Name: transfer functions in mechanical motion
Uploaded: 2024-09-26T13:17:18.000Z
Duration: 1 min 22 s
Channel: JoVE

K12 Standards

Texas Essential Knowledge and Skills

High School

Physics

PHYS.1

IPC.1.G

Mechanical Systems

في الأنظمة الميكانيكية ، تشبه الينابيع والكتل أدوار المحاثات والمكثفات في الشبكات الكهربائية ، مع وظيفة تبديد الطاقة للمخمد اللزج المقابل لوظيفة المقاومة الكهربائية.
تشمل القوى المؤثرة على الكتلة قوة مطبقة تتحرك في نفس الاتجاه والقوى المنبعثة من الزنبرك والمثبط اللزج والتسارع الذي يعمل ضده.
مثل شبكة RLC ، يتم تعريف الأنظمة الميكانيكية الانتقالية من خلال معادلة تفاضلية فريدة تمت صياغتها من خلال تطبيق قانون نيوتن الذي ينص على أن مجموع جميع القوى المؤثرة على الكتلة يجب أن يساوي صفرا.
علاوة على ذلك ، يتم استخدام تحويل لابلاس في المعادلة في ظل ظروف أولية صفرية. ينتج عن هذا التعبير ، عند تبسيطه ، دالة النقل.
تعكس المكونات في الأنظمة الميكانيكية الدورانية مكونات الأنظمة الانتقالية ولكنها تختبر الدوران.
تتم إدارتها بشكل مشابه للتلك الانتقالية ، مع قوة استبدال عزم الدوران ، والإزاحة الزاوية التي تحل محل الإزاحة الانتقالية ، والقصور الذاتي الذي يحل محل الكتلة.
في المعادلة التفاضلية من الدرجة الثانية لنظام الدوران ، يتم تطبيق تحويل لابلاس ثم تبسيطه بشكل أكبر لإنتاج دالة النقل.

الأنظمة الميكانيكية

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الأنظمة الميكانيكية تشبه الشبكات الكهربائية حيث تلعب الزنبركات والكتل أدوارًا مماثلة للمحثات والمكثفات على التوالي. يعمل المثبط اللزج في الأنظمة الميكانيكية بشكل مشابه للمقاوم في الشبكات الكهربائية، حيث يبدد الطاقة. تشمل القوى المؤثرة على الكتلة في مثل هذه الأنظمة قوة مطبقة في اتجاه الحركة، تقابلها قوى من الزنبرك، ومثبط لزج، وتسارع الكتلة. يُوصَف هذا التفاعل بين القوى رياضيًا باستخدام قانون نيوتن الثاني، الذي ينص على أن مجموع جميع القوى المؤثِّرة على الكتلة يجب أن يكون صفرًا.

في الأنظمة الميكانيكية الانتقالية، يتم التقاط السلوك بواسطة معادلة تفاضلية فريدة مستمدة من قانون نيوتن. تأخذ هذه المعادلة في الاعتبار جميع القوى المؤثرة على الكتلة. لحل النظام تحليليًا، يتم تطبيق تحويل لابلاس على هذه المعادلة التفاضلية في ظل ظروف أولية صفرية. يحول تحويل لابلاس، وهو أداة رياضية قوية، معادلة التفاضل في مجال الزمن إلى معادلة جبرية في مجال لابلاس. يؤدي تبسيط هذه المعادلة إلى الحصول على دالة نقل النظام، وهو مفهوم بالغ الأهمية يربط بين استجابة الخرج والقوة المدخلة في مجال التردد. تعد دالة التحويل ضرورية لتحليل استقرار النظام وديناميكياته.
الأنظمة الميكانيكية الدورانية توازي الأنظمة الانتقالية ولكنها تنطوي على حركة دورانية. في هذه الأنظمة، يحل عزم الدوران محل القوة، ويحل الإزاحة الزاوية محل الإزاحة الانتقالية، ويحل القصور الذاتي الدوراني محل الكتلة. تصف معادلة التفاضل التناظرية لنظام دوراني، والتي تم اشتقاقها بشكل مماثل باستخدام قانون نيوتن الثاني للدوران، ديناميكيات الحركة الدورانية. من خلال تطبيق تحويل لابلاس على معادلة التفاضل من الدرجة الثانية هذه، وتبسيطها، يتم الحصول على دالة التحويل للنظام الدوراني. توفر هذه الدالة رؤى حول سلوك النظام الدوراني، على غرار الطريقة التي تساعد بها دالة التحويل في الأنظمة الانتقالية في فهم ديناميكيات الحركة الخطية.

