Centroid Calculation: Methods and Applications | 质心位置的计算方法与应用

📚 Centroid Calculation: Methods and Applications | 质心位置的计算方法与应用

The centroid, often called the geometric centre, is the average position of all points in a shape or system. It is one of the most powerful concepts in mathematics and physics, appearing in everything from beam bending to satellite orbits. In this article, we explore how to calculate centroids for discrete particle systems and continuous bodies, examine the centroid positions of common shapes, and discuss the real-world applications that make this quantity essential across engineering, architecture and physics.

质心，通常也称为几何中心，是形状或系统中所有点的平均位置。它是数学和物理学中最有力的概念之一，出现在从梁的弯曲到卫星轨道的方方面面。在本文中，我们探讨如何计算离散质点系统和连续物体的质心，考察常见形状的质心位置，并讨论使这一量在工程、建筑和物理学中至关重要的实际应用。

1. What Is a Centroid? | 什么是质心？

For a set of discrete particles, the centroid is the point where the whole system would balance perfectly if suspended. Mathematically, it is the weighted average of particle positions, where each position is weighted by its mass. The centroid is not necessarily a point that lies inside the material; for example, the centroid of a hollow ring lies at its geometric centre, in the empty space.

对于一组离散质点，质心是如果整个系统被悬挂时能够完美平衡的点。在数学上，它是质点位置的加权平均值，其中每个位置按其质量加权。质心不一定位于材料内部；例如，空心圆环的质心位于其几何中心，即在空洞中。

The centroid is closely related to the centre of mass in physics. When gravity is uniform, the two coincide exactly. In a purely geometric context, we treat density as constant and reduce the problem to one of pure geometry.

质心与物理学中的质心紧密相关。当重力均匀时，两者完全重合。在纯几何情境下，我们将密度视为常数，从而将问题简化为纯粹的几何问题。

2. Centroid of a System of Particles | 质点系的质心

Consider n particles with masses m₁, m₂, …, mₙ located at position vectors r₁, r₂, …, rₙ. The centroid position vector R is given by:

考虑 n 个质量分别为 m₁, m₂, …, mₙ 的质点，其位置矢量分别为 r₁, r₂, …, rₙ。质心位置矢量 R 由下式给出：

R = (m₁r₁ + m₂r₂ + … + mₙrₙ) / (m₁ + m₂ + … + mₙ)

In Cartesian coordinates, this gives three separate scalar equations: one for each coordinate axis. The x-coordinate of the centroid is x̄ = Σ(mᵢxᵢ) / Σmᵢ, and similarly for ȳ and z̄.

在笛卡尔坐标系中，这给出了三个独立的标量方程：每个坐标轴一个。质心的 x 坐标为 x̄ = Σ(mᵢxᵢ) / Σmᵢ，ȳ 和 z̄ 的计算方式类似。

If the total mass is labelled M, we can write the numerator as the moment of the system about each axis. For example, the sum Σ(mᵢxᵢ) is called the first moment of mass about the yz-plane, and the centroid is the point where this moment divided by M is located.

如果将总质量记为 M，则可以将分子视为系统关于各轴的矩。例如，求和 Σ(mᵢxᵢ) 称为关于 yz 平面的质量一次矩，而质心就是该矩除以 M 后所处的位置。

3. Extension to Continuous Objects | 推广到连续物体

When the mass is distributed continuously over a volume, the sums become integrals. For a three-dimensional object with variable density ρ(r), the centroid coordinates are:

当质量连续分布在体积上时，求和变为积分。对于密度为 ρ(r) 的三维物体，质心坐标为：

x̄ = ∫ x ρ dV / ∫ ρ dV, ȳ = ∫ y ρ dV / ∫ ρ dV, z̄ = ∫ z ρ dV / ∫ ρ dV

If the density is constant, ρ cancels from numerator and denominator, simplifying the expressions. The centroid then depends only on the shape, not on the material. This is why, in geometry, we can speak of the centroid of a lamina (a thin flat plate) or of a solid region without specifying its composition.

如果密度恒定，ρ 会从分子和分母中约去，从而简化表达式。此时质心只取决于形状，而与材料无关。这就是为什么在几何学中，我们可以直接讨论薄板或实心区域的质心，而无需指定其材质。

For a two-dimensional region R in the xy-plane, the centroid coordinates are x̄ = ∫ x dA / A and ȳ = ∫ y dA / A, where dA is the area element and A is the total area. The integrals in the numerators are the first moments of area about the y- and x-axes respectively.

