Single-Degree-Freedom Vibration Simulation

Assignment Requirements Summary The task is to model and simulate a single-degree-of-freedom vibration system representing a machine mounted on a resilient support and subjected to a harmonic force. The system consists of: * Machine mass: m = 125 kg * Spring stiffness: ke = 60.5 kN/m * Damping coefficient: ce = 950 kg/s * Harmonic force: F = F₀ sin(Ωt) * Force amplitude: F₀ = 150 N * Excitation frequencies: * f₁ = 3.5 Hz, so Ω₁ = 21.991 rad/s * f₂ = 14 Hz, so Ω₂ = 87.965 rad/s The motion of the machine is governed by the equation: m\ddot{x}(t) + c_e\dot{x}(t) + k_e x(t) = F_0 \sin(\Omega t) where x(t) is the vertical displacement response of the machine. Required Work 1. Develop a MATLAB Model Create a MATLAB script that solves the equation of motion using a suitable ODE solver, such as ode45. The MATLAB model should include: * Definition of system parameters: mass, damping, stiffness, force amplitude, and excitation frequency. * Conversion of the second-order differential equation into first-order equations. * An m-function to represent the equation of motion. * Simulation of the system response over the time range 0 to 10 seconds. * Separate simulations for both excitation frequencies: * Ω = Ω₁ * Ω = Ω₂ 2. Develop a Simulink Model Create a Simulink block diagram to solve the same equation of motion. The Simulink model should include: * A sinusoidal force input block. * Blocks representing the mass, damping, and spring terms. * Integrator blocks to obtain velocity and displacement. * Output blocks or scopes to display the displacement response. * Simulations for both excitation frequencies. 3. Simulate the Dynamic Response Use both the MATLAB and Simulink models to simulate the vertical displacement of the machine over the interval: 0 \leq t \leq 10 \text{ s} The simulations must be carried out for: * Low-frequency excitation: 3.5 Hz * High-frequency excitation: 14 Hz 4. Plot the Results Produce displacement-time plots showing the machine vibration response x(t). The results should include: * MATLAB displacement response at 3.5 Hz * MATLAB displacement response at 14 Hz * Simulink displacement response at 3.5 Hz * Simulink displacement response at 14 Hz The plots should clearly show how the machine displacement changes with time under each excitation frequency. 5. Discuss the Results Explain the differences in vibration response between the two excitation frequencies. The discussion should include: * Why the machine vibrates differently at 3.5 Hz and 14 Hz. * The effect of excitation frequency on displacement amplitude. * The relationship between excitation frequency and the system’s natural frequency. * The role of damping in reducing vibration. * Whether the system response is larger near resonance or lower at higher frequency. Overall Aim The overall aim is to compare MATLAB and Simulink modelling approaches for a damped single-degree-of-freedom vibration system and explain how excitation frequency affects the machine’s vibration response.

Project ID: 40426895