(b) (c) Consider a machine room containing a compressor which can be approximated as a rotating...

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(b) (c) Consider a machine room containing a compressor which can be approximated as a rotating machine with a mass of M= 14 kg. The damping ratio can be determined by observing the recorded displacement curve under free vibration as shown in Figure Q1(a). () DOS DOL 0.03 002 0.01 -0.01 0.02 -0.03 -0.04 0.01 0.02 m n 0.03 0.04 0:05 0.06 0.07 0.00 0.00 0.1 time (x) Figure Q1(a) (i) If the (n)-th and (+)-th peak values of the curve are x, and x,.,, respectively, where j is the number of cycles between the two peaks, show that the logarithmic decrement can be expressed as 8 = In T₁ = 215 In- (ii) Find the damping coefficient c of the system from Figure 1(a) (4 marks) The compressor is switched on and undergoes forced vibration due to the unbalanced rotating force. The amplitude is measured to be 0.08 m at resonance. Knowing that the unbalanced mass m is 15% of M, calculate the eccentricity e of the unbalanced mass. (4 marks) (4 marks) Show that the force transmitted to the floor takes the form (r is the frequency ratio) 1+(25x)² √(1-²)²+(257)² (b) (c) Consider a machine room containing a compressor which can be approximated as a rotating machine with a mass of M= 14 kg. The damping ratio can be determined by observing the recorded displacement curve under free vibration as shown in Figure Q1(a). () DOS DOL 0.03 002 0.01 -0.01 0.02 -0.03 -0.04 0.01 0.02 m n 0.03 0.04 0:05 0.06 0.07 0.00 0.00 0.1 time (x) Figure Q1(a) (i) If the (n)-th and (+)-th peak values of the curve are x, and x,.,, respectively, where j is the number of cycles between the two peaks, show that the logarithmic decrement can be expressed as 8 = In T₁ = 215 In- (ii) Find the damping coefficient c of the system from Figure 1(a) (4 marks) The compressor is switched on and undergoes forced vibration due to the unbalanced rotating force. The amplitude is measured to be 0.08 m at resonance. Knowing that the unbalanced mass m is 15% of M, calculate the eccentricity e of the unbalanced mass. (4 marks) (4 marks) Show that the force transmitted to the floor takes the form (r is the frequency ratio) 1+(25x)² √(1-²)²+(257)²

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a i From the definition of logarithmic decrement we have lnxnxn... View the full answer