I made a stiffness change to the tip of a cantilever beam but I can only shift the frequency so far.  

What’s up? 

Now this needs to be discussed.  

改变了悬臂梁顶部刚度，但仅能改变频率这么多点。

怎么回事？

喔，这需要讨论。

 

OK. This is another one of those problems that I see many people get confused about. Let’s start with a simple cantilever beam and explain some basic properties that are inherent in the system.

好的。这是诸多问题中的又一个，我看到很多人对之感到困惑。让我们从一个简单的悬臂梁开始，并解释系统一些内在的基本特性。

 

First, let’s start with a simple finite element model to investigate the effects of stiffness at the tip of the cantilever beam. Figure 1 shows the cantilever beam along with the cantilever beam with a spring at the tip and the cantilever beam with the end pinned. A finite element model of the beam will be used to lend some insight into what happens when the spring at the tip of the beam is varied from low stiffness to high stiffness.

首先，我们从简单的有限元模型开始，来研究悬臂梁顶部刚度的影响。悬臂梁、顶部带弹簧的悬臂梁以及顶部铰支条件下的悬臂梁如图1所示。我们将利用梁的有限元模型来深入领会一下当梁顶部的弹簧刚度由低变到高时发生了什么。

 

Table 1 shows the first three modes of the cantilever beam and then the change in frequency as the stiffness is increased along with the final pinned result if the spring was infinitely stiff. It is very important to notice that as the spring stiffness is increased, the final frequencies converge towards the final result where the cantilever is pinned at the tip.

表1列出了悬臂梁前3阶模态，接下来列出了随着刚度增加时的频率变化情况、以及如果弹性刚度无穷大，最后为铰支边界条件下的频率。值得注意的是，随着弹簧刚度增加，最终频率趋向于悬臂梁顶部铰支边界条件下的最终结果。

So this implies that the no matter how much stiffness you add at the end of the cantilever beam, the frequency can only shift so far and then any additional increases in stiffness have very little effect at all – it is a point of diminishing returns.

那么这意味着，不管你在悬臂梁顶部加了多少刚度，频率只能移动这么多点，并且后面刚度增加的再多，也几乎没有什么影响 — 这是一个收益递减点。

 

Now let’s further consider the simple cantilever beam and let’s look at the tip response. The frequency response function is shown in Figure 2 with a drive point measurement at the tip of the beam where the stiffness is to be added to the beam.

现在让我们进一步考虑这个简单的悬臂梁，看看其顶部的响应。频响函数如图2所示，在梁的顶部测量驱动点频响，顶部也是刚度施加的地方。

 

So now let’s look at the frequency response function and discuss the different parts of this function.  At the natural frequencies, there is a peak in the function. Basically, this is a region in frequency where it takes very little force to cause large response. At the resonant frequency it appears that the structure has no apparent stiffness.  

那么现在我们来看一看频响函数，并讨论这个频响的不同部分。这个函数在固有频率上有峰值。大致说来，在频域内有一个区域，在这个区域内用很小的力可以引起很大的响应。在共振频率上，结构好像没有明显的刚度。

 

Now at the antiresonances, this is a region in frequency where it takes excessive force and there is very little to essentially no response. At the antiresonant frequency, it appears that the structure is infinitely stiff. That is to say that at the antiresonant frequencies, there is no displacement and it appears that the cantilever is pinned at that point at that antiresonant frequency.

现在来看反共振频率，在频域范围内有一段区域，在这个区域上施加很大的力，却基本上没有响应。好像结构刚度无穷大。也就是说在反共振频率上没有位移，并且悬臂梁在那个点上好像被铰支固定了。

Now if there would be a change in stiffness at the tip of the cantilever beam, then there would be a shift in the peaks of this function. If stiffness is added to the tip of the beam then the peaks will shift upward. This is shown in Figure 3.

你看，如果悬臂梁顶部刚度变化，那么这个函数的峰会发生迁移。如果梁顶部刚度增加，峰值会向上迁移。如图3所示。

But as the stiffness is increased, there will be some limit to the shift in the frequency of the system.  Now if we realize that the antiresonance is actually the frequency at which the cantilever beam tip displacement is zero, then it is obvious that this is the frequency where the beam appears to be pinned at the tip. This is shown in Figure 4. From that schematic it is easy to realize that the peaks of the unconstrained cantilever beam can never shift past the antiresonances of the cantilever beam because this is essentially the cantilever constrained at the tip which is the pinned condition.

但是随着刚度增加，系统频率的迁移会有一些限制。现在如果我们认识到反共振频率实际上是悬臂梁顶部位移为零处的频率，那么显然，这是梁仿佛在顶部被铰支固定时的频率。如图4所示。根据这个示意图，很容易认识到，无约束悬臂梁的峰值永远不会迁移超出悬臂梁的反共振频率，因为这本质上是顶部约束为铰支边界条件的悬臂梁。

So from this discussion, it should be clear that the cantilever beam frequencies can only shift so far when a spring is considered at the tip of the beam. Further, we can actually identify how far those frequencies can shift by looking at the antiresonances at the tip of the unconstrained cantilever beam.

那么综上所述，很显然，当在梁的顶部考虑一个弹簧时，悬臂梁的频率仅能迁移这么多。更进一步讲，通过观察无约束悬臂梁顶部的反共振频率，我们实际上可以确定这个频率能迁移多远。

 

I hope that this discussion clears up the mystery as to why the frequencies can only shift so far before there is no further change in the frequencies. The best way to prove it to yourself is to make a simple finite element model and check out the results. If you have any more questions on modal analysis, just ask me.

我希望这个讨论澄清了这个谜团，关于固有频率没有进一步变化之前，为什么它们只能移动这么多的谜团。为你自己证明这点的最好方法是生成一个简单的有限元模型，并检查输出结果。如果你有关于模态分析的任何其他问题，尽管问我好了。