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Hysteresis in a simple V-shaped spring-mass system Simulation created using Claude and GPT4o and GPTo1-preview

 v3 added hysteresis graph and other enhancements

link
https://sg.iwant2study.org/ospsg/index.php/1208

 v2 enhanced as a teaching aid using gpt 01 preview, largely 95%, 5% human made.

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https://sg.iwant2study.org/ospsg/index.php/1208

The Simulation Setup

The Spring Mass Hysteresis Simulation models two identical linear springs connected symmetrically to a mass in a "V" shape.

link
https://sg.iwant2study.org/ospsg/index.php/1208


The springs exert a force on the mass that varies with displacement according to Hooke's Law, , where is the spring constant, and is the displacement from the equilibrium position. In this simulation, users can apply an adjustable force to the mass and observe its behavior as the force is varied over time. The simulation allows you to:

  1. Adjust the Applied Force: Users can vary the magnitude and direction of the applied force over time, allowing a detailed investigation of how the system responds to changes.

    link
    https://sg.iwant2study.org/ospsg/index.php/1208

  2. Visualize the Mass Motion: The simulation provides a visual representation of the mass movement, capturing both linear displacement and oscillatory motion.

  3. Analyze Hysteresis Loop Graphs: The primary feature of this simulation is the ability to display the force-displacement hysteresis loops that characterize the energy losses due to non-conservative forces in the system.

    link
    https://sg.iwant2study.org/ospsg/index.php/1208

    Users can plot and analyze these graphs, which are crucial for understanding hysteresis behavior.

Physics Behind the Hysteresis Phenomenon

The term hysteresis refers to a system's history-dependent response, meaning that the behavior of the system depends on the history of the applied force rather than just the current value. In this setup, as the force applied to the mass is varied, the resulting displacement is not simply a direct function of the current force. Instead, the response depends on the previous states of the system, leading to the formation of a hysteresis loop in the force vs. displacement graph.

In the case of the spring-mass system in a "V" configuration, the hysteresis occurs due to factors like internal friction, energy dissipation, and possible non-linearities in the springs. The hysteresis loop formed in the force-displacement graph is indicative of energy loss during the cycle of loading and unloading.

Features of the Simulation

  1. Real-Time Graphing: The simulation includes a real-time graph that plots force vs. displacement, allowing users to observe the formation of hysteresis loops. This visualization is essential for identifying energy losses in the system.

  2. Adjustable Spring Constants and Damping:

    link
    https://sg.iwant2study.org/ospsg/index.php/1208

    Users can change the spring natural length  and introduce damping, which can significantly impact the shape of the hysteresis loop. By experimenting with these parameters, users can investigate how damping influences the energy dissipation in the system.

  3. Step-by-Step Force Application: The simulation allows for a stepwise application of force, making it easier to analyze the system's response to gradual or sudden changes. This is particularly useful for observing the onset of hysteresis and comparing different loading paths.

  4. Data Export for Analysis: Users can export the force-displacement data for further analysis. This feature is useful for students preparing for SYPT 2025, as it allows for a more in-depth exploration of how different parameters influence hysteresis behavior.

Applying the Simulation to SYPT 2025 B5

In the SYPT 2025 B5 problem, participants are tasked with investigating the motion of a mass connected to two identical springs in a "V" shape, focusing on how the mass's response depends on the history of applied forces. This simulation provides an ideal platform to:

  • Visualize Historical Dependence: By applying forces in different sequences, participants can observe how the mass's motion depends on the history of the force application, demonstrating the essence of hysteresis.

  • Study Energy Dissipation: The area enclosed by the hysteresis loop in the force-displacement graph represents the energy dissipated in the system. By analyzing this loop, students can gain insights into the factors contributing to energy losses.

  • Experiment with Variables: Participants can alter variables such as spring natural length initial conditions, and damping factor to explore their effects on the hysteresis behavior, helping to understand the parameters that lead to significant hysteresis effects.

Conclusion

The Spring Mass Hysteresis Simulation is a powerful tool for visualizing and understanding the complex phenomenon of hysteresis in a spring-mass system. With features like adjustable force, real-time graphing, and parameter customization, it provides an excellent platform for students preparing for SYPT 2025 B5 to experiment and deepen their understanding of spring hysteresis.

By leveraging the insights gained from this simulation, students can build a solid foundation for investigating the history-dependent response of physical systems, an essential concept in understanding many real-world applications of physics.

Ready to explore spring hysteresis? Try out the Spring Mass Hysteresis Simulation here and start your journey into mastering hysteresis for SYPT 2025!

Modeling a Spring-Mass System with Hysteresis

The spring-mass system is a classic physics problem that demonstrates the interaction between forces, motion, and energy. In this blog, we will dive into the mathematical modeling of a V-shaped spring-mass system with hysteresis. This includes equations of motion, force breakdown, and how this system can be simulated interactively using the provided JavaScript and HTML code.

