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

Wenn die Position eines Systems durch eine unabhängige Variable definiert ist, hat es einen Freiheitsgrad. Seine potentielle Energiefunktion kann dann gegen diese Variable aufgetragen werden.
Wenn das System das Gleichgewicht erreicht, ist die Steigung der Funktion an dieser Position Null.
Die Stabilität der Gleichgewichtskonfiguration kann mit Hilfe der zweiten Ableitung der potentiellen Energiefunktion in Bezug auf die Variable untersucht werden.
Wenn die zweite Ableitung positiv ist, ist die potentielle Energie minimal und das System befindet sich in einer stabilen Gleichgewichtskonfiguration.
Umgekehrt, wenn die zweite Ableitung negativ ist, ist die potentielle Energie maximal. Dies impliziert eine instabile Gleichgewichtskonfiguration.
Wenn die zweite Ableitung Null ist, müssen Derivate höherer Ordnung bewertet werden.
Wenn die erste Ableitung ungleich Null positiv ist und eine gerade Ordnung hat, ist die Gleichgewichtskonfiguration stabil.
Wenn die erste Ableitung ungleich Null negativ ist und eine ungerade oder gerade Ordnung hat, ist sie instabil.
Wenn alle Ableitungen höherer Ordnung Null sind, hat das System ein neutrales Gleichgewicht erreicht. Dabei bleibt die potentielle Energiefunktion um die Gleichgewichtsposition konstant.

<ol>
	<li>Hibbeler, R.C. (2016). Technische Mechanik: Statik. Vierzehnte Auflage, New Jersey: Pearson. Abschnitt 11.7, S. 601-602.</li>
	<li>Bier, F.P., Johnston, E.R., Mazurek, D.F., Cornwell, P.J. und Self, B.P. (2016). Vektormechanik für Ingenieure. Elfte Auflage, New York: McGraw-Hill Education. S. 599-600.</li>
</ol>

ein-freiheitsgrad-system

In der Maschinenbautechnik bilden Ein-Freiheitsgrad-Systeme die Grundlage für eine Vielzahl von elektrischen und mechanischen Komponenten. Mit Hilfe dieser Modelle können Ingenieure das Verhalten verschiedener Teile in einem größeren System vorhersagen, was ihnen Einblick gibt, wie sich verschiedene Kräfte gegenseitig beeinflussen.

Ein Ein-Freiheitsgrad-System wird durch eine unabhängige Variable definiert, die seinen Zustand und sein Verhalten bestimmt. Ein Beispiel für ein Ein-Freiheitsgrad-System ist ein einfacher harmonischer Oszillator, wie beispielsweise eine Masse, die an einer Feder befestigt ist. Die Position der Masse entlang einer Achse bestimmt ihren Zustand und ihr Verhalten, und die von der Feder ausgeübte Kraft verursacht eine hin- und herpendelnde Bewegung. Die potentielle Energie (V) solcher Systeme kann auf einem Graphen dargestellt werden, um zu zeigen, wie viel Energie der Gegenstand je nach seiner Position relativ zu seinem Ausgangspunkt besitzt.

Durch das Verständnis dieser Ein-Freiheitsgrad-Systeme können Ingenieure wertvolles Wissen gewinnen bei der Konstruktion komplexer Designs. Dies ermöglicht ihnen, ihre Funktionsweise unter unterschiedlichen Bedingungen präzise zu modellieren und vor dem Bau der Designs in der Realität vorherzusagen. Zudem können sie Einblicke in die grundlegenden Gleichungen gewinnen, die Kräfte und Beschleunigungen verbinden, was dazu beiträgt, physikalische Eigenschaften wie Trägheit zu verstehen.

Bei Untersuchungen zur Stabilität des Gleichgewichts verschiedener Konfigurationen analysieren Ingenieure die zweite Ableitung der potentiellen Funktion, um festzustellen, welche Gleichgewichtskonfigurationen stabil, instabil oder neutral sind. Stabile Gleichgewichte entsprechen den Mindestwerten auf dem Energiepotenzial-Diagramm, während maximale Werte auf Instabilität hinweisen und konstante Werte Neutralität an diesen Punkten anzeigen. Dies hilft den Designern ein besseres Verständnis dafür zu geben, welche Designs für gegebene Umgebungen am besten geeignet sind, wo äußere Faktoren im Laufe der Zeit dynamische Veränderungen verursachen könnten, die den Betrieb einer Maschine' stören könnten, wenn sie nicht gut im Voraus vorhergesagt werden.

Das Gesamtwissen über Ein-Freiheitsgrad-Systeme ist ein unschätzbares Werkzeug für Ingenieure, die robuste Maschinen oder Fahrzeuge konstruieren möchten, angefangen bei kleinen Weckern bis hin zu modernen Raketen, die ins Weltall fliegen.

Ein-Freiheitsgrad-System

Im Maschinenbau bilden Systeme mit einem Freiheitsgrad die Basis für eine Vielzahl von elektrischen und mechanischen Komponenten. Mit diesen Modellen ...

Education

Lehre

JoVE Core

Mechanical Engineering

Maschinenbau

Unit 1

Einheit 1

Virtual Work

Virtuelle Arbeit

KEY TERMS AND CONCEPTS

KERNBEGRIFFE UND KONZEPTE

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 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

Chemistry and Physics

IPC.1

Observational Research

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PHYS.1.A

Study Design in Statistics

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Experimental Designs

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Scientific Method

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Categories and Inductive Inferences

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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

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Electrical Safety

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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

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Cleaning Glassware

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Proper Lab Notebook Keeping

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

	Nitration of Methyl BenzoateExpand
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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

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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

A Pareto chart is a bar graph or a combination of both line and bar graphs. The bar lengths represent the individual values or the frequency, while the ...

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.
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Contingency Table

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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

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Problem Solving in Statics

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Free-body Diagrams: Problem Solving

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Equilibrium and Free-body Diagrams

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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 ...

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Dynamics of Structures

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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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