Name: resultant force and wrench motion
Uploaded: 2023-09-22T11:49:22.000Z
Duration: 1 min 23 s
Channel: JoVE

3次元システムには、互いに垂直ではない等価な合力と合力のカップルモーメントがあります。
結果として生じるカップルモーメントは、合力の作用線に平行および垂直なコンポーネントに分解できます。
垂直成分は、合力が垂直な距離だけ移動する場合に交換できます。
この合力と同一線上のカップルモーメントの組み合わせは、レンチまたはねじと呼ばれるボディをその軸を中心に移動および回転させます。
レンガに作用する3つの力を考えてみましょう。それらをレンチと交換し、レンガと交差するポイントを見つけます。
まず、合力とその大きさを求めます。
次に、力の作用点から交点までの位置ベクトルを考えます。
それらの外積と対応する力により、結果として生じるカップルモーメントが得られます。
次に、合力の単位ベクトルを決定します。
結果として得られるモーメントは、その大きさと単位ベクトルの積と等しくなります。
成分を等価化して方程式を解くと、交点の座標が得られます。

<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。179、189ページ。</li>
</ol>

力とカップルシステムの簡略化 ii

三次元のシステムでは、オブジェクトに複数の力が働くことがあります。これらの力は、合力としてまとめることができます。同様に、これらの力によって生成されるモーメントは、合同力矩としてまとめることができます。一部の場合では、これらの二つの値は互いに直交しない場合があります。つまり、それらの間には90度の角度がありません。この特殊な状況は、合同力と合同力矩の相互作用をより深く理解する必要があります。

この状況をさらに分析するために、合同トルクは、合力の作用線に平行な成分と垂直な成分の2つに解析することができます。この分解により、各成分に個別に焦点を当て、それらのシステムへの個別の影響を理解することができます。合同力矩の平行成分は、合力に垂直な軸周りの回転を引き起こしません。これは、力の方向に沿った「ひねり」の効果と考えることができます。

一方、合同トルクの垂直成分は、力とモーメントの両方に垂直な軸周りの回転を引き起こします。この垂直成分は、合力が垂直な距離で移動される場合に置き換えることができます。つまり、元の力と合同トルクの垂直成分の組み合わせは、異なる位置で作用する単一の力に置き換えられ、等価なシステムが得られます。

合同力とコリニアなトルク（平行成分）の組み合わせは、ボディを両方の軸周りで移動および回転させるレンチやネジのように視覚化することができます。言い換えると、剛体は力の方向に沿った同時の直線運動と、力に平行な軸周りの回転運動を経験します。この特殊な運動は、レンチやネジと呼ばれ、工学、ロボット工学、および生物力学に多くの応用があります。

たとえば、ロボット工学では、レンチの概念は、エンドエフェクタが並進運動と回転運動の両方を必要とする複雑なタスクを実行するための運動学と力学の理解に不可欠です。同様に、生体力学では、レンチの概念は、さまざまな活動中に骨や関節に作用する力とモーメントを研究するために適用され、研究者がより良い義肢や整形外科デバイスの設計に役立てられます。

力とカップルシステムの簡略化 II

3次元システムでは、複数の力がオブジェクトに作用する可能性があります。これらの力は、合力として知られる単一の等価な力に組み合わせることができます...

simplification-of-a-force-and-couple-system-ii

Education

JoVE Core

Mechanical Engineering

機械工学

Unit 1

ユニット1

Force System Resultants

力の系の合力

KEY TERMS AND CONCEPTS

主要用語と概念

Simplification of a Force and Couple System: II

resultant force and wrench motion

Resultant force and resultant couple moment describe how several forces and moments act together on a rigid body in three dimensions. When many forces act on an object, they can be replaced by one equivalent force. The moments from those forces can also be combined into one equivalent couple moment.

In some cases, the resultant force and the resultant couple moment are not mutually perpendicular. That means the angle between them is not 90 degrees. To study this situation, the couple moment is split into two parts. One part is parallel to the line of action of the resultant force, and the other part is perpendicular to it.

The parallel component acts like a twisting effect along the direction of the force. It does not create rotation about an axis perpendicular to the resultant force. The perpendicular component causes rotation about an axis that is perpendicular to both the force and the moment.

The perpendicular component can be replaced by moving the resultant force a certain perpendicular distance. This lets the original force and the perpendicular moment component be represented by a single force at a new location. The remaining force and parallel couple moment then form a wrench, also called a screw.

A wrench or screw motion combines translation and rotation at the same time. The body moves along the direction of the force and rotates about an axis parallel to that force. This idea is useful in engineering, robotics, and biomechanics. In robotics, it helps describe the motion of robotic manipulators and their end-effectors. In biomechanics, it helps researchers study forces and moments on bones and joints during movement and support better prosthetics and orthopedic devices.

Resultant Force and Wrench Motion

moment of a force: scalar formulation

Torque on a Flywheel: Direction and Size

Torque, or the moment of a force, describes how a force can make an object turn about an axis. It is a vector quantity, so it has both size and direction. This idea is important in physics, mechanics, and engineering. A simple flywheel example shows how torque works. The flywheel rotates about point O when a force is applied to it. The moment arm is the perpendicular distance from point O to the line of action of the force. The moment, in newton-meter (N-m), is found by multiplying the force...

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Calculating Force Moments in Problem Solving

Calculating force moments in problem solving helps students analyze rotation in mechanical systems. The scalar form of the moment of a force is used to work through these problems step by step. Start by drawing a free-body diagram, or FBD. An FBD is a sketch that shows every force acting on the system. Each force must be identified and included. Next, identify the axis or point about which the moment is calculated. In some problems, that axis or point is given. In other cases, you must choose...

resultant moment: scalar formulation

Net Moment Sign Convention in 2D Forces

Net moment in two-dimensional force systems shows how multiple forces can cause rotation about a fixed point. The scalar formulation helps analyze equilibrium in structures that are acted on by several forces. It focuses on the x-y plane and uses a simple sign rule to describe the direction of rotation. To find the resultant moment, the moments from all forces in the system are considered about a chosen point A. Each moment is assigned a sign based on its sense of rotation. Positive moments...

