Name: spring-mass motion and energy stability
Uploaded: 2023-09-22T11:49:37.000Z
Duration: 1 min 24 s
Channel: JoVE

K12 Standards

Texas Essential Knowledge and Skills

High School

Physics

PHYS.1

IPC.1.G

One-Degree-of-Freedom System

システムの位置が 1 つの独立変数によって定義される場合、その自由度は 1 です。その後、そのポテンシャルエネルギー関数をその変数に対してプロットできます。
システムが平衡状態に達すると、その位置での関数の傾きはゼロになります。
平衡配置の安定性は、変数に対するポテンシャルエネルギー関数の2次導関数を使用して調査できます。
2次導関数が正の場合、ポテンシャルエネルギーは最小になり、システムは安定した平衡配置にあります。
逆に、2次導関数が負の場合、ポテンシャルエネルギーは最大になります。これは、不安定な平衡配置を意味します。
2次導関数が0の場合、高次導関数を評価する必要があります。
最初の非ゼロ導関数が正で偶数次数の場合、平衡配置は安定しています。
あるいは、最初の非ゼロ導関数が負で、奇数または偶数の次数を持つ場合、不安定です。
すべての高次微分がゼロの場合、システムは中立平衡を達成しています。ここでは、ポテンシャルエネルギー関数は平衡位置の周りで一定のままです。

<ol>
	<li>ヒベラー、RC(2016)。エンジニアリングメカニクス:静力学。第14版、ニュージャージー州:ピアソン。セクション11.7、601-602ページ。</li>
	<li>ビール、FP、ジョンストン、ER、マズレック、DF、コーンウェル、PJおよびセルフ、BP(2016)。エンジニアのためのベクトル力学。第11版、ニューヨーク:マグロウヒル教育。599-600ページ。</li>
</ol>

一自由度システム

機械工学では、一自由度システムは広範な電気および機械部品の基礎を形成しています。これらのモデルを使用することで、エンジニアは大規模なシステム内の様々な部品の振る舞いを予測することができ、異なる力がどのように相互作用するかを理解することができます。

一自由度システムは、状態と振る舞いを決定する独立変数によって定義されます。一自由度システムの一例は、ばねに接続された質量などの単振り子です。質量の位置が一軸上で振動の状態と振る舞いを決定し、ばねによって力が加えられることで振動が発生します。このようなシステムでは、潜在エネルギー（V）をグラフにプロットすることで、オブジェクトの位置に応じてどれだけのエネルギーを保持しているかを示すことができます。

一自由度システムを理解することにより、エンジニアは複雑な設計を構築する際に貴重な知識を得ることができます。これにより、設計を具現化する前に、さまざまな条件下での動作を正確にモデル化および予測することができます。さらに、力と加速度を結び付ける基本的な方程式を理解することにより、慣性などの物理特性を理解するのにも役立ちます。

異なる構成の平衡の安定性を調査するために、エンジニアは潜在関数の二階導関数を分析し、平衡の安定、不安定、または中立性を決定します。安定な平衡は潜在エネルギープロット上の最小値に対応し、最大値は不安定を示し、一定値はその点での中立性を示します。これにより、外部要因によって時間の経過とともに動的な変化が起こる可能性のある環境で最適な設計を把握することができます。これにより、機械の動作が事前に十分に予測されない場合に、機械の動作が妨害されることを防ぐのに役立ちます。

一自由度システムに関する全体的な知識は、アラーム時計から外宇宙に向かうモダンなロケットまで、頑丈な機械や乗り物を構築しようとするエンジニアにとって貴重なツールです。

機械工学では、1自由度システムは、さまざまな電気部品や機械部品の基礎を形成しています。これらのモデルを使用して、...

Education

JoVE Core

Mechanical Engineering

機械工学

Unit 1

ユニット1

Virtual Work

仮想仕事

KEY TERMS AND CONCEPTS

主要用語と概念

spring-mass motion and energy stability

A one-degree-of-freedom system uses a single independent variable to describe its state and behavior. In mechanical engineering, this kind of model helps explain how parts in a larger system respond to force. It also gives engineers a way to predict motion before a design is built in real life.

A simple harmonic oscillator is one example of a one-degree-of-freedom system. A mass attached to a spring moves along one axis, and that position determines the system’s state. The spring force makes the mass move back and forth, which creates oscillation.

