This assay aims to explain the phase characteristics of driven harmonic motion using an example without relying on equations. Although the mathematical explanation of the phase relation in this motion is straightforward, an intuitive understanding might be achieved through relatable everyday experiences.
This is the property where the phase of the driven subject can be in the same phase of the driving mechanism when the driving frequency is below the natural frequency of the system and can be in the opposite phase otherwise. Here we assume the system is frictionless, otherwise these phase figures can deviate depending on the friction amount and on the ratio of frequencies.
To illustrate this, let's consider a classical piston-spring-mass system. In this setup, a mass is connected to a piston via a weightless spring, allowing the mass to move back and forth along a single axis, denoted as the x-axis.
We know that a spring-mass system as a simple harmonic oscillator, it can oscillate at a specific frequency called natural frequency when the mass is released from a non-equilibrium position, such as when the spring is stretched or compressed. The oscillation frequency can be changed by using different lengths of the spring. For instance, if we attach one end of the spring to a non-moving base and connect the mass to the other end, the oscillation frequency will be lower compared to using a shorter length of the spring.
In a driven harmonic motion scenario, we have a mass attached to a spring while the other end of spring is attached to a mechanism called the piston that can cyclically move back and forth along the length of the spring. In this system, the mass may exhibit a combined motion carrying both the natural frequency of the spring-mass system and the frequency of the piston. However, we can initiate the system in such a way that the mass oscillates solely at the frequency of the piston's motion. Let assume this is obtained by initially introducing a damping factor that decays the natural frequency components and subsequently progressively reducing and removing the damping factor.
Now, let's proceed with the experiment: Assuming we can run the piston at various speeds, performing a smooth sinusoidal motion, we can set a speed at which we observe the piston and the mass moving in opposite directions all the time. In other words, during one-half of a cycle, the piston and the mass approach each other, while during the other half, they move away from each other.
Under these conditions, we expect to observe a stationary point along the spring. Since the system is assumed to have negligible or a small friction, we can expect that the system will continue oscillating in this manner for a while even if we physically fix the spring to a base at this stationary point. By doing so, the spring-mass part of the system effectively becomes a simple harmonic oscillator and oscillates at a frequency determined by this part of the spring, which is higher than the natural frequency of the full-length (spring) system. The crucial point here is that this higher natural frequency (of the mass-spring system beginning from the stationary point) corresponds to the actual frequency of the piston's motion. This leads us to the following conclusion:
If we observe a stationary point in a driven harmonic oscillator based on a piston-spring-mass system, or if we consistently notice the ends of the spring moving in opposite directions, we can infer that the piston is operating at a frequency higher than the natural frequency of the system.
It can be also said: If one forces to oscillate a spring-mass system above its natural frequency by driving it from other end of the spring, the result is that the ends of the spring move always in opposite directions.
Note that if the system has some friction (damping), there would be no truely stationary point along the spring however, it would be not hard to spot the point along the spring where the motion is minimum.
A complete evaluation of the driven harmonic motion/oscillator can be found at http://farside.ph.utexas.edu/teaching/315/Waves/node13.html
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