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Understanding the System

This simulation involves:

  1. A Mass: Hanging from two V-shaped springs.
  2. Springs: Exert forces proportional to their extension or compression, governed by Hooke's Law.
  3. External Forces: Such as gravity, external upward forces, and damping.
  4. Hysteresis Effect: Captures the energy lost in the system due to non-conservative forces like damping.

The system is visualized dynamically on a canvas where forces like spring force, gravity, damping, and external force are displayed interactively.

Equations of Motion

The motion of the mass is governed by Newton's Second Law:

Fnet=ma

Where:

  • Fnet: Net force acting on the mass (sum of all forces).
  • m: Mass of the object.
  • a: Acceleration of the object.

Forces Acting on the Mass

  1. Spring Forces (Fspring): Each spring follows Hooke’s Law:

    Fspring=−k(L−L0)⋅ΔxL

    Where:

    • k: Spring constant.
    • L: Current length of the spring.
    • L0: Natural length of the spring.
    • Δx: Displacement of the spring’s attachment point.

    The spring force is divided into components (Fx, Fy) along the x and y axes.

  2. Gravitational Force (Fgravity):

    Fgravity=mg

    Acts downward, where:

    • g: Acceleration due to gravity (9.81 m/s2).

  3. Damping Force (Fdamping): A force opposing motion, proportional to velocity:

    Fdamping=−bv

    Where:

    • b: Damping coefficient.
    • v: Velocity of the mass.

  4. External Force (Fexternal): An adjustable external force applied to the mass.

Net Force and Acceleration

The total forces in the x and y directions:

Fx=Fspring,1,x+Fspring,2,x+Fdamping,xFy=Fspring,1,y+Fspring,2,y+Fdamping,y+Fexternal−Fgravity

The acceleration components are:

ax=Fxmay=Fym

Modeling the System

Updating Position and Velocity

Using the acceleration values from above, the velocity and position are updated iteratively using Euler’s Method:

vx(t+Δt)=vx(t)+axΔtvy(t+Δt)=vy(t)+ayΔtx(t+Δt)=x(t)+vxΔty(t+Δt)=y(t)+vyΔt

Where:

  • Δt: Time step.

Spring Length and Direction

For each spring, the displacement vector (Δx,Δy) determines the length L and direction of the spring force:

L=(Δx)2+(Δy)2

The force components are then:

Fx=F⋅ΔxLFy=F⋅ΔyL

Interactive Features in the Simulation

Sliders

The simulation provides sliders to control:

  1. Mass: Adjust the weight of the mass.
  2. Spring Natural Length: Change the rest length of the springs.
  3. Damping Factor: Alter the system's energy dissipation.
  4. External Force: Apply an upward force.
  5. Gravity: Modify the gravitational acceleration.

Force Visualizations

  • Each force (spring, gravity, damping, external) is displayed with arrows and labels.
  • Background colors of sliders match force arrow colors for intuitive learning.

Applications of the Model

  1. Physics Education:

    • Understand equilibrium positions where the net force is zero.
    • Explore how damping affects oscillatory motion and leads to eventual rest.

  2. Engineering:

    • Simulate material properties and energy dissipation in mechanical systems.

  3. Interactive Learning:

    • Visualize forces and their impact dynamically.

Further Exploration

To enhance your understanding:

  1. Experiment with different values for damping and observe how it affects hysteresis.
  2. Adjust the natural length of the springs to simulate different initial configurations.
  3. Test equilibrium by applying different external forces and observing the steady state.

 

Enhanced Blog Post: Spring-Mass Hysteresis Simulation

Visualizing the Spring-Mass System

The image showcases the Spring-Mass Hysteresis Simulation in action. Below, we’ll break down its features and link them to the physics concepts it demonstrates.

Key Features of the Simulation

1. Force Visualizations

Each force acting on the mass is represented with a distinct arrow and label:

  • F_spring (Green and Blue): The forces exerted by the left and right springs.
  • F_gravity (Red): The downward gravitational force.
  • F_damping (Cyan): The damping force opposing motion.
  • F_external (Purple): An optional upward external force.

These arrows scale dynamically to represent the magnitude of the forces, giving an intuitive understanding of how they interact.

2. User Interface

The controls at the top allow users to toggle the visibility of specific forces:

  • Show Spring Forces
  • Show Gravity Force
  • Show External Force
  • Show Damping

This interactivity helps focus on individual components for a deeper analysis.

3. Sliders with Color-Coded Labels

The sliders below the force toggles are color-coded to match the force arrows, enhancing the correlation between input and output:

  • Spring Natural Length (Orange): Adjusts the equilibrium length of the springs.
  • External Force (Purple): Modifies the upward force applied to the mass.
  • Damping Factor (Cyan): Controls the energy dissipation in the system.
  • Gravity (Red): Changes the gravitational acceleration.
  • Mass (No color): Sets the mass of the object.