对于 xy 平面中的二维区域 R，质心坐标为 x̄ = ∫ x dA / A 和 ȳ = ∫ y dA / A，其中 dA 是面积微元，A 是总面积。分子中的积分分别是关于 y 轴和 x 轴的面积一次矩。

4. Centroids of Standard 2D Shapes | 常见二维图形的质心

Knowing the centroid of a few basic shapes saves enormous time in practice. Here is a table of the most frequently encountered laminae:

在实践当中，记住几个基本形状的质心可以节省大量时间。下表列出了最常遇到的薄板形状：

Shape 形状	Centroid Location 质心位置
Rectangle 矩形	Intersection of diagonals 对角线交点
Triangle 三角形	Intersection of medians 中线交点
Semicircle 半圆	4r/(3π) from diameter 距直径 4r/(3π)
Quarter circle 四分之一圆	4r/(3π) from each radius 距每条半径 4r/(3π)
Circle 圆	Centre 圆心
Parallelogram 平行四边形	Intersection of diagonals 对角线交点

These results follow directly from integration or symmetry. For instance, the centroid of a semicircle of radius r lies along its axis of symmetry at a distance of 4r/(3π) ≈ 0.424r above the diameter.

这些结果直接由积分或对称性得出。例如，半径为 r 的半圆的质心位于其对称轴上，距直径 4r/(3π) ≈ 0.424r。

5. The Centroid of a Triangle | 三角形的质心

The centroid of any triangle is the point where the three medians intersect. A median is the line segment drawn from a vertex to the midpoint of the opposite side. The centroid divides each median in the ratio 2:1, with the longer segment adjacent to the vertex.

任何三角形的质心都是三条中线相交的点。中线是从顶点到对边中点的线段。质心将每条中线按 2:1 的比例分割，较长的线段靠近顶点。

If the three vertices have coordinates (x₁, y₁), (x₂, y₂) and (x₃, y₃), then the centroid is simply the arithmetic mean of the coordinates:

若三个顶点的坐标分别为 (x₁, y₁)、(x₂, y₂) 和 (x₃, y₃)，则质心为坐标的简单算术平均值：

x̄ = (x₁ + x₂ + x₃)/3, ȳ = (y₁ + y₂ + y₃)/3

This is a special case of the general formula with equal masses at each vertex. The result is independent of the shape of the triangle: an equilateral, scalene or obtuse triangle all follow the same rule.

这是每个顶点具有相等质量的一般公式的特例。结果与三角形的形状无关：等边三角形、不等边三角形或钝角三角形都遵循相同的规则。

6. Composite Shapes: The Decomposition Method | 组合图形：分解法

Most real-world objects are not simple geometric shapes. The decomposition method handles these by breaking the object into simpler pieces whose centroids are known, then combining them using a weighted average based on area (or mass).

大多数现实物体并不是简单的几何形状。分解法通过将物体拆分成质心已知的简单部分，然后基于面积（或质量）进行加权平均来加以处理。

For a lamina made of k parts with areas A₁, A₂, …, Aₖ and centroid coordinates (x̄ᵢ, ȳᵢ), the overall centroid is:

对于由 k 个部分组成、面积分别为 A₁, A₂, …, Aₖ 且质心坐标为 (x̄ᵢ, ȳᵢ) 的薄板，总体质心为：

x̄ = (A₁x̄₁ + A₂x̄₂ + … + Aₖx̄ₖ) / (A₁ + A₂ + … + Aₖ)

Notice the subtle connection to section 2: the formula is structurally identical to the centre-of-mass formula, with area replacing mass as the weighting factor. This parallel makes it easy to remember and apply.

请注意这与第二节公式之间的微妙联系：该公式在结构上与质心公式完全相同，只是用面积代替质量作为权重因子。这种平行性使其易于记忆和应用。

A critical practical detail is the handling of holes. A hole can be treated as a piece with negative area. This ‘negative area method’ is widely used in mechanical engineering when analysing cross-sections of girders and brackets.