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Moment of Force and Direction Rules

The moment of force describes the twisting effect a force can have on an object. It shows why a force may cause rotation instead of straight-line motion. The moment arm is the perpendicular distance from the force’s line of action to the axis of rotation. The moment of force is a vector quantity, not a scalar. Its vector form comes from the cross product of the position vector and the force vector. A cross product gives a vector that is perpendicular to both input vectors. The moment about a...

cartesian form for vector formulation

Calculating Moment with Cartesian Vectors

Calculating moment with Cartesian vectors shows how position and force vectors work together. The moment of force is the cross product of these vectors, so it is a vector quantity. In Cartesian form, unit vectors help write the vectors in component form. The cross product can be arranged in determinant form. That determinant is then expanded to find the Cartesian vector form of the moment of force. This method gives a clear way to work through the calculation step by step. A revolving door...

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Calculating Torque from Multiple Forces

Resultant moment describes the total turning effect of multiple forces acting on an object. It shows how several forces can combine to make a body rotate about a point. In physics, moment is the tendency of a force to cause rotation. The resultant moment of a force system is a vector quantity. That means it can be added and subtracted like other vectors. For a system with two forces, F1 and F2, acting on a pole at points A and B, the resultant moment is written as MR = (rA x F1) + (rB x F2).

principle of moments

Resolving Forces in Moment Calculations

The principle of moments, also called Varignon's theorem, explains how a force creates rotation about a point. It is a key idea in physics and engineering. It also helps describe when a rigid body stays in equilibrium under external forces. The moment of a force is found by multiplying the force by the perpendicular distance from the point of application to the point being studied. A larger force or a larger distance gives a larger moment. This makes the idea useful for checking how strongly a...

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Torque About an Axis in 3D Systems

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Bicycle Torque and Moment About an Axis

The moment of a force about an axis describes how a force can make an object turn around a fixed line. In scalar analysis, the moment is found from the perpendicular distance between the axis of rotation and the line of action of the force. That distance is called the moment arm. A bicycle gives a clear example of this idea. When the cyclist pushes on the pedal, the chain applies a force to the wheel’s axle and makes it rotate. The axis of rotation passes through the axle, and the moment arm...

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Torque on a Bicycle Wheel: Vector Method

Torque, or the moment of a force, explains how a force can make an object rotate about an axis. A bicycle is a clear example. When a cyclist presses on the pedal, the crankshaft turns and helps drive the wheel. To analyze this motion, a coordinate system is set up first. The x-axis is tangential to the wheel, the y-axis is perpendicular to the x-axis and points upward, and the z-axis points outward from the wheel. In this setup, the force on the wheel axle lies along the x-axis. The axle...

couple

Couple Moment in Force Systems

A couple is a pair of parallel forces that are equal in magnitude and opposite in direction. The two forces are separated by a perpendicular distance called the couple arm. Together, they create a turning effect on a body instead of a net push. This turning effect is called the couple moment. It rotates the body about an axis that is perpendicular to the plane of the forces. The SI unit for a couple moment is the Newton-meter (N-m). A common example is tightening a bolt with a lug wrench. In...

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Couple Moments and the Right-Hand Rule

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equivalent couples

Equivalent Couples in Mechanical Systems

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moment of a couple: problem solving

Torque from a Couple: Solving Forces

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system of forces and couples

Equivalent Force and Moment Systems

Structural systems often experience several forces and couple moments at the same time. Engineers simplify these loading cases by replacing them with an equivalent force and moment system at a chosen point O. This makes the analysis easier while keeping the same external effects on the member. A key idea in this process is the principle of transmissibility. A force is a sliding vector, which means it can be moved anywhere along its line of action without changing how the body behaves...

simplification of a force and couple system i

simplification-of-a-force-and-couple-system-i

Resultant Forces in Force and Couple Systems

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simplification of a force and couple system: ii

Resultant force and resultant couple moment describe how several forces and moments act together on a rigid body in three dimensions. When many forces act on an object, they can be replaced by one equivalent force. The moments from those forces can also be combined into one equivalent couple moment. In some cases, the resultant force and the resultant couple moment are not mutually perpendicular. That means the angle between them is not 90 degrees. To study this situation, the couple moment is...

distributed loads

Load Distribution on Shelves and Dams

Load distribution helps engineers describe how force spreads across a surface or structure. A distributed load is a continuous force per unit area, rather than a single point force. It is a common idea in many practical situations. A bookshelf full of stacked books is one simple example. The weight of the books is spread along the shelf, so the pressure changes across the surface. This pressure is measured in Newtons per square meter, or Pascals. A hydroelectric dam gives another clear...

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Finding Resultant Loads on a Beam

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distributed loads: problem solving

Equivalent Resultant Loads on Beams

Equivalent resultant loads on beams can be found by breaking a distributed load into smaller regions. This makes the beam easier to analyze. Each region can then be handled on its own before the results are combined. The magnitude of the equivalent resultant load for each region comes from the area of that region. In this setting, area represents the force applied by the distributed load over that part of the beam. After finding the force for each region, the position of each resultant load...

key terms and concepts

Learn how a resultant force and couple moment can be split into parallel and perpendicular parts, then combined into wrench motion in 3D systems.

Resultant Force and Wrench Motion | 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

Resultant Force and Wrench Motion

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