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

Dans les systèmes mécaniques, les ressorts et les masses s’apparentent aux rôles des inductances et des condensateurs dans les réseaux électriques, la fonction de dissipation d’énergie d’un amortisseur visqueux correspondant à celle de la résistance électrique.
Les forces agissant sur la masse comprennent une force appliquée se déplaçant dans la même direction et les forces du ressort, de l’amortisseur visqueux et de l’accélération agissant contre lui.
Comme le réseau RLC, les systèmes mécaniques translationnels sont définis par une équation différentielle unique formulée en appliquant la loi de Newton qui stipule que la somme de toutes les forces agissant sur la masse doit être égale à zéro.
De plus, la transformée de Laplace est utilisée sur l’équation dans des conditions initiales nulles. Cette expression, lorsqu’elle est simplifiée, donne la fonction de transfert.
Les composants des systèmes mécaniques rotationnels reflètent ceux des systèmes translationnels, mais subissent une rotation.
Ceux-ci sont gérés de la même manière que les translationnels, le couple remplaçant la force, le déplacement angulaire remplaçant le déplacement de translation et l’inertie remplaçant la masse.
Dans l’équation différentielle du second ordre pour le système de rotation, la transformée de Laplace est appliquée, puis simplifiée pour produire la fonction de transfert.

systèmes mécaniques

Les systèmes mécaniques sont analogues aux réseaux électriques, où les ressorts et les masses jouent des rôles similaires à ceux des inducteurs et des condensateurs, respectivement. Un amortisseur visqueux dans les systèmes mécaniques fonctionne de la même manière qu'une résistance dans les réseaux électriques, en dissipant l'énergie. Les forces agissant sur une masse dans de tels systèmes comprennent une force appliquée dans la direction du mouvement, contrecarrée par les forces du ressort, un amortisseur visqueux et l'accélération de la masse. Cette interaction de forces est décrite mathématiquement à l'aide de la deuxième loi de Newton, qui stipule que la somme de toutes les forces agissant sur la masse doit être nulle.

Dans les systèmes mécaniques translationnels, le comportement est capturé par une équation différentielle unique dérivée de la loi de Newton. Cette équation tient compte de toutes les forces agissant sur la masse. Pour résoudre le système de manière analytique, la transformée de Laplace est appliquée à cette équation différentielle dans des conditions initiales nulles. La transformée de Laplace, un outil mathématique puissant, convertit l'équation différentielle du domaine temporel en une équation algébrique dans le domaine de Laplace. La simplification de cette équation donne la fonction de transfert du système, un concept crucial qui relie la réponse de sortie à la force d'entrée dans le domaine fréquentiel. La fonction de transfert est essentielle pour analyser la stabilité et la dynamique du système.

Les systèmes mécaniques de rotation sont parallèles aux systèmes de translation mais impliquent un mouvement de rotation. Dans ces systèmes, le couple remplace la force, le déplacement angulaire remplace le déplacement de translation et l'inertie de rotation remplace la masse. L'équation différentielle analogue pour un système de rotation, dérivée de manière similaire en utilisant la deuxième loi de Newton pour la rotation, décrit la dynamique du mouvement de rotation. En appliquant la transformée de Laplace à cette équation différentielle du second ordre et en la simplifiant, on obtient la fonction de transfert du système de rotation. Cette fonction permet de comprendre le comportement du système de rotation, de la même manière que la fonction de transfert des systèmes de translation permet de comprendre la dynamique des mouvements linéaires.

Systèmes mécaniques

Dans les systèmes mécaniques, les ressorts et les masses s’apparentent aux rôles des inductances et des condensateurs dans les réseaux électriques, avec la fonction de dissipation d’énergie ...

Education

Enseignement

JoVE Core

Electrical Engineering

Génie électrique

Unit 1

Unité 1

Modeling in Time and Frequency Domain

Modélisation dans les domaines temporel et fréquentiel

KEY TERMS AND CONCEPTS

TERMES ET CONCEPTS CLÉS

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

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PHYS.1.B

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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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PHYS.1.E

Review and Preview

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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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