Matlab Sessions Laboratory 5: The Mass Spring 145870

Matlab Sessions Laboratory 5mat 275 Laboratory 5the Mass Spring Syst

This assignment entails examining harmonic oscillation through modeling the motion of a mass-spring system with differential equations. The objectives include investigating how system parameters influence the solutions and behaviors of the system, analyzing the motion via graphical representation, and exploring the effects of parameters such as mass and spring stiffness on the oscillations and energy conservation. The primary MATLAB command used in this context is the ode45 function, which will be employed to solve the relevant differential equations, both in the case of undamped and damped systems. Students will perform simulations to analyze periods, amplitudes, maximum velocities, and energy conservation, and will extend these analyses to explore the effect of damping coefficients and parameter variations through both numerical and analytical methods.

Paper For Above instruction

Harmonic oscillations are fundamental phenomena in classical mechanics, exemplified by the motion of a mass attached to a spring. The mathematical modeling of such systems involves differential equations that describe the displacement of the mass over time, facilitating a comprehensive understanding of oscillatory behavior. In this paper, we analyze the dynamics of a mass-spring system without damping and with damping, leveraging MATLAB's ode45 solver to simulate and visualize oscillations under various parameter conditions.

Modeling the Mass-Spring System without Damping

The classical mass-spring system's governing equation stems from Newton's second law, considering the forces acting on the mass. When a mass m is suspended from a spring with spring constant k, the equilibrium position corresponds to a spring length ℓ0. If the mass is displaced downward by an amount y, the resulting forces balance according to mg - k(ℓ + y - ℓ0) = m d²(ℓ + y)/dt². Simplifying leads to the second-order differential equation: m d²y/dt² + ky = 0. Dividing through by m and introducing ω₀² = k/m results in the standard simple harmonic oscillator form: d²y/dt² + ω₀² y = 0.

To analyze such oscillations numerically, the equation is converted into a system of first-order equations: letting v = dy/dt, then dv/dt = -ω₀² y. This system can be efficiently solved using MATLAB's ode45 function, with initial conditions such as y(0) = 0.1 meters and v(0) = 0, representing an initial downward displacement of 10 centimeters with no initial velocity.

The MATLAB implementation involves defining a function for the derivatives, passing parameters to allow for easy variation, and plotting the displacement and velocity over time. The solution demonstrates characteristic sinusoidal functions, confirming theoretical expectations of harmonic motion.

Analysis of Harmonic Motion

From the generated plots, several key features are discerned. The displacement y(t) displays periodic oscillations with a specific period T, which can be read graphically and computed analytically as T = 2π/ω₀. For ω₀ = 2, the period is approximately π radians or about 3.14 seconds. The motion lattice shows that the mass does not "come to rest" due to perpetual oscillations without damping; however, it remains within a fixed amplitude around the equilibrium position.

The amplitude of the oscillations is directly given by the initial displacement, here 0.1 meters. The maximum velocity occurs at the equilibrium point where the displacement crosses zero, and can be computed as v_max = ω₀ * amplitude, i.e., approximately 0.2 m/s in the example.

By varying system parameters m and k, it is observed that increasing mass m lowers ω₀, thereby increasing the period T, whereas increasing k (stiffness) increases ω₀ and reduces T. MATLAB simulations with different m and k values reinforce this inverse relationship, and the phase plots (v vs y) show elliptical trajectories characteristic of harmonic oscillators. These plots reveal the energy exchange between kinetic and potential forms, with total energy remaining constant in the ideal case.

Energy Conservation and Phase Space Analysis

In the absence of damping, the total mechanical energy E = ½ m v² + ½ k y² remains conserved. Plotting E(t) over time confirms that the energy remains constant within numerical accuracy. This conservation aligns with the theoretical derivation where dE/dt = 0, reflecting no energy loss. The phase plot (v vs y) exhibits elliptical geometry, indicating continuous oscillations in the phase space without convergence to a point, consonant with undamped harmonic motion.

Damped Mass-Spring System Dynamics

Real-world systems often experience damping due to viscous forces, modeled by adding a damping term c dy/dt to the equation. The resulting differential equation d²y/dt² + 2p dy/dt + ω₀² y = 0 captures the damping effect, with p = c/(2m). Numerical simulations using MATLAB's ode45, incorporating the damping coefficient c, elucidate how increased damping influences oscillatory behavior.

Simulations portray that higher damping coefficients result in faster decay of oscillations, eventually approaching equilibrium without oscillation if damping is sufficiently large. To quantify this, the minimal time t₁ for the solution to satisfy |y(t)| 1, the system is overdamped, preventing oscillations entirely.

Mathematically, the energy E(t) diminishes over time due to damping, as shown by the relation dE/dt 0. Phase plots reveal spiraling trajectories towards the origin, illustrating energy loss, in contrast with undamped systems where the ellipse persists indefinitely. The critical damping value c_critical can be computed analytically as c = 2√(m k), marking the boundary between oscillatory and non-oscillatory responses.

Impact of Damping Coefficient c

Numerical experiments with various damping coefficients (c = 2, 4, 6, 8) confirm that increasing damping reduces oscillation amplitude more rapidly; the graphs show increasingly swift decay of displacement over time. The phase space trajectories also become more tightly spiraled, approaching the origin more directly. For c ≥ c_critical, oscillations cease, and the system smoothly returns to equilibrium.

Energy Dissipation and Phase Space Behavior in Damped Systems

The energy E(t) in damping cases decreases monotonically, consistent with the negative energy derivative: dE/dt 0. In the phase plot, the trajectory spirals inward, reflecting energy loss. Unlike undamped oscillations, the system in the damped case asymptotically approaches rest, illustrating the dissipative nature of damping forces. When pruning the system at large times, the phase trajectory approaches the origin, denoting the mass at equilibrium with zero velocity.

Conclusions

The investigated models highlight the crucial influence of system parameters on harmonic oscillations. The stiffness k and mass m determine the frequency and period of oscillation, with stiffer springs or lighter masses resulting in faster oscillations. Damping introduces energy dissipation, reducing oscillation amplitude and eventually halting motion for sufficiently large damping coefficients. These insights are vital in designing mechanical systems to achieve desired dynamic responses, ensuring stability or controlled oscillation as required.

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