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

I still don't understand curvefitting ...

How do you get mode shapes from FRFs?

Well . . . let's see if we can clear this up.

我还是不理解曲线拟合…

你是怎样从FRFs中得到模态振型的？

嗯…那我们看看能否澄清这个问题。

Modal parameter estimation (commonly referred to as curvefitting) is probably, by far, the hardest part of experimental modal analysis for most people to understand. I know I can write out all the equations to explain this. But I will probably bore you to death. Not only do I have to write out all the equations relating to the modal parameter estimation process, I also have to show the equations relating the residue to the mode shape. And, of course, the concept of a residue is another abstract concept. (Oh, how I wished they had called it a mode shape rather than a residue since this only confuses everyone.)

到目前为止，对大多数人来讲，模态参数估计（通常称为曲线拟合）可能是试验模态分析中最难以理解的部分。我知道我可以写出所有的公式来解释这点。但是，我可能会烦死你。我不但必须写出与模态参数估计过程有关的所有公式，而且我也必须指出这些公式建立了留数与模态振型之间的关系。另一方面，留数当然是另一个抽象的概念。（噢，我多么希望人们以前就称它为模态振型，而不是留数，因为那只会搞得大家一头雾水。）

Last time (Feb 1999), we talked about the curvefitting model and the basic equation we use for estimating parameters, of which one form is

上次（1999年1月），我们曾经讨论了曲线拟合模型，以及用于估计参数的基本公式，其中一种形式表示如下

Now those terms in the matrix, [A], are the residues which are obtained from the curvefitting process; we also get the poles, or frequency and damping, from the denominator of the equation. Now these residues can be shown to be related to the mode shapes. Without going through all the steps, the resulting relationship is shown below (with some terms expanded)

现在，矩阵[A]中的那些项是留数，是从曲线拟合过程中得到的；我们同时从公式的分母中得到极点，或频率和阻尼。现在，可以证明这些留数与模态振型有关。不去深入到每一步，我们可以得到如下的关系式（将某些项展开）

And if we were to look at each of the columns we would see the mode shape is contained in the column with some scalar multipliers; we would also see that due to reciprocity, the rows also contain the mode shapes. If we were to look at one column, such as the first column, then we would see

好了，如果我们来观察矩阵的每一列元素，可以看出列元素中包含模态振型，带有某些标量乘数；同时也可以看出，因为互易性，各行也包含有模态振型。如果观察某一列，例如第1列，那么，可以看到

The residues are, therefore, nothing more than the mode shape multiplied by a scalar which is the value of the mode shape at the reference location, u, and the scaling constant, q. (The q scale constant allows for mode shapes to be represented with different scale constants (unit modal mass, unit length, etc.)

因此，留数无非是模态振型乘以一个标量因数 — 模态振型在参考点位置的数值u，同时乘以归一常数q。（归一常数q允许模态振型可以用不同的归一常数来表示（单位模态质量，单位长度等。）

Great, so here are some equations that you may or may not fully understand or appreciate. Maybe a better way to explain the concept is through some simple pictures. Let's go back to that simple plate that we discussed some time ago (Feb 1998) and explain very simply how we can get mode shapes from measurements (then maybe you'll appreciate what the math is doing for us).

太好了，这样，有些公式你可能能完全理解或领会，也可能不能完全理解或领会。也许解释这个概念的更好办法是借助于一些简单的图形。让我们回过头来看看先前讨论过的那个简单平板（1998年1月），同时，也非常简单地解释一下我们是怎样从测量结果中得到模态振型的（这样，你或许可以领会数学可以为我们做什么？）。

Now let's take some measurements on the plate so that we get a total of 6 FRFs - at the 4 corners and at the 2 mid-points. We want to be able to determine what the first two mode shapes look like from these measurements. Now we could look at the log magnitude of the FRFs but this is not very useful since all the peaks would be positive in this plot.

现在，我们在平板上进行测量，结果是共得到6个FRFs — 在4个角以及2个中点位置。从这些测量结果中，我们希望能够确定前两阶模态看起来是什么样子。现在，可以观察FRFs的对数幅值，但这不是很有用，因为在这个图上，所有的峰值都将是正的。

A more informative plot is the imaginary part of the FRF. This shows both amplitude and, most importantly, the direction of the response. Without getting into all of the technical math, we know that the peak amplitude of the imaginary part of the FRF is directly related to the residue (and the residue is related to the mode shape). This approximate equation is shown below

FRF的虚部是一个包含更多信息的图形。这不但显示了幅值，而且更重要的是，显示了响应的方向。不去深入研究所有的专业数学知识，我们知道，FRF虚部的峰值幅度与留数直接相关（同时，留数与模态振型相关）。这个近似公式如下所示

This very simplistic approach to determining mode shapes is commonly referred to as peak picking since we are picking the peak of the FRF. Now let's look at some of the peaks for each of the measurements at each of the points. (In all of the plots shown, the amplitude of the scale ranges from minus one to plus one and the dashed line is one-half. In addition, the frequency axis has been removed.) Now let's just concentrate on mode 1 first and then go on to mode 2.

