Please solve whatever problems you are able to. I am happy to post additional questions to answer further parts of the above problems.
Vibration analysis of a simply-supported beam with an attached mass and spring. A simply-supported beam has an attached mass and a spring at its midpoint, as shown. Assume that the beam is uniform and its geometric and material properties are specified, including the moment of inertia of the cross-section about its neutral axis. The attached mass has a mass of m and the spring constant of the attached spring is k. The beam is subjected to a point harmonic load Fsn at x=0.75L. Damping is ignored in this study. Assume that the motion is small from its static equilibrium position and the beam is originally at rest.
F sin(ωt)
4
m
x
k
E, P, L, A, I
2
The exact solution is very difficult to find and we will use the Rayleigh-Ritz approach to find an approximate solution. A one-term approximation is employed, i.e. vx(t) = T where the trial function is selected as x(t) = sin(L). The problem is thus reduced to an IVP of an SDOF system for O(t) as
m(d^2x/dt^2 + kx) = Fsin(ωt), x(0) = 0, dx/dt(0) = 0
1. Determine the equivalent mass meg based on the total system kinetic energy. Both the beam and the attached mass should be included.
2. Determine the equivalent stiffness constant keg based on the total system potential energy. Both the beam and the attached spring should be included.
3. Determine the equivalent force fegt based on the work done by the given force.
4. Find the lowest natural frequency of the system.
5. Find the vibration response of the attached mass.
