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Imagine you have a child’s swing in your backyard. Giving it a gentle push every once in a while, it swings back and forth effortlessly. This is similar to what physicists call 'natural frequency' – the frequency at which a system tends to vibrate when not subjected to an external force.

Every physical system possesses a natural frequency, dependent on its physical properties like mass and stiffness. When a system, like a bridge or a building, vibrates on its own accord, it does so at this inherent frequency. The concept reminds us of a tuning fork, vibrating naturally when struck. The pitch you hear is a direct result of this frequency. Understanding natural frequency is critical because it plays a key role in engineering and safety considerations, helping to prevent destructive resonances, which can lead to catastrophic failures.

When you whisper softly or shout loudly, the difference isn't in the pitch, but in the loudness of your voice, or in the context of vibrations, the 'amplitude'. In physical systems, amplitude of vibration refers to the maximum extent of a vibration or oscillation, measured from the position of equilibrium.

The higher the energy of a vibrating system, the larger its amplitude. This can be seen on a guitar string, which vibrates with a small amplitude when plucked gently and with a larger amplitude when plucked strongly. Furthermore, at resonance, the amplitude reaches its peak because the system is absorbing energy from the constant external force at a rate that matches its natural dissipation rate, causing oscillations with greater intensity.

An external force, such as your hand pushing on a swing, can set a system in motion at a frequency decided by that force. This concept is critical when considering forced vibrations where an external periodic force drives a system.

If the frequency of this external force aligns closely with the system's natural frequency, the system will begin to resonate. This phenomenon can be helpful or harmful; it allows cellphone signals to transmit via resonance with certain frequencies, but it also has the potential to cause bridges to sway and even collapse when winds cause forced vibrations at the right frequency. Therefore, engineers meticulously calculate these frequencies during design to ensure safety and efficiency in the built environment.