Dynamics and Control

- The purpose of the anti-vibration mounts is to isolate the instrument from vibrations of the basement. A sensitive instrument with a mass of 2 is supported on ten bubble mounts each of stiffness valid for low deformation amplitudes. The mount exhibits viscous damping with coefficient and has a nonlinear stiffness at high amplitude deformation as it is shown in Figure A1.

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Figure A1. Schematic of bubble mount and deformation response

- Determine the steady state vibration amplitude of the instrument if the basement vibrates with a frequency of and an amplitude of . Assume that the buble mounts are operating in the linear zone of their deformation.

(3 marks)

- Determine the range of excitation frequency where the isolation effect takes place. Explain the importance of isolation effect in this case?

(2 marks)

- Comment qualitatively how the system response behaves. You can use the relevant matlab scrip in the module shell to demonstrate your response. (3 marks)

- How the response of system change if the bubble mounts operate outside the range of linear deformation (assume bubble defelction was over than 4 mm).

(2 marks)

- An engine with mass of M=2000 kg is supported on a rubber mount with stiffness of k=1.5 MN/m (Figure A2). To reduce the engine vibrations a dashpot is fitted in the system. When the dashpot tested without the engine running, it reduced the amplitude of vibrations to one-sixth of the initial value in three complete oscillation’s cycles. When the engine is runing at 550 rev/min the unbalance force is F= 2.5 kN.

Figure A2, Schematic of engine and rubber mount

- Determine the damping and frequency ratios

(2 marks)

- The amplitude of the engine’s vertical displacement

(3 marks)

- Comment qualitatively how the system response behaves. You can use the relevant matlab scrip in module shell to represent the vibration response.

(3 marks)

- What are the difference between under damp, critical damp and overdamp vibration responses? You should explaim key differences in your response.

(2 marks)

- A trolley of mass is connected to the wall by a spring with stiffness coefficient, and a dashpot with viscous damping coefficient, as shown in Figure A3. There is an eccentric mass inside the trolley rotating with angular velocity being shifted from the centre of rotation on distance .

Figure A3. Mass-spring-damper system with rotating imbalance

Determine:

- the system natural frequency and damping ratio z.

(2 marks)

- the steady state amplitude of vibration of the trolley

(3 marks)

- the maximum acceleration amplitude of the trolley while the rotation rate of imbalance mass increases very slowly from rest till very high values

(3 marks)

- Comment qualitatively how the system response behaves if the damper increases very slowly from into .You can use the relevant matlab scrip in the module shell to solidfy your response.

(2 marks)

**A4.** In following mechanism, the mass of each block is m= 4 kg. The blocks moving together without slipping and coupled by six identical springs of stiffness as shown in Figure A4.a.

- b)

Figure A4. Schematic of Two Degree of Freedom System

- Detrmine the governing equations for oscillations of this coupled trolley system

(2 marks)

- Derive the characteristic equation for given value of the prameters and find out two frequencies of oscillations specific to this system

(3 marks)

c) Describe the two modes of vibrations in this system and calculate the amplitudes ratio for these modes

(2 marks)

d) Explain qualitatively how system behavior will change when between two trolleys will be additionally installed a dashpot (see Figure A.2. b).

(3 marks)

**A5.** The following questions come from our experimental sessions, rectilinear machine (ECP M210) and gyroscop.

a) In general terms explaion why the dynamic system parameters (Wn, c, r, z, MF, ..) are important factors to desgin of mechanical system.

(2 marks)

b) Consider the resopnase of undamped two degree of ECP M201 system where a second trolley is connected by using an additional spring, after that a sine sweep control signal is applied as depicted in Figure A5. In gernal term explain which dynamic information you can get from the provided graphs.

(3 marks)

**Figure A5. Two degree of undamped system response **

- In general terms explain the gyroscopic effect and the right-hand rule to find the direction of precession angular velocity, angular momentum, and torque. (3 marks)

- Describe an example of gyroscopic apparatus in industry.

(2 marks)

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