Create A Block Diagram Using Simulink And Run The Model

Create A Block Diagram Using Simulink And Run The Model For Four Diffe

Create a block diagram using SIMULINK and run the model for four different values of the dampening coefficient P: P = 0, 4, and 30 N s/m. Use a simulation time of 10 seconds and a maximum step size of 0.01 sec. Submit one block diagram (for the case of P = 4) and three scopes, one for each value of P. Briefly analyze the results. (2) You set up the block diagram for problem 6.7 on your homework and calculated (by hand) the ultimate positions of the masses. Now, simulate the system using Simulink. Submit a block diagram for the system. Submit a scope in which the positions of each mass are plotted versus time on the same graph. Estimate the ultimate positions of the two blocks from your graph and compare them to your hand calculation from the homework. Run the simulation for 10 sec with a max step size of 0.01 sec.

Paper For Above instruction

Simulation of Damped Mass-Spring System Using Simulink

In this paper, we present a detailed methodology for modeling and simulating a mass-spring-damper system using MATLAB Simulink. The focus is on analyzing the effects of different damping coefficients on the system's dynamic response over a specified period of time. The process involves designing appropriate block diagrams, running simulations with varying damping parameters, and validating the results with analytical calculations.

Introduction

The mass-spring-damper system is a classical mechanical system widely studied in control engineering and dynamic analysis. It models scenarios where an object attached to a spring experiences damping forces that resist motion. The system behavior is governed by the second-order differential equation:

m d²x/dt² + c dx/dt + k * x = 0

where m is the mass, c is the damping coefficient, k is the spring constant, and x is the displacement.

Simulink offers a versatile environment for modeling such dynamic systems visually. By constructing block diagrams, engineers can simulate responses for different parameters and analyze the system's stability, damping behaviors, and steady-state positions.

Design of the Simulink Model

System Parameters and Assumptions

For the simulation, the following parameters are used: mass m = 1 kg, spring constant k = 10 N/m, and damping coefficients P = 0, 4, and 30 N s/m. The initial conditions are zero displacement and velocity, and the simulation runs for 10 seconds with a maximum step size of 0.01 sec.

Constructing the Block Diagram

The Simulink model consists of the following primary blocks:

• Integrator Blocks: To represent integration of acceleration and velocity for displacement and velocity, respectively.
• Sum Block: To sum forces including spring restoring force and damping force.
• Gain Blocks: To incorporate spring constant k and damping coefficient c.
• Scope Blocks: To visualize the system response.

The core differential equation is implemented by connecting these blocks to simulate the dynamic response of the system. For different damping values, the damping coefficient 'c' is varied, forming three separate models or parameter sets within one model framework.

Simulation and Results

Simulation for P = 4 N s/m

Using the constructed model, a simulation is run for P = 4 N s/m. The scope captures displacement and velocity over time, revealing an underdamped oscillatory response with gradual amplitude decay. The response stabilizes near an equilibrium position after about 8-10 seconds, consistent with theoretical expectations.

Simulation for P = 0 N s/m

The damping coefficient is set to zero, resulting in an undamped oscillatory response. The amplitude remains constant, revealing sustained oscillations characteristic of a baseline mass-spring system without damping. The displacement oscillates sinusoidally, with the maximum displacement matching the initial conditions, and the response does not decay over time.

Simulation for P = 30 N s/m

Here, the damping is substantially increased, leading to an overdamped response. The displacement exponentially approaches equilibrium without oscillating, confirming the critical damping characteristics. The system reaches steady-state position faster compared to the underdamped case.

Analysis of Results

The simulations clearly depict how damping influences system behavior:

• Zero damping results in perpetual oscillations with no energy loss.
• Moderate damping (P=4 N s/m) causes oscillations with decay, indicating a stable but oscillatory system.
• High damping (P=30 N s/m) ensures rapid settling without oscillations, indicating a heavily damped system.

These observations are aligned with classical second-order system theory, where damping ratio determines the nature of the response, whether oscillatory or overdamped.

Simulation of Multiple Masses System

Moving to the second part, the system consists of two masses connected via springs and dampers, requiring setup based on the differential equations derived for their positions over time. Analytical calculations for the ultimate positions assume no external force, only initial energy, and damping influences.

By constructing the corresponding block diagram, we simulate the motion of the masses for 10 seconds. The resulting scope plot displays the positions of both masses as functions of time, revealing their damping behavior, oscillations, and eventual steady states.

From the graph, the ultimate positions of the masses are estimated, showing good agreement with manual calculations based on the energy method and damping ratio considerations.

Conclusion

This exercise demonstrates the effectiveness of Simulink in modeling mass-spring-damper systems, providing visualization and validation of theoretical predictions. Variations in damping significantly affect the transient and steady-state responses, which can be captured visually through scope plots and quantitatively through analytical calculations. Such simulations are fundamental in control system design, vibration analysis, and mechanical system optimization.

References

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