# on 30-Aug-2018 (Thu)

#### Annotation 3261637987596

 一般来说，绝大多数场地的反应谱都是这个形状的，只不过具体数值有所不同。这也就是所 谓的「超高层建筑是地震中最安全的地方」。我们也说过了，结构的自振周期跟结构高度相 关，一般的二三层的房子，周期 0.3 秒左右，刚好在「共振」的范围内。而二三十层的高层 结构，周期大概 1 秒，对应的地震加速度已经下降了很多。至于超高层建筑，自振周期甚至 能达到 5 秒、6 秒甚至更大。在反应谱上，对应于 6 秒的加速度已经非常小了，带来的侧向 效应可能甚至不如风荷载大。

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#### Annotation 3261664726284

[Guide] How to Anki Maths the right way. : Anki

### Definition

In a definition card, there is usually three deletions. The first is the name of the defined object. The second is the notation of this object. The third one is the definition. If an object admits many equivalent definitions, all definitions appears on the same card. Each definition being a different cloze deletion. Indeed, if your card is «A square is ... » and you answer «A diamond with a 90° angle», you don't want to be wrong because it is written «A rectangle with two adjacent side of same length». Therefore, the card is: «{{c1::A square}} is: equivalently -{{c2::A diamond with a 90° angle}} or -{{c3::A rectangle with two adjacent side of same length}}»

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are often similar. For example, in a linear algebra book, many results begin by «Let U,V be two vector space and T a morphism from u to V». It is helpful not to have to retype it in each cards. <span>Definition In a definition card, there is usually three deletions. The first is the name of the defined object. The second is the notation of this object. The third one is the definition. If an object admits many equivalent definitions, all definitions appears on the same card. Each definition being a different cloze deletion. Indeed, if your card is «A square is ... » and you answer «A diamond with a 90° angle», you don't want to be wrong because it is written «A rectangle with two adjacent side of same length». Therefore, the card is: «{{c1::A square}} is: equivalently -{{c2::A diamond with a 90° angle}} or -{{c3::A rectangle with two adjacent side of same length}}» Beware, sometime it is better if the name and the notation belong to the same deletion. For example, if X is a subset of a vector space, it is easy to guess that «Aff(X)» is «the Affine

Question

### Definition

In a definition card, there is usually three deletions. The first is [...]. The second is the notation of this object. The third one is the definition. If an object admits many equivalent definitions, all definitions appears on the same card. Each definition being a different cloze deletion. Indeed, if your card is «A square is ... » and you answer «A diamond with a 90° angle», you don't want to be wrong because it is written «A rectangle with two adjacent side of same length». Therefore, the card is: «{{c1::A square}} is: equivalently -{{c2::A diamond with a 90° angle}} or -{{c3::A rectangle with two adjacent side of same length}}»
the name of the defined object

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Definition In a definition card, there is usually three deletions. The first is the name of the defined object. The second is the notation of this object. The third one is the definition. If an object admits many equivalent definitions, all definitions appears on the same card. Each definition b

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are often similar. For example, in a linear algebra book, many results begin by «Let U,V be two vector space and T a morphism from u to V». It is helpful not to have to retype it in each cards. <span>Definition In a definition card, there is usually three deletions. The first is the name of the defined object. The second is the notation of this object. The third one is the definition. If an object admits many equivalent definitions, all definitions appears on the same card. Each definition being a different cloze deletion. Indeed, if your card is «A square is ... » and you answer «A diamond with a 90° angle», you don't want to be wrong because it is written «A rectangle with two adjacent side of same length». Therefore, the card is: «{{c1::A square}} is: equivalently -{{c2::A diamond with a 90° angle}} or -{{c3::A rectangle with two adjacent side of same length}}» Beware, sometime it is better if the name and the notation belong to the same deletion. For example, if X is a subset of a vector space, it is easy to guess that «Aff(X)» is «the Affine

