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'm still overwhelmed by all this modal stuff

Laplace, Fourier, FRFs, and all that!

Can you put the big picture together for me?

我仍然对所有这些模态内容感到茫然

拉普拉斯，傅里叶、频响、所有这一切！

你能为我汇总一下重点吗？

Sure ... sometimes it helps to stand back and look at everything from a complete picture. I have a figure that I have used for many years now to help people see things more clearly. I call it "The Big Picture". Let's just look at this picture and discuss all the pieces individually.

当然没有问题 … 有时它可以帮助我们往后站，从一幅完整的画面来观察一切。我有一幅图，用了很多年，现在可以帮人更明白地看清事物。我称之为“重点图”。让我们看看这幅图，并单独讨论图中的各个部分吧。

First let's start with an analytical representation such as the finite element model shown. Basically, we use the FEM to approximate a lumped mass system that is interconnected by springs to represent the physical system. Since the analytical approximation is described in terms of a force balance for each mass that is described in the system, we end up with one equation for each mass (or degree of freedom) used to approximate the system. Since we need many small little finite elements to accurately describe the system, I end up with many equation and unknowns. Right away, it becomes convenient to describe all these equations using matrices. Now once I have assembled all these equations, a mathematical routine called an eigensolution is used to represent the system in simpler terms - the system's frequencies and mode shapes. This is what we do in the finite element process.

首先，我们从一个解析表达式，如所示的有限元模型FEM入手。大致说来，我们利用FEM来近似描述一个通过弹簧互相联系在一起的集中质量系统，以表示物理系统。因为对系统中描述的每一个质量来说，可根据力平衡方程来描述这种解析意义上的近似，因此最终对于每一个质量（或自由度）都得到了一个方程，用来近似描述这个系统。因为需要很多非常小的微元来精确描述系统，所以最后可以得到很多方程，具有很多未知数。可以很方便地立即用矩阵来描述所有这些方程。现在一旦将所有这些方程组织在一起，就可以利用一个称之为特征解的数学形式，按照几个更简单的项 — 系统频率和模态振型，来表示系统 。这就是在有限元方法中我们所做的事情。

Well, without getting into all the details, I can take those same equations and transform them into the Laplace domain. (No - we don't convert to the Laplace domain to make your life miserable - we do it to make some of the equations easier to handle. Please believe me on this one!) Now in the Laplace domain, we have, [B(s)], the system equation and its inverse, [Hs)], the system transfer function.  Now we know that this inverse is the adjoint of the system matrix (or the cofactors of the system matrix) divided by the determinant of the system matrix. This inverse is described in all vibrations text books (usually in Appendix A).

好了，不去深究所有细节，我可以得到那些相同的方程，并且将它们变换到拉氏域。（没有 — 我们并没有切换到让你的生活充满痛苦的拉氏域 — 做变换是为了让某些方程更加易于处理。这一点，请相信我！）现在，在拉氏域内得到了系统方程[B(s)]以及它的逆矩阵[H(s)]，也即系统传递函数。目前我们知道，这个逆矩阵等于系统矩阵的伴随矩阵（或系统矩阵的余因子矩阵）除以系统矩阵的行列式。在所有的振动教科书中，都会介绍这个逆矩阵（通常是在附录A中）。

So big deal! What's that mean to you! Well, it turns out that the adjoint matrix contains the modal vectors and we call this the Residue Matrix. The determinant of [B(s)] contains the roots, or poles of the system. Well, this is the same basic information that is obtained from the analytical model. So we could determine the system dynamic characteristics from either the analytical model or from the Laplace domain representation - they both will give the same results.

没什么大不了的！对你来说这意味着什么！嗯，可以证明，伴随矩阵含有模态振型，而且我们称之为留数矩阵。[B(s)]的行列式含有系统的根，或称系统的极点。好了，这跟得自于解析模型的基本信息是一模一样的。所以我既可以根据解析模型，也可以根据拉氏域表达式，来确定系统的动力学特性 — 这二者将给出相同的结果。

Now another important relationship is the Frequency Response Function, FRF. This is the system transfer function evaluated along the jω axis. The FRF is actually a matrix of terms, [H(jω)]. Well, since we are dealing with a matrix, it is convenient to identify input-output measurements with a subscript. So a particular output response at point 'i' due to an input force at point 'j' is called hij (jω).

现在，另一个重要的关系式是频响函数FRF。它是系统传递函数沿jω轴求值。FRF实际上是一个矩阵形式的项，[H(jω)]。嗯，因为我们正在处理矩阵，所以用下标来标识输入-输出测量结果会很方便。因而‘j’测点处的输入激励引起的‘i’测点处的一个特定响应称为hij (jω)。

Now remember that the system transfer function has been defined up to this point from mass, damping and stiffness quantities. This function can be computed or synthesized for any input-output combination over any frequency band desired. So if we wanted, we could synthesize several FRFs that make up either one full row or one full column of the FRF matrix if needed or desired as shown in the figure.