Engineers also study the potential energy, or V, of these systems. They can plot V on a graph to show how much energy the object stores at different positions relative to its resting point. This helps connect position, force, and motion in a clear way.

These models are useful for building complex designs because they make behavior easier to predict under different conditions. They also help explain the basic equations that link force and acceleration, including ideas connected to inertia. For that reason, one-degree-of-freedom systems are a valuable tool when designing machines and vehicles.

Engineers also use the second derivative of the potential function to check equilibrium stability. Stable equilibrium points appear at minimum values on the potential energy plot. Maximum values show unstable equilibrium, and constant values indicate neutral equilibrium. This kind of analysis helps designers judge which configurations are likely to work best in environments where outside forces may change over time, from small alarm clocks to modern rockets heading into space.

Spring-Mass Motion and Energy Stability

work

Work in Force and Motion

Work connects force, motion, and energy in mechanics. It happens when a force acts on an object and the object moves along the line of action of that force. Work can also be described as energy transfer caused by applying a force, which leads to a change in the object’s motion or position. Work is calculated as the force multiplied by the part of the displacement in the same direction as the force. It can also be written as the dot product of the applied force and the object’s displacement. In...

work of a couple moment

Couple Moment Work in Rotation

Mechanical engineering studies motion, energy, and force, and it uses the idea of a couple moment to describe how two equal and opposite forces act on a rigid body. A couple moment is created when those forces act at two points on the body that are separated by a distance. When a rigid body moves because of a couple, its displacement can be split into two parts. First, the two points move by the same amount to their final positions. Then the body rotates about one of those points. During the...

virtual work

Virtual Work in Force Equilibrium

Virtual work is a mechanics principle used to check equilibrium with imaginary motion. If a body is in static or dynamic equilibrium, the total virtual work from external forces and couple moments must be zero for any virtual displacement. A virtual displacement is an imagined small movement. It can be a shift or a rotation. The virtual work done by a force equals the force dotted with the virtual displacement in the force’s direction. For a couple moment, virtual rotational work is found by...

virtual work for a system of connected rigid bodies

virtual-work-for-a-system-of-connected-rigid-bodies

Finding Forces with Virtual Work in Linked Bodies

Virtual work helps find forces in a system of connected rigid bodies. It is a useful method when several links move together and the system is in equilibrium. In that case, the total virtual work is zero. To analyze the system, start with a free-body diagram. Use one independent coordinate to describe the link configuration. Then mark the deflected position that comes from a positive virtual displacement. This makes the small change in position clear before any calculations begin. Next,...

principle of virtual work: problem solving

principle-of-virtual-work-problem-solving

Virtual Work for Finding Unknown Forces

The principle of virtual work helps find unknown forces and moments in equilibrium problems. It is an important idea in mechanics and engineering. The method uses a small, hypothetical motion called a virtual displacement. If the system is in equilibrium, the work done by the forces during that displacement is zero. To apply this principle, the positions of the internal and external forces must be written in terms of one common coordinate parameter. That parameter can be an angle, a distance,...

mechanical efficiency of real machines

Friction’s Role in Machine Efficiency

Mechanical efficiency shows how well a machine turns input work into output work. It is calculated as the ratio of output work to input work. An ideal machine has an efficiency of one, which means the input work equals the output work. Real machines are never perfectly ideal because some energy is lost. Friction is one of the main causes of that loss. In a toggle vise used to compress wooden blocks, the sliding block rubs against the horizontal plane and creates friction. To find the...

conservative forces

Conservative Forces and Path-Independent Work

Conservative forces are forces whose work depends only on an object’s starting and ending positions. The path between those positions does not change the amount of work done. This idea is important in mechanical engineering because it helps describe how energy moves through a system. A conservative force also conserves energy in the sense that its work is path independent. Gravity is one example. Electrostatic forces and springs modeled by Hooke’s law are also examples. A ball rolling down a...

potential energy

Gravitational and Elastic Energy Stored

Gravitational and elastic potential energy are forms of stored energy from conservative forces. A conservative force, such as gravity or an elastic force, gives a body the capacity to do work. This capacity is measured as potential energy and depends on the body’s position relative to a fixed reference point, or datum. For gravitational potential energy, the datum is the level where the energy is taken as zero. If a body is above this reference plane, its weight has positive gravitational...