This setup enables users to test various configurations and observe the resulting dynamics.

Understanding the Display

The Canvas

  • Central Red Circle: Represents the mass.
  • Anchored Points (Blue Dots): The fixed ends of the V-shaped springs.
  • Dynamically Adjusting Arrows: Indicate forces acting on the mass.

Equilibrium Positions

At the bottom of the simulation, the time, current mass position, and equilibrium positions are displayed. These equilibrium positions are calculated by solving for where the net force acting on the mass equals zero:

Fnet=0

In this example:

  • ye1=−128.01
  • ye3=91.40

This tells us the two stable positions where the mass can remain at rest, depending on the applied external force.

Interactive Exploration

This simulation invites students to:

  1. Explore Energy Loss: Enable damping to see how the system eventually comes to rest due to energy dissipation.
  2. Investigate Hysteresis: Observe how the system’s response to forces depends on its motion history.
  3. Visualize Forces: Study how forces balance at equilibrium positions and change dynamically during motion.

How It Helps Learning

  • Visual Learning: Links equations and physical concepts to visual elements like arrows and motion.
  • Hands-On Experimentation: Sliders let users explore how changes in parameters affect system behavior.
  • Real-Time Feedback: The dynamic display helps solidify intuition about forces and motion.

Try It Yourself

Load the simulation, adjust the parameters, and observe how the mass responds to the forces. Pay attention to how equilibrium positions and motion change with damping, gravity, and external forces.

Physics concepts come alive when visualized—experience it firsthand!

 

 

 

About

 

 

Theory and Equations for the V-Shaped Spring-Mass Hysteresis Simulation

Introduction

This simulation models a mass connected to two springs arranged in a V-shape, demonstrating the hysteresis effect in mechanical systems. Hysteresis refers to the phenomenon where the system's response depends not only on its current state but also on its history. In this context, hysteresis is observed when the path of deformation and recovery of the spring-mass system differs, leading to energy dissipation, often as heat.

System Description

  • Mass (\( m \)): A point mass that can move freely in two dimensions.
  • Springs: Two identical linear springs with spring constant \( k \) and natural (unstretched) length \( L_0 \), connected from fixed anchor points to the mass.
  • Anchor Points: Fixed points located at \( \mathbf{A}_1 = (-a, -b) \) and \( \mathbf{A}_2 = (a, -b) \), forming the V-shape.
  • Forces Acting on the Mass:
    • Spring Forces (\( \mathbf{F}_1 \) and \( \mathbf{F}_2 \)): Elastic forces from the two springs.
    • Damping Force (\( \mathbf{F}_{\text{damping}} \)): Viscous damping proportional to the velocity.
    • Gravity (\( \mathbf{F}_g \)): Gravitational force acting downward.
    • External Force (\( \mathbf{F}_{\text{ext}} \)): User-adjustable force applied in the vertical direction.

Coordinate System

The origin is set at the center of the canvas.

  • Mass Position (\( \mathbf{r} \)): \( \mathbf{r} = (x, y) \)
  • Anchor Points:
    • Left Anchor: \( \mathbf{A}_1 = (-a, -b) \)
    • Right Anchor: \( \mathbf{A}_2 = (a, -b) \)
  • Constants from the Simulation:
    • \( a = 100 \) units
    • \( b = 100 \) units

Equations of Motion

Spring Forces

For each spring, the force is calculated using Hooke's Law for linear springs.

  1. Displacement Vectors:
    • Left Spring: \( \mathbf{d}_1 = \mathbf{r} - \mathbf{A}_1 \)
    • Right Spring: \( \mathbf{d}_2 = \mathbf{r} - \mathbf{A}_2 \)
  2. Lengths of Springs:
    • \( L_1 = \| \mathbf{d}_1 \| \)
    • \( L_2 = \| \mathbf{d}_2 \| \)
  3. Unit Vectors Along Springs:
    • \( \hat{\mathbf{u}}_1 = \dfrac{ \mathbf{d}_1 }{ L_1 } \)
    • \( \hat{\mathbf{u}}_2 = \dfrac{ \mathbf{d}_2 }{ L_2 } \)
  4. Spring Forces:
    • \( \mathbf{F}_1 = -k ( L_1 - L_0 ) \hat{\mathbf{u}}_1 \)
    • \( \mathbf{F}_2 = -k ( L_2 - L_0 ) \hat{\mathbf{u}}_2 \)

Damping Force

The damping force is proportional to the velocity of the mass.