现在记住，目前为止系统传递函数是根据质量、阻尼、和刚度物理量来定义的。对任意的输入-输出组合，在任意想得到的频带范围内，都可以计算出或综合出这个函数。所以，如果我们想要，我们就能够综合出一些FRFs，组成所需要的或想要得到的FRF矩阵的一个完整的行或一个完整的列，如图所示。

Now what we need to realize is that those FRFs that were generated (synthesized) contain information relative to the system characteristics. Remember that the FRFs can be generated from residues and poles. And that the residues are directly related to the mode shapes and the poles are the frequency and damping of the system.

现在我们需要认识到的是，这些生成的（综合而来的）FRFs含有与系统特性相关的信息。记住，FRFs可以根据留数和极点来生成。另外，留数与振型直接相关，而极点是系统的频率和阻尼。

So the parameters that make up the FRFs, are the parameters that we wish to extract from the FRFs.  This is what modal parameter estimation is all about. Basically, we use the FRFs in a mathematical algorithm to extract the generic information that makes up the FRFs - the frequency, damping and mode shapes. We often refer to this process as curvefitting. The basic information that is extracted is the mode shapes which are related to information contained in the adjoint matrix or residue matrix and the poles which relate to information in the determinant of the system matrix.

所以，组成FRFs的参数就是我们希望从FRFs中提取出来的参数。这就是模态参数估计的全部内容所在。大致说来，我们在数学算法中利用FRFs来提取组成FRF的基本信息 — 频率、阻尼、和模态振型。常常称这个过程为曲线拟合。提取出来的基本信息是模态振型，它与包含在伴随矩阵或留数矩阵里的信息相关；以及极点，它与系统矩阵的行列式里的信息相关。

This pretty much summarizes the process - except one important thing needs to be addressed. Up until now we have only discussed using the mass, damping and stiffness approximations to compute system characteristics from the finite element model or from the Laplace domain representation of the system. Both these approaches use approximations of the physical parameters of mass, damping and stiffness to describe the system and so they will both provide the same basic information. If there were some other way to estimate those FRFs without assuming physical properties then I could employ the modal parameter estimation techniques to extract the desired information.

这很好地总结了这个过程 — 除了一个重要的需要解决的问题之外。目前为止，我们只讨论了利用质量、阻尼、和刚度近似项，根据有限元模型或者根据系统拉氏域表达式来计算系统特性。这两种方法都是利用质量、阻尼、和刚度这些物理参数的近似项来描述系统，因此，它们将提供相同的基本信息。如果能有其他不需假定物理属性的方法来估计这些FRFs，那我就能够利用模态参数估计技术来提取想要得到的信息。

This is where modal testing comes in. Basically, my structure is excited with some measured force.  The response of the system due to the applied force is measured along with the force. Now this time data is transformed to the frequency domain using the FFT and basically a ratio of output response to input force is computed to form an approximation of the FRF.

这就是模态试验的由来。总的说来用某种可测的激振力去激励结构。测量激振力引起的系统响应，与此同时测量激振力。现在利用FFT将这个时域数据转换到频域，大致说来计算输出响应与输入激振力的比值来形成FRF的一个近似。

There are many implications of making these measurements which involve digital signal processing concepts which are much too involved to discuss in detail right now (but I think you get the idea where I'm going with all this).

进行这些测量有很多含意，它包含的数字信号处理的概念太多，多到现在不能立即进行详尽的讨论（但是根据这些内容，我认为你知道我将要讲什么）。

So we could measure one input-output FRF based on this approach. If we used a shaker to excite the structure and move the accelerometer to many points then we could measure a column of the FRF matrix. (If we collected the data using impact techniques then we would measure one row of the FRF matrix). So the big advantage of making measurements is that I measure the response of the system due to the applied force – I don't ever make any assumptions as to the mass, damping and stiffness of the system - and I avoid any erroneous approximations I may make. Of course, I need to make sure that I make very good measurements otherwise I will distort my system characteristics.

所以基于这个方法，我们能够测量某个输入-输出FRF。如果利用激振器来激励结构，并且移动加速度计到很多测点，就能测得FRF矩阵的一列。（如果利用锤击技术来采集数据，那么能够测得FRF矩阵的一行）。所以进行测量的巨大优势在于，我测量激振力引起的结构响应 — 我并没有对系统的质量、阻尼、和刚度作任何假设 — 并且避免了可能做出的任何错误近似。当然要确保得到非常好的测量结果，否则会对系统特性造成失真。

So I hope this clears some things up for you. If you have any other questions about modal analysis, just ask me.

因此我希望这个讨论为你澄清了某些东西。如果你有关于模态分析的任何其他问题，尽管问我好了。