MatLab The differential equation for a damped harmonic oscil

((MatLab))

The differential equation for a damped harmonic oscillator can be written as x + 2 omega_n x + omega^2_n x = 0 Where the natural frequency for mass m and stiffness k can be calculated as: omega_n = squareroot k/m and the damping ratio for damping constant c can be calculated as: =c/2mOmega_n MATLAB provide tools for simulating such differential equations. One of which is the function ODE45(), and it requires that you pass it the name of a function that it can call to calculate the acceleration. There is a critical value of damping that will cause a damped harmonic oscillator to return to it equilibrium position in the shortest time. This critical damping value can be calculated as: c_c = 2m squareroot k/m = 2squareroot km Use m = 10 kg, k = 30 N/m, X_0 = 1 m, and V_0 = 0 m/s Write two MATLAB functions Called ME_101_Hwk_5 that: calculates the critical value of damping c_c. calls ODE45 to simulate the motion of the oscillator with critical damping calls ODE45 to simulate the motion of the oscillator with half the critical damping value calls ODE45 to simulate the motion of the oscillator with twice the critical damping value plots the results in two separate subplots, arranged in a 2 times 1 grid. Each plot should have correct axis labels, with proper units, a legend indicating which line style corresponds to which damping value, and a title that indicates some or all of the other parameters responsible for the particular curves plotted. The first plot contains the position vs time for all three damping values the second lot contains the velocity vs time for all three damping values Called ode_fun that: accepts the current time and a column vector of the velocity and position calculates the acceleration of the mass at the current instance returns the acceleration and velocity in a column vector Submit the two MATLAB functions and the two plots all formatted in 10 pt Courier and arranged in portrait mode with headers containing your name and footers containing page numbers similar to this page.

Solution

Proposals for a parking ramp having been defeated, we plan to build a parking lot in the downtown urban renewal section. The cost of land is 200W + 100D, where W is the width along the street and D the depth of the lot in meters. The available width along the street is 100m, while the maximum depth available is 200m. We want to have at least 10,000m2 in the lot. To avoid unsightly lots, the city requires that the longer dimension of any lot be no more than twice the shorter dimension. Formulate the minimum cost design problem.