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我们知道，振型（Mode Shapes）关于质量矩阵和刚度矩阵都具有正交性。振型的正交性(Orthogonality of Modes )指的是，如果 The mode shapes have the orthogonalities about the mass and stiffness matrices, which means if
利用振型（Mode Shapes）的正交性，我们可以将一个多自由度体系(Mdof System)的结构，转化为以振型为坐标的若干个单自由度体系(Sdof System)进行求解，求得单自由度的地震响应后，再将振型响应组合得到结构的总的响应，因此振型（Mode Shapes）是结构动力学中非常重要的一个概念，然而又是非常抽象的一个概念。如何理解振型（Mode Shapes）的含义呢？ Based on the orthogonality of mode shapes, we can transfer a MDOF structure system to many SDOF systems and get their seismic responses. The seismic response of MDOF system can be acquired by combining the modal response of SDOF system. The mode shape is such an important concept in structural dynamics, however it is so abstract. How to understand the concept of mode shapes? 振型（Mode Shapes）的重要意义在于将结构的响应分解到若干个独立的坐标轴上，坐标轴必须是独立的，即具有正交性(Orthogonality)。这样类似的概念，在我们生活中经常会用到。 The significance of mode shapes is that the structural response can be decomposed to many independent coordinates. The independence of coordinate axis is the orthogonality. The similar concept is also used frequently in our regular lives. 例如，下图中一盘宫爆鸡丁，我们看它的组成(Contents)，大部分人看到的是鸡肉(Chicken)、黄瓜(Cucumber)、花生(Peanut)、胡萝卜(Carrot)、青椒(Green Pepper)、辣椒(Chili)等。然而，在营养(Nutrition)学家的眼中，这盘菜的组成将会是脂肪(Fat)、维生素(Vitamin)、蛋白质(Protein)、胆固醇(Cholesterol)及碳水化合物(Carbohydrate)等。可见，对于同样一件东西，如果观察者的角度不同（即坐标系选取得不同），得到的成份将会不同。对于普通人眼中，鸡肉、黄瓜、花生都是“正交”的，因为一件东西不可能即是黄瓜又是鸡肉。而在营养学家看来，脂肪、维生素、蛋白质这些也都是“正交”的，某种物质不可能既是脂肪又是维生素。因此，对于同一盘菜，我们可以说它是由多少肉和蔬菜组成，也可以说是由多少脂肪、维生素、碳水化合物等组成。 For example, a dish of Kung Pao Chicken as shown in the following figure, concern its content, most people think that it involves chicken, cucumber, peanut, carrot, green pepper, chili etc. However, from the nutritionist's view, it may be composed of fat, vitamin, protein, cholesterol, carbohydrate and so on. Hence, for the same item, its contents may be different from different viewers (or selecting different coordinate axes). From the views of general people, the chicken, peanut and cucumber are orthogonal, because one item cannot be chicken and cucumber at the same time. From the views of nutrition, the fat, vitamin and protein are also orthogonal, due that one material cannot befat and vitamin at the same time. So, you can say that the dish involves how much fat and vegetable. You can also say that the dish involves how much fat,vitamin and carbohydrate. 事物外表所呈现出来的特性，取决于内在的各阶“振型”（Mode Shapes）特性。因此，为了研究事物为何呈现这样或那样的特性，我们需要根据研究目的，确定各阶正交的“振型”。例如，为了研究由若干人组成的群体所表现出来的特征，我们将这个群体中的人划分成不同的“振型”，“振型”的取法是多种多样的，下面举例说明。 The exterior characteristics of the item aredetermined by the interior properties of “modes”. So, we should select different orthogonal modes according to the research objective. For example, we can divide a group of people into many “modes” to study the characteristics of the group. There many different methods to determine modes, as the following example. 我们可以根据性别确定群体的两阶“振型”，第1阶“振型”为男，第2阶“振型”为女。这两阶振型一定是正交的，因为一个人不可能既是男又是女。数学是如此精确和完善，然而现实生活却是如何复杂，导致数学中的概念在现实面前总显得无能为力。