This question gets asked often. There are a lot of different aspects relating to this so let's start with a simple explanation without using too much math and explain all of this with a simple schematic. Let's use the figure to discuss all these different aspects of the time domain, frequency domain, modal space and physical space. Now there are a lot of parts to discuss in the figure, so let's take them in pieces - one at a time - and then summarize everything at the end. You might also want to remember the discussion we had before when you asked me about what modal analysis was all about ("Could you explain modal analysis for me?") to help with the discussion here.

经常有人问这个问题。关于这个问题，不同的方面有很多，所以我们先从简单解释入手，不运用过多的数学知识，而是用一个简单的示意图来解释所有这一切。我们利用这幅图来讨论时域、频域、模态空间和物理空间的所有这些不同的方方面面。图中现在要讨论的很多，所以我们化整为零 — 每次讨论一个方面 — 后面进而化零为整。你或许还记得我们先前的讨论，那时你问过我模态分析是什么（“你能为我解释模态分析吗？”），它有助于理解下面的讨论。

First, let's consider a simple cantilever beam and imagine that the beam is excited by a pulse at the tip of the beam. The response at the tip of the beam will contain the response of all the modes of the system (shown in the black time response plot); notice that there appears to be response at several different frequencies. This time response at the tip of the beam can be converted to the frequency domain by performing a Fourier Transform of the time signal. There is a significant amount of math that goes along with this process but it is a common transformation that we perform all the time. The frequency domain representation of this converted time signal is often referred to as the frequency response function or FRF for short (shown in the black frequency plot); notice that there are peaks in this plot which correspond to the natural frequencies of the system.

首先，考虑一个悬臂梁，并假定用一个脉冲在梁端部进行激励。梁端部响应会含有系统所有阶模态的响应（如黑色时域响应图所示）；注意到，在几个不同频率上好像都有响应。对时域信号进行傅立叶变换，将梁端部的时域响应变换到频域。这个过程伴随有相当多的数学知识，但它一直是我们要进行的一个常见变换。这个变换的时域信号的频域表达式常称为频响函数或简写为FRF（如黑色频域图所示）；注意到，图上有多个峰，对应于系统的固有频率。

Before we discuss the time and frequency plots any further, let's talk about the physical model in the upper left part of the figure. We know that the cantilever beam will have many natural frequencies of vibration. At each of these natural frequencies, the structural deformation will take on a very definite pattern, called a mode shape, as described previously [1]. For this beam, we see that there is a first bending mode shown in blue, a second bending mode shown in red and a third bending mode shown in green. Of course, there are also other higher modes not shown and we will only discuss the first three modes here but it could easily be extended to higher modes.

在进一步讨论时域和频域图之前，让我们先讨论图中左上部的物理模型。我们知道悬臂梁具有多个振动固有频率。在每个固有频率上，结构具有特定的变形形式，称为模态振型，如前所述[1]。对此梁，我们看到，第一阶弯曲模态如蓝色所示，第二阶弯曲模态如红色所示，第三阶弯曲模态如绿色所示。当然，还有其他更高阶模态没有显示出来，而此处我们将仅限于讨论前三阶模态，但它很容易推广到更高阶的模态。

Now the physical beam could also be evaluated using an analytical lumped mass model or finite element model (shown in black) in the upper right part of the figure. This model will generally be evaluated using some set of equations where there is an interrelationship, or coupling, between the different points, or degrees of freedom (dof), used to model the structure. This means that if you pull on one of the dofs in the model, the other dofs are also affected and also move. This coupling means that the equations are more complicated in order to determine how the system behaves.  As the number of equations used to describe the system gets larger and larger, the complication in the equations becomes more involved. We often use matrices to help organize all of the equations of motion describing how the system behaves which looks like

不过，也可以利用图中右上部的解析集中质量模型或者有限元模型（黑色所示）来计算这个实物梁。此模型通常用某个方程组来求解，用方程组对结构进行建模，其中不同的点或自由度（dof）之间相互影响或耦合。这表明，如果你在模型的一个自由度上拉，其他自由度也会受影响，也会活动。这种耦合意味着为了确定系统如何反应，方程组更加复杂了。当描述系统的方程数越来越大时，方程的复杂性愈甚。我们通常用矩阵将所有的运动方程组织在一起，以描述系统是如何反应的，如下所示：

where [M], [C], [K] are the mass, damping and stiffness matrices respectively, along with the corresponding acceleration, velocity and displacement and the force applied to the system. Usually the mass is a diagonal matrix and the damping and stiffness matrices are symmetric with off-diagonal  terms indicating the degree of coupling between the different equations or dofs describing the system.  The size of the matrices is dependent on the number of equations that we use to describe our system.  Mathematically, we perform something called an eigensolution and use the modal transformation equation to convert these coupled equations into a set of uncoupled single dof systems described by diagonal matrices of modal mass, modal damping and modal stiffness in a new coordinate system called modal space described as

其中[M]、[C]、[K]分别为质量、阻尼和刚度矩阵，除此之外还有相应的加速度、速度和位移以及施加于系统上的激振力。通常质量矩阵为对角阵；阻尼矩阵和刚度矩阵为对称阵，具有非对角元素，以表明描述系统的不同方程或自由度之间的耦合程度。矩阵的大小依赖于我们用以描述系统的方程数。从数学上讲，我们求得所谓的特征解，并利用模态变换方程，将这些耦合的方程组转换为一组解耦的单自由度系统，在新坐标系统中，它由模态质量、模态阻尼和模态刚度等对角矩阵来描述，这个新坐标系统称为模态空间，表示如下：

So we can see that the transformation from physical space to modal space using the modal transformation equation is a process whereby we convert a complicated set of coupled physical equations into a set of simple uncoupled single dof systems. And we see in the figure that the analytical model can be broken down into a set of single dof systems where the single dof describing mode 1 is shown in blue, mode 2 is shown in red and mode 3 is shown in green. Modal space allows us to describe the system easily using simple single dof systems.