Question

### Definition

In a definition card, there is usually three deletions. The first is the name of the defined object. The second is [...]. The third one is the definition. If an object admits many equivalent definitions, all definitions appears on the same card. Each definition being a different cloze deletion. Indeed, if your card is «A square is ... » and you answer «A diamond with a 90° angle», you don't want to be wrong because it is written «A rectangle with two adjacent side of same length». Therefore, the card is: «{{c1::A square}} is: equivalently -{{c2::A diamond with a 90° angle}} or -{{c3::A rectangle with two adjacent side of same length}}»
the notation of this object

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Definition In a definition card, there is usually three deletions. The first is the name of the defined object. The second is the notation of this object. The third one is the definition. If an object admits many equivalent definitions, all definitions appears on the same card. Each definition being a different cloze deletion. Indeed, if

#### Original toplevel document

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are often similar. For example, in a linear algebra book, many results begin by «Let U,V be two vector space and T a morphism from u to V». It is helpful not to have to retype it in each cards. <span>Definition In a definition card, there is usually three deletions. The first is the name of the defined object. The second is the notation of this object. The third one is the definition. If an object admits many equivalent definitions, all definitions appears on the same card. Each definition being a different cloze deletion. Indeed, if your card is «A square is ... » and you answer «A diamond with a 90° angle», you don't want to be wrong because it is written «A rectangle with two adjacent side of same length». Therefore, the card is: «{{c1::A square}} is: equivalently -{{c2::A diamond with a 90° angle}} or -{{c3::A rectangle with two adjacent side of same length}}» Beware, sometime it is better if the name and the notation belong to the same deletion. For example, if X is a subset of a vector space, it is easy to guess that «Aff(X)» is «the Affine

Question

### Definition

In a definition card, there is usually three deletions. The first is the name of the defined object. The second is the notation of this object. The third one is [...]. If an object admits many equivalent definitions, all definitions appears on the same card. Each definition being a different cloze deletion. Indeed, if your card is «A square is ... » and you answer «A diamond with a 90° angle», you don't want to be wrong because it is written «A rectangle with two adjacent side of same length». Therefore, the card is: «{{c1::A square}} is: equivalently -{{c2::A diamond with a 90° angle}} or -{{c3::A rectangle with two adjacent side of same length}}»
the definition

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Definition In a definition card, there is usually three deletions. The first is the name of the defined object. The second is the notation of this object. The third one is the definition. If an object admits many equivalent definitions, all definitions appears on the same card. Each definition being a different cloze deletion. Indeed, if your card is «A square is ... »

#### Original toplevel document

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are often similar. For example, in a linear algebra book, many results begin by «Let U,V be two vector space and T a morphism from u to V». It is helpful not to have to retype it in each cards. <span>Definition In a definition card, there is usually three deletions. The first is the name of the defined object. The second is the notation of this object. The third one is the definition. If an object admits many equivalent definitions, all definitions appears on the same card. Each definition being a different cloze deletion. Indeed, if your card is «A square is ... » and you answer «A diamond with a 90° angle», you don't want to be wrong because it is written «A rectangle with two adjacent side of same length». Therefore, the card is: «{{c1::A square}} is: equivalently -{{c2::A diamond with a 90° angle}} or -{{c3::A rectangle with two adjacent side of same length}}» Beware, sometime it is better if the name and the notation belong to the same deletion. For example, if X is a subset of a vector space, it is easy to guess that «Aff(X)» is «the Affine

Question

### Definition

In a definition card, there is usually three deletions. The first is the name of the defined object. The second is the notation of this object. The third one is the definition. If an object admits many equivalent definitions, [...]. Indeed, if your card is «A square is ... » and you answer «A diamond with a 90° angle», you don't want to be wrong because it is written «A rectangle with two adjacent side of same length». Therefore, the card is: «{{c1::A square}} is: equivalently -{{c2::A diamond with a 90° angle}} or -{{c3::A rectangle with two adjacent side of same length}}»
all definitions appears on the same card. Each definition being a different cloze deletion

