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

Are you sure you can get mode shapes from one row or column of the H matrix?

Sure!  

Let's walk through an example.

你确信可以从H矩阵的一行或一列得到模态振型吗？

当然确信无疑!

那我们来看一个例子吧。

Let's use the beam that we have discussed before as an example. For this beam, we considered three measurement points. There are a total of nine possible input-output FRFs that can be measured.  Remember we discussed that these measurements can be obtained from either shaker or impact testing.  So that we have some numbers to discuss, the beam mode shape values are shown in Figure 1. (The values will be kept simple for discussion purposes.)

我们还是利用以前讨论过的梁为例子。对这个梁，我们考虑3个测点。可以测量的可能存在的输入-输出FRFs共有9个。记住，我们曾经讨论过，这些测量结果可以从激振器试验或者锤击试验中得到。这样一来，我们有几个数字可以讨论，梁的模态振型数字如图1所示。（为了讨论的目的，将简单设定数值。）

Awhile ago we described how the peak amplitude of the imaginary part of the FRF is directly related to the residue (which is directly related to the mode shape). In fact, we said that the residue was approximated by

先前时候，我们曾经说过，FRF的虚部峰值是如何与留数直接相关的（留数与模态振型直接相关）。事实上，我们说过留数可以根据下式来近似

and that the individual values of the mode shape can be obtained from

并且，模态振型的每个数值可以从下式得到

Now let's plot the FRF matrix for this beam with three measurement points. I could show any one of the different parts of the FRF, but it turns out that the imaginary part of the FRF is most informative for this discussion since it shows both magnitude and direction; all the plots have the same -10 to +10 scale. This is shown in Figure 2.

现在，我们用图形来表示这个具有3个测点的梁的FRF矩阵。我可以去显示FRF的任意的不同的部分。但是事实证明，对于本次讨论，FRF虚部的信息最丰富，因为它不但指出了幅值而且指出了方向。所有的图形具有相同的刻度范围-10～+10。如图2所示。

Now let's use the third row of measurements to determine the mode shape for mode 1; this implies that point three is the reference location. Now if I were to pick the peak of the FRF for mode 1, the amplitudes are proportional to the shape of the cantilever beam first mode as seen in Figure 3

现在，利用测量结果的第3行来确定1阶模态的振型；这意味着测点3为参考点。现在，对1阶模态，如果我要拾取FRF的峰值，则峰值的幅度与悬臂梁第1阶模态的振型成比例，如图3所示。

 If you look at the values of the amplitude for mode 1 for points 1, 2 and 3, you will see that they are -2, -5, -8, respectively. These values are the values of the mode shape shown in Figure 1. (Notice that I have arbitrarily scaled the values to maintain an easy interpretation of the data. Also notice that the shape could be either plus or minus since the "shape" is the same.)

如果你观察第1阶模态的测点1、2、和3的幅值，会发现它们分别是-2、-5和-8。这些数值是图1所示的模态振型的数值。（注意，为了易于解释数据，我已经将这些数值进行了任意的归一。另外也要注意，振型可以是正的，也可以是负的，因为“形状”是一样的。）

Now let's use the second row of the FRF matrix. If I pick the peak of the FRF for mode 1, the amplitudes are again proportional to the shape of the cantilever beam first mode as seen in Figure 4.

现在，利用FRF矩阵的第2行。如果拾取1阶模态的FRF峰值，那么这些幅值又一次与悬臂梁第1阶模态的振型成比例，如图4所示。

If you look at the values of the amplitude for mode 1 for points 1, 2 and 3, you will see that they are approximately -1.2, -3.13, -5, respectively. At first glance, these values look different but we can notice that the "ratio" or "shape" is exactly the same as the previous case.

如果你观察1阶模态的测点1、2、和3的幅值，会发现它们分别近似等于-1.2、-3.13、和-5。乍看之下，这些数值看上去不一样，但是我们可以注意到，“比值”或“形状”与以前的情况完全相同。

In fact, if I scale the values of the mode shape from the third row by the ratio of the value of the mode shape at reference point 2 (5.0) to the value of the mode shape at reference 3 (8.0), then I will get the mode shape listed above for the 2nd row of the FRF matrix [ 2 (5/8)=1.2, 5 (5/8)=3.13, 8 (5/8)=5 ].  This is exactly what I expect to get based on the theory relating mode shapes to residues, so I'm actually not surprised. (We could also look at the first row of the FRF matrix and arrive at the same results.)

