Laboratory Objective: Introduction to Analog Computers and Hands-On Simulation

Experiment #1: Introduction to Analog Computers

Objective: Introduction to Analog Computers and its operation

Introduction

Analog computers, the predecessors of modern digital computers, played a vital role during World War II and continue to find applications in various fields. Mimicking the behavior of differential equations, analog computers utilize electrical circuits to solve complex problems. By employing active electrical networks composed of resistors, capacitors, and operational amplifiers (OP amps), analog computers can simulate linear and nonlinear systems efficiently.

The laboratory employs the COMDYNA GP-6 Analog computers, versatile systems capable of simulating mathematical models of up to four state variables.

These computers facilitate hands-on experiments, allowing students to gain practical insights into systems engineering, mathematical modeling, and simulation.

The COMDYNA GP-6 Analog Computer

The GP-6 features internal components such as operational amplifiers, summer resistor networks, integrator capacitor sets, and coefficient potentiometers. A patch panel facilitates analog programming, while front panel banana jacks enable easy interfacing with external instrumentation. Additional features include precision reference voltages and compute-time readouts, enhancing usability and convenience in laboratory settings.

Experimental Procedures

To demonstrate basic operations, students connect the analog computer as a summer or integrator, facilitating summing and integration operations. The analog block diagrams guide the setup, ensuring proper connections and configurations. Once operational, students observe the output signals on display panels and record results for analysis.

Review of Solutions of Linear Ordinary Differential Equations

First-order Equations

A linear first-order equation with constant coefficients, representing system behavior, requires a thorough understanding of its solution techniques. Comprising complementary and particular solutions, these equations govern transient and steady-state responses.

By applying methods such as integrating functions or Laplace transforms, students can derive accurate solutions for practical applications.

Second-order Equations

Second-order equations, common in mechanical systems like mass-spring systems, offer valuable insights into system dynamics. Analyzing solutions through analog simulation provides students with a deeper understanding of system behavior under varying parameters. By comparing experimental and theoretical results, students can assess system stability, response times, and oscillatory behavior.

Experiment #2: Basic Operations of the Analog Computer

Objectives: To learn basic analog computer operations, including summing and integration.

Introduction

Summing and integration operations are fundamental to analog computing, allowing the synthesis of complex systems from simpler components. By configuring operational amplifiers and resistors, students can perform mathematical operations efficiently. Analog computers facilitate hands-on learning, enabling students to grasp abstract concepts through tangible experiments.

Equipment

The GP-6 Analog Computer, equipped with power supplies and connecting wires, provides the necessary infrastructure for conducting experiments. Its modular design and user-friendly interface make it ideal for educational purposes.

Experimental Setup

Students configure the analog computer as a summer or integrator, following prescribed block diagrams. By adjusting potentiometers and input signals, students observe the output voltages corresponding to mathematical operations. Through manual and repetitive operations, students explore different modes of operation and gain insights into analog computing principles.

Experiment #3: Analog Simulation of a First Order System (RC Circuit)

Objectives: To simulate a first-order system, specifically an RC circuit, and study its behavior.

Introduction

The RC circuit serves as a fundamental example of a first-order system, exhibiting transient and steady-state responses to input signals. By analyzing analog simulations, students can observe capacitor charging and discharging dynamics, influenced by resistor and capacitor values. These experiments deepen understanding of system time constants and stability.

Equipment

The GP-6 Analog Computer, in conjunction with an X-Y plotter, enables the simulation and visualization of RC circuit responses. Its versatility allows for the exploration of various circuit configurations, providing valuable insights into system behavior.

Experimental Procedure

Students derive system equations and draw analog block diagrams corresponding to specified resistor and capacitor values. By connecting the analog circuit and monitoring output voltages, students observe capacitor charging and plot response curves. Comparing experimental and theoretical results enhances comprehension of system dynamics.

Experiment #4: Analog Simulation of a Second Order Mass-Spring Mechanical System

Objectives: To simulate a second-order mass-spring system and analyze its behavior.

Introduction

The mass-spring system exemplifies a second-order mechanical system, offering insights into oscillatory behavior and energy dissipation. By configuring analog simulations, students explore the effects of mass, spring constant, and damping on system response. These experiments foster a deeper understanding of mechanical dynamics and stability.

Equipment

The GP-6 Analog Computer, in conjunction with an X-Y plotter, facilitates the simulation and visualization of mass-spring system responses. Its flexible configuration allows students to explore various parameter values, enriching their learning experience.

Experimental Procedure

Students derive system equations and draw analog block diagrams for specified mass, damping, and spring constant values. By connecting the analog circuit and monitoring displacement responses, students observe transient and oscillatory behavior. Comparative analysis of experimental data enhances understanding of system dynamics.

Experiment #5: Function Generation Using the Analog Computer

Objectives: To synthesize various time functions using the analog computer.

Introduction

Analog computers enable the generation of diverse time functions, essential in signal processing and control applications. By formulating differential equations corresponding to desired functions, students simulate and visualize waveforms using analog circuits. These experiments enhance proficiency in mathematical modeling and signal synthesis.

