Name: equivalent force and moment systems
Uploaded: 2023-09-22T11:49:21.000Z
Duration: 1 min 16 s
Channel: JoVE

考虑一个承受多种力和力偶矩系统的结构构件。
为了将力平移到不在该力作用线上的点 O，需附加一个力偶，其力矩等于该力对点 O 的力矩。
根据可传性原理，作用在物体上的力是一个滑动矢量，可沿其作用线施加于任意一点。
由于力偶矩是一个自由矢量，因此可直接移至点 O。
合力等于所有力的总和，且与点的位置无关。
合力偶矩等于所有力偶矩以及各力对点 O 的力矩之和，且其值取决于该点的位置。
作用在点 O 的合成力和力偶矩所产生的外部效应，与原始的力和力偶矩系统相同。
该方法可推广应用于多种力、力的分量以及力偶矩。

<ol>
	<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。第 167 -168 页。</li>
</ol>

作用力和力偶系统

在结构系统的分析中，常常会遇到承受各种力和力矩的构件。简化这些系统可以使分析更易于处理和理解。其中一种简化的方法是将一个力转移到一个不在其作用线上的O点，并为其添加一个力矩，其大小等于力相对于O点的力矩。 传递性原理在这个过程中起着关键作用。根据这个原理，作用在物体上的力是一个滑动向量，可以沿着其作用线的任意点进行力的施加而不改变其外力产生的效应。这个性质允许我们将一个力转移到结构构件上的不同位置，同时保持系统的整体行为是不变的。 由于力矩是一个自由向量，可以直接移动到O点而不对系统产生影响的外力产生的效应。这个简化的过程需要计算作用在O点的合力和力矩。合力等于所有力的和，与点的位置无关。相反的，合力矩取决于点的位置，等于所有力矩以及有关O点的力矩和。 通过应用这些原理，在点O处产生作用的合力和力矩与原始的力和力矩系统产生了相同的外力产生的效应。这种等价性确保了简化的系统能够准确地代表了原始的、更为复杂的系统行为。 这种方法可以推广到任意数量的力、它们的分量和力矩。通过系统地结合力和力矩，工程师和科学家们能够有效地分析复杂的结构系统，并预测其在各种附加条件下的行为。 研究结构系统中的力和力矩对于理解其行为和性能至关重要。通过运用传递性原理并将力系统简化为等效系统，工程师和科学家可以更容易地分析和预测复杂系统的行为。这种理解有助于设计高效和有效的结构，并促进了技术和创新的进步。

在结构系统的分析中，常会遇到承受各种力和力偶矩的构件。简化这些系统可以……

Education

教育

JoVE Core

Mechanical Engineering

机械工程

Unit 1

第一单元

Force System Resultants

力系合力

KEY TERMS AND CONCEPTS

关键术语与概念

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 externally. A couple moment is a free vector, so it can be shifted directly to point O without changing the system’s external effects.

To build the equivalent system at point O, the resultant force is found by adding all the forces. This resultant force does not depend on where point O is located. The resultant couple moment does depend on point O, and it equals the sum of all couple moments plus the moments of the forces about point O.

This same method works for any number of forces, force components, and couple moments. The final force and couple moment at point O produce the same external effects as the original loading system. By combining forces and moments in this way, engineers can analyze complex structures more efficiently and predict how they will behave under loading conditions.

Equivalent Force and Moment Systems

moment of a force: scalar formulation

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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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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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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

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resultant moment: vector formulation

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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...

principle of moments: problem solving

Torque About an Axis in 3D Systems

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moment of a force about an axis: scalar

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...

moment of a force about an axis: vector

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

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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

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

Force and couple systems can be reduced to a simpler equivalent system. This makes rigid body analysis easier because the external effect of the forces is captured with fewer elements. When the resultant force and resultant couple moment have perpendicular lines of action, the system may be reduced further to a single resultant force along a new line of action. Three common force systems can be simplified this way: concurrent, coplanar, and parallel. A concurrent force system has lines of...

simplification of a force and couple system: ii

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...

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...

load along a single axis

Finding Resultant Loads on a Beam

Structural engineering often requires finding the resultant load on a beam with changing force patterns. Here, a beam of length L carries a varying load made from parabolic and trapezoidal load distributions along the x-axis. The goal is to find the total load, where it acts, and the centroid of the combined area so the beam’s response can be predicted. The parabolic load is handled first. A small force element, dR, acts over a short distance, dx, and these elements are integrated across the...

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 forces and couple moments can be replaced by an equivalent system at point O using transmissibility, resultant force, and resultant moment.

Equivalent Force and Moment Systems | 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

Equivalent Force and Moment Systems

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