Assignment -

Vibrations are everywhere, so too are the eigenvalues (or frequencies) associated with them. This project offers a glimpse into the wide range of eigenvalue analysis. Many other applications, such as the VLSI, can essentially be written into the same vibration problem.

Consider a vibrating system interconnected as in the following figure,

where m_i, k_j, and c_l represent the mass, stiffness, and damping constants, respectively. Let the column vector x = [x₁, x₂, x₃, x₄]^T stands for vertical displacements (see the drawing) of the masses from their equilibria. Assume that the Newton's second law prevails, that the restoring force of a spring is negative proportional to its displacement, and that the damping force is negative proportional to its velocity.

1. Assume that the Newton's second law prevails, that the restoring force of the sprint is negative proportional to its displacement, and the damping force is negative proportional to the velocity. Derive the equation of motion in the matrix form

M d²x/dt² + C dx/dt + Kx = 0. (1)

Express each 4 × 4 matrix explicitly in terms of the parameters m_i, k_j, and c_l. (Hint: Use the basic physics force = m ∗ (acceleration), but pay attention to the direction where the force is applied and the net force.)

2. By assuming that a fundamental solution is of the form

x(t) = e^λtv (2)

for some scalar λ and nontrivial vector v, show that λ and v must satisfy the quadratic eigenvalue problem

Q(λ)v := (Mλ² + Cλ + K)v = 0. (3)

(Hint: Substitute x(t) from (2) into (1) and simplify.)

3. A conventional way to solve the quadratic eigenvalue problem (eq:eigenvalue) is to linearize it in the following way. Define

Consider the generalized eigenvalue problem

Show that y solves (3). Also, what is z?

Please provide me solutions with complete explanation.