一个关键的实践细节是如何处理孔洞。孔洞可以被视为具有负面积的组成部分。这种”负面积法”在机械工程分析梁和支架的截面时被广泛使用。

7. The Power of Symmetry | 对称性的力量

Whenever a shape has an axis or point of symmetry, the centroid lies on that axis or at that point. This observation often eliminates an entire coordinate calculation at once.

只要一个形状具有对称轴或对称点，质心就一定位于该轴或该点上。这一观察往往可以立即消除一整条坐标轴的计算。

For example, the centroid of a rectangle lies at its centre; the centroid of an isosceles triangle lies on its altitude; and the centroid of any regular polygon coincides with its circumcentre. In each case, symmetry alone gives one coordinate immediately.

例如，矩形的质心位于其中心；等腰三角形的质心位于其高线上；任何正多边形的质心都与它的外心重合。在每种情况下，仅凭对称性就能立即得到一个坐标。

It is essential to remember that symmetry must apply to the geometry, not to the placement of the coordinate axes. A symmetric shape with an arbitrarily chosen origin does not lose its symmetry; the centroid simply appears at the symmetric location relative to that origin.

需要强调的是，对称性必须适用于几何本身，而不是坐标轴的放置位置。一个具有对称性的形状即使原点选择任意，其对称性也不会消失；质心只是出现在相对于该原点的对称位置上。

8. The First Moment of Area Explained | 面积一次矩解析

The first moment of area (often just called the first moment) about a given axis is the product of the region’s area and the perpendicular distance from the region’s centroid to that axis. It is denoted by Q and has units of length cubed (m³).

面积一次矩（通常简称为一次矩）关于某给定轴的定义是：区域的面积乘以该区域质心到该轴的垂直距离。它记作 Q，单位为长度的三次方（m³）。

For the y-axis, the first moment is Q_y = x̄A, and for the x-axis, Q_x = ȳA. These quantities appear naturally in beam theory: the shear stress at a point in a beam cross-section is proportional to the first moment of the area above that point.

关于 y 轴的一次矩为 Q_y = x̄A，关于 x 轴的一次矩为 Q_x = ȳA。这些量自然地出现在梁理论中：梁截面上某点的剪应力与该点以上面积的一次矩成正比。

This interpretation helps explain why the centroid formula works: to find the centroid, we solve for the point at which all first moments balance. If we place a pivot at the centroid of a lamina, the clockwise and anticlockwise moments about any axis through that pivot are exactly equal.

这种解释有助于理解质心公式为何有效：求质心就是求解所有一次矩达到平衡的点。如果我们将一个支点放在薄板的质心处，则关于通过该支点的任意轴的顺时针和逆时针矩恰好相等。

9. Engineering and Architectural Applications | 工程与建筑应用

In structural engineering, the centroid of a beam’s cross-section defines the neutral axis. When a beam bends under load, the neutral axis experiences zero longitudinal stress. All bending calculations in steel and concrete design begin by locating this axis correctly.

在结构工程中，梁截面的质心定义了中性轴。当梁在荷载作用下弯曲时，中性轴处的纵向应力为零。钢材和混凝土设计中的所有弯曲计算都从正确定位该轴线开始。

The stability of a ship depends on the relative positions of the ship’s centre of mass and its centre of buoyancy. Naval architects shift ballast to adjust the overall centroid so that the vessel floats upright and rightens itself after heeling in waves.

船舶的稳定性取决于船舶质心与其浮心之间的相对位置。造船工程师通过调整压载物来移动总体质心，使船舶能够正直漂浮，并在波浪中倾斜后自我扶正。

In civil engineering, retaining walls and dams are designed so that the resultant force of water pressure passes through the base within a certain distance from the centroid of the base cross-section. This prevents tension cracks from developing in the concrete.

在土木工程中，挡土墙和水坝的设计要保证水压力的合力通过基底时，距基底截面质心的距离处于某一范围内。这样可以防止混凝土中产生拉伸裂缝。

10. Applications in Physics and Sports | 在物理和运动中的应用

In classical mechanics, the motion of a rigid body separates into translation of the centre of mass and rotation about the centre of mass. The trajectory of a thrown hammer, a diver performing somersaults, or a spinning ice skater can all be analysed using this decomposition.