通常称这个非常简单的确定模态振型的方法为峰值拾取法，因为我们拾取了FRF的峰值。现在，针对每个测点位置的各个测量结果，我们观察某些峰值。（在所有显示的图中，幅值的刻度范围从-1到+1，并且短划线是1/2。另外去掉了频率轴。）现在我们首先集中精力考察模态1，然后再考察模态2。

Look at the FRF for mode 1 for point 1. Notice that this amplitude is 0.5 and it is negative. If we look at point 2, then we see that the amplitude is also 0.5 and it is also negative. This means that point 1 and 2 are moving with the same amplitude and in the same direction for mode 1. If we look at point 5 and 6, we see the same thing as point 1 and 2. So we can see that points 1, 2, 5, and 6 are all moving with the same amplitude in the same direction.

观察1阶模态的测点1的FRF。注意到，这个幅值是0.5，并且是负的。如果观察测点2，会那么我们看到，幅值也是0.5，并且也是负的。这意味着，对1阶模态，测点1和2按照相同幅度、相同方向运动。如果观察测点5和6，我们可以看到跟测点1和2相同的情况。这样一来，我们可以看出测点1、2、5、和6都是按照相同幅度，相同方向运动。

An important point to make here is that if I only measure these four points, then it would appear to me that the mode shape of the plate would be a rigid body mode (all four points moving together with equal amplitude). This is a common problem encountered when too few points are used to describe the mode shape of a system.

此处要说明的非常重要的一点是，如果仅仅测量这4个点，那么在我看来，平板的模态振型将是刚体模态（所有4个点按照相同的幅度一起运动）。用于描述系统模态振型的测点太少时，这是个常见的问题。

Now look at point 3 for mode 1. Notice that since amplitude is 0.5 but that it is positive. Then same can be said for point 4. So we see that point 3 and 4 have the same amplitude and move in the same direction together. But we also notice that points 3 and 4 are moving in the opposite direction from the rest of the points. Now, while we haven't measured more than 6 points, we start to see that the plate is deflecting into a pattern that is plate bending in characteristic. If we measured more points, then we would see a much better defined mode shape.

对于1阶模态，现在观察测点3。注意到，尽管幅值是0.5，但它是正的。接下来，测点4也是如此这般。因此，我们看到，测点3和4具有相同的幅度，并且按照相同的方向一起运动。但是我们也注意到，测点3和4按照与其他测点的相反的方向运动的。现在，尽管测量的点不超过6个，但我们开始看到了平板正在变形成一种形式，一种平板弯曲的特征形式。如果我们测量了更多的点，那么我们将会看到一个更加明确的模态振型。

Now if we look at mode 2 we can step through and look at all the points and what we will see is that point 1 and 2 have the same amplitude but now they are moving in opposite directions. The same is true for points 5 and 6. But we notice that points 1 and 5 are also moving in opposite directions; the same is true for points 2 and 6. So we see that there is some type of twisting or torsional type deformation pattern for mode 2. If we look at points 3 and 4, we notice that these points have zero value. This is because points 3 and 4 are node points for the torsional mode of the plate. Again, adding more points better defines the shape.

现在，如果我们观察模态2#，我们可以一步一步来考察，并且观察所有测点，这样，我们将看到的是，测点1和2具有相同的幅度，但是现在它们按相反的方向运动。对于测点5和6，情况也是如此。但是，我们注意到，测点1和5也是按照相反的方向运动；测点2和6同样如此。这样，我们看出，对于模态2，存在某种扭曲或扭转类型的变形形式。如果我们观察测点3和4，我们注意到，这些点具有零值。这是因为测点3和4是平板扭转模态的节点。同样，添加更多点可以更好地确定振型。

So now we can see that the peaks of the imaginary part of the FRF are directly related to the mode shape of the plate for each of the modal peaks. Without going through all the math, the residues are terms that are extracted from the curvefitting process and these residues are directly related to the modes shapes of the plate. This was shown pictorially to keep things simple.

所以现在我们可以看出，对于每阶模态的峰值，FRF虚部的峰值与平板模态振型直接相关。不必深细所有的数学内在根由，留数是从曲线拟合过程中提取出来的因项，并且这些留数与平板的模态振型直接相关。为使情况简单起见，已经按照图形方式说明了这一点。

I hope that this helps to clear up the mystery as to how we get mode shapes from FRFs. Think about it and if you have any more questions about modal analysis, just ask me.

我希望，这个解释有助于澄清关于我们是如何从FRFs中得到模态振型的这个谜团。好好思考一下这个问题，如果你有关于模态分析的如何其他问题，问我好了。