Case-Study-No. 2

The governing differential equations of motion are given as:

[25 Bonus Marks for deriving these equations. Use either the 2nd law of Newton or Lagrange’s equations £to derive the above equations].

Solve this set of differential equations using matrix method (eigenvalues & eigen vectors)

Solve this set of differential equations numerically using the fourth-order Runge-Kutta method. [The details of two-iterations are required. Then it can be programmed for additional iterations. Both hard and electronic copies of the source code of the computer program are required!]

(usex ⃗(t=0)=0 and x ̇ ⃗(t=0)=0 and ∆t=T/10 where is the smallest time period T=2π/ω)

Plot (x_1,x_2,x_3,and x_4 ),velocity,and accelerations in terms of time
Note that:
In cases (b and c), the expressions for x1(t), x2(t), x3(t), and x4(t) are derived analytically, then the velocity and acceleration can be determined by taking the first and second derivative of x(t).
In cases (d and e), the Runge-Kutta method (both 4th-order and 5th-order) results in displacement and velocity vectors. Acceleration is to be determined using numerical differentiation (central difference)

Proposed References for item (a):