Name: spring-mass-damper laplace block diagram
Uploaded: 2024-11-21T14:54:07.000Z
Duration: 1 min 20 s
Channel: JoVE

スプリング・マス・ダンパー・システムの 2 次微分方程式について考えてみます。システムは、初期条件がゼロでラプラス領域に変換されます。
次に、方程式を再配置して出力を分離し、特定の伝達関数を持つブロックに入る信号として解釈できます。
出力は、2 回積分するか、それに応じて後乗算することによって得られます。
簡単に言うと、右側の信号が接続され、システムの最終的なブロック線図表現につながります。
内部フィードバックループから項を因数分解することで、さらに単純化することができ、その結果、代替のブロック線図が得られます。
ブロック線図モデルには、加速度と速度を表す内部変数を組み込むこともできます。
1/sはラプラス領域での積分に対応するため、最初に加速度を積分して速度を求め、その後、速度を積分して変位信号を生成します。
システムの伝達関数は、入力信号とフィードバック信号のブロックをコンパレータの右側に移動し、内部フィードバックループを単純化することで求められます。結果として得られる方程式は、システムの伝達関数です。

数式とブロック図の関係

Relation between Mathematical Equations and Block Diagrams

バネ・質量・ダンパー系では、二次微分方程式がシステムの動的動作を記述します。この方程式を初期条件がゼロのラプラス領域に変換すると、効果的に分析および操作できます。ラプラス領域への変換により、微分方程式が代数方程式に変換され、出力を分離するプロセスが簡略化されます。

<img alt="Equation1" src="/files/ftp_upload/16105/16105_PT_Equation_001.svg" style="height: 36px;">

バネ・質量・ダンパー系の標準微分方程式にラプラス変換を適用すると、次の出力が得られます。

<img alt="Equation2" src="/files/ftp_upload/16105/16105_PT_Equation_002.svg" style="height: 24px;">

ブロック図を作成する際、右側の信号を接続して表現を簡略化できます。ブロック図をさらに改良して、加速度や速度などの内部変数を組み込むことができます。1/s はラプラス領域での積分に対応するため、加速度信号を積分して速度を取得し、速度を積分して変位信号を生成します。

ブロック図の簡略化には、内部フィードバックループからの項の因数分解が含まれます。このプロセスにより、加速度、速度、変位の関係を視覚的に表す代替ブロック図が作成されます。

システムの伝達関数を導出するには、入力信号とフィードバック信号を表すブロックをコンパレータの右側に移動し、内部フィードバックループを簡略化します。結果として得られる式は、伝達関数を示します。

<img alt="Equation3" src="/files/ftp_upload/16105/16105_PT_Equation_003.svg" style="height: 38px;">

<img alt="Equation4" src="/files/ftp_upload/16105/16105_PT_Equation_004.svg" style="height: 120px;">

この伝達関数は、システムの動作を分析し、さまざまな入力に対する応答を予測し、望ましいパフォーマンス特性を実現するための制御戦略を設計するために不可欠です。

スプリング・マス・ダンパー・システムの 2 次微分方程式について考えてみます。システムは、ゼロの初期値の下でラプラスドメインに変換されます...

relation-between-mathematical-equations-and-block-diagrams

Education

JoVE Core

Electrical Engineering

電気工学

Unit 1

ユニット1

Diagrams and Signal Flow Graphs

ダイアグラムと信号流れ線図

KEY TERMS AND CONCEPTS

主要用語と概念

spring-mass-damper laplace block diagram

A spring-mass-damper system can be written as a second-order differential equation. This equation describes how the system changes over time. When the equation is transformed into the Laplace domain with zero initial conditions, it becomes easier to analyze and rearrange.

The Laplace transform changes a differential equation into an algebraic equation. That makes it simpler to isolate the output. For the standard spring-mass-damper model, the transformed equation gives the output in a form that can be used directly for block diagram construction.

The block diagram is built by connecting the signals on the right-hand side of the equation. It can also include internal variables such as acceleration and velocity. In the Laplace domain, 1/s means integration. So acceleration is integrated to get velocity, and velocity is integrated to get displacement.

The diagram can be simplified by factoring terms from the internal feedback loop. This creates another block diagram that shows the links between acceleration, velocity, and displacement more clearly. To find the transfer function, the block for the input and feedback signals is moved to the right side of the comparator, and the internal feedback loop is reduced. The final transfer function helps describe system behavior, predict the response to inputs, and support control design.

Spring-Mass-Damper Laplace Block Diagram

elements of block diagrams

Block Diagrams in Control Systems

Block diagrams show the input-output relationships in a system. They help students see how one signal affects another. A heating system is a simple example. The set temperature turns the furnace on, and the room warms to the desired level. Block diagrams can model both linear and nonlinear systems. Linear systems often use Laplace transform variables. Nonlinear systems are usually described with time domain variables. This makes block diagrams useful for many kinds of control problems. A...

relation between mathematical equations and block diagrams

A spring-mass-damper system can be written as a second-order differential equation. This equation describes how the system changes over time. When the equation is transformed into the Laplace domain with zero initial conditions, it becomes easier to analyze and rearrange. The Laplace transform changes a differential equation into an algebraic equation. That makes it simpler to isolate the output. For the standard spring-mass-damper model, the transformed equation gives the output in a form...

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Simplifying Control System Block Diagrams

Simplifying control system block diagrams helps make a system easier to analyze. The goal is to reduce several connected blocks to one block so the transfer function can be found more easily. This process keeps the system behavior the same while making the diagram simpler to work with. A branch point is the place where one signal splits into multiple paths. To simplify the diagram, the branch point can be moved to a new location. The new position must preserve the mathematical relationship...

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Mason’s Rule in Signal Flow Graphs

Signal-flow graphs are a clear way to represent control systems. They offer an alternative to traditional block diagrams. In these graphs, branches stand for systems and nodes stand for signals. Arrows show the direction of signal flow, and the transfer function is written next to each arrow. Signal-flow graphs also handle subtraction in a different way from block diagrams. In a block diagram, summing junctions often use negative signs. In a signal-flow graph, those negative signs are built...

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key terms and concepts

Learn how a spring-mass-damper equation becomes a Laplace-domain block diagram, with acceleration, velocity, displacement, and transfer function steps.

Spring-Mass-Damper Laplace Block Diagram | JoVE K12

JoVE Core 電気工学 は、電気回路の基本概念と回路解析テクニックを網羅したアニメーションビデオシリーズです。オペアンプと回路変換および 1 次・ 2 次回路の概念を網羅しています。実験を撮影した実写動画では、設計例や現代社会での応用が含まれており、複雑な概念も分かりやすく学習いただけます。

electrical engineering

Spring-Mass-Damper Laplace Block Diagram

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