\( \mathbf{F}_{\text{damping}} = -\gamma \mathbf{v} \)

  • \( \gamma \): Damping coefficient (damping factor)
  • \( \mathbf{v} \): Velocity of the mass (\( \mathbf{v} = \dfrac{ d\mathbf{r} }{ dt } \))

Gravitational Force

The gravitational force acts downward.

\( \mathbf{F}_g = m g \hat{\mathbf{j}} \)

  • \( g \): Gravitational acceleration (negative value indicates downward direction)
  • \( \hat{\mathbf{j}} \): Unit vector in the vertical direction

External Force

The external force is applied in the vertical direction.

\( \mathbf{F}_{\text{ext}} = F_{\text{ext}} \hat{\mathbf{j}} \)

  • \( F_{\text{ext}} \): User-adjustable external force

Total Force

The total force acting on the mass is the sum of all forces.

\( \mathbf{F}_{\text{total}} = \mathbf{F}_1 + \mathbf{F}_2 + \mathbf{F}_{\text{damping}} + \mathbf{F}_g + \mathbf{F}_{\text{ext}} \)

Equations of Motion

Using Newton's second law, the acceleration of the mass is:

\( m \mathbf{a} = \mathbf{F}_{\text{total}} \)

Therefore,

\( \mathbf{a} = \dfrac{ \mathbf{F}_{\text{total}} }{ m } \)

Breaking this into components:

\( a_x = \dfrac{ F_{\text{total}, x} }{ m } \)

\( a_y = \dfrac{ F_{\text{total}, y} }{ m } \)

Where:

\( F_{\text{total}, x} = F_{1, x} + F_{2, x} - \gamma v_x \)

\( F_{\text{total}, y} = F_{1, y} + F_{2, y} - \gamma v_y + m g + F_{\text{ext}} \)

Numerical Integration

The simulation uses the Euler method for numerical integration to update velocity and position:

\( v_x(t + \Delta t) = v_x(t) + a_x \Delta t \)

\( v_y(t + \Delta t) = v_y(t) + a_y \Delta t \)

\( x(t + \Delta t) = x(t) + v_x(t + \Delta t) \Delta t \)

\( y(t + \Delta t) = y(t) + v_y(t + \Delta t) \Delta t \)

Hysteresis in the System

Hysteresis in this V-shaped spring-mass system arises due to the energy dissipation caused by damping. As the mass oscillates, the damping force removes energy from the system, leading to a difference in the path of deformation (loading) and recovery (unloading).

Energy Considerations

  • Elastic Potential Energy of Springs:

    \( U_{\text{spring}} = \dfrac{1}{2} k ( L_1 - L_0 )^2 + \dfrac{1}{2} k ( L_2 - L_0 )^2 \)

  • Kinetic Energy of the Mass:

    \( K = \dfrac{1}{2} m \left( v_x^2 + v_y^2 \right) \)

  • Mechanical Energy:

    \( E_{\text{mechanical}} = K + U_{\text{spring}} + m g y \)

Due to damping, the mechanical energy decreases over time, resulting in hysteresis when plotting force versus displacement or when observing the system's response over multiple cycles.

Visualization of Hysteresis

  • Force-Displacement Loops: By plotting the total force acting on the mass against its displacement, a hysteresis loop can be observed, indicating the energy dissipated per cycle.
  • Phase Space Trajectories: Plotting velocity versus displacement shows the damping effect and the system's approach to equilibrium.

Parameters and Their Effects

  • Spring Constant (\( k \)): Determines the stiffness of the springs. Higher \( k \) leads to stronger restoring forces.
  • Natural Length (\( L_0 \)): Affects the equilibrium position of the mass. Adjusting \( L_0 \) changes the initial tension or compression in the springs.
  • Mass (\( m \)): Influences the inertia of the system. A larger mass results in slower acceleration for the same force.
  • Damping Coefficient (\( \gamma \)): Controls the rate of energy dissipation. Higher damping reduces oscillations more quickly.
  • Gravity (\( g \)): Alters the weight of the mass. Adjusting \( g \) can simulate different gravitational environments.
  • External Force (\( F_{\text{ext}} \)): Allows for external manipulation of the system, such as applying an upward or downward force.

Practical Implications

Understanding hysteresis in mechanical systems is crucial for designing materials and structures that experience cyclic loading. Energy dissipation due to hysteresis affects the fatigue life and performance of mechanical components.

References

For a detailed exploration of the hysteresis in a V-shaped spring-mass system, refer to the work by Christopher Ong:

Ong, C. "Hysteresis in a simple V-shaped spring-mass system." American Journal of Physics, 89(7), 663 (2021). Link to paper

Conclusion

This simulation provides a visual and interactive way to study the hysteresis phenomenon in a mechanical system. By adjusting parameters and observing the system's response, one can gain insights into the interplay between forces, energy dissipation, and motion in damped oscillatory systems.