potential-energy criterion for equilibrium

potential-energy-criterion-for-equilibrium

Stable Equilibrium Through Potential Energy

Potential energy helps determine when a mechanical system is stable at equilibrium. When a system is acted on by gravitational and elastic forces, its total potential function is the algebraic sum of gravitational potential energy and elastic potential energy. If the system is already in equilibrium and is given a small displacement, the work done on the system equals the negative of the change in potential energy between the starting and ending positions. For a virtual displacement, which is...

stability of equilibrium configuration

Types of Equilibrium in Mechanical Systems

Types of equilibrium in mechanical systems are stable, neutral, and unstable. These three cases help explain how a structure or object responds when it is pushed slightly away from balance. They are important for predicting whether a system will return to its original position, stay where it is, or move farther away. Stable equilibrium occurs when a system returns toward its original position after a small displacement. In this case, the potential energy is at a minimum. A cantilever beam...

one-degree-of-freedom system

A one-degree-of-freedom system uses a single independent variable to describe its state and behavior. In mechanical engineering, this kind of model helps explain how parts in a larger system respond to force. It also gives engineers a way to predict motion before a design is built in real life. A simple harmonic oscillator is one example of a one-degree-of-freedom system. A mass attached to a spring moves along one axis, and that position determines the system’s state. The spring force makes...

stability of equilibrium configuration: problem solving

stability-of-equilibrium-configuration-problem-solving

Equilibrium Stability Through Potential Energy

Stability of equilibrium in physics and engineering can be studied by using a system’s potential energy. An equilibrium configuration is a state where the system can remain at rest. Stability describes whether the system tends to return to that state after a small disturbance. Problem solving begins by finding the equilibrium points of the system. This is done by setting the derivative of the potential energy function equal to zero and solving for the independent variable values that satisfy...

key terms and concepts

Study one-degree-of-freedom systems through a spring-mass example, potential energy graphs, and stability of stable, unstable, and neutral equilibrium.

Spring-Mass Motion and Energy Stability | 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

Chemistry and Physics

IPC.1

Observational Research

Source: Laboratories of Gary Lewandowski, Dave Strohmetz, and Natalie Ciarocco&#8212;Monmouth University

If you want to know how someone thinks or feels, ...

Solving Problems in Physics

Problem-solving is the ability to apply general physical principles to specific situations, usually expressed by equations. It is an essential skill in ...

how-children-solve-problems-using-causal-reasoning

How Children Solve Problems Using Causal Reasoning

Source: Laboratories of Judith Danovitch and Nicholaus Noles&#8212;University of Louisville
Imagine a young child hears an adult say, &#8220;I want to ...

Naturalistic Observations

If you want to understand how behavior occurs, one of the best ways to gain information is to simply observe the behavior in its natural context. However, ...

Models, Theories, and Laws

Scientists frequently use models to help them comprehend a specific collection of phenomena. In physics, a model is a condensed version of a physical ...

/files/media/science-education/science-education-thumbs/14686.jpg

The Scientific Method

The scientific method is a detailed, empirical problem-solving process used by biologists and other scientists. This iterative approach involves ...

Observational Learning

Observational learning, also known as imitation or modeling, occurs when an individual observes and replicates another&#39;s behavior.
The Bobo doll ...

Perspectives on Cognitive Psychology

Source: Laboratory of Jonathan Flombaum&#8212;Johns Hopkins University

As an Assistant Professor of Psychology and Cognitive Science at Johns Hopkins ...

Case Studies

There are many research methods available to psychologists in their efforts to understand, describe, and explain behavior and the cognitive and biological ...

Ask questions and define problems based on observations or information from text, phenomena, models, or investigations

PHYS.1.A

Study Design in Statistics

A study design is a set of techniques that allow a researcher to collect and analyze data from different variables defined for a specific research ...

Experimental Designs

An experimental design is a systematic process that allows researchers to evaluate the relationship between dependent and independent variables. There are ...

Realism in Experimentation

Source: Laboratories of Gary Lewandowski, Dave Strohmetz, and Natalie Ciarocco&#8212;Monmouth University

In an ideal world researchers would conduct ...

Data Collection by Experiments

Data collection is a systematic method of obtaining, observing, measuring, and analyzing accurate information. An experimental study is a standard method ...

