Engineering Analysis Assignment -

For every problem, provide The MATLAB script/function files that solve the problems.

Problem 1:

Plot the function f(t) = (x+5)²/(4+3x²) for -3 ≤ x ≤ 5.

using plot command. Use the array for variable x with spacing Δx = 0.01. Label the axis of the plot with names "Variable x" and "Function f(x)"

Problem 2:

A parametric equation is given by

x = 3t/(1+t³), y = 3t²/(1+t³)

(Note that the denominator approaches 0 when t approaches -1) Plot the function (the plot is called the Folium of Descartes) by plotting two curves in the same plot-one for -30 ≤ t ≤ -1.6 and the other for -0.6 ≤ t ≤ 40.

Problem 3:

The position x as a function of time of a particle that moves along a straight line is given by

x(t) = 0.41t⁴ - 10.80t³ + 64t² - 8.2t+ 4.4 ft

The velocity v(t) of the particle is determined by the derivative of x(t) with respect to t, and the acceleration a(t) is determined by the derivative of v(t) with respect to t.

Derive the expressions for the velocity and acceleration of the particle, and make plots of the position, velocity, and acceleration as functions of time for 0 ≤ t ≤ 8 s.

Label the axis appropriately with correct units.

Problem 4:

Write a function that solves problem. Provide all math derivations that demonstrate how the polar coordinates of the sum vector are related to the polar coordinates of r₁ and r₂.

In polar coordinates a two-dimensional vector is y given by its radius and angle (r, θ). Write a user-defined MATLAB function that adds two vectors that are given in polar coordinates. For the function name and arguments use [r th]= AddVecPol (r1 f th1, r2 th2), where the input arguments are (r₁, θ₁) and (r₂, θ₂), and the output arguments are the radius and angle of the result. Use the function to carry out the following additions:

(a) r₁ = (5, 23^o), r² = (12, 40^o). (b) r₁ = (6, 80^o), r2 = (15, 125^o).

Problem 5:

1. Write a MATLAB function TrigFun that returns value of a function given by the equation

f(x) = a₁ cos(ω₁x) + a2 sin(ω₂x)

at given a₁, a₂, ω₁, ω₂, and x. The list of input parameters of function TrigFun should include only three parameters: array a = [a₁, a₂], array ω = [ω₁, ω₂], and value x.

2. Prepare a script that uses TrigFun to plot values of f(x) in the interval 0 ≤ x ≤ 10 with equal spacing Δx = 0.05 at a₁ = 1, a₂ = 0.25, ω₁ = π, ω₂ = 4π.

Problem 6:

A two-dimensional state of stress at a point in a loaded material is defined by three components of stress σ_xx, σ_yy, and τ_xy. The maximum and minimum normal stresses (principal stresses) at the point, σ_max and σ_min, are calculated from the stress components by:

σ_max/min = (σ_xx+σ_yy)/2 ± √(((σ_xx-σ_yy)/2)² + τ²_xy)

Write a user-defined NIATLAB function that determines the principal stresses from the stress components. For the function name and arguments use [Smax, Smin] = princstress (Sxx, Syy, Sxy). The input arguments are the three stress components, and the output arguments are the maximum and minimum stresses.

Use the function to determine the principal stresses for the following states of stress:

(a) σ_xx = -190MPa, σ_yy = 145 MPa, and τ_xy = 110 MPa.

(b) σ_xx = 14ksi, σ_yy = -15 ksi, and τ_xy = 8 ksi.