Source: Chinese Min Education Press, Author: He Lieyun.
When the simple harmonic movement does not consider the influence of friction and other resistance and other factors, the mechanical energy of the system is conserved during the vibration process, so the amplitude of the oscillator remains unchanged during the vibration process, and this vibration is called unduminated vibration. However, the actual vibration is always affected by the resistance, and the mechanical energy of the vibration system is constantly decreasing due to the need to overcome the resistance to do the work. At the same time, the vibration system interacts with the surrounding medium, the vibration propagates outward to form a wave, and as the wave propagates, the mechanical energy of the system is continuously reduced, so the amplitude is gradually reduced. This vibration with a gradually decreasing amplitude is called a damping vibration, and the image of the damping vibration is shown in Figure 1.
Figure 1
In these two aspects, there are generally such doubts:
- First, whether the damping vibration is "periodic";
- The second is whether the damping vibration has "isochronization" (the time interval between oscillators passing through the equilibrium position twice in a row is the same).
1. Qualitative analysis
To know if damping vibrations are "periodic", we must first know what the period of mechanical vibration is. The definition is: the time it takes for an object to complete a full vibration, called the cycle of vibration. In the definition of the cycle, there is the concept of full vibration, which means that the object that does mechanical vibration starts from a certain point, and the state of motion when it returns to that point next time is exactly the same as the state of motion when it begins to vibrate, and the time taken is the shortest. Therefore, the mechanical vibration that can repeat the original state of motion (displacement, velocity, acceleration, etc.) is full vibration, and the non-equal-amplitude damping vibration is not full vibration, so it has no cycle.
There are two different theories about whether damped vibrations are "isochronous":
- The first statement is that it has "isochronousness", the reason is that although the amplitude of the damping vibration is decreasing, it can be seen as a superposition of many simple harmonic movements with decreasing amplitudes, because the simple harmonic motion has isochronicity, its period has nothing to do with the amplitude, so the phase of the damping vibration and the simple harmonic motion is consistent, and the rhythm is the same, so it has "isochronism".
- The second argument is that there is no "isochronousness" on the grounds that when the object does damping vibrations, it loses mechanical energy. When the oscillator passes through the same point twice before and after, the speed of the latter time is definitely smaller than the previous one. In this way, the average speed from the equilibrium position to the maximum displacement is always greater than the average speed at the time of return, so the return is slower and the corresponding time is longer. According to this reasoning, the vibrational rhythm of the damped vibration will become slower and slower, and finally stop, and the period becomes infinite, so it does not have "isochronousness".
2. Quantitative analysis
The slow motion of a spring oscillator in oil or a viscous liquid is a typical example of damping vibration, as shown in Figure 2, as can be seen by fluid mechanics, the size of the resistance of the spring oscillator when moving in oil or in a viscous liquid is proportional to the size of the speed, as obtained by Newton's second law
Figure 2
where Υ is the damping coefficient. Divide the sides by m, to get it
cause
ω0 is the natural circular frequency of the vibration system; the β is the damping coefficient, which is related to the nature of the vibration system and the nature of the medium. Thus, the equation can be written as
Here we are discussing the damping vibration of the under-damped state with little resistance, that is, β< ω0, and the kinematic equation of the particle in the spring oscillator can be found from the above equation
where A and a are pending constants, determined by the initial conditions. There are two factors in this equation, Ae-βt represents the amplitude that decays over time, cos(ω't+a) indicates that the vibration changes periodically at a circular frequency of ω', and the two-factor multiplication indicates that the particles do reciprocating motion with a shrinking range of motion. Since the state of motion of a particle cannot be completely repeated every time that has elapsed, damping vibration is not a periodic motion. However, cos(ω't+a) is cyclical, it guarantees that the time interval between every two consecutive times a particle passes through the equilibrium position and moves in the same direction is the same, which shows that "isochronism" has very strict conditions. And since
Greater than the inherent period of the spring oscillator system T=2π/ω0, it can be seen that the rhythm of damping vibration has become slower.
III. Conclusions
Comparing the conclusions drawn from qualitative and quantitative analysis, the view that damped vibrations are not "periodic" is the same, but the results of the analysis on whether damped vibrations are "isochronic" are different.
Only conclusions through quantitative analysis are reliable and correct, and in both statements of careful analysis of qualitative analysis, the reasoning is not rigorous and one-sided, leading to erroneous conclusions.
- The first statement in the qualitative analysis does not take into account that when the oscillator does the damping vibration, the size of the recovering force due to the influence of the resistance is no longer proportional to the displacement, and the wrong conclusion is that the damping vibration and the simple harmonic motion phase are consistent, and the rhythm is the same" is also the same.
- The second statement in the qualitative analysis only considers that the oscillator starts from the flat position to the maximum displacement, and returns to the equilibrium position from the maximum displacement, that is, the motion of the time 0 to t2 or t2 to t4 in Figure 1, and the motion process is not analyzed as a whole, thus ignoring that while the average speed decreases, the distance traveled by the oscillator from the equilibrium position back and forth is also shortened, resulting in the wrong conclusion.
In summary, the damping vibration is not a periodic motion, but the time interval required for the particle to pass through the equilibrium position and move in the same direction twice in a row is the same. When discussing damping vibration, we cannot copy the law of simple harmonic motion, nor can we do simple qualitative analysis, but also do quantitative analysis if necessary, otherwise it is easy to draw wrong conclusions.
bibliography:
Qi Anshen, Du Danying. mechanics. Beijing:Higher Education Press,1997.] 280～281
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