Scientific Method

Observation and Background Research
ExpandNOTE: The first two steps in the scientific process involve studying and observing natural phenomena of ...

The Scientific Method in Nursing Process

The scientific method provides the foundation for any research. It is the most reliable and objective of all forms of gaining knowledge and guides in ...

Observational Studies

Observational studies are a type of analytical study where researchers observe events without any interventions. In other words, the researcher does not ...

Study Designs in Epidemiology

Consider an example of testing calcium&#39;s effect on a woman&#39;s bone weight.
In an ideal study design, the same woman would be observed in two ...

What is an Experiment?

An experiment is a planned activity carried out under controlled conditions. The purpose of an experiment is to investigate the relationship between two ...

Categories and Inductive Inferences

Source: Laboratories of Nicholaus Noles and Judith Danovitch&#8212;University of Louisville
It might be possible for the human brain to keep track of each ...

from-theory-to-design-the-role-of-creativity-in-designing-experiments

From Theory to Design: The Role of Creativity in Designing Experiments

Source: Laboratories of Gary Lewandowski, Dave Strohmetz, and Natalie Ciarocco&#8212;Monmouth University

Research studies come into being when a ...

Apply scientific practices to plan and conduct descriptive, comparative, and experimental investigations, and use engineering practices to design solutions to problems

PHYS.1.B

guidelines-in-case-of-a-laboratory-emergency

Guidelines in Case of a Laboratory Emergency

Robert M. Rioux &amp; Zhifeng Chen, Pennsylvania State University, University Park, PA
The most common laboratory emergencies include chemical spills, ...

electrical-safety-precautions-and-basic-equipment

Electrical Safety Precautions and Basic Equipment

Source: Ali Bazzi, Department of Electrical Engineering, University of Connecticut, Storrs, CT.
Electric machines and power electronics experiments ...

Electrical Safety

Robert M. Rioux &amp; Suprita Jharimune, Pennsylvania State University, University Park, PA
Among the many hazards present in the laboratory, electrical ...

Emergency Eyewash and Shower Stations

Robert M. Rioux, Pennsylvania State University, University Park, PA
The Occupational Safety and Health Administration (OSHA) mandates that an emergency ...

Use appropriate safety equipment and practices during laboratory, classroom, and field investigations as outlined in Texas Education Agency-approved safety standards

PHYS.1.C

Common Lab Glassware and Uses

Source: Laboratory of Dr. Neal Abrams &#8212; SUNY College of Environmental Science and Forestry
Glassware is a regular appearance in the professional ...

Lab Techniques

Source: Smaa Koraym at Johns Hopkins University, MD, USA

	Determining the Density of an EggExpand
	Density is a measure of how compact a substance or ...

Scientific Measurement and Lab Skills

Source: Smaa Koraym at Johns Hopkins University, MD, USA

	Measuring Diameter and CircumferenceExpand
	Error can be introduced in any experiment, even ...

Cleaning Glassware

Source: Vy M. Dong and Daniel Kim, Department of Chemistry, University of California, Irvine, CA
Organic synthesis is about transforming a readily ...

Proper Lab Notebook Keeping

Source: Lara Al Hariri and Ahmed Basabrain at the University of Massachusetts Amherst, MA, USA

	Nitration of Methyl BenzoateExpand
	From this lab onward, ...

sample-preparation-for-analytical-characterization

Sample Preparation for Analytical Characterization

Source: Laboratory of Dr. B. Jill Venton - University of Virginia
Sample preparation is the way in which&#160;a sample is treated to prepare&#160;for ...

Making Solutions in the Laboratory

The ability to successfully make solutions is a basic laboratory skill performed in virtually all biological and chemical experiments.  A solution is a ...

Use appropriate tools such as balances, ballistic carts or equivalent, batteries, computers, constant velocity cars, convex lenses, copper wire, discharge tubes with power supply (H, He, Ne, Ar), data acquisition probes and software, dynamics and force demonstration equipment, electrostatic generators, electrostatic kits, friction blocks, graph paper, graphing technology, hand-held visual spectroscopes, inclined planes, iron filings, lab masses, laser pointers, magnets, magnetic compasses, metric rulers, motion detectors, multimeters (current, voltage, resistance), optics bench, optics kit, photogates, plane mirrors, prisms, protractors, pulleys, resistors, rope or string, scientific calculators, stopwatches, springs, spring scales, switches, tuning forks, wave generators, or other equipment and materials that will produce the same results.