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sually three deletions. The first is the name of the defined object. The second is the notation of this object. The third one is the definition. If an object admits many equivalent definitions, <span>all definitions appears on the same card. Each definition being a different cloze deletion. Indeed, if your card is «A square is ... » and you answer «A diamond with a 90° angle», you don't want to be wrong because it is written «A rectangle with two adjacent side of same len

#### Original toplevel document

Unknown title
are often similar. For example, in a linear algebra book, many results begin by «Let U,V be two vector space and T a morphism from u to V». It is helpful not to have to retype it in each cards. <span>Definition In a definition card, there is usually three deletions. The first is the name of the defined object. The second is the notation of this object. The third one is the definition. If an object admits many equivalent definitions, all definitions appears on the same card. Each definition being a different cloze deletion. Indeed, if your card is «A square is ... » and you answer «A diamond with a 90° angle», you don't want to be wrong because it is written «A rectangle with two adjacent side of same length». Therefore, the card is: «{{c1::A square}} is: equivalently -{{c2::A diamond with a 90° angle}} or -{{c3::A rectangle with two adjacent side of same length}}» Beware, sometime it is better if the name and the notation belong to the same deletion. For example, if X is a subset of a vector space, it is easy to guess that «Aff(X)» is «the Affine

### Theorem

A theorem usually admits two deletions. Hypothesis and conclusion. It sometime admits a third deletion if the theorem has a classical name. For example you may want to remember that the theorem stating «In a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. » is called «the Pythagorean theorem».
Beware, no hypothesis in a deletion should introduce an object. «If, ... , then the center of P is not trivial» is hard to understand, whereus «Let P be a group. If ..., then the center of P is non trivial» is more clear.

Unknown title
ted «Aff(X)». What you really want is to recall that «the set of vectors of the form Sum of x_ir_i, with x_i\in X and r_i in the field, where the sum of the r_i is 1» is «the affine hull of X». <span>Theorem A theorem usually admits two deletions. Hypothesis and conclusion. It sometime admits a third deletion if the theorem has a classical name. For example you may want to remember that the theorem stating «In a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. » is called «the Pythagorean theorem». Beware, no hypothesis in a deletion should introduce an object. «If, ... , then the center of P is not trivial» is hard to understand, whereus «Let P be a group. If ..., then the center of P is non trivial» is more clear. Hypothesis A first important thing is to always write all hypothesis. Sometime, some hypothesis are given in the beginning of a chapter and are assumed to be true in the whole chapter.

Question

### Theorem

A theorem usually admits [...] deletions. Hypothesis and conclusion. It sometime admits a third deletion if the theorem has a classical name. For example you may want to remember that the theorem stating «In a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. » is called «the Pythagorean theorem».
Beware, no hypothesis in a deletion should introduce an object. «If, ... , then the center of P is not trivial» is hard to understand, whereus «Let P be a group. If ..., then the center of P is non trivial» is more clear.
two

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Theorem A theorem usually admits two deletions. Hypothesis and conclusion. It sometime admits a third deletion if the theorem has a classical name. For example you may want to remember that the theorem stating «In a right

#### Original toplevel document

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ted «Aff(X)». What you really want is to recall that «the set of vectors of the form Sum of x_ir_i, with x_i\in X and r_i in the field, where the sum of the r_i is 1» is «the affine hull of X». <span>Theorem A theorem usually admits two deletions. Hypothesis and conclusion. It sometime admits a third deletion if the theorem has a classical name. For example you may want to remember that the theorem stating «In a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. » is called «the Pythagorean theorem». Beware, no hypothesis in a deletion should introduce an object. «If, ... , then the center of P is not trivial» is hard to understand, whereus «Let P be a group. If ..., then the center of P is non trivial» is more clear. Hypothesis A first important thing is to always write all hypothesis. Sometime, some hypothesis are given in the beginning of a chapter and are assumed to be true in the whole chapter.

