Q1 Consider the flexural vibration of a beam, of length l, cross section area A, density ρ, second moment of area I and Young’s Modulus E, that is simply-supported at both ends but with a torsional spring, of stiffness kt attached at the right-hand end, as shown in figure Q1.
(a) The general solution for the flexural vibration of a beam is given by
where are constants and Clearly stat-ing the boundary conditions, derive the frequency equation for the beam in terms of kt and αl. (7 marks)
(b) It is proposed that an appropriate function for a guessed modeshape to be used in Rayleigh’s method is x
4 + ax3 + bx2 + cx + d. For the case where kt = , using the boundary conditions suggest appropriate values for a, b, c and d. Comment on whether this would also be an appropriate guess for the beam if a lumped mass was added to the righthand end of it.
Note that you are not expected to apply Rayleigh’s method. (4 marks)
(c) For the case where kt =, using a single element FE model, calculate an estimate of the first natural frequency. Suggest how this could be used alongside the answer to part (a) to give an accurate estimate for the first natural frequency of the beam. Comment on the likely accuracy of this estimated natural frequency compared to the Rayleigh’s method estimate that could be calculated using the guessed modeshape in part (b). (9 marks)
The mass and stiffness matrices for an element of beam of length L subject to flexural vibration are:
Q2 Consider the following open loop transfer function (OLTF) where a plant Gp (s) is controlled by a proportional controller Gp (s) = K.
(a) For K = 1 sketch the polar plot of the OLTF and determine whether the closed loop is stable by applying the Nyquist criterion (note: consider s = 0 a stable pole). In addition, estimate, using your sketch and/or calculations, the gain margin and phase margin of the closed loop.
For full marks you need to include the expressions of the OLTF gain and phase and fill in the table below:
(10 marks)
(b) Discuss how the phase margin and gain margin will evolve if the chosen K is greater than 1. Reflect on the possibility of achieving non-oscillatory response for any value of K and suggest, if needed, changes in the controller that could deliver over-damped behaviour. (10 marks)