PHYS.1.D

Units and Standards of Measurement

A physical quantity is defined either by specifying its measurement method or by stating how it is calculated from other measurements. For example, ...

International System of Units

The International System of Units, known as the SI system, is a universally accepted measurement system recognized and used worldwide. The SI system is ...

Base Quantities and Derived Quantities

In any system of units, the units for some physical quantities must be specified through a measurement process. These measurements are the base quantities ...

How Data are Classified: Numerical Data

Data that are countable or measurable in specific units are called numerical or quantitative data. Quantitative data are always numbers. Quantitative data ...

Quantitative Analysis

Quantitative analysis is a technique for measuring the amount of specific constituents in a sample. When the sample&#39;s composition is unknown, ...

Qualitative Analysis

Qualitative analysis is the process of identifying elements, ions, or compounds in an unknown sample. It is the first and most fundamental type of ...

Collect quantitative data using the International System of Units (SI) and qualitative data as evidence

PHYS.1.E

Review and Preview

Data are individual items of information obtained from a population or sample. Data may be classified as qualitative (categorical), quantitative ...

Ogive Graph

An ogive graph is sometimes called a cumulative frequency polygon. It is one type of frequency polygon that shows cumulative frequency. In other words, ...

Histogram

The histogram is a graphical representation in the x-y form of data distribution in a data set. The horizontal x-axis is labeled with what the data ...

Relative Frequency Histogram

The relative frequency depicts the proportion of data points that have each value. The frequency tells the number of data points that have each value. ...

Scatter Plot

The most common and easiest way to display the relationship between two variables, x and y, is a scatter plot. A scatter plot shows the direction of a ...

Time-Series Graph

A time-series graph is a line graph with repeated measurements taken at successive intervals of time. It is also called a time series chart. To construct ...

Bar Graph

A bar graph is also called a bar chart and consists of bars that are separated from each other. It either uses horizontal or vertical bars to show ...

Multiple Bar Graph

As the name suggests, a multiple bar graph is the same as a bar graph but has multiple bars to depict relationships between different data values. One can ...

Pareto Chart

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Pie Chart

A pie chart (or a pie graph) is a circular graphical chart or a pictorial representation of categorical data. It is divided into slices of pie each ...

Life Tables

A life table or a mortality table represents the survival patterns of a population.
It comprises multiple columns, each serving a practical purpose, ...

Contingency Table

A contingency table provides a way of portraying data that can facilitate calculating probabilities. It is a method of displaying a frequency distribution ...

Organize quantitative and qualitative data using bar charts, line graphs, scatter plots, data tables, labeled diagrams, and conceptual mathematical relationships

PHYS.1.F

Basic Concept

Engineering mechanics is a branch of engineering that studies motion and the forces acting on objects. It is a fundamental subject and forms the basis of ...

Problem Solving in Statics

Problem-solving in statics is a crucial aspect of engineering and physics that involves resolving issues associated with bodies in a state of equilibrium. ...

Free-body Diagrams: Problem Solving

Free-body diagrams are essential tools for physicists and engineers studying the motion of objects. Free-body diagrams are graphical representations of ...

Equilibrium and Free-body Diagrams

Source: Ketron Mitchell-Wynne, PhD, Asantha Cooray, PhD, Department of Physics &#38; Astronomy, School of Physical Sciences, University of California, ...

Mechanical Systems

In mechanical systems, springs, and masses are akin to the roles of inductors and capacitors in electrical networks, with the energy-dissipating function ...

In mechanical engineering, one-degree-of-freedom systems form the basis of a wide range of electrical and mechanical components. Using these models, ...

Dynamics of Structures

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Conservation of Energy Approach to System Analysis

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Develop and use models to represent phenomena, systems, processes, or solutions to engineering problems

Scientific and engineering practices. The student, for at least 40% of instructional time, asks questions, identifies problems, and plans and safely conducts classroom, laboratory, and field investigations to answer questions, explain phenomena, or design solutions using appropriate tools and models. The student is expected to:

Spring-Mass Motion and Energy Stability

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