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

床に置かれた三脚を考えてみましょう。その脚はAでボールとソケットのジョイントを使用して接続され、もう一方の端Bは床にあります。
脚に作用する合力と合力カップルモーメントがゼロの場合、脚は平衡状態にあると言われ、平衡方程式の助けを借りて表現できます。
脚が平衡状態にあるため、平衡のベクトル方程式とスカラー方程式を満たします。
平衡のベクトル方程式によれば、脚に作用するすべての外力はベクトルの合計がゼロでなければなりません。すべてのカップルモーメントと点に関するすべての力のモーメントのベクトル合計もゼロに等しくなければなりません。
平衡のスカラー方程式によれば、外力がデカルト形式で表される場合、それぞれの方向に沿った力の成分の合計はゼロでなければなりません。
同様に、x、y、z軸のモーメントの成分のスカラー合計もゼロでなければなりません。

<ol>
	<li>ヒベラー、RC(2016)。エンジニアリングメカニクス \u2012 静力学と力学。 ニュージャージー州ホーボーケン:ピアソンプレンティスホール。250ページ</li>
</ol>

三次元における平衡の方程式

Equations of Equilibrium in Three Dimensions

構造物や静止したシステムを分析する際、それらが平衡状態にあることを確認する必要があります。この時、ベクトルとスカラーの平衡の方程式が重要です。これらの方程式は、構造物が安定し崩れたり崩壊したりしないことを保証するために不可欠です。ベクトルとスカラーの平衡の方程式は、物体にかかる力を分析するための枠組みを提供します。

ベクトルの平衡の方程式によれば、物体にかかる全ての外力のベクトル和はゼロに等しくなければなりません。さらに、カップルモーメントのベクトル和と全ての力のモーメントが特定の点においてゼロに等しくなければなりません。

<img alt="Equation 1" src="/files/ftp_upload/14275/14275_Equation_1v2.svg" style="text-align: center; height: 33px;" />

スカラーの平衡の方程式によれば、外力がデカルト形式で表現されている場合、それぞれの方向に沿った力の成分の合計はゼロでなければなりません。

<img alt="Equation 2" src="/files/ftp_upload/14275/14275_Equation_2v2.svg" style="text-align: center; height: 55px;" />

同様に、モーメントの成分のスカラーの合計もx軸、y軸、z軸においてゼロでなければなりません。

<img alt="Equation 3" src="/files/ftp_upload/14275/14275_Equation_3v2.svg" style="text-align: center; height: 55px;" />

これらの方程式は、機械工学において基礎となり、構造物が安全で安定した状態にあることを保証します。

静止している構造やシステムを解析する際には、それらが平衡状態にあることを確認する必要があります。ここで、ベクトル方程式とスカラー方程式...

equations-of-equilibrium-in-three-dimensions

Education

JoVE Core

Mechanical Engineering

機械工学

Unit 1

ユニット1

Equilibrium of a Rigid Body

剛体のつり合い

KEY TERMS AND CONCEPTS

主要用語と概念

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 機械工学は、平衡、力、トラス解析、構造力学などの重要なトピックをカバーする簡潔でわかりやすいアニメーション ビデオを提供します。実際のアプリケーションを探索し、科学者主導の実験を通じてこれらのコンセプトが実際に動作しているのを目撃してください。

mechanical engineering

Checking Force Balance in 3D Structures

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