Harmonic Motion

content

Linear differential equations
Harmonic motion

Concepts

Harmonic Motion
Period
Frequency
Peak value
Average value
Mean-square value (or variance)
Root mean-square (RMS) value
Decibel (dB)

1 Linear differential equations

In the study of physics, usually the course is divided into a series of subjects, such as mechanics, electricity, optics, etc., and one studies one subject after the other. For example, this course has so far dealt mostly with mechanics. But a strange thing occurs again and again: the equations which appear in different fields of physics, and even in other sciences, are often almost exactly the same, so that many phenomena have analogs in these different fields. To take the simplest example, the propagation of sound waves is in many ways analogous to the propagation of light waves. If we study acoustics in great detail we discover that much of the work is the same as it would be if we were studying optics in great detail. So the study of a phenomenon in one field may permit an extension of our knowledge in another field. It is best to realize from the first that such extensions are possible, for otherwise one might not understand the reason for spending a great deal of time and energy on what appears to be only a small part of mechanics.[1]

The harmonic oscillator, which we are about to study, has close analogs in many other fields; although we start with a mechanical example of a weight on a spring, or a pendulum with a small swing, or certain other mechanical devices, we are really studying a certain differential equation. This equation appears again and again in physics and in other sciences, and in fact it is a part of so many phenomena that its close study is well worth our while. Some of the phenomena involving this equation are the oscillations of a mass on a spring; the oscillations of charge flowing back and forth in an electrical circuit; the vibrations of a tuning fork which is generating sound waves; the analogous vibrations of the electrons in an atom, which generate light waves; the equations for the operation of a servosystem, such as a thermostat trying to adjust a temperature; complicated interactions in chemical reactions; the growth of a colony of bacteria in interaction with the food supply and the poisons the bacteria produce; foxes eating rabbits eating grass, and so on; all these phenomena follow equations which are very similar to one another, and this is the reason why we study the mechanical oscillator in such detail. The equations are called linear differential equations with constant coefficients. [1]

在振动研究中，最最基本的就是系统的运动微分方程，如果该方程是线性微分方程(Linear differential equations)的话，那么就很容易解得系统的响应。

下图中，三种貌似不同的系统，但是它们的运动微分方程都是线性微分方程，其解也类似。就如Feynman 所言，这类线性微分方程在其他领域中亦常见。此是一种模型，可以适用不同领域。

01-window_1_1 SDF system_small displacements [2].JPG

该类方程的解描述如下。

02-window_1_2 Summary of the Description of SHM [2].JPG

2 Harmonic Motion

在图2 中求得的自由响应 x(t) 是 Harmonic Motion.

The fundamental kinematic properties of a particle moving in one dimension are displacement, velocity, and acceleration.[2]

所以得到的自由响应的位移，速度和加速度的关系如下图所示，它们之间相差的就是一个相位。

03-Relationship between Disp_Vel_Acce for SHM [2].JPG

angular natural frequency wn 为系统固有频率, 系统周期Period和频率Frequency为

04-Period [2].JPG

05-Frequency [2].JPG

上面说到振动中最基本的就是线性微分方程。那么来看看 SHM吧。

06-Three Equivalent Representations of HM [2].JPG

Peak Value.

峰值就是系统响应的幅值A。

Average value

07-Average value [2].JPG

Mean-square value

Since the square of displacement is associated with a system's potential energy, the average of the displacement squared is sometimes a useful vibration property to discuss.[2]

08-Mean-Square-value [2].JPG

Root mean-square (RMS) value

上式1.21 开平方根的值就是 RMS value, which is commonly used in specifying vibration.

Decibel (dB).

以参考信号为基准，才有dB嘛。

09-Decibel Eq. [2].JPG

The decibel is used to quantify how far the measured signal x1 is above the reference signal x0. Note that if the measured signal is equal to the reference signal, then this corresponds to 0 dB. The decibel is used extensively in acoustics to compare sound levels.[2]

Reference

[1] Feynman R P, Leighton R B, Sands M, et al. The Feynman lectures on physics.[M]// The Feynman lectures on physics. Addison-Wesley Pub. Co. 1963
[2] Inman D J, Singh R C. Engineering vibration[M]. Upper Saddle River: Prentice Hall, 2001.

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2017-01-05 09:14:26