Reference no: EM132419666

Vibration of Mechanical System Assignment - Multi-DOF system, Vibration Absorption, Vibration of an Euler-Bernoulli beam (on MATLAB)

A) Free Vibration of 3-DOF system

Consider the above 3 DOF system (Fig.1). The numerical values of masses (m_i) and stiffnesses (k_i) will be generated by a random number generator in your program (using function rand), and the subsequent results must follow.

1) Establish the set of governing differential equation for free vibration in the matrix form.

2) Establish the frequencies of free vibration. (Eigen Value method).

3) Find the modeshapes (Eigen Vectors). Draw them.

4) Find the generalized masses and generalized stiffnesses.

B) Forced Vibration of 3-DOF system

Consider a force F₁ = f cos(ωt) acts on m₁, and F₂ = f sin(ωt) acts on the m₂. No force acts on the m₃.

1) Find the generalized forcing vector. Plot as a function of time.

2) Find the principal coordinates as functions of time. Plot them.

3) Establish the expressions for the modal displacements. Plot them as function of time.

C) 2 DOF system: Vibration attenuator

The vibration amplitude of Mass M (Fig.2) is found to be unacceptably large. A small mass m is attached through a spring to it, in order to reduce its oscillations. The mass 'm' is a variable. Plot the resultant amplification factor of M vs. the tuning factor of m.

Plot the resultant amplification factor of M vs. the mass m; for an external forcing F(t) = F₀cos(ω_et) acting on M.

D) Vibration of a Floating Euler-Bernoulli beam

1) Consider a Mild Steel (ρ = 7850 kg/m³, E = 209 GPa, L = 100m) Euler-Bernoulli beam. For rectangular cross section : B = 2 m, D = 1m.

2) Generate the first 10 transverse vibration modeshapes for the given boundary conditions (Check attached EXCEL Sheet). Check their orthogonality.

3) Establish the inertia matrix. Mesh it. Establish the stiffness matrix. Mesh it.

4) Write the Modal GDE in the matrix form, for free vibration. Include 5% proportional damping.

5) Plot the principal coordinates as function of time (both C.F. and P.I., first separately and then combined).

6) Generate the total deflection as a function of space and time and mesh it.

7) Assume a harmonic Froude-Krylov forcing F(t) α cos(kx - ωt), obeying the deep water dispersion relation. The amplitude of the wave is derived from the wave-breaking limit (Derive the amplitude of the Froude-Krylov force). The wavelength equals the length of the beam. Establish the expression for the generalized forcing and plot them as a function of time. All formulations must be shown in the hard copy.

8) Write the Modal GDE in the matrix form, for forced vibration.

9) Plot the principal coordinates as function of time, including the complimentary functions.

10) Generate the total deflection as a function of space and time and mesh it.

11) Generate the total bending moment, bending stress, shear stress as functions of space and time, and mesh them. Mention the location, instant and magnitude of its maximum.