The flexible modes shifted down! What’s up?

Now this needs to be discussed.  

我对一个自由-自由系统做了刚度修改。

弹性体模态却下移了！ 怎么回事？

喔，这需要讨论。

  

Alright – that’s a pretty bold statement that will turn almost anyone’s head. I think we need to first describe what actually happened in this particular case. But rather than using the specific data originally presented, a simple beam can be used to describe what happened in this case.  

完全正确 — 那是一个非常大胆的说辞，几乎会让所有人为之侧目。我认为我们需要首先叙述一下，在这个特定情况下实际上发生了什么。但是与其利用原来提供的具体数据，倒不如用一个简单的梁来说明在这种情况下发生了什么。

 

The way this problem was described was that a free-free system was tested and then the system was constrained to fix or constrain the corners of the system. When the actual modification was performed to constrain the free-free system, the modes obtained were lower than the flexible modes of the unconstrained system.

这个问题的描述方式是，对一个自由-自由系统进行试验，接着系统被约束固定或者约束系统的各个角。当进行了实际修改来约束自由-自由系统时，得到的模态比无约束系统的弹性体模态还要低。

 

Of course, the first thing that everyone stated was that if you increase the stiffness of any system, then the modes must shift upwards because

当然首先每个人都说过，如果你增加任意系统的刚度，那么模态必然向上迁移，因为

and if the stiffness is increased then

并且如果刚度增加，那么

So it stands to reason that the frequencies must shift upwards – and the fact that the frequencies were lower just doesn’t make sense.

所以频率必然向上迁移，此中道理显而易见，不用多费口舌解释 — 并且事实上频率要是降低真是没有道理。

 

So let’s start with a simple beam that was analyzed and tested in a free-free condition. The first several free-free modes were 164 Hz, 452 Hz and 888 Hz. The unconstrained modes of the planar beam are shown in Figure 1 for reference.

那么我们从一个简单的梁开始，在自由-自由的条件下对梁进行分析和试验。前几阶自由-自由模态是164Hz、452Hz、和888Hz。作为参考，这个平面梁的无约束模态如图1所示。

Then the simple beam was tested in a pinned (or constrained) condition. The first several free-free modes were 72 Hz, 288 Hz and 647 Hz. The constrained modes of the planar beam are shown in Figure 2 for reference. Clearly, the modes did not shift up as expected.

接下来，在铰支（或有约束）的条件下对这个简单的梁进行试验。前几阶自由-自由模态是72H、288Hz、和647Hz。作为参考，这个平面梁的有约束模态如图2所示。显然，这些模态没有如期那样向上迁移。

So exactly what happened here???  

那么这里到底发生什么了？？？

 

This is a very simple problem. But I have heard this many times over many years and inevitably the same problem exists.

这是一个非常简单的问题。但是这么多年以来我听到这个问题很多次，并且这个同样的问题不可避免地存在着。

 

Typically, people are concerned about the flexible modes of the system because those are the modes that generally cause all the noise and vibration problems that we encounter. But those are not the only modes that are needed to describe the entire system.

通常，人们关心系统的弹性体模态，因为主要是由那些模态导致了我们遇到的所有噪声和振动问题。但是，那些模态并不是需要用来描述整个系统的仅有的模态。

 

The basic problem here is that everyone forgot that an unconstrained system has flexible modes AND the rigid body modes. Now most times people don’t measure the rigid body modes in test and they don’t include them as part of the flexible modes measured during a modal test. And from an analytical standpoint, many times the eigensolution is performed but either a shifted eigenproblem is solved or only the flexible modes are obtained.

这里的根本问题是，人人都忘记了一个无约束系统具有弹性体模态以及刚体模态。现在大多数时候，人们在试验中不测量刚体模态，并且在模态试验过程中，他们不包含刚体模态作为测量的弹性体模态的组成部分。另外，从解析的角度来看，很多时候进行特征值求解，却要么求解了一个迁移特征值问题，要么只能得到弹性体模态。

 

While the rigid body modes exist, many times people ignore them – mainly because they are not the source of the noise and vibration problems that are of concern. So Figure 3 shows the set of modes for the beam to more correctly include the rigid body modes as well as the flexible modes. So now once we realize that the first modes are actually at 0 Hz from the analytical model or a very low frequency from a test, then the intuition that adding stiffness shifts the modes upwards makes proper sense.  Table 1 shows the frequencies of the free-free beam with rigid body modes along with the pinned modes.

尽管刚体模态存在，但是很多时候人们对它们视而不见 — 主要是因为它们不是所关心的噪声和振动问题的根源。因此图3展示了梁的模态集合，更准确地包含了刚体模态以及弹性体模态。所以现在，一旦我们认识到根据解析模型，第1阶模态实际上是0Hz，或者根据试验，第1阶模态位于一个非常低的频率上，那么，增加刚度会使模态向上迁移的直觉是对的。表1给出了自由-自由梁的各阶频率，以及刚体模态和铰支模态。

 

So the basic problem is that the rigid body modes can’t be ignored; they are a part of the total description for the beam. Notice now that all the frequencies in Table 1 do shift upwards as the stiffness is increased.

所以根本问题在于，刚体模态不能被忽略掉；它们是梁的总体描述的一个组成部分。现在注意到在表1中，随着刚度增加，所有的频率的确向上迁移了。

 

One way to easily prove this to yourself is to make a simple free-free beam model. The next model to develop is the beam with two very soft springs. Then make subsequent beam models where the spring stiffness is increased until ultimately the spring is so stiff that it is an approximation of a pinned end condition.

很容易为自己证明这点的一个方法是制做一个简单的自由-自由梁模型。下一个要做的模型是一个带有两个非常软弹簧的梁。接下来再做一个梁模型，其中，增加弹簧刚度，直至最后弹簧是如此之硬，以至于逼近铰支的边界条件。

 

Along the way, it would be very beneficial to look at the mode shapes. When the springs supporting the beam are very soft, then the mode shape for the beam looks very much like a rigid body.

按照这种方法进行下去，来观察模态振型将是非常有意义的。当支撑梁的弹簧非常软时，梁的模态振型看起来非常像刚体。

 

But as the stiffness of the support beam increases, the frequencies will increase and the mode shapes will start to migrate from rigid body modes to modes that have some rigid body mode component but also start to develop some more flexible attributes.  

但是，随着支持梁的刚度增加，频率会提高，并且模态振型开始从刚体模态向有点儿刚体模态成分的模态迁移，但也开始生成更多些个弹性体属性。

 

When the support spring stiffness gets larger and larger, the rigid body mode characteristic will diminish as the flexible characteristic becomes more pronounced. Ultimately, the rigid body characteristic will disappear and the flexible characteristic will completely dominate the mode shape.

当支持弹簧刚度变得越来越大时，随着弹性体属性变得越来越突出，刚体模态属性将减少。最后刚体属性将消失不见，而弹性体属性将完全支配模态振型。

 

This little exercise will then clearly show that the rigid body modes are critically needed to describe the modes of the system.

那么这个小练习将清楚地表明为了描述系统模态，刚体模态必不可少。

 

I hope this simple example clears up any misconceptions that you may have had. If you have any more questions on modal analysis, just ask me.

我希望这个简单的例子澄清了你或许曾经有过的错误概念。如果你有关于模态分析的任何其他问题，尽管问我好了。