 

 

Sample Learning Goals

    1. Physics Equations:

      • \section*{Physics Equations} The physics equations used in the simulation include: \subsection*{Hooke's Law} \begin{equation} F = -k \cdot x \end{equation} \subsection*{Damping Force} \begin{equation} F_d = -b \cdot v \end{equation} \subsection*{Gravitational Force} \begin{equation} F_g = m \cdot g \end{equation} \subsection*{Net Force} \begin{equation} F_{net} = F_{spring1} + F_{spring2} + F_{external} + F_{gravity} \end{equation} \subsection*{Acceleration} \begin{equation} a = \frac{F_{net}}{m} \end{equation} \subsection*{Velocity Update} \begin{equation} v = v + a \cdot dt \end{equation} \subsection*{Position Update} \begin{equation} x = x + v \cdot dt \end{equation}

    Activities to Try with the Simulation:

    1. Exploring Hysteresis:

      • Vary the external force and observe how the mass oscillates. Note the difference in the path of deformation and recovery.
      • Adjust the damping factor to see how it affects the energy loss and the hysteresis loop.

    2. Effect of Mass:

      • Change the mass of the system and observe how it affects the oscillation frequency and amplitude.

    3. Gravity Variations:

      • Experiment with different gravity values (positive and negative) to understand the impact of gravity on the spring-mass system.

    4. Spring Natural Length:

      • Modify the natural length of the springs and see how it affects the system's equilibrium position and oscillation behavior.

    5. Force vs. Position Graph:

      • Plot the force vs. position graph to visualize the hysteresis loop. This can be done by recording the position and force data over time.

    These activities will help in understanding the concept of hysteresis and the dynamics of a V-shaped spring-mass system.

  

For Teachers

Exploring Hysteresis with a V-Shaped Spring-Mass System Simulation

Understanding complex physical phenomena can be made easier and more engaging through interactive simulations. Today, we delve into the hysteresis effect using a V-shaped spring-mass system simulation, inspired by the work of Christopher Ong. This simulation allows you to visualize how hysteresis occurs in such a system, illustrating the energy loss typically observed due to this phenomenon.

What is Hysteresis?

Hysteresis is a phenomenon where the state of a system depends not only on its current conditions but also on its history. In other words, the path of deformation and recovery of the system differs, leading to energy dissipation, usually in the form of heat. This is commonly seen in magnetic materials, elastic hysteresis in rubbers, and mechanical systems like the one we are exploring here.

The Simulation

This simulation models a V-shaped spring-mass system, where the mass is connected to two springs anchored at fixed points. By adjusting various parameters, you can see how the system behaves under different conditions and observe the hysteresis effect in action.

Features of the Simulation

  1. External Force Adjustment: Apply an upward force to the mass to see how it affects the system.
  2. Spring Natural Length: Change the natural length of the springs to see how the system's equilibrium changes.
  3. Mass: Adjust the mass of the object to observe how inertia influences the system's behavior.
  4. Damping Factor: Modify the damping factor to see how energy dissipation affects the system.
  5. Gravity: Change the gravity to simulate different gravitational environments.

How to Use the Simulation

  • Play/Pause: Start or pause the simulation.
  • Step: Advance the simulation step by step to observe gradual changes.
  • Reset: Reset all parameters to their default values.

The controls are intuitive and allow for a hands-on approach to learning about hysteresis.

Description of the Simulation

This simulation demonstrates the hysteresis effect in a V-shaped spring-mass system based on the work of Christopher Ong. For more details, you can refer to the original paper.

Hysteresis occurs when the path of deformation and recovery of the spring-mass system differ, leading to energy loss typically as heat. The simulation models the forces acting on the mass, including spring forces, damping, external force, and gravity.

Created by This email address is being protected from spambots. You need JavaScript enabled to view it. using Claude (initial code) and GPT-4 (finishing the code).

Visual Representation

 

Code Implementation

Here's a snippet of the code that powers the simulation. Feel free to explore and modify it to suit your learning needs.