Equipment

The GP-6 Analog Computer, coupled with an X-Y plotter, facilitates the synthesis and visualization of time functions. Its versatile capabilities empower students to explore waveform generation and understand the underlying principles of analog signal processing.

Experimental Procedure

Students derive differential equations representing target time functions and draw analog block diagrams for simulation. By configuring analog circuits and monitoring output signals, students observe waveform generation and plot response curves. Comparative analysis with theoretical functions enriches understanding and reinforces theoretical concepts.

Experiment #6: Analog Simulation of a System of Coupled Masses

Introduction

In engineering, many systems have multiple inputs and outputs, presenting challenges in modeling and simulation. Systems with such characteristics are known as multiple-input, multiple-output (MIMO) systems. Studying the behavior of each subsystem separately and then combining them is one approach to understanding such complex systems. Alternatively, one can analyze the effect of each input on the overall system outputs individually and then combine these effects to determine the system's overall response. This experiment aims to introduce students to the modeling and simulation of MIMO systems using an analog computer.

Background

The system under consideration consists of two mass-spring subsystems coupled side-by-side. The equations governing the behavior of the system are provided below:

The system parameters are specified in Table 1. The coupling between the subsystems is represented by terms \(f_1\) and \(f_2\).

Equipment

• GP-6 Analog Computer
• X-Y Plotter

Procedure

Step 1: Rewrite Dynamical Equations

Rewrite the dynamical equations governing the behavior of the system in a form suitable for simulation.

Step 2: Analog Simulation

Simulate the system for each set of parameter values and plot the response of the system for both \(x_1\) and \(x_2\) using the X-Y plotter.

Step 3: Analytical Solution

Analyze \(x_1(t)\) and \(x_2(t)\) analytically assuming \(f_1(t) = 0\) and \(f_2(t) = 10u(t)\).

Step 4: Comparison

Compare the analytical results with the values obtained from the simulation.

Report

Your report should include:

• Free-body diagram for the different masses
• Analog simulation diagrams with labels
• Plots of the displacements of the variables \(x_1\) and \(x_2\)
• Discussion of results and conclusions

Experiment 7: Introduction to Digital Computer Simulation (MATLAB & SIMULINK)

Introduction

This lab introduces MATLAB, a powerful numerical simulation software, along with its graphical user interface (GUI) Simulink. These tools are used for solving modeling equations and obtaining system responses to different inputs.

Exploring the dynamics of coupled mass-spring systems and their simulations are fundamental in engineering education. In this experiment, we delve into MATLAB functionalities, particularly MATLAB m-files, to facilitate the simulation of systems with varying parameters. Understanding the behavior of such systems under different conditions is crucial in engineering design and analysis.

Programming in MATLAB using M-files

MATLAB's M-files offer a streamlined approach to executing simulations. These files contain MATLAB statements that can be saved and re-executed effortlessly. Particularly useful for Simulink simulations are:

• Setting and changing parameter values
• Executing model simulations
• Plotting simulation results

Modifying Simulink Models

Building upon the mass-spring system model developed previously, we introduce modifications to enable parameterization. This modification involves replacing specific values with symbolic variables like M, K, and B. By saving the model as an M-file, we enhance its reusability and flexibility.

Simulink programs can seamlessly integrate with MATLAB's environment, allowing for the loading of variable values directly from MATLAB. By defining variables in MATLAB and executing Simulink programs, the simulation automatically inherits these values, fostering a cohesive workflow.

Post-simulation analysis often involves visualizing output data. MATLAB's plotting capabilities empower users to create insightful visualizations effortlessly. By executing dedicated M-files, users can generate plots of simulation results, enhancing data interpretation.

For comprehensive analysis, MATLAB supports subplots and multiple runs within a single simulation. Subplots enable the simultaneous visualization of multiple variables, while multiple runs facilitate the exploration of parameter variations and their impact on system behavior.

Experiment #9: Simulation of Systems with Relative Displacements

Introduction

In systems where relative displacements between moving bodies are significant, traditional modeling approaches may fall short. This experiment focuses on simulating a two-mass system with relative displacements, challenging students to adapt their modeling techniques accordingly.

Procedure

1. Draw the free-body diagram for the two-mass system.
2. Write the modeling equation based on the free-body diagram, considering relative displacements.
3. Solve the equations and compare them with predefined equations.
4. Develop the Simulink diagram reflecting the system's behavior.
5. Execute the simulation and plot the desired responses using To Workspace blocks.

Experiment #10: Simulation of Systems Represented by State Variable Model

Introduction

Incorporating state variables enhances the modeling accuracy of complex systems. This experiment focuses on simulating a two-mass system using state variable models, providing students with insights into dynamic system analysis.

Procedure

1. Draw the free-body diagram for the two-mass system.
2. Write the modeling equations in state variable form, considering displacements and velocities.
3. Develop the corresponding Simulink diagram.
4. Execute the simulation and plot the desired responses.

Conclusion

The laboratory experiments on analog computers provide valuable insights into systems engineering, mathematical modeling, and simulation techniques. By conducting hands-on experiments, students deepen their understanding of complex systems and gain practical skills applicable in various engineering disciplines. Analog computing remains relevant in modern engineering education, offering a unique perspective on computational methods and system analysis.