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

I showed some mode shapes to someone

They asked me if the structural design was ok

What should I tell them?

我展示模态振型给人看

人家问我结构设计的好吗？

我该告诉他们什么？

If I could have a dollar for every time I have heard that question, I'd be rich! The basic answer is you just don't have enough information to answer that question. People who ask that question have no idea what they are asking about. You have to be very diplomatic in telling them that the question is a silly one to ask.

如果我每听到这个问题一次，能得到一美元，那我就发财了！这个问题的基本回答是，你还没有足够的信息来回答这个问题。问这个问题的人也不知道他们在问什么。你不得不非常外交辞令地告诉他们，这个问题问得有点傻。

One of the reasons why they are apt to ask the question is because you probably showed them an animation and their impression is that the structure is deforming (since they see the deflections on the computer screen). Of course, you know that this is only a characteristic shape that the structure will undergo when subjected to a force that excites that mode. Sometimes I've been known to say "well let's increase the amplitude of the animation and see if we get the structure to break on the screen". (Of course, this is ridiculous!!!  This can't happen.) I use this statement to start to explain what shapes are all about. Animation is only a mechanism to understand how the structure may deform if that mode is excited by the forcing function.

他们为什么动不动就问这个问题的一个原因是，因为你可能给他们展示了一个振型动画，他们的印象是结构正在变形（因为他们在计算机屏幕上看到了形变）。当然你知道，这仅仅是结构受到一个激振力，力激起了那阶模态时所呈现出一个特征形状。我有时听到有人说“嗯，让我们加大动画振幅，看看在屏幕上我们能否破坏结构”。（当然，这也太荒谬了！！！不可能发生）我用这句话开端来解释振型到底是什么。动画仅仅是一种手段，用来理解如果激振力激起了那阶模态，结构将会如何变形。

One of the key points here is that we need to know the applied force. For some reason, people forget that we need a force applied to the system to get a response. The physical equation of motion is

这里的一个关键点是我们需要知道作用力。出于某些原因，人们忘记了要得到响应，我们需要一个力作用到系统上。物理运动方程是

and the equivalent modal space representation is

而等价的模态空间表达式是

Notice that there is a force on the right hand side of this equation. When we solve for the characteristic equation of the system, we assume that there is no force on the right hand side. This is how we obtain the dynamic characteristics of the system. One way to look at it is that the modes of the system are nothing more than a very elaborate set of filters which have the ability to amplify and attenuate an input signal on a frequency basis. If we just look at the filters themselves, can we make any assessment whether the filters are good or bad for a particular application? Of course not! All we can say is that the filters have some characteristics which relate to a center frequency, rolloff and some gain settings as seen in Figure 1.

注意，在这个方程的右侧有一个力。当我们求解系统特征方程时，假定在右侧没有力。这就是我们如何获得系统动态特性的。看待这个问题的一种方法是，系统模态只不过是一组非常复杂的滤波器，在频率基础上具有将输入信号进行放大和衰减的能力。对于一个特定的应用，如果仅仅观察这些滤波器本身，我们能够对它们做出是好还是坏的任何评价吗？当然不能！我们所能说的就是，这些滤波器具有与中心频率、滚降、和某些增益设置有关的一些特性，如图1所示。

Well ... the dynamic characteristics of a structural system are quite the same. We can identify each mode (each filter) as having a natural frequency (center frequency), damping (rolloff) and residue/mode shape (gain). We need to very clearly understand that the mode shapes are only characteristics and we cannot determine the goodness or badness of a mode unless we know the forcing function - that is, the right hand side of the equation.

嗯 … 结构系统的动力学特性也完全相同。我们可以识别每一阶模态（每一个滤波器），它具有一个固有频率（中心频率）、阻尼（滚降）、和留数/模态振型（增益）。我们需要非常清楚地认识到，模态振型仅仅是属性，并且我们无法确定某一阶模态的好坏，除非我们知道了激振力 — 也即方程的右侧。

As another example, let's say we wanted to determine the stiffness of a cantilever beam. Well, we could go out in the lab and apply a force to the tip of the cantilever beam and measure the resulting displacement. We know that we could determine the stiffness as K = F / X. Now this stiffness is an important parameter or characteristic of the beam. But once I determine the stiffness, do I know if the beam will fail or not? Of course not! I would need to know the actual force that was applied to the beam - wouldn't I? You see, in the test lab we applied an arbitrary force and measured the displacement due to that force in order to determine the character of the beam. Someone needs to identify the actual real world force before I can compute the actual displacement. And then I need to have some specification defined as to how to assess the acceptability of the structure due to the design or real world forces - which brings me to another important point.

另一个例子，比如说我们想要确定一个悬臂梁的刚度。嗯，我们可以到试验室里，在悬臂梁的顶部施加一个力，并测量最终的位移。我们知道可以确定刚度为K = F / X。现在这个刚度是梁的一个重要参数或属性。但是当我确定了刚度，我就能知道梁会失效与否吗？当然不能！我应该需要知道施加到梁上的实际作用力 — 难道不是吗？你看，在试验室里，为了确定梁的属性，我们施加了一个任意力并测量了这个力引起的位移。在我们能够计算实际位移之前，需要有人确定这个真实世界的实际作用力。并且，接下来我需要有一些规范，规定如何评估结构在设计上或真实世界作用力条件下的可接受性 — 这给我带来了另外一个重要的问题。

One thing that people often forget is that once the mode shapes are obtained and a dynamic design force is specified, the response can be computed, but someone needs to identify a specification defining what is acceptable and unacceptable for the response. This, at times, can be one of the most frustrating parts of the structural dynamic response modeling process. The responses can be computed but no one has defined what the level of acceptance is. Many times this very important detail is overlooked in the process of extracting pretty animated mode shapes. Then everyone asks... how much deflection is acceptable, how long will the component life be, does it "feel" good, is the response too noisy, etc.