html
<!DOCTYPE html> <html lang="en"> <head> <meta charset="UTF-8"> <meta name="viewport" content="width=device-width, initial-scale=1.0"> <title>Spring-Mass Hysteresis Simulation</title> <style> /* Styling for the body, container, and canvas */ body { font-family: Arial, sans-serif; display: flex; flex-direction: column; align-items: center; height: 100vh; margin: 0; background-color: #f0f0f0; } .container { background-color: rgb(110, 100, 100); padding: 20px; border-radius: 10px; box-shadow: 0 0 10px rgba(0,0,0,0.1); width: 100%; box-sizing: border-box; position: relative; } canvas { border: 1px solid #ddd; width: 100%; height: 60vh; } .controls { display: flex; align-items: center; flex-wrap: wrap; justify-content: center; margin-bottom: 10px; } .controls h1 { margin: 0; margin-right: 20px; font-size: 1.5em; color: white; flex: 1 100%; text-align: center; } button { margin-right: 5px; margin-top: 5px; display: inline-block; } .slider-container { margin-bottom: 10px; flex: 1 100%; text-align: center; } .slider-container label { display: inline-block; width: 150px; } .slider-container input { width: 50%; } .value-display { display: inline-block; width: 50px; text-align: right; } #info { position: relative; text-align: center; color: white; margin-bottom: 10px; } #footer { margin-top: 10px; color: rgb(110, 100, 100); text-align: center; } #description { color: rgb(110, 100, 100); text-align: center; margin-top: 10px; margin-bottom: 10px; } @media (max-width: 768px) { .slider-container label { width: auto; display: block; margin-bottom: 5px; } .slider-container input { width: 100%; } .controls { flex-direction: column; } } </style> </head> <body> <div class="container"> <!-- Control buttons for play/pause, step, and reset --> <div class="controls"> <h1>Spring-Mass Hysteresis Simulation</h1> <div> <button id="playPauseBtn">Play</button> <button id="stepBtn">Step</button> <button id="resetBtn">Reset</button> </div> </div> <!-- Slider for adjusting external force --> <div class="slider-container"> <label for="forceSlider">External Force (Upward):</label> <input type="range" id="forceSlider" min="-40" max="40" step="1" value="0"> <span>Force: <span id="forceValue" class="value-display">0</span></span> </div> <!-- Slider for adjusting spring natural length --> <div class="slider-container"> <label for="lengthSlider">Spring Natural Length:</label> <input type="range" id="lengthSlider" min="50" max="150" step="1" value="100"> <span>Length: <span id="lengthValue" class="value-display">100</span></span> </div> <!-- Additional sliders --> <div class="slider-container"> <label for="massSlider">Mass:</label> <input type="range" id="massSlider" min="0.5" max="5" step="0.1" value="1"> <span>Mass: <span id="massValue" class="value-display">1</span></span> </div> <div class="slider-container"> <label for="dampingSlider">Damping Factor:</label> <input type="range" id="dampingSlider" min="0" max="1" step="0.01" value="0.1"> <span>Damping: <span id="dampingValue" class="value-display">0.1</span></span> </div> <div class="slider-container"> <label for="gravitySlider">Gravity:</label> <input type="range" id="gravitySlider" min="-9.81" max="9.81" step="0.1" value="-9.81"> <span>Gravity: <span id="gravityValue" class="value-display">-9.81</span></span> </div> <!-- Canvas for drawing the simulation --> <canvas id="simCanvas"></canvas> <!-- Info section to display time and mass position --> <div id="info">Time: 0.00s | Mass Position: (0.00, 0.00)</div> </div> <!-- Description of the simulation --> <div id="description"> <p>This simulation demonstrates the hysteresis effect in a V-shaped spring-mass system based on the work of Christopher Ong. For more details, you can refer to the <a href="https://pubs.aip.org/aapt/ajp/article-abstract/89/7/663/1056905/Hysteresis-in-a-simple-V-shaped-spring-mass-system?redirectedFrom=fulltext" target="_blank">original paper</a>.</p> <p>Hysteresis occurs when the path of deformation and recovery of the spring-mass system differ, leading to energy loss typically as heat.</p> <p>The simulation models the forces acting on the mass, including spring forces, damping, external force, and gravity.</p> </div> <!-- Footer for credits --> <div id="footer"> Created by This email address is being protected from spambots. You need JavaScript enabled to view it. using Claude (initial code) and GPT-4 (finishing the code) </div> <!-- Google Analytics and Ads scripts --> <script async="true" src="https://www.googletagmanager.com/gtag/js?id=G-S9EWRY1CPJ"></script> <script> window.dataLayer = window.dataLayer || []; function gtag(){dataLayer.push(arguments);} gtag('js', new Date()); gtag('config', 'G-S9EWRY1CPJ'); </script> <script data-ad-client="ca-pub-0121577198857509" async="true" src="https://pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script> <script> // Get references to HTML elements const canvas = document.getElementById('simCanvas'); const ctx = canvas.getContext('2d'); const playPauseBtn = document.getElementById('playPauseBtn'); const stepBtn = document.getElementById('stepBtn'); const resetBtn = document.getElementById('resetBtn'); const forceSlider = document.getElementById('forceSlider'); const forceValue = document.getElementById('forceValue'); const lengthSlider = document.getElementById('lengthSlider'); const lengthValue = document.getElementById('lengthValue'); const massSlider = document.getElementById('massSlider'); const massValue = document.getElementById('massValue'); const dampingSlider = document.getElementById('dampingSlider'); const dampingValue = document.getElementById('dampingValue'); const gravitySlider = document.getElementById('gravitySlider'); const gravityValue = document.getElementById('gravityValue'); const info = document.getElementById('info'); let isPlaying = false; // Variable to track if the simulation is playing let externalForce = 0; // External force acting on the mass let massPosition = { x: 0, y: 0 }; // Position of the mass let massVelocity = { x: 0, y: 0 }; // Velocity of the mass let time = 0; // Simulation time // Constants for the spring and mass system const k = 0.5; // Spring constant let m = 1; // Mass let L0 = 100; // Natural length of springs let dampingFactor = 0.1; // Damping factor for the system let gravity = -9.81; // Gravitational acceleration const dt = 0.1; // Time step for the simulation // Function to resize canvas dimensions function resizeCanvas() { canvas.width = canvas.clientWidth; canvas.height = canvas.clientHeight; drawSystem(); } // Function to draw an arrow on the canvas function drawArrow(fromx, fromy, tox, toy, color) { const