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في الأنظمة الميكانيكية ، تشبه الينابيع والكتل أدوار المحاثات والمكثفات في الشبكات الكهربائية ، مع وظيفة تبديد الطاقة ...

Education

JoVE Core

Electrical Engineering

الهندسة الكهربائية

Unit 1

الوحدة الأولى

Modeling in Time and Frequency Domain

النمذجة في النطاقين الزمني والترددي

KEY TERMS AND CONCEPTS

المصطلحات والمفاهيم الأساسية

transfer functions in mechanical motion

Mechanical motion can be modeled with springs, masses, and viscous dampers. In these models, springs behave like inductors and masses act like capacitors in an electrical network. A viscous damper works like a resistor because it removes energy from the system.

A force acts on the mass in the direction of motion. It is balanced by the spring force, the damping force, and the force from the mass’s acceleration. Newton’s second law is used to describe this balance, and the sum of the forces on the mass must be zero.

For translational mechanical systems, this force balance leads to a second-order differential equation. The equation captures the motion of the mass under the applied force. To solve it analytically, the Laplace transform is applied under zero initial conditions. This changes the time-domain equation into an algebraic equation in the Laplace domain.

After simplification, the transfer function is obtained. A transfer function relates the output response to the input force in the frequency domain. It is a key tool for studying stability and system dynamics.

Rotational mechanical systems follow the same overall idea, but the motion is rotational. In this case, torque replaces force, angular displacement replaces translational displacement, and rotational inertia replaces mass. A similar second-order equation comes from Newton’s second law for rotation.

Applying the Laplace transform to the rotational equation gives the rotational transfer function. This function helps describe how the rotational system behaves. It serves the same purpose for rotational motion that the translational transfer function serves for linear motion.

Transfer Functions in Mechanical Motion

transfer function in control systems

Transfer Function from Input to Output

The transfer function describes how a linear time-invariant, or LTI, system maps an input to an output in the frequency domain. It gives a compact way to study system behavior without working only in the time domain. It also connects the differential equations of the system to a form that is easier to handle mathematically. For an nth-order LTI system, the behavior is written as a differential equation with constant coefficients. In that equation, r(t) is the input and c(t) is the output. The...

electrical systems

RLC Circuit Transfer Functions

RLC circuits use resistors, capacitors, and inductors to show how electrical signals change over time. In circuit analysis, these passive linear components are arranged so the link between an input and an output can be described with a transfer function. For this topic, the output is the voltage across the capacitor, compared with the input voltage. A series RLC circuit can be analyzed with Kirchhoff's Voltage Law, or KVL. KVL says that the sum of all voltages around a closed loop is zero.

mechanical systems

Mechanical motion can be modeled with springs, masses, and viscous dampers. In these models, springs behave like inductors and masses act like capacitors in an electrical network. A viscous damper works like a resistor because it removes energy from the system. A force acts on the mass in the direction of motion. It is balanced by the spring force, the damping force, and the force from the mass’s acceleration. Newton’s second law is used to describe this balance, and the sum of the forces on...

electro-mechanical systems

DC Motor Modeling in Electromechanical Systems

DC motor modeling is a key part of electromechanical systems. These systems combine electrical parts and mechanical parts to produce a useful output. A DC motor is one common example because it turns electrical energy into mechanical motion. It is used in many places, from simple fans to complex robotic mechanisms. Inside the DC motor, the armature is a rotating circuit placed in a magnetic field. When current flows through the armature, it experiences a force from the magnetic field. That...