Question

### Theorem

A theorem usually admits two deletions. [...]. It sometime admits a third deletion if the theorem has a classical name. For example you may want to remember that the theorem stating «In a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. » is called «the Pythagorean theorem».
Beware, no hypothesis in a deletion should introduce an object. «If, ... , then the center of P is not trivial» is hard to understand, whereus «Let P be a group. If ..., then the center of P is non trivial» is more clear.
Hypothesis and conclusion

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Theorem A theorem usually admits two deletions. Hypothesis and conclusion. It sometime admits a third deletion if the theorem has a classical name. For example you may want to remember that the theorem stating «In a right triangle, the square of the hypotenus

#### Original toplevel document

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ted «Aff(X)». What you really want is to recall that «the set of vectors of the form Sum of x_ir_i, with x_i\in X and r_i in the field, where the sum of the r_i is 1» is «the affine hull of X». <span>Theorem A theorem usually admits two deletions. Hypothesis and conclusion. It sometime admits a third deletion if the theorem has a classical name. For example you may want to remember that the theorem stating «In a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. » is called «the Pythagorean theorem». Beware, no hypothesis in a deletion should introduce an object. «If, ... , then the center of P is not trivial» is hard to understand, whereus «Let P be a group. If ..., then the center of P is non trivial» is more clear. Hypothesis A first important thing is to always write all hypothesis. Sometime, some hypothesis are given in the beginning of a chapter and are assumed to be true in the whole chapter.

Question

### Theorem

A theorem usually admits two deletions. Hypothesis and conclusion. It sometime admits a third deletion if the theorem has [...]. For example you may want to remember that the theorem stating «In a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. » is called «the Pythagorean theorem».
Beware, no hypothesis in a deletion should introduce an object. «If, ... , then the center of P is not trivial» is hard to understand, whereus «Let P be a group. If ..., then the center of P is non trivial» is more clear.
a classical name

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Theorem A theorem usually admits two deletions. Hypothesis and conclusion. It sometime admits a third deletion if the theorem has a classical name. For example you may want to remember that the theorem stating «In a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares

#### Original toplevel document

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ted «Aff(X)». What you really want is to recall that «the set of vectors of the form Sum of x_ir_i, with x_i\in X and r_i in the field, where the sum of the r_i is 1» is «the affine hull of X». <span>Theorem A theorem usually admits two deletions. Hypothesis and conclusion. It sometime admits a third deletion if the theorem has a classical name. For example you may want to remember that the theorem stating «In a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. » is called «the Pythagorean theorem». Beware, no hypothesis in a deletion should introduce an object. «If, ... , then the center of P is not trivial» is hard to understand, whereus «Let P be a group. If ..., then the center of P is non trivial» is more clear. Hypothesis A first important thing is to always write all hypothesis. Sometime, some hypothesis are given in the beginning of a chapter and are assumed to be true in the whole chapter.

#### Annotation 3261679930636

 我们前面的例子里，相当于地面加速度是 0.16g，也就是说，我们近似认为， 地震到来的这 48 秒内，地面加速度一直是 0.16，没有变化。但实际上呢，真实地震加速度 的变化非常不规律，其实是上面的这个图形，一会儿是+0.1，一会儿+0.5，一会儿又是 0， 一会儿又是-0.4，随时间变化非常剧烈

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#### Annotation 3261681503500

 房子的自振频率是 ，变换一 下形式，也就是 。也就是说，我们的地震作用力其实是 。作 用力除以质量等于加速度，我们把质量除到等号左边，这个式子就变成了 。 这又是什么意思呢？我们已经知道，对于我们的这个小房子，自振频率 ω 是一个定值而房 子的位移 u 就是我们上面的红色曲线。把这条曲线等比例放大 ω 的平方倍，我们就得到 了房子的地震加速度曲线

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#### Annotation 3261683076364

 ，在地震作用下，房子 的位移越大，其等效的加速度也就越大，换言之，所受的地震力也就越大。 也就是说，我们所应该考虑的地震力的大小，就是最下图这条黑色曲线再乘上房子的质量。

