Name: checking force balance in 3d structures
Uploaded: 2023-09-22T11:49:23.000Z
Duration: 1 min 30 s
Channel: JoVE

考虑一个放置在地面上的三脚架；其支腿在A处通过球窝关节连接，另一端B位于地面上。
当作用在腿部的合力与合力矩为零时，称该腿部处于平衡状态，可通过平衡方程加以表达。
由于腿部处于平衡状态，因此满足矢量和标量的平衡方程。
根据平衡的矢量方程，作用在腿部的所有外力的矢量和必须为零。所有力偶矩以及所有力对任意一点的力矩的矢量和也必须等于零。
根据标量平衡方程，若将外力表示为笛卡尔形式，则沿各个方向的力的分量之和必须为零。
同理，力矩在 x、y 和 z 轴方向上的分量的标量和也必须为零。

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	<li>希伯勒，RC（2016 年）。工程力学 \u2012 静力学和动力学。 新泽西州霍博肯:Pearson Prentice Hall。页码 250</li>
</ol>

在三维平衡中的方程

Equations of Equilibrium in Three Dimensions

在分析静止的结构或系统时，必须确保它们能够处于平衡状态。这就是矢量和标量平衡方程能够发挥作用的地方。这些方程对确保结构的稳定、不会坍塌或崩溃至关重要。平衡的矢量和标量方程为物体上力的框架提供了分析作用。 根据矢量的平衡方程可以得知，作用在物体上所有外力的矢量和都必须等于零。此外，偶力矩的矢量和以及关于某一点处所有力矩的矢量和也都必须等于零。 <img alt="方程1" src="/files/ftp_upload/14275/14275_Equation_1v2.svg" style="text-align: center; height: 33px;"> 标量的平衡方程表明了，如果以笛卡尔形式表示外力，则力在各个方向上的分量之和必须为零。 <img alt="方程2" src="/files/ftp_upload/14275/14275_Equation_2v2.svg" style="text-align: center; height: 55px;"> 同样地，力矩在x、y和z轴上的分量之和也必须为零。 <img alt="方程3" src="/files/ftp_upload/14275/14275_Equation_3v2.svg" style="text-align: center; height: 55px;"> 这些方程在机械工程中是基础，以确保结构的安全稳定。

在分析静止状态下的结构或系统时，必须确保其处于平衡状态。这需要用到……的矢量与标量方程。

equations-of-equilibrium-in-three-dimensions

Education

教育

JoVE Core

Mechanical Engineering

机械工程

Unit 1

第一单元

Equilibrium of a Rigid Body

刚体的平衡

KEY TERMS AND CONCEPTS

关键术语与概念

checking force balance in 3d structures

Checking force balance in 3D structures is a key step when a system is at rest. The vector and scalar equations of equilibrium are used to test whether a body is stable and will not collapse or fall apart. These equations give a clear way to study the forces acting on a body.

The vector equations of equilibrium require the vector sum of all external forces on a body to equal zero. They also require the vector sum of the couple moments and the moments of all forces about a point to equal zero. A couple moment is a turning effect produced by a pair of forces.

The scalar equations of equilibrium use Cartesian form, which means the forces are written in x, y, and z components. In that form, the sum of the force components along each direction must be zero. The scalar sum of the moment components about the x, y, and z axes must also be zero. These conditions are fundamental in mechanical engineering because they help show that structures are safe and stable.

Checking Force Balance in 3D Structures

conditions of equilibrium

Rigid Body Force Balance in Equilibrium

Rigid body equilibrium depends on a balance of external forces and moments. A rigid body is in equilibrium when it has no translational motion and no rotational motion. This happens when the net force and the net couple acting on the body are both zero. Internal forces are not used when checking equilibrium. These forces act in equal and opposite pairs inside the body, so they cancel each other. Because of that, they do not affect the overall motion or change in motion of the rigid body. The...

support reactions

Types of Structural Supports

Types of structural supports control how a coplanar force system moves and reacts under load. A coplanar force system is a set of forces that all lie in the same plane. The supports create reactions at the point of contact. These reactions help keep a structure stable under external forces. Supports work by stopping translation with an equal and opposite force. They can also stop rotation by applying a couple moment in the opposite direction. In structural design, these reactions are important...

internal forces and center of gravity

Forces Inside a Body and Balance

Internal forces and the center of gravity help explain how objects stay in balance. These mechanics ideas matter for structures, machines, and vehicles that must remain stable under different loads and conditions. Internal forces are created inside a body when its particles interact. They can appear as tension, compression, or shear forces. These forces act in collinear pairs with equal magnitude and opposite direction. Between connected parts, they cancel each other and help keep the body in...

alternative sets of equilibrium equations

alternative-sets-of-equilibrium-equations

Equilibrium Conditions for a Supported Plate

Equilibrium conditions for a supported plate are used to describe when forces and moments balance. In this state, the plate has no net movement and no rotation. Engineers use these equations to study how structures behave under external loads. In many cases, the standard equilibrium equations are enough. But some systems need an alternative set of equations to match the geometry of the support points. A rectangular plate supported at points A, B, and C is one example. For this plate, the...

two force member

Two-Force Body Equilibrium

A two-force body is a rigid body acted on by only two external forces. In equilibrium, those two forces must have the same magnitude, the same line of action, and opposite directions. This balance is what keeps the body at rest. For a plate with forces F1 and F2 applied at points A and B, equilibrium means the sum of moments about any axis is zero. If moments are summed about point A, the moment from F2 must be zero. That means the line of action of F2 must pass through point A. The same idea...

three force member

Solving a Three-Force Member Example

A three-force member is a rigid body acted on by three forces at three points. The forces must have concurrent lines of action. If the forces are parallel, their lines of action stay parallel instead. A dumpster provides the example used here. It is connected to a pin support at point A and to a pin on a hydraulic cylinder at point B. The hydraulic cylinder acts as a two-force member in equilibrium. The dumpster weight acts through the center of gravity, and the support reaction acts at point...

equations of equilibrium in three dimensions

Checking force balance in 3D structures is a key step when a system is at rest. The vector and scalar equations of equilibrium are used to test whether a body is stable and will not collapse or fall apart. These equations give a clear way to study the forces acting on a body. The vector equations of equilibrium require the vector sum of all external forces on a body to equal zero. They also require the vector sum of the couple moments and the moments of all forces about a point to equal zero.

support reactions in three dimensions

3D Support Types for Stability and Motion

Support reactions in three dimensions keep structures stable and in equilibrium. They stop unwanted translation and rotation, so a design can handle external forces safely and work as intended. These reactions are important in many structures and systems. A ball and socket joint is one support that allows free rotation about any axis. It does not provide a moment that resists turning. At the same time, it restricts motion in space with three reaction forces along the three dimensions. A...

constraints and statical determinacy

Constraints in Structural Equilibrium

Constraints in structural engineering control how a body can move and help keep it in mechanical equilibrium. A constraint is a restriction on motion. When the right number of constraints is used, the system can stay balanced with the fewest supports needed. A system with the proper number of supports is called statically determinate. In that case, the unknown reaction forces can be found from equations of equilibrium. These systems are the easiest to analyze and design because the forces can...

key terms and concepts

Learn how vector and scalar equilibrium equations keep 3D structures stable by balancing forces and moments in x, y, and z directions.

Checking Force Balance in 3D Structures | JoVE K12

JoVE Core Mechanical Engineering offers concise, easy-to-understand animated videos covering key topics such as equilibrium, forces, truss analysis, and structural mechanics. Explore their real-world applications, and witness these concepts in action through scientist-led experiments.

mechanical engineering

Checking Force Balance in 3D Structures

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