In the ideal case of zero damping, the amplitude of simple harmonic motion (SHM) can become theoretically infinite at resonance due to the specific way in which the driving force and the natural frequency of the system interact.
Resonance occurs when the driving frequency of an external force matches the natural frequency of the system. In the case of zero damping, the natural frequency is solely determined by the system's mass and the properties of the restoring force acting on it, such as the spring constant in the case of a mass-spring system.
When the driving frequency matches the natural frequency, the system absorbs energy from the external force most efficiently. At resonance, the amplitude of the system's motion can increase without bound if the driving force continues to provide energy to the system. This occurs because the driving force is in phase with the system's displacement, resulting in constructive interference that amplifies the motion.
Mathematically, the amplitude of SHM at resonance in the absence of damping can be expressed as:
A = (F_0 / k) / sqrt((1 - (f / f_0)^2)^2 + (2ξ(f / f_0))^2)
Where: A is the amplitude of SHM F_0 is the amplitude of the driving force k is the spring constant f is the frequency of the driving force f_0 is the natural frequency of the system ξ is the damping ratio (which is zero in the case of zero damping)
As f approaches f_0, the term (1 - (f / f_0)^2) approaches zero, and the denominator of the expression tends toward zero. Therefore, the amplitude A approaches infinity, indicating an unbounded increase in amplitude at resonance.
It's important to note that in practical systems, damping is always present to some extent, and energy losses due to damping prevent the amplitude from becoming truly infinite. However, in the idealized scenario of zero damping, the amplitude at resonance can theoretically grow without limit.