Q1 Consider the flexural vibration of a beam, of length 7. density p. cross-sectional area A, second moment of area I and Young's Modulus E that is simply supported at one

end. At the other end the beam is constrained by rollers and connected to a spring of stiffness k as shown in the Figure Q1. beam Figure Q1 k 77777777777 ΩΩ (a) Using the general solution for flexural vibration of a uniform Euler-Bemoulli beam y(x, t) = [Ã sin(ax) + B cos(ax) + C sinh(ax) + Dcosh (ax)] x x [E sin(wt) + Fcos (wt)]/n(c) Discuss how the solution to part (a) might be sued to assess the accuracy of the FE natural frequency. Also discuss how you might simplify the FE model if you wished to find the second natural frequency. (d) For the beam in Figure Q1, comment on the likely accuracy of the predicted first natural frequency using Rayleigh's method and the guessed deflection. y(x, t) = sin cos (wt) The mass and stiffness matrices for an element of beam of length L subject to flexural vibration are: M-PAL 420 156 22L 54 22L 54 -13L 4L² 13L -312 13L 156 -22L -13L-31²-221 EI 12 6L -12 GL 6L AL² -6L 2L² -12 -6L 12 -6L 6L 2L2 -6L AL²/n(c) Discuss how the solution to part (a) might be sued to assess the accuracy of the FE natural frequency. Also discuss how you might simplify the FE model if you wished to find the second natural frequency. (d) For the beam in Figure Q1, comment on the likely accuracy of the predicted first natural frequency using Rayleigh's method and the guessed deflection. y(x, t) = sin cos (wt) The mass and stiffness matrices for an element of beam of length L subject to flexural vibration are: M-PAL 420 156 22L 54 22L 54 -13L 4L² 13L -312 13L 156 -22L -13L-31²-221 EI 12 6L -12 GL 6L AL² -6L 2L² -12 -6L 12 -6L 6L 2L2 -6L AL²