人们经常忘记的一件事情是，一旦获得了模态振型并确定了动态设计力，就可以计算响应了，但需要有人确定一个规范，规定对于响应，什么是可以接受的，什么是不可以接受的。有时，这是结构动力学响应建模过程中一个最令人沮丧的部分。响应可以计算，但还没人确定什么水平是可以接受的。在提取非常生动的模态振型的过程中，很多时候，人们对这个非常重要的细节视而不见。接下来每个人都问 … 多大程度的变形是可以接受的，部件的寿命有多长，它“感觉”好吗，响应太吵吗？等等。

So now that we have discussed a few of these things, let's go back to our plate example that we have discussed before concerning different aspects of modal analysis. Figure 2 shows a schematic of a typical forced vibration problem. There is some force which is applied in the time domain. Well, this time signal is very confusing so it helps to identify some important characteristics of this force if it is transformed to the frequency domain using the FFT process. Now I know that this force is multiplied times the frequency response function in order to get the output of the system. That output could then be transformed back to the time domain if desired. Well, the important point to make here is that the FRF is multiplied by the force spectrum.

既然我们已经讨论了不少这方面的内容，我们再回过头来看看我们的平板例子，先前我们讨论过关于模态分析的各方面内容。图2展示了一个典型的强迫激励振动问题的示意图。有某种力作用在时域。嗯，这个时域信号让人非常困惑，那么将这个力用FFT方法转换到频域，有助于确定力的某些重要特性。现在我知道，为了得到系统输出，将这个力乘以频响函数。接下来，如果想要的话，那个输出可以变换回时域。对了，这里要做的非常重要的一点是力谱乘以FRF。

That means that the input force spectrum is amplified and attenuated by this multiplication. The FRF controls how this force is amplified and attenuated on a frequency basis. In the figure above, the FRF appears to have contribution for all four modes shown. That assumes that the applied force and response location exists at a point where there is participation of each of the four modes of the system.

这意味着，这个乘积将输入力谱进行了放大和衰减。在频率基础上，FRF决定了这个力是怎样放大和衰减的。上图中，FRF好像对图中显示的全部4阶模态都有贡献。这假定了作用力和响应位于某个点上，在这个点上系统4阶模态的每一阶都有参与。

But what if the force was applied at a location of a node of a mode. Let's say that the force was applied along the symmetry line along the length of the plate. Then, the applied force would not excite any of the torsional modes from that location; then we say that those modes don't participate in the response of the plate due to that force. The same is true for the response location. So we can see that both the input and output locations will have an effect on the response of the system. (In fact, the mode shape amplitudes have a strong influence on how much a particular mode contributes to the overall response.)

但是如果力作用在某一阶模态的节点上，情况又会怎样呢？比方说，力作用在沿着平板长度方向的对称线上。那么，从这个位置，这个作用力将激不起任何扭转模态；则我们说，由于那个力的原因，这些模态没有参与到平板响应中去。对于响应位置，情况也同样如此。所以我们可以看出，不但输入位置而且输出位置都会对系统的响应有影响。（事实上，在某一阶特定模态对总体响应有多大程度的贡献方面，模态振型的振幅有着强烈的影响。）

While we could say that certain modes may not participate in the response of the system, that does not imply that those modes don't exist - they just are not needed to compute the response of the system.  But the modes still exist - they define the dynamic characteristics of the system. Depending on the location of the applied force and the point where response needs to be measured (as well as the frequency content of the signal), will determine how the structure responds. Some modes may be more dominant in the level of response and others may be less dominant in the response - again depending on the particular input-output location selected. But all the modes exist – they just may not all be activated on a uniform basis.

尽管我们可以讲，某些阶模态可能没有参与到系统响应中去，这并不意味这这些阶模态不存在 — 只是不需要它们来计算系统响应。但是这些模态仍然存在 — 它们确定了系统的动力学特性。根据作用力施加的位置以及需要测量响应的位置（还有信号的频率成分），将决定结构如何响应。某些模态在响应水平中可能更占优势，而其他阶模态在响应中可能不占优势 — 同样取决于所选的特定输入-输出位置。但是所有的模态都是存在的 — 可能只是没有在一个统一的基础上将它们激发起来。

So what we need to remember is that a modal test only defines the character of the system. We apply an arbitrary force which is measured along with the response of the system due to the applied force.  This enables us to determine the dynamic characteristics of the system - the frequency, damping and mode shapes. These are only characteristics of the system. We display the mode shape (animate them) to better understand how the structure may deform if a force is applied to the system that excites one or more modes of the system. Remember, modal analysis doesn't use the force on the right hand side of the equation - the mode shapes are independent of the force.

所以我们需要记住的一点是，模态试验仅仅确立了系统的特性。我们施加一个任意力，测量力的同时测量力引起的系统响应。这使得我们能够确定系统的动力学特性 — 频率、阻尼和模态振型。这些仅仅是系统的属性。如果一个力作用到系统上，激起了一阶或更多阶系统模态，那么我们展示模态振型（对它们进行动画）是为了更好地理解结构可能会怎样变形。记住，模态分析没有利用方程右侧的力 — 模态振型独立于力。

Now I hope you understand why you can't answer the question that you asked. If you have any other questions about modal analysis, just ask me.

现在我希望你理解了为什么你不能回答你问得那个问题了。如果你有关于模态分析的任何其他问题，尽管问我好了。