Matlab Sessions Laboratory 5: The Mass Spring
Matlab Sessions Laboratory 5mat 275 Laboratory 5the Mass Spring Syst
Examine harmonic oscillation in a mass-spring system modeled by differential equations. Determine how parameters influence solutions, analyze the behavior of the system from graph data, and investigate effects of parameters on system dynamics, including damping effects. Use MATLAB's ode45 function to simulate and analyze these behaviors, focusing on undamped and damped harmonic motions, energy conservation, and phase relationships.
Paper For Above instruction
Harmonic oscillation phenomena in mass-spring systems serve as foundational models in classical mechanics, illustrating key principles of differential equations, energy conservation, and dynamic response analysis. This paper discusses the modeling, simulation, and analysis of such systems, particularly emphasizing the effects of system parameters and damping on motion behavior, using MATLAB as a computational tool.
Introduction and Theoretical Background
The mass-spring system is a classical example of simple harmonic motion (SHM), characterized by restoring forces proportional to displacement. The undamped system typically involves a mass attached to an ideal spring governed by Newton's second law, leading to a second-order ordinary differential equation (ODE):
m \dfrac{d^2 y}{dt^2} + k y = 0,
where m is the mass, k is the spring constant, and y(t) is the displacement from equilibrium.
Rearranged, the ODE becomes:
\[\frac{d^2 y}{dt^2} + \omega_0^2 y = 0,\]
with \(\omega_0 = \sqrt{\frac{k}{m}}\) being the natural angular frequency of oscillation. Its solutions describe sinusoidal behavior characterized by amplitude and period, with the period \(T = \frac{2\pi}{\omega_0}\).
Simulation of Undamped Harmonic Motion
Using MATLAB, the second-order differential equation is converted into a system of first-order equations:
\[
\begin{cases}
y' = v, \\
v' = -\omega_0^2 y,
\end{cases}
\]
where v(t) = y'(t). The MATLAB function ode45 is employed to numerically solve this system given initial conditions (e.g., y(0)=0.1m, v(0)=0). The results include displacement and velocity over time, which can be plotted for analysis.
Parameter Effects on System Behavior
The system's natural frequency \(\omega_0\) depends on m and k. Larger m results in a lower \(\omega_0\), longer period, and slower oscillations; larger k produces higher \(\omega_0\), resulting in faster oscillations. MATLAB simulations with varied m and k provide visual verification, illustrating their influence on period and amplitude.
Energy Conservation in Undamped Systems
The total mechanical energy:
\[
E = \frac{1}{2} m v^2 + \frac{1}{2} k y^2,
\]
should remain constant in an ideal undamped system. Numerical simulations confirm this by plotting E over time, demonstrating negligible fluctuation when damping is absent.
Damped Harmonic Motion
Introducing damping involves an additional term proportional to velocity, c y', leading to the modified equation:
\[
m y'' + c y' + k y = 0,
\]
or in normalized form:
\[
y'' + 2 p y' + \omega_0^2 y = 0,
\]
where \(p = c/(2m)\). MATLAB simulations with added damping coefficient c reveal how damping influences oscillation, causing energy decay and eventual cessation of motion for sufficiently large c.
Analysis of Damping Effects
The damping coefficient c affects the frequency and amplitude decay, with critical damping occurring at a specific value \(c_{crit}\), where oscillations cease without overshoot. MATLAB simulations across varied c values illustrate transition from underdamped to overdamped behavior. The smallest c eliminating oscillations is derived analytically as:
\[
c_{crit} = 2 \sqrt{m k}.
\]
Energy Decay in Damped Systems
The energy in damped systems diminishes over time, as shown by plotting the sum of kinetic and potential energies. The rate of decrease aligns with the damping factor c, where an increase in c results in more rapid energy loss. Mathematically, from the energy derivative:
\[
\frac{dE}{dt} = - c v^2,
\]
indicating that energy is continually dissipated when c > 0, and increases (hypothetically) when c
Phase Space and System Behavior
Phase plot v vs y reveals the system's energy exchange, with elliptical trajectories for underdamped cases, shrinking spirals for damping, and non-oscillatory responses for critical or overdamped states. These plots visually demonstrate how damping modifies the system's phase relationship and energy transfer.
Conclusion
Modeling and simulation of harmonic oscillators demonstrate the profound influence of parameters such as mass, spring constant, and damping coefficient on system behavior. MATLAB serves as an effective tool to visualize solutions, analyze energy conservation, and understand the dynamic transitions from oscillatory to non-oscillatory regimes. These insights have applications across engineering, physics, and biological systems exhibiting oscillatory dynamics.
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