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#### Annotation 3261684649228

 这两者之间的关系是这样的： 我们在第二篇里的 Matlab 代码，事实上就是在解这个微分方程，已知每个时刻的 ，求 解每个时刻的 u。这里面的 m 是质量，k 是刚度，而 c 则是阻尼。质量与加速度相关， 阻尼与速度相关，刚度和位移相关，这三者之和应该与外力平衡，而外力就是地面的加速度 引起的惯性力。 阻尼 ，其中的 就是我们第一篇里提到的阻尼比。而我们上面也得出了 。把 c 和 k 的表达式都带进微分方程，我们得到 两边都消去质量 m，我们就得到了 。 真相已经浮出水面了，在地面加速度和房子加速度之间的微分方程里，参数只有两个，一个 是自振频率 ω ，另一个是阻尼比 ζ。我们已经知道，对于大多数建筑结构，阻尼比都是 0.05， 因此，起作用的独立参数只有一个，就是自振频率

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#### Annotation 3261686222092

 同样的地震来了，同样的地面加速度，为什么不同的房子反应不同？为什么不同的 房子的加速度不同？原因只有一个，那就是它们的自振频率不同。频率是周期的倒数，也就 是说，自振周期直接决定了房子的地震作用的大小。

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#### Annotation 3261687794956

 这时候，最大位移为 391.93 毫米，而最大地震力与自身重力的比值是 0.394。 有的看官说了，最大位移 391 毫米，都快 40 厘米了，真的会这么大吗？ 这是 1971 年美国 San Fernando 地震中 Olive View 医院一楼的角柱，混凝土几乎全部破坏， 纵筋严重变形，整个二层以上的建筑物发生了 60 厘米的水平位移。怎么样？地震的力量真 的是超乎想象的。

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#### Annotation 3261689367820

 1985 年墨西哥城大地震就属于这种比较少见的类型，震害偏向高周期段。所有的低层建 筑几乎毫发未损，而高层建筑的震害非常严重，多座钢结构高层建筑发生整体垮塌。震后的 景象令人乍舌，小砖房、破瓦房都好好的，钢结构的高层反而倒了。

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#### Annotation 3261690940684

 上一篇我们讨论了如何从地震的实测地面加速度入手，得到房子的等效地震加速度，也就是 所需要承受的地震力。地震作用的大小，其实只跟房子的自振周期有关。

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#### Annotation 3261692513548

 最终的结果，我们得到了大量的自振周期与等效地震加速度相对应的数据

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#### Annotation 3261694086412

 对于同一个自振 周期而言，位移、速度、加速度是等比例缩放的，所以我们可以把这三个反应谱画在同一张 图形里

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#### Annotation 3261697494284

 首先，我们看一层发生单位位移的情况，也就是上图最左边这种情况。请问这个时候有多少 质量发生了移动，有多少质量没有发生移动？ 如果只有一层发生单位位移，那么其实只有一层的 300 吨发生了移动，二层、三层的质量并 没有移动。也就是说，对于一层发生单位位移的这种情况来说，二层、三层的质量没有跟着 移动，相当于有效质量是 0。 我们把这种情况的三个楼层的有效质量写成一行，按一层、二层、三层的顺序，也就是 300 吨、0、0。 同样的，如果只有二层发生位移，那一层和三层的质量也不会移动，同样写成一行，0、300 吨、0。 只有三层发生位移，一层和二层的质量不移动，这时候的情况是 0、0、300 吨。 然后，我们按照一层、二层、三层发生单位位移的顺序，把这三行排列起来，组成一个三乘 三的矩阵： 300 0 0 0 300 0 0 0 300

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#### Annotation 3261700640012