在现实生活中，有些男人选择了变性变成了女人，如著名的舞蹈家金星；有些女人变性成男人，如英国的艾德里安·达尔顿生；更为荒谬的是英国西伦敦房产开发商查尔斯·凯恩，先从男性变成了女性，几年后厌倦了当女人又变成了男人。这些变性人的出现，让本来满足“正交”条件的男性和女性这两阶“振型”不再成立。当然，排除了这些个例，数学的公式和原理还是比较完美的。 We can determine two modes of the peopleaccording to the gender, first mode for male and the second for female. Themodes of gender must be orthogonal, because one cannot be male and female atthe same time. The mathematics is so precisely, and the real life is socomplex, which result that the precise mathematics failed to describe the reallife. In real life, some man chooses to change to be woman by transsexualoperation, such as Chinese famous dance star Jin Xing. Some woman changes to beman, such as Adrian Dalton of UK. And more ridiculously, Charles Caineo of UKchanges to be female from male. After several years, “she” feels boring as awoman, so “she” changes back to be a man. The transgender makes the modes ofmale and female non-orthogonal. Except these particular cases, the formula andtheory of mathematics are still perfect.
当然，对于同样这个群体，除了按照性别来确定振型外，我们还可以按照籍贯来划分“振型”，一个人不可能同时有两个出生地，因此籍贯应该是满足“振型”的正交条件的。然而，拥有4个户口的“房姐”的出现，让按照“籍贯”划分出来的各阶振型不再“正交”，现实生活的残酷再一次打败了严谨的数学。 For the same group of people, besides determining the modes by gender, we can also divide modes by their birth places.One cannot have two birth places, so the modes of birthplace should be orthogonal. However, someone have many fake IDs, which result that the modes of birthplaces are non-orthogonal. The ruthlessness of real life defeats the strict math again. 对于住在学校的学生，我们还可以根据他的房间号来划分出不同的“振型”，理论上来讲，一个学生不应该同时拥有两个房间号，因此也应该满足正交条件。除非他梦游，前半夜睡在这个房间，后半夜睡在另外房间。但是，我们绝对不能按照“爱好”来划分“振型”，因为一个人可以同时拥有多种爱好，因此“爱好”不满足振型的正交条件。 For students living in the dormitory of the university, we can determine the modesby room number. Theoretically, one student cannot have two room numbers, so the modes of room number should be orthogonal. Except that one is a sleepwalker,which he sleeps in this room some time and another room other time. However, we absolutely cannot determine the modes by their hobbies. The hobbies are non-orthogonal,for one may have many hobbies. 群体整体表现出来的特性，是与“振型”（Mode Shapes）的特性密切相关，群体中某阶“振型”占的比重越大，群体整体的特性越接近于该阶“振型”（Mode Shapes）的特性。例如，这群人一起去购物并将所选购的商品放在同一个购物框内，如果这个群体中男性占了多数，男人好动的“振型特性”将会导致购物筐中充满了运动器材，相反，如果这个群体中女性占了多数，女生天性爱美的“振型特征”将会导致购物筐中充满了化妆品。这个群体一起去吃饭，如果其中四川人占了多数，他们将会消耗掉更多的辣椒，如果山西人占了多数，他们会消耗掉更多的醋，因为“四川”这阶振型在吃上将表现为辣的特征，而“山西”这阶振型将表现为酸的特征。 The characteristics of the group are related to one of the mode shapes. If more people in the group have one particular mode, the characteristic of the group will show the one of this particular mode. For example, the group of the people are shopping and put their goods into the same cart. If more males in the group, the activity“mode” of the male will result that the cart is full of sporting goods. However, if more females in the group, the beauty “mode” of female will result that the cart is full of makeups. 综上可知，“振型”的概念如此重要，应用如此广泛。 Based on the above analysis, we realized that the concept of modes is so important and applied so extensively.
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