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By J. E. Rubio

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Extra resources for Control and Optimization: The Linear Treatment of Nonlinear Problems

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The rotor is carefully balanced, the maximum out-of-balance being lo-' kg m. In previous applications the turbo-charger has been run successfully at 35 OOO rev/min and 55 OOO rev/min. Calculate the amplitude of the deflection of the mid-point of the shaft, and its phase with respect to the out-of-balance excitation when it is running at these speeds. In a new application it was proposed to run the turbo-charger at 45 OOO rev/min. Comment on the implications of this proposal. 1 m. 71 x lo6 N/m. The effective mass, me,, of the system can be taken to be the mass of the rotor plus one half of the mass of the shaft.

134. 579 mm. ~ Example 11 A gun is designed so that when fired the barrel recoils against a spring. At the end of the recoil a viscous damper is engaged which allows the barrel to return to its equilibrium position in the minimum time without overshoot. 6 m. 46 The vibrations of systems having one degree of freedom [Ch. 1 m from its initial position. 62 = i500(30)2, that is k = 175 kN/m. Since critical damping is required, the damping coefficient is 2J(500 x 175 x lo3) = 18700 Ns/m. For critical damping x =(A If x + Bt)e-"' = xo and i= 0 x = xo(1 at t = 0, + wt)e-"?

Since the viscous damping force c i is dependent on the frequency of oscillation, it is not a suitable way of modelling the internal damping of solids and structures. t + k x = 0. However, it has been observed from experiments carried out on many materials and structures that under harmonic forcing the stress leads the strain by a constant angle, 2. Thus for an harmonic strain, E = E~ sin vt, where v is the forcing frequency, the induced stress is G = o0 sin (vt + z). Hence G = oo cos 2 sin vt + G~ sin x cos vt 42 The vibrations of systems having one degree of freedom cos = a.

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