linear approximation in frequency domain

Linearizing Nonlinear RL Circuits

Linearizing nonlinear RL circuits helps turn a hard circuit problem into a simpler linear one. Linear systems follow superposition, which means the total response equals the sum of the responses to separate inputs. They also follow homogeneity, which means scaling the input scales the output by the same amount. Nonlinear systems do not automatically have those properties. For small changes around an operating point, though, a nonlinear system can often be treated as linear. This is done with a...

state space representation

RLC Circuit State Equations

RLC circuit state equations show how a circuit changes over time using state variables. This time-domain method is useful for linear, time-invariant systems, but it also helps when frequency-domain analysis is not enough. It is especially useful for nonlinear systems, time-varying systems, and multiple-input multiple-output systems. The state-space approach turns an nth-order system into a set of simultaneous first-order differential equations called state equations. For a common second-order...

transfer function to state space

Phase-Variable State Equations from Transfer Functions

Phase-variable state equations show how a transfer function can be rewritten for computer simulation. In state-space form, the system is described with state variables, which are the output and its first several derivatives. This makes it easier to model a physical system on a digital computer. The starting point is an nth-order linear differential equation with constant coefficients. Systems like an RLC circuit can be written this way. To build the state equations, the transfer function is...

state space to transfer function

Finding a Transfer Function from State Space

Finding a transfer function from state-space form is a key step in system analysis. It changes a time-domain model into a frequency-domain form. That makes it easier to study and design control systems. The process starts with the state-space equations. These include the state equation and the output equation. In these equations, x(t) is the state vector, u(t) is the input vector, y(t) is the output vector, and A, B, C, and D are the matrices that define the system dynamics. Next, the Laplace...

linear approximation in time domain

Linearizing Nonlinear State-Space Models

Linearizing nonlinear state-space models helps describe systems that change with time or operating conditions. State-space representation is often effective for this kind of modeling. It is especially useful for systems such as a simple pendulum or a translational mechanical system with nonlinear springs. A simple pendulum can be modeled by summing torques around the pivot point. In this case, the mass is evenly distributed along the pendulum, and the center of mass is located at half its...

key terms and concepts

Learn how springs, masses, and dampers are modeled with Newton’s law and the Laplace transform to find transfer functions in translational and rotational motion.

Transfer Functions in Mechanical Motion | JoVE K12

JoVE Core Electrical Engineering offers easy-to-understand animated videos that cover fundamental concepts of electric circuits along with circuit analysis techniques. It incorporates first-order and second-order circuits using operational amplifiers and circuit transformation concepts. The inclusion of practical design examples and real-world applications through scientist-in-action videos adds significant educational value, making complex concepts accessible.

electrical 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

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How Children Solve Problems Using Causal Reasoning

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Naturalistic Observations

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Models, Theories, and Laws

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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.
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Perspectives on Cognitive Psychology

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

Case Studies

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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

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.
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, ...

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

One-Degree-of-Freedom System

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

Source:&#160;Roberto Leon, Department of Civil and Environmental Engineering, Virginia Tech, Blacksburg, VA
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Conservation of Energy Approach to System Analysis

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Stability of Equilibrium Configuration: Problem Solving

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

What is a Hypothesis?

A hypothesis can be a simple sentence or statement about a property or any phenomenon observed or predicted for a population. It is usually a claim about ...

Null and Alternative Hypotheses

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Types of Hypothesis Testing

There are three types of hypothesis tests: right-tailed, left-tailed, and two-tailed.
When the null and alternative hypotheses are stated, it is observed ...

Hypothesis: Accept or Fail to Reject?

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Distinguish among scientific hypotheses, theories, and laws.

PHYS.1.H

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:

Transfer Functions in Mechanical Motion

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