 第一种情况，只有第一层发生 1 毫米的位移，这时候负责一层擎天柱需要付出 400 千牛的推 力，因为不仅仅需要把一层的柱子推弯，还得把二层的柱子也推弯。注意，这时候负责二层 的大黄蜂并不只是看热闹，为了把二层保持在原来的位置，大黄蜂需要施加反方向的 200 千牛推力，这样二层才不会跟着一层一起移动。而因为大黄蜂已经把二层顶住了，三层就不 会移动了，所以负责三层的铁皮边儿上看着就行了。 同样，第二种情况，大黄蜂需要 400 千牛才能把二层推动 1 毫米。而为了防止一层和三层跟 着二层一起移动，擎天柱和铁皮都需要付出反方向的 200 千牛，保证把一层和三层固定在原 来的位置。 第三种情况，因为三层就是顶层了，再往上就没有柱子了，所以铁皮只需要付出 200 千牛的 推力。为了防止二层跟着一块儿动，大黄蜂需要反方向 200 千牛把二层顶住。因为大黄蜂已 经把二层固定了，所以负责一层的擎天柱就可以歇一歇了。 跟质量一样，我们把这三种情况下三个楼层的受力写成矩阵的形式： 400 -200 0 -200 400 -200 0 -200 200 这也就是我们这个三层房子的刚度矩阵。

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#### Annotation 3261703523596

 r earlier tweeting, deleting and re-posting unsupported claims that Google is "suppressing voices of Conservatives," this afternoon the President's personal Twitter account posted a new attack video. It claimed that the company failed to promote his State of the Union address with a link on its homepage a

) Tweet (102) Share (30) Save Capture of Google.com from The Internet Archive's Wayback machine on the evening of January 30th, 2018 during Donald Trump's SOTU address. The Internet Archive Afte<span>r earlier tweeting, deleting and re-posting unsupported claims that Google is "suppressing voices of Conservatives," this afternoon the President's personal Twitter account posted a new attack video. It claimed that the company failed to promote his State of the Union address with a link on its homepage after doing so for President Barack Obama, but several things about it don't hold up. In a statement to the media, Google explained that isn't quite true, since a President's first addres

#### Annotation 3261705620748

 有了自振频率，我们就可以求自振周期了。对应这三个自振频率，我们也有三个自振周期值。 最最简单的理解，我们可以认为第一周期可以近似代表房子的特性。也就是说，我们这个三 层房子的自振周期可以近似看作 0.547 秒。而我们一层房子的自振周期是 0.243 秒，注意到 我们三层房子的周期大概是一层房子的 2.25 倍，这也就是我们在第三篇里提到的房子的周 期可以近似认为与层数或者高度成正比。

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#### Annotation 3262253501708

 Ganze Zahlen (Int) umfassen in Haskell nur einen bestimmten Zahlbereich. Wie in den meisten Programmiersprachen wird der Zahlbereich durch die Bin ¨ arzahl b begrenzt, die in ein Halbwort (2 Byte), Wort (4 Byte), Doppel- wort (8 Byte) passen, so dass der Bereich −b, b darstellbar ist: −2 31 , 2 31 −1. Die Darstellung ist somit f ¨ ur die Operatoren nicht ausreichend, da die Er- gebnisse außerhalb dieses Bereichs sein k ¨ onnten. Das bedeutet, man muss damit rechnen, dass bei Operationen (+, −, ∗, /) das Ergebnis nicht in den Zahlbereich passt, d.h. dass ein ¨ Uberlauf stattfindet.

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#### Annotation 3262255861004

 Unbeschr ¨ ankte ganze Zahlen (Integer) Diese kann man in Haskell ver- wenden, es gibt diese aber nicht in allen Programmiersprachen. Hier ist die Division problematisch, da die Ergebnisse nat ¨ urlich nicht immer ganze Zahlen sind. Verwendet wird die ganzzahlige Division. Die Division durch 0 ergibt einen Laufzeitfehler.

