Eigenvalues of Matrices in Vibration Systems | 矩阵特征值在振动系统中的应用

📚 Eigenvalues of Matrices in Vibration Systems | 矩阵特征值在振动系统中的应用

Linear algebra is often perceived as a purely abstract discipline, yet one of its most profound applications lies in the physical world of mechanical vibration. When a bridge sways, a car absorbs a bump, or a molecule vibrates, the underlying mathematics is driven by the eigenvalues and eigenvectors of carefully constructed matrices.

线性代数常被视为一门纯粹的抽象学科，然而它最深远的应用之一却存在于机械振动的物理世界之中。当桥梁晃动、汽车吸收路面颠簸、或分子发生振动时，其背后的数学正是由精心构造的矩阵的特征值与特征向量所驱动的。

1. From Physics to Matrices | 从物理到矩阵

Consider a simple mass-spring system. According to Newton’s second law, the force acting on a mass equals the product of its mass and acceleration: F = ma. For a spring, Hooke’s law states that the restoring force is proportional to the displacement from equilibrium: F = -kx, where k is the spring constant measured in newtons per metre.

考虑一个简单的质量-弹簧系统。根据牛顿第二定律，作用在质量上的力等于其质量与加速度的乘积：F = ma。对于弹簧，胡克定律指出恢复力与偏离平衡位置的位移成正比：F = -kx，其中 k 为劲度系数，单位为牛顿每米。

When multiple masses and springs are coupled together, the displacement of each mass affects the motion of the others. This interdependence creates a system of coupled differential equations—and this is precisely where matrices emerge as a natural and powerful language. The number of independent displacement coordinates is called the number of degrees of freedom of the system.

当多个质量与弹簧相互耦合时，每个质量的位移都会影响其他质量的运动。这种相互依赖关系构成了一组耦合微分方程——这正是矩阵作为一种自然而强大的语言登场的时刻。独立位移坐标的数目称为系统的自由度数。

For a single mass attached to a wall by a spring, the equation of motion m x” + k x = 0 is a simple second-order differential equation whose solution is sinusoidal with angular frequency ω = √(k/m). But for systems with two, three, or hundreds of coupled masses, this scalar approach breaks down, and matrices become indispensable.

对于单个质量通过弹簧连接墙壁的情形，运动方程 m x” + k x = 0 是一个简单的二阶微分方程，其解为正弦函数，角频率为 ω = √(k/m)。但对于具有两个、三个乃至数百个耦合质量的系统，这种标量方法不再适用，矩阵因此变得不可或缺。

2. Deriving the Matrix Form | 推导矩阵形式

Let us examine a system of two masses, m₁ and m₂, connected by springs with constants k₁, k₂ and k₃, anchored to fixed walls at both ends. Let x₁ and x₂ represent the displacements of the two masses from their equilibrium positions.

让我们考察一个由两个质量 m₁ 和 m₂ 通过弹簧连接的系统，三根弹簧的劲度系数分别为 k₁、k₂ 和 k₃，且系统两端锚定在固定墙壁上。设 x₁ 和 x₂ 分别代表两个质量相对于平衡位置的位移。

Applying Newton’s second law to each mass and using Hooke’s law for each spring yields two coupled second-order differential equations. The force on mass 1 comes from spring 1 pulling left and spring 2 acting between the masses; the force on mass 2 comes from spring 2 and spring 3:

对每个质量应用牛顿第二定律，并对每根弹簧使用胡克定律，得到两个耦合的二阶微分方程。质量 1 所受的力来自弹簧 1 的向左拉力以及弹簧 2 在两质量之间的作用；质量 2 所受的力来自弹簧 2 与弹簧 3：

m₁x₁” = -k₁x₁ + k₂(x₂ – x₁)

m₂x₂” = -k₂(x₂ – x₁) – k₃x₂

Rearranging and grouping terms, we express this system in matrix form as Mx” + Kx = 0, where M is the mass matrix, K is the stiffness matrix, and x = (x₁, x₂)ᵀ is the displacement vector:

重新整理并合并同类项后，我们将该系统以矩阵形式表达为 Mx” + Kx = 0，其中 M 为质量矩阵，K 为刚度矩阵，x = (x₁, x₂)ᵀ 为位移向量：

M = [[m₁, 0], [0, m₂]], K = [[k₁ + k₂, -k₂], [-k₂, k₂ + k₃]]

Notice the symmetric structure of K: the off-diagonal entries are equal and negative, reflecting the mutual coupling through the middle spring. This symmetry is not accidental; it mirrors the physical principle of Newton’s third law, where the force exerted by mass 1 on mass 2 is equal and opposite to that exerted by mass 2 on mass 1.

注意 K 的对称结构：非对角元相等且为负值，反映了中间弹簧的相互耦合。这种对称性并非偶然；它体现了牛顿第三定律的物理原理，即质量 1 对质量 2 的力与质量 2 对质量 1 的力大小相等、方向相反。

3. The Eigenvalue Problem Emerges | 特征值问题的出现

To solve the system Mx” + Kx = 0, we look for solutions in which all masses oscillate with the same angular frequency ω. Physically, this corresponds to a synchronized motion where the relative amplitudes of the masses remain constant over time. We therefore try a solution of the form:

为了求解系统 Mx” + Kx = 0，我们寻找所有质量以相同角频率 ω 振荡的解。物理上，这对应于一种同步运动，各质量的相对振幅在时间上保持恒定。因此我们尝试如下形式的解：

x(t) = ve^(iωt)

where v is a constant vector describing the pattern of motion and i = √(-1). The complex exponential e^(iωt) is a compact way to represent sinusoidal oscillation, since by Euler’s formula e^(iωt) = cos(ωt) + i sin(ω

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