MODAL SPACE - IN OUR OWN LITTLE WORLD

模态空间–在我们自己的小世界中   

Pete Avitabile 著  KINGSCI INSTRUMENTS-KSI科尚仪器 组织 westrongmc 译

Could you explain modal analysis for me?

Well...it will take a little bit but here's one that anyone can understand.

你能为我解释模态分析吗？

嗯…说来有点话长，但下面的解释人人都可理解。

You're not the first one to ask me to explain modal analysis in simple terms so anyone can understand it. In a nutshell, we could say that modal analysis is a process whereby we describe a structure in terms of its natural characteristics which are the frequency, damping and mode shapes - its dynamic properties. Well that's a mouthful so let's explain what that means. Without getting too technical, I often explain modal analysis in terms of the modes of vibration of a simple plate. This explanation is usually useful for engineers who are new to vibrations and modal analysis.

请我用简单的概念来解释模态分析，以便任何人都可以理解它，你不是第一个人。简言之，模态分析是一种方法，籍此，可以根据结构的频率、阻尼和振型等固有属性 — 其动态特性 — 来描述结构。这真够绕口的，那我们来解释这是什么意思。不钻技术牛角尖，我经常用一个简单平板的振动模态来解释模态分析。对于刚接触振动及模态分析的工程师们来讲，这种解释向来有益。

 

Let’s consider a freely supported flat plate. Let's apply a constant force to one corner of the plate. We usually think of a force in a static sense which would cause some static deformation in the plate. But here what I would like to do is to apply a force that varies in a sinusoidal fashion. Let's consider a fixed frequency of oscillation of the constant force. We will change the rate of oscillation of the frequency but the peak force will always be the same value - only the rate of oscillation of the force will change.  We will also measure the response of the plate due to the excitation with an accelerometer attached to one corner of the plate.

考虑一个自由支撑平板，施加常力于平板一角。我们通常从静态的意义上来看待一个力，它在平板内引起某种静态变形。但这里我要做的是施加一个按正弦方式变化的力，振荡频率固定的常力。我们将改变振荡频率，但不改变力的峰值 — 仅是力的振荡频率改变。同时在平板一角安装一加速度计来测量激励引起的平板响应。

Now if we measure the response on the plate we will notice that the amplitude changes as we change the rate of oscillation of the input force. There will be increases as well as decreases in amplitude at different points as we sweep up in time. This seems very odd since we are applying a constant force to the system yet the amplitude varies depending on the rate of oscillation of the input force. But this is exactly what happens - the response amplifies as we apply a force with a rate of oscillation that gets closer and closer to the natural frequency (or resonant frequency) of the system and reaches a maximum when the rate of oscillation is at the resonant frequency of the system. When you think about it, that's pretty amazing since I am applying the same peak force all the time - only the rate of oscillation is changing!

如果现在测量平板响应，会注意到当改变输入力的振荡频率时，响应幅值也发生变化。频率升高过程中，不同时刻点上，幅值有增也有减。这好像很奇怪，因为我们施加常力于系统，响应幅值却随输入力的振荡速率而变化。但这确确实实发生了 —  当施加的力的振荡速率越来越接近于系统固有频率（或共振频率）时，响应增大；当振荡速率为系统固有频率时，响应达到最大值。想想看，这真令人惊奇，因为我每时每刻都施加了相同幅值的力 — 仅仅是振荡速率改变而已！

This time data provides very useful information. But if we take the time data and transform it to the frequency domain using the Fast Fourier Transform then we can compute something called the frequency response function. Now there are some very interesting items to note. We see that there are peaks in this function which occur at the resonant frequencies of the system. Now we notice that these peaks occur at frequencies where the time response was observed to have maximum response corresponding to the rate of oscillation of the input excitation.

这个时域数据提供了非常有用的信息。但是如果采集到时域数据，并利用快速傅立叶变换将它变换到频域，则可以求得所谓的频响函数。现在有几点要关注：系统共振频率处，这个函数上有峰值。输入激励的振荡速率等于峰值频率的位置，观察到了时域最大响应。

Now if we overlay the time trace with the frequency trace what we will notice is that the frequency of oscillation at the time at which the time trace reaches its maximum value corresponds to the frequency where peaks in the frequency response function reach a maximum. So you can see that we can use either the time trace to determine the frequency at which maximum amplitude increases occur or the frequency response function to determine where these natural frequencies occur. Clearly the frequency response function is easier to evaluate.