headlen = 10; // Length of arrowhead in pixels const dx = tox - fromx; // Change in x const dy = toy - fromy; // Change in y const angle = Math.atan2(dy, dx); // Angle of the arrow ctx.strokeStyle = color; // Set arrow color ctx.beginPath(); ctx.moveTo(fromx, fromy); ctx.lineTo(tox, toy); ctx.lineTo(tox - headlen * Math.cos(angle - Math.PI / 6), toy - headlen * Math.sin(angle - Math.PI / 6)); ctx.moveTo(tox, toy); ctx.lineTo(tox - headlen * Math.cos(angle + Math.PI / 6), toy - headlen * Math.sin(angle + Math.PI / 6)); ctx.stroke(); } // Function to draw a spring between two points function drawSpring(x1, y1, x2, y2, naturalLength, segments = 20) { const dx = x2 - x1; // Change in x const dy = y2 - y1; // Change in y const length = Math.sqrt(dx * dx + dy * dy); // Length of the spring const angle = Math.atan2(dy, dx); // Angle of the spring ctx.save(); // Save the current context state ctx.translate(x1, y1); // Translate the context to the starting point of the spring ctx.rotate(angle); // Rotate the context to align with the spring const segmentLength = length / segments; // Length of each segment const amplitude = segmentLength / 2; // Amplitude of the zigzag ctx.beginPath(); ctx.moveTo(0, 0); for (let i = 1; i <= segments; i++) { const x = i * segmentLength; const y = (i % 2 === 0 ? 1 : -1) * amplitude; ctx.lineTo(x, y); } // Set the color based on spring length if (length < naturalLength) { ctx.strokeStyle = 'orange'; // Compression } else { ctx.strokeStyle = 'yellow'; // Extension } ctx.stroke(); ctx.restore(); // Restore the previous context state } // Function to draw the entire system function drawSystem() { ctx.clearRect(0, 0, canvas.width, canvas.height); // Clear the canvas const centerX = canvas.width / 2; // Center x-coordinate const centerY = canvas.height / 2; // Center y-coordinate // Draw left spring drawSpring(centerX - 100, centerY - 100, centerX + massPosition.x, centerY + massPosition.y, L0); // Draw right spring drawSpring(centerX + 100, centerY - 100, centerX + massPosition.x, centerY + massPosition.y, L0); // Draw the mass ctx.beginPath(); ctx.arc(centerX + massPosition.x, centerY + massPosition.y, 10, 0, 2 * Math.PI); ctx.fillStyle = '#f00'; ctx.fill(); // Draw anchor points ctx.beginPath(); ctx.arc(centerX - 100, centerY - 100, 5, 0, 2 * Math.PI); ctx.arc(centerX + 100, centerY - 100, 5, 0, 2 * Math.PI); ctx.fillStyle = '#00f'; ctx.fill(); // Draw the external force vector if any if (externalForce !== 0) { const forceScale = 5; // Scale factor for force visualization drawArrow( centerX + massPosition.x, centerY + massPosition.y, centerX + massPosition.x, centerY + massPosition.y - externalForce * forceScale, 'purple' ); } } // Function to update the position of the mass function updatePosition() { const dx1 = massPosition.x + 100; // Distance from left spring anchor const dy1 = massPosition.y + 100; // Distance from left spring anchor const dx2 = massPosition.x - 100; // Distance from right spring anchor const dy2 = massPosition.y + 100; // Distance from right spring anchor const L1 = Math.sqrt(dx1 * dx1 + dy1 * dy1); // Length of left spring const L2 = Math.sqrt(dx2 * dx2 + dy2 * dy2); // Length of right spring const Fx1 = -k * (L1 - L0) * dx1 / L1; // Force by left spring in x const Fy1 = -k * (L1 - L0) * dy1 / L1; // Force by left spring in y const Fx2 = -k * (L2 - L0) * dx2 / L2; // Force by right spring in x const Fy2 = -k * (L2 - L0) * dy2 / L2; // Force by right spring in y const Fx = Fx1 + Fx2; // Total force in x const Fy = Fy1 + Fy2 - externalForce - m * gravity; // Total force in y, include external force and gravity const ax = Fx / m - dampingFactor * massVelocity.x; // Acceleration in x const ay = Fy / m - dampingFactor * massVelocity.y; // Acceleration in y massVelocity.x += ax * dt; // Update velocity in x massVelocity.y += ay * dt; // Update velocity in y massPosition.x += massVelocity.x * dt; // Update position in x massPosition.y += massVelocity.y * dt; // Update position in y time += dt; // Increment time drawSystem(); // Redraw the system updateInfo(); // Update the info display } // Function to update the information display function updateInfo() { info.textContent = `Time: \({time.toFixed(2)}s | Mass Position: (\){massPosition.x.toFixed(2)}, ${massPosition.y.toFixed(2)})`; } // Function to toggle the simulation play/pause state function toggleSimulation() { isPlaying = !isPlaying; playPauseBtn.textContent = isPlaying ? 'Pause' : 'Play'; if (isPlaying) { simulationLoop(); } } // Simulation loop to update the position continuously function simulationLoop() { if (isPlaying) { updatePosition(); requestAnimationFrame(simulationLoop); } } // Function to perform a single step of the simulation function step() { isPlaying = false; playPauseBtn.textContent = 'Play'; updatePosition(); } // Function to reset the simulation to its initial state function reset() { isPlaying = false; playPauseBtn.textContent = 'Play'; massPosition = { x: 0, y: 0 }; massVelocity = { x: 0, y: 0 }; time = 0; externalForce = 0; forceSlider.value = 0; forceValue.textContent = '0'; L0 = 100; lengthSlider.value = 100; lengthValue.textContent = '100'; m = 1; massSlider.value = 1; massValue.textContent = '1'; dampingFactor = 0.1; dampingSlider.value = 0.1; dampingValue.textContent = '0.1'; gravity = -9.81; gravitySlider.value = -9.81; gravityValue.textContent = '-9.81'; drawSystem(); updateInfo(); } // Event listeners for the buttons and sliders playPauseBtn.addEventListener('click', toggleSimulation); stepBtn.addEventListener('click', step); resetBtn.addEventListener('click', reset); forceSlider.addEventListener('input', () => { externalForce = parseFloat(forceSlider.value); forceValue.textContent = externalForce.toFixed(1); drawSystem(); }); lengthSlider.addEventListener('input', () => { L0 = parseFloat(lengthSlider.value); lengthValue.textContent = L0.toFixed(0); drawSystem(); }); massSlider.addEventListener('input', () => { m = parseFloat(massSlider.value); massValue.textContent = m.toFixed(1); drawSystem(); }); dampingSlider.addEventListener('input', () => { dampingFactor = parseFloat(dampingSlider.value); dampingValue.textContent = dampingFactor.toFixed(2); drawSystem(); }); gravitySlider.addEventListener('input', () => { gravity = parseFloat(gravitySlider.value); gravityValue.textContent = gravity.toFixed(2); drawSystem(); }); // Initial draw and info update drawSystem(); updateInfo(); // Resize canvas when window is resized window.addEventListener('resize', resizeCanvas); resizeCanvas(); // Initial resize to set canvas dimensions </script> </body> </html>

