency.
For beginners, the main idea is that the force can act now, while the largest response may happen a little later.
Energy Transfer
A driving force transfers energy to the oscillator. This transfer is most effective when the force acts in step with the motion. For example, pushing a swing at the right moments increases its amplitude. Pushing at the wrong moments can reduce the motion instead.
The rate of energy transfer depends on both the force and the velocity. This is related to power, since instantaneous power is
$$
P = Fv
$$
So even if a force is present, it only gives positive energy to the system when it acts in a way that matches the motion appropriately.
A driving force can increase, maintain, or decrease oscillation amplitude depending on its timing relative to the motion.
Energy transfer is related to
$$
P = Fv
$$
If force and motion are properly timed, the oscillator gains energy effectively.
Mathematical Model
For a mass-spring system with damping and an external periodic driving force, a common equation is
$$
m\frac{d^2x}{dt^2} + b\frac{dx}{dt} + kx = F_0 \cos(\omega t)
$$
Here, $m$ is the mass, $b$ represents damping, $k$ is the spring constant, and the term on the right is the driving force.
This equation shows clearly that the motion is determined by three influences, inertia, restoring tendency, and damping, together with the external forcing.
You do not need to solve this equation yet to understand the driving force. The important part is that the external force appears as a separate term that continuously acts on the system.
Comparison with Free Oscillation
In free oscillation, the system is displaced and then left alone. It oscillates because of its restoring force. In forced oscillation, the system is continuously acted on by an external force.
The difference is summarized below.
| Type of motion | External periodic force present | Energy supplied continuously | Frequency of long-term motion |
|---|---|---|---|
| Free oscillation | No | No | Natural frequency |
| Forced oscillation | Yes | Yes | Driving frequency |
Visual Picture
The idea of a periodically driven oscillator can be sketched as a block attached to a spring, with an external force pushing it back and forth.
In this picture, the spring provides the restoring force, and the blue arrow represents the external driving force that changes with time.
Main Point
A driving force is the repeated external cause of forced oscillation. It supplies energy, sets the frequency of the steady motion, and determines how strongly the system responds. Its strength, timing, and frequency are the key factors that shape the motion.
In forced oscillations, the long-term steady motion occurs at the driving frequency, because the external force continually controls the motion.