实际上，如果我对根据第3行得到的模态振型数值进行归一，比例因子等于参考点2处的模态振型值（5.0）比上参考点3处的模态振型值（8.0），那么我可以得到上面列出的根据FRF第2行得到的模态振型[ 2 (5/8) =1.2, 5 (5/8)=3.13, 8 (5/8)=5 ]。基于模态振型与留数相关的理论，这正是我所期望得到的那样。所以我并不惊讶。（我们也可以观察FRF矩阵的第1行，同样可以得到相同的结果）

So we can see that we can get the mode shape of the beam from any row of the FRF matrix. If we remember that the reciprocity holds true, then we know that the rows and columns contain the same information. So now I can also see that the mode shape is in every column of the FRF matrix. So this is why we say that you can use any row or column of the FRF matrix to estimate the mode shape. Of course, I can write out all the equations to show this but the pictorial description is sufficient (and I know how you hate it when I start writing equations!)

于是可以看到，我们能够从FRF矩阵的任意一行得到梁的模态振型。如果我们记得互易性成立的话，那么，我们知道，不同的行或列包含了相同的信息。因此现在我还可以看到，模态振型存在于FRF矩阵的每一列中。这就是为什么我们说，你可以利用FRF矩阵的任意一行或列来估计模态振型。当然，我可以写成所有的公式来说明这点，但是这个按图形方式表示的描述是足够的了（并且我知道，当我开始写公式时，你是怎样地讨厌它！）

Now let's look at mode 2 and use the third row of the FRF matrix. If I pick the peak of the FRF for mode 2, the amplitudes are proportional to the shapes of the cantilever beam second mode as seen in Figure 5.

现在观察2阶模态，并且利用FRF矩阵的第3行。如果拾取2阶模态的FRF峰值，这些峰的幅值与悬臂梁第2阶模态的振型成比例，如图5所示。

If you look at the values of the amplitude for points 1, 2 and 3, you will see that they are 3, 0, -8, respectively. They are the values of the mode shape shown in Figure 1.

如果观察测点1、2、和3的幅值，会看到它们分别是3、0、-8。它们是图1所示的模态振型的数值。

But when I look at the second row of the FRF matrix for mode 2, there is no information pertaining to mode 2. How could this happen? Well, the value of the mode shape for mode 2 at point 2 is zero – it’s the node of the mode. Anytime we use an input location or response location that is located at a node point (zero shape value) then we will not be able to see the mode from that reference location.

但是对2阶模态，当观察FRF矩阵的第2行时，却没有2阶模态的相关信息。这又是怎么发生的？怎么，2阶模态在测点2位置的模态振型的数值为零 — 它是模态的节点。无论什么时候，当我们使用的输入位置或响应位置位于节点（零振型值）时，那么我们将无法从那个参考点位置观察到模态。

One last picture may help to put it all together for you. Figure 7 shows a waterfall plot of the imaginary part of 15 measurements taken on the beam; the three measurements corresponding to the ones in Figure 2 are shown in color. In this plot, the information pertaining to mode 1 is shown in blue, mode 2 in red and mode 3 in green. We see that the mode shapes can be obtained from the peak of the imaginary part of the FRF. From these plots we can see the first, second and third bending shapes for the cantilever beam.

最后一张图或许有助于为你将所有信息汇总在一起。图7显示了在梁上采集到的15个频响测量结果虚部的瀑布图；图2中响应的3个测量结果用红色显示出来。在这个图中，与1阶模态相关的信息用蓝色显示，2阶模态用红色，而3阶模态用绿色。我们看到，可以从FRF的虚部峰值得到模态振型。从这些图中，我们可以看出这个悬臂梁的第1阶、第2阶、和第3阶弯曲振型。

So, in conclusion, we can say that you can use any row or any column of the FRF matrix to estimate any mode of the system, provided that the reference is not located at the node of a mode. I hope this answers your question. If you have any other questions about modal analysis, just ask me.

因此综上所述，我们可以说，你可以利用FRF矩阵的任意一行或一列来估计系统的任意一阶模态，只要参考点没有位于某一阶模态的节点之上。我希望这个讨论解答了你的问题。如果你有关于模态分析的任何其他问题，尽管问我好了。