Activities to Try

  1. Observe Hysteresis: Apply an external force and vary it to see how the system’s path differs during loading and unloading.
  2. Damping Effects: Adjust the damping factor and observe how quickly the system returns to equilibrium. Note how energy dissipation changes with different damping values.
  3. Mass Variation: Change the mass and see how inertia affects the system’s response to external forces and damping.
  4. Gravity Simulation: Experiment with different gravity values to simulate the system on different planets or in a microgravity environment.

Conclusion

Interactive simulations like this provide a powerful tool for visualizing and understanding complex physical phenomena. By adjusting parameters and observing outcomes, you gain a deeper insight into concepts like hysteresis and the dynamics of spring-mass systems. Explore the simulation, modify the code, and share your findings!

Feel free to reach out with any questions or feedback on the simulation. Happy learning!

Software Requirements

[SIMU_SWREQ]

Translation

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Research

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Video

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Credits

https://pubs.aip.org/aapt/ajp/article-abstract/89/7/663/1056905/Hysteresis-in-a-simple-V-shaped-spring-mass-system?redirectedFrom=fulltext

Version:

https://claude.ai/chat/6df14257-3c20-46ae-b2e5-86df175e6c07

https://chatgpt.com/c/042b8321-ad20-4b80-867e-3965ce9b985a

https://weelookang.blogspot.com/2024/06/hysteresis-in-simple-v-shaped-spring.html

Other Resources

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