An electric motor drives a pulley with radius R via a stiff shaft that can be...

  

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An electric motor drives a pulley with radius R₁ via a stiff shaft that can be modelled as rigid. The pulley is connected to a larger pulley of radius R₂ via a belt of stiffness k of the section joining the pulleys on each side. The total moment of inertia of the motor and the pulley on its shaft is IT. The other pulley and what rotates with it has a moment of inertia of IP. 2.1 Show using Lagrange's equations that the equations of motion for small rotational oscillations can be written as 001 R² -R₁ R₂ CH** +2k IP. -R₁ R2 R₂ 2.2 Why must one natural frequency of this system be zero? (2 marks) 2.3 The other natural frequency, is to be designed to be 10 Hz, to avoid low frequency vibration problems. Show that the stiffness k will achieve this is such that w₂² = (2017)² = 2k (+³) (4 marks) 2.4 Show that the mode shape associated with this natural frequency can be given by 0₂ = - 2110₁ (4 marks) R₁ lp 0 S = = wn 2k (6 marks) 2.5 If each side of the belt has damping c, by using the fact that for Rayleigh damping of mode i the damping ratio is such that 25;wi = a + Bw? or otherwise show that the mode of vibration of part (4) will have a damping ratio of (2 marks) 2.6 Give an equation for a damped natural frequency that could be used to correct h to allow for damping. (2 marks) e Ir R₁ k R₂ le 0₂ An electric motor drives a pulley with radius R₁ via a stiff shaft that can be modelled as rigid. The pulley is connected to a larger pulley of radius R₂ via a belt of stiffness k of the section joining the pulleys on each side. The total moment of inertia of the motor and the pulley on its shaft is IT. The other pulley and what rotates with it has a moment of inertia of IP. 2.1 Show using Lagrange's equations that the equations of motion for small rotational oscillations can be written as 001 R² -R₁ R₂ CH** +2k IP. -R₁ R2 R₂ 2.2 Why must one natural frequency of this system be zero? (2 marks) 2.3 The other natural frequency, is to be designed to be 10 Hz, to avoid low frequency vibration problems. Show that the stiffness k will achieve this is such that w₂² = (2017)² = 2k (+³) (4 marks) 2.4 Show that the mode shape associated with this natural frequency can be given by 0₂ = - 2110₁ (4 marks) R₁ lp 0 S = = wn 2k (6 marks) 2.5 If each side of the belt has damping c, by using the fact that for Rayleigh damping of mode i the damping ratio is such that 25;wi = a + Bw? or otherwise show that the mode of vibration of part (4) will have a damping ratio of (2 marks) 2.6 Give an equation for a damped natural frequency that could be used to correct h to allow for damping. (2 marks) e Ir R₁ k R₂ le 0₂