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Assignment Question 3 in Applied Mechanics

Fig. Q3 shows a rigid light beam which is hinged at O to a wall. The beam carries a point mass of m at the free end. The beam is connected at its mid-point to a spring (stiffness k) which is also connected to a support that moves with a vertical motion y = Y sinwt. A damper (coefficient c) connects the mid-point of the beam to a fixed support.

(i) By taking moments about O of the forces acting on the beam show that an equation of motion can be obtained with the form

4mx + cx + kx = kY sin ωt

Please note that in the equation above the first x should have two dots above it and the second x should have one dot above it only. The reason why they haven't is because I am unsure how to type them in, my apologises.

where x is the displacement of the mid-point of the beam.

(ii) Determine the maximum value for w if the amplitude ratio of the x displacement (i.e. X) to Y is not to exceed 10%.

Take m = 2 kg; k = 6.25 kN/m; c = 45 Ns/m.
(The Answer to part (ii) is given as 87.8 rad/s)

Please could you respond to this question in its simplest form, with any additional notes for me to understand the process of the answer. I look forward to your reply. Many Thanks for your time and effort. Mr P Stones.
PS. I have also included an attachment of the question and sketch.