因此我们可以看出，利用模态变换方程，从物理空间到模态空间的变换是一个过程，籍此，我们把一个复杂的耦合物理方程组转换为一组简单的解耦的单自由度系统。并且，在图中我们看到，这个解析模型可以分解为一组单自由度系统，其中描述第1阶模态的单自由度如蓝色所示，第2阶模态如红色所示，第3阶模态如绿色所示。模态空间允许我们方便地利用简单的单自由度系统来描述系统。

Now let's go back to the time and frequency responses shown in black. We know that the total response can be obtained from the contribution of each of the modes. The total response shown in black comes from the summation of the effects of the response of the model shown in blue for mode 1, red for mode 2 and green for mode 3. This applies whether I describe the system in the time domain or the frequency domain. Each domain is equivalent and just presents the data from a different viewpoint. It's a lot like money - as I go from country to country, the money in each country looks different but it's really the same thing. So we can see that the total time response is made up of the part of the time response due to the contribution of the time response of mode 1 shown in blue, mode 2 in red and mode 3 in green.  We can also see that the total FRF is made up of the part of the FRF due to the contribution of the FRF of mode 1 shown in blue, mode 2 in red and mode 3 in green. (We have only shown the magnitude part of the FRF here; this function is actually complex which is correctly displayed using both magnitude and phase or real and imaginary parts of the FRF).                   

现在我们回过头来讨论时域和频域响应，如黑色所示。我们知道，可以根据每阶模态的贡献求得总响应。黑色所示的总响应是由各模型响应结果求和得到的，如蓝色第1阶，红色第2阶，和绿色第3阶模态所示。不论我是在时域还是在频域来描述系统，这都是对的。各域等价，只是呈现数据的角度不同而已。这与货币很相似 — 当我从一个国家到另一个国家时，每个国家的货币看起来都不一样，但它其实是一回事。所以，我们可以看出，时域总响应是由各部分时域响应所组成，各部分时域响应由蓝色1阶、红色2阶、和绿色3阶模态的时域响应的贡献而来。我们也可以看出，总频响是由各部分频响所组成，各部分频响由蓝色1阶、红色2阶、和绿色3阶模态的频响贡献而来。（此处只显示了频响函数的幅值部分；这个函数实际上是复数的，需要用频响的幅值和相位或者实部和虚部两部分来正确表示）。

Since we can break the analytical model up into a set of single dof systems, we could determine the FRF for each of the single dof systems as shown with mode 1 in blue, mode 2 in red, and mode 3 in green. We could also determine the time response for each of these single dof systems through a closed form solution for the response of a single dof system due to the pulse input or we could simply inverse Fourier Transform the FRF for each of the single dof systems. We could also measure the response of the beam at the tip due to the pulse and filter the response of each of the modes of the system, and we would see the response of each of the modes of the system with mode 1 shown in blue, mode 2 in red and mode 3 in green. (Of course, I'm simplifying a lot of theory here so we can understand the concepts.)

因为可以将解析模型分解为一组单自由度系统，所以我们能够确定每个单自由度系统的频响函数，如蓝色1阶、红色2阶、和绿色3阶模态所示。也能够确定每个单自由度系统的时域响应，对于脉冲输入引起的单自由度系统的响应，可通过闭式解求得或者对每个单自由度系统的频响函数进行简单的傅立叶逆变换得到。也能够测量脉冲输入引起的梁端部响应，并且对系统每阶模态的响应进行滤波，这样我们可以观察系统每阶模态的响应，如蓝色1阶、红色2阶、和绿色3阶模态示。（当然，这里我对很多理论进行了简化处理，以便我们能够理解这些概念。）

Now that we have pulled apart all the pieces of the figure, I think it should be much clearer that there is really no difference between the time domain, frequency domain, modal space and physical space.  Each domain is just a convenient way for presenting or viewing data. However, sometimes one domain is much easier to see things than another domain. For instance, the total time response does not clearly identify how many modes there are contributing to the response of the beam. But the total FRF in the frequency domain is much clearer in showing how many modes are activated and the frequency of each of the modes. So often we transform from one domain to another domain simply because the data is much easier to interpret.

至此我们已经将图中各部分进行了单独分析，时域、频域、模态空间和物理空间之间真的没有区别，我认为这点应该是相当明了的了。每个域仅仅是呈现或观测数据的一种方便手段。但是有时，一个域比另一个域更容易看清事物。例如，时域总响应不能清楚地表明有多少阶模态于梁的响应有所贡献。然而频域中的总频响函数却更清楚地表明了激起了多少阶模态以及各阶模态的频率。所以我们经常从一个域变换到另外一个域，只不过是因为数据更加容易解释。

While there is a lot more to it all, I hope this simple schematic and explanation helps to put everything in better perspective. Think about it and if you have any more questions about modal analysis, just ask me.

尽管还有其他很多内容，但我希望这个简图及解释有助于更好地理清头绪。好好考虑一下吧，如果你有关于模态分析的如何其他问题，只管问我好了。