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

Se la posizione di un sistema è definita da una variabile indipendente, ha un grado di libertà. La sua funzione di energia potenziale può quindi essere tracciata rispetto a quella variabile.
Quando il sistema raggiunge l'equilibrio, la pendenza della funzione in quella posizione è zero.
La stabilità della configurazione di equilibrio può essere studiata utilizzando la derivata seconda della funzione di energia potenziale rispetto alla variabile.
Se la derivata seconda è positiva, l'energia potenziale è minima e il sistema è in una configurazione di equilibrio stabile.
Al contrario, se la derivata seconda è negativa, l'energia potenziale è massima. Ciò implica una configurazione di equilibrio instabile.
Se la derivata seconda è zero, devono essere valutate le derivate di ordine superiore.
Se la derivata prima diversa da zero è positiva e ha un ordine pari, la configurazione di equilibrio è stabile.
In alternativa, se la prima derivata diversa da zero è negativa e ha ordine pari o dispari, è instabile.
Se tutte le derivate di ordine superiore sono zero, il sistema ha raggiunto l'equilibrio neutro. Qui, la funzione potenziale-energia rimane costante attorno alla posizione di equilibrio.

<ol>
	<li>Hibbeler, R.C. (2016). Ingegneria Meccanica: Statica. Quattordicesima edizione, New Jersey: Pearson. Sezione 11.7, pp. 601-602.</li>
	<li>Birra, F.P., Johnston, E.R., Mazurek, D.F., Cornwell, P.J. e Self, B.P. (2016). Meccanica vettoriale per ingegneri. Undicesima edizione, New York: McGraw-Hill Education. Pagine 599-600.</li>
</ol>

sistema a un grado di libertà

Nell'ingegneria meccanica, i sistemi a un grado di libertà formano la base di una vasta gamma di componenti elettrici e meccanici. Utilizzando questi modelli, gli ingegneri possono prevedere il comportamento di diverse parti all'interno di un sistema più ampio, ottenendo così una comprensione di come le diverse forze interagiscono tra loro.

Un sistema a un grado di libertà è definito da una variabile indipendente che determina il suo stato e comportamento. Un esempio di sistema a un grado di libertà è un oscillatore armonico semplice, come una massa attaccata a una molla. La posizione della massa lungo un asse determina il suo stato e comportamento, e la forza esercitata dalla molla la fa oscillare avanti e indietro. L'energia potenziale (V) in tali sistemi, può essere tracciata su un grafico per mostrare quanta energia l'oggetto possiede, in base alla sua posizione rispetto al suo punto di riposo.

Comprendendo questi sistemi a un grado di libertà, gli ingegneri possono acquisire una conoscenza preziosa durante la progettazione di design complessi. Ciò consente loro di modellare e prevedere con precisione il funzionamento dei design in diverse condizioni prima di costruire i design nella realtà. Inoltre, possono anche ottenere una comprensione delle equazioni di base che collegano le forze e le accelerazioni, aiutando a comprendere proprietà fisiche come l'inerzia.

Per le indagini sulla stabilità dell'equilibrio di diverse configurazioni, gli ingegneri analizzano la seconda derivata della funzione potenziale, per determinare quali configurazioni di equilibrio sono stabili, instabili o neutrali. Gli equilibri stabili corrispondono ai valori minimi sul grafico dell'energia potenziale, mentre i valori massimi indicano l'instabilità e i valori costanti denotano la neutralità in quei punti. Ciò aiuta i progettisti ad avere una migliore comprensione di quali design funzionano meglio per determinati ambienti, in cui fattori esterni possono causare cambiamenti dinamici nel tempo che potrebbero interrompere il funzionamento di una macchina se non previsti in anticipo.

La conoscenza complessiva dei sistemi a un grado di libertà rappresenta uno strumento prezioso per gli ingegneri che cercano di costruire macchine o veicoli robusti, dagli orologi da sveglia ai moderni razzi diretti verso lo spazio esterno.

Sistema a un grado di libertà

Nell'ingegneria meccanica, i sistemi con un grado di libertà costituiscono la base di un'ampia gamma di componenti elettrici e meccanici. Utilizzando questi modelli, ...

Education

Didattica

JoVE Core

Mechanical Engineering

Ingegneria Meccanica

Unit 1

Unità 1

Virtual Work

Lavoro virtuale

KEY TERMS AND CONCEPTS

TERMINI E CONCETTI CHIAVE

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

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

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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

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PHYS.1.C

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PHYS.1.D

Units and Standards of Measurement

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International System of Units

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How Data are Classified: Numerical Data

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Quantitative Analysis

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Qualitative Analysis

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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

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Relative Frequency Histogram

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Scatter Plot

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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

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Multiple Bar Graph

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Pareto Chart

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Pie Chart

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Life Tables

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PHYS.1.F

Basic Concept

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Spring-Mass Motion and Energy Stability

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