在经典力学中，刚体的运动可以分解为质心的平动和绕质心的转动。投掷的链球、做空翻动作的跳水运动员、旋转的滑冰运动员等轨迹都可以用这种分解来分析。

Athletes intuitively adjust their body position to control their centre of mass. A high jumper uses the Fosbury flop to arch their back so that the body passes over the bar while the centre of mass passes just below it, reducing the height they must physically lift themselves.

运动员直觉地调整身体姿势来控制自己的质心。跳高运动员使用背越式技术弓起背部，使身体越过横杆而质心恰好从横杆下方通过，从而减小需要实际举升的高度。

In vehicle dynamics, a low centre of mass increases stability. Sports cars are designed with a low, central centroid to reduce body roll in corners, while SUVs, with their higher centroids, face a greater risk of rollover during sharp manoeuvres.

在车辆动力学中，低质心能提高稳定性。跑车在设计时追求低而居中的质心以减少过弯时的车身侧倾，而质心较高的 SUV 在急转时则面临更大的侧翻风险。

11. Worked Example: Composite L-Shaped Lamina | 例题：L 形组合薄板

A uniform L-shaped lamina is formed by removing a 2 cm × 2 cm square from a 5 cm × 3 cm rectangle, as shown in the coordinate system where the bottom-left corner of the large rectangle is at the origin. Find the centroid.

一个均匀的 L 形薄板是通过从 5 cm × 3 cm 的矩形中挖去一个 2 cm × 2 cm 的正方形形成的，坐标系中长方形的左下角位于原点。求该薄板的质心。

Step 1: Treat the shape as the large rectangle A plus the missing square B with negative area.

步骤 1： 将该形状视为大矩形 A 加上具有负面积的正方形 B。

Rectangle A: area A₁ = 5 × 3 = 15 cm², centroid at (2.5, 1.5).

矩形 A：面积 A₁ = 5 × 3 = 15 cm²，质心位于 (2.5, 1.5)。
Removed square B: area A₂ = −2 × 2 = −4 cm², centroid at the centre of the removed square. If the square is at the top-right corner with its left edge at x = 3 and bottom edge at y = 1, its centroid is at (4, 2).

被挖去的正方形 B：面积 A₂ = −4 cm²，质心位于被挖去正方形的中心。若正方形位于右上角，其左边缘在 x = 3 且下边缘在 y = 1，其质心位于 (4, 2)。

Step 2: Apply the decomposition formula.

步骤 2： 应用分解公式。

x̄ = (15 × 2.5 + (−4) × 4) / (15 − 4) = (37.5 − 16) / 11 = 21.5 / 11 ≈ 1.95 cm

ȳ = (15 × 1.5 + (−4) × 2) / 11 = (22.5 − 8) / 11 = 14.5 / 11 ≈ 1.32 cm

Step 3: Interpret the result. The L-shaped lamina has its centroid at approximately (1.95, 1.32) cm, slightly shifted left and downward from the centre of the original rectangle because the removed mass was in the top-right region.

步骤 3： 解释结果。L 形薄板的质心约为 (1.95, 1.32) cm，相比原矩形中心略微向左下方偏移，这是因为被挖去的质量位于右上区域。

12. Summary and Key Takeaways | 总结与要点

The centroid of a discrete system is found by weighting each position by mass; for a continuous body, sums become integrals.

离散系统的质心通过以质量为权重对各个位置求加权平均得到；对于连续物体，求和变为积分。

For laminae, the same technique applies using areas as weights. Resources such as the centroid locations of basic shapes, symmetry arguments, and the negative area method for holes turn complex problems into simple arithmetic.

对于薄板，使用面积作为权重即可应用相同的技术。常见基本形状的质心位置、对称性论证以及处理孔洞的负面积法等工具，可以把复杂的问题转化为简单的算术运算。

Applications are everywhere: from neutral axes in beam design and vessel stability in naval architecture, to vehicle dynamics and athletic performance. A solid grasp of centroid calculation is not merely a mathematical exercise; it is a foundational skill for engineers, physicists and designers alike.

应用无处不在：从梁设计中的中性轴和船舶工程中的稳性，到车辆动力学和运动表现。牢固掌握质心计算不仅仅是一项数学练习；对于工程师、物理学家和设计师而言，这都是一项基础技能。

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Centroid Calculation: Methods and Applications | 质心位置的计算方法与应用