现在如果将时域波形跟频响图形叠加在一起，会注意到时域波形达到最大值时的振荡频率与频响函数最大峰处的频率相一致。所以，既可以利用时域波形来确定幅值达到最大值处的频率，也可以用频响函数来确定固有频率何处发生。显然，用频响函数更容易求。

 

You thought it was pretty amazing how the structure has these natural characteristics. Well, the deformation patterns at these natural frequencies also take on a variety of different shapes depending on which frequency is used for the excitation force.

结构为何具有这些固有属性，你感到大为惊奇。对了，在这些固有频率处，变形形式也大为不同，依赖于激振力用哪个频率。

 

Now let's see what happens to the deformation pattern on the structure at each one of these natural frequencies. Let's place 45 evenly distributed accelerometers on the plate and measure the amplitude of the response of the plate with different excitation frequencies. If we were to dwell at each one of the frequencies - each one of the natural frequencies - we would see a deformation pattern that exists in the structure. The figure shows the deformation patterns that will result when the excitation coincides with one of the natural frequencies of the system. We see that when we dwell at the first natural frequency, there is a first bending deformation pattern in the plate shown in blue. When we dwell at the second natural frequency, there is a first twisting deformation pattern in the plate shown in red. When we dwell at the third and fourth natural frequencies, the second bending and second twisting deformation patterns are seen in green and magenta, respectively.  These deformation patterns are referred to as the mode shapes of the structure. (That's not actually perfectly correct from a pure mathematical standpoint but for the simple discussion here, these deformation patterns are very close to the mode shapes, for all practical purposes.)

好了，我们来看一看，在每个固有频率处，结构上的变形形式是怎样的。在平板上均布45个加速度计，测量不同激振频率的平板响应幅值。如果在每个频率处驻留 — 每次一个固有频率 — 可以观察结构上的变形形式。图中显示了按某一阶系统固有频率激励时，所得到的变形形式。在第一阶固有频率驻留时，平板具有第一阶弯曲变形形式，如蓝色所示。在第二阶固有频率驻留时，平板具有第一阶扭转变形形式，如红色所示。在第三、四阶固有频率驻留时，第二阶弯曲和第二阶扭转变形形式分别如绿色和紫红色所示。这些变形形式称为结构的模态振型。（从纯粹数学角度讲，这不完全正确。但此处简单讨论起见，事实上，这些变形形式非常接近于模态振型。）

      

Now these natural frequencies and mode shapes occur in all structures that we design. Basically, there are characteristics that depend on the weight and stiffness of my structure which determine where these natural frequencies and mode shapes will exist. As a design engineer, I need to identify these frequencies and know how they might affect the response of my structure when a force excites the structure. Understanding the mode shape and how the structure will vibrate when excited helps the design engineer to design better structures. Now there is much more to it all but this is just a very simple explanation of modal analysis.

你看，我们设计的所有结构都具有这些固有频率和模态振型。从根本上讲，这些特性依赖于结构的质量和刚度，它决定了固有频率和模态振型何处存在。作为设计工程师，需要识别这些频率，并且需要知道当力激励结构时，它们是如何影响结构响应的。理解模态振型和受激结构如何振动，有助于设计工程师设计出更优的结构。然而模态分析的内容还有很多，这只是一个非常简单的解释。

 

Now we can better understand what modal analysis is all about - it is the study of the natural characteristics of structures. Both the natural frequency and mode shape (which depends on the mass and stiffness distributions in my structure) are used to help design my structural system for noise and vibration applications. We use modal analysis to help design all types of structures including automotive structures, aircraft structures, spacecraft, computers, tennis rackets, golf clubs ... the list just goes on and on.

现在我们能够更好地理解模态分析是什么 — 它研究结构的固有特性。利用固有频率和模态振型（依赖于结构的质量和刚度分布）有助于设计在噪声和振动方面应用的结构系统。我们利用模态分析来帮助设计所有类型的结构，包括汽车、飞机、航天器、计算机、网球拍、高尔夫球杆...这个清单完全可以一直列下去。

 

I hope this very brief introduction helps to explain what modal analysis is all about. I know I explained modal analysis to my Mom using the example above and I think for the first time she actually understood what I actually do. Since then, she has been heard explaining it to her friends using a variety of words closely resembling modal analysis, of which the best one was the time she referred to it as noodle analysis ... but that's another story!

我希望这个非常简明的介绍有助于解释模态分析到底是什么。我曾经利用上面的例子向我的妈妈解释模态分析，并且我认为她第一次真正了解我到底是从事什么工作的了。从此以后，听到妈妈给她的朋友们解释模态分析，使用了不少非常接近模态分析的词，其中最接近的一次是她称之为面条分析（noodle analysis）... 好了，扯远了！