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#### Annotation 3262258220300

 Rationale Zahlen Werden in einigen Programmiersprachen unterst ¨ utzt; meist als Paar von Int, in Haskell auch exakt als Paar von Integer

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#### Annotation 3262260841740

 我们三层房子的第一周期是 0.547 秒，同样的地点，同样的反应谱，当横坐标周期等于 0.547 的时候，纵坐标地震影响系数等于 0.134。也就是说，三层房子总的地震力等于总的重力的 0.134 倍。 三层房子的总质量是 900 吨，总重力就是 900 吨乘以重力加速度，总的地震力就是总重力乘 以 0.134，等于 1185 千牛

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#### Annotation 3262262414604

 对于一层房子来说，只有一个可以水平移动的楼层，所以求出来的 471 千牛的地震力只能全 部作用在这个楼层上。而对于三层房子来说，三个楼层都可以单独水平移动，也就是说，每 个楼层都要承担一部分的水平地震力，三个楼层加起来，刚好等于总的地震力 1185 千牛。

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#### Annotation 3262263987468

 ，对于楼层来说，地震作用的大小取决于两个参数：楼层的质量和楼 层的高度。作为地面运动引起的惯性力，质量越大，惯性力越大；高度越高，惯性作用越强。 所以，地震力分配到各个楼层是按照楼层的质量与高度的乘积进行分配的。

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#### Annotation 3262266084620

 比如说，质量以每层的质量为单位，那么每层的质量都是 1，高度以层高为单位，那么一层 的高度是 1，二层的高度是 2，三层的高度是 3。质量乘以高度，一层是 1，二层是 2，三层 是 3。三个数加起来等于 6，所以一层的地震力是 6 分之 1，也就是 0.167，二层是 6 分之 2， 也就是 0.333，而三层是 6 分之 3，也就是 0.5。 按照求出来的分配系数，把总的 1185 千牛分配到三个楼层。一层分得六分之一，也就是 198 千牛；二层分得三分之一，也就是 395 千牛；三层分得二分之一，也就是 593 千牛。

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#### Annotation 3262267657484

 第三层柱子要承受三层楼板处的 593 千牛，而第二层的柱子要承受第三层的 593 再加上第二 层的 395 千牛，总共 988 千牛，而第一层柱子要承担其上的 593+395+198 千牛，也就是 1185 千牛。这也就是每一个楼层的柱子需要承担的水平力

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#### Annotation 3262275521804

 对于我们的这个三层房子来说，以上就是最最简单的确定地震力的方法。因为我们根据重力和地震影响系数求出来的总地震力，其实也就是底层柱子需要承受的水平力，也就是例子里 的 1185 千牛，所以中国的抗震规范把这种方法叫做「底部剪力法」。在 ASCE7 里，它叫做 Equivalent Lateral Force Method，等效侧向水平力法。

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#### Annotation 3262277881100

 如果我们取特征矩阵的任意不相同的两列，比如第一列和第二列，第二列和第三列，其中之一转置之后，乘以质量矩阵或者刚度矩阵，再乘以剩下的那一列，那么结果都是 0。 只有当我们取同样的列的时候，比如第一列的转置，乘以质量矩阵，再乘以第一列，这时候结果才不为 0这也就是正交的概念，所谓的 orthogonality。把特征矩阵的转置，乘以质量矩阵或者刚度矩 阵，再乘以特征矩阵，我们就能得到 principal 质量矩阵或者刚度矩阵。

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#### Annotation 3262285483276

 其实等于 0.084 倍第一振型，加 0.882 倍第二振型，再加上 0.035 倍第三振型。 我们可以对比一下上面的这两个例子。虽然都是由三种振型叠加而成，但这三种振型占的份 额并不相同。第一个例子里第一振型占了绝大部分份额，而第二个例子里则是第二振型起到 了主导作用。这是因为，我们的第一个例子的振动模式非常接近于第一振型，而第二个例子 的振动模式则是接近于第二振型。

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#### Annotation 3262287056140

 问题又来了，如何知道这三种振型在地震下的位移是多少呢？如何知道每种振型占多大的份 额呢？怎么确定振型的叠加系数呢？众位看官，且听我们下回分

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#### Flashcard 3266903936268

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