Show that the steady state complex amplitude of a damped oscillator driven by an external force Fexp (iot) is given by the expression F A = M(w – u) + iwb Explain the meanings of the symbols used and the condition for resonance. A machine of total mass 100 kg is supported by a spring resting on the floor and its motion is constrained to be in the vertical direction only. The system is lightly damped with a damping constant of 942 N sm-. The machine contains an eccentrically mounted shaft which, when rotating at an angular frequency w, produces a vertical force on the system of Fow cos wr, where Fo is a constant. It is found that resonance occurs at 1200 r.p.m. (revolutions per minute) and the amplitude of vibration in the steady state is then 1 cm. Estimate the amplitude of vibration in the steady state when the driving frequency is (a) 2400 r.p.m. and (b) very large. You may assume that gravity has a negligible effect on the motion.
Show that the steady state complex amplitude of a damped oscillator driven by an external force Fexp (iot) is given by the expression F A = M(w – u) + iwb Explain the meanings of the symbols used and the condition for resonance. A machine of total mass 100 kg is supported by a spring resting on the floor and its motion is constrained to be in the vertical direction only. The system is lightly damped with a damping constant of 942 N sm-. The machine contains an eccentrically mounted shaft which, when rotating at an angular frequency w, produces a vertical force on the system of Fow cos wr, where Fo is a constant. It is found that resonance occurs at 1200 r.p.m. (revolutions per minute) and the amplitude of vibration in the steady state is then 1 cm. Estimate the amplitude of vibration in the steady state when the driving frequency is (a) 2400 r.p.m. and (b) very large. You may assume that gravity has a negligible effect on the motion.
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Transcribed Image Text:Show that the steady state complex amplitude of a damped oscillator
driven by an external force Fexp (iot) is given by the expression
F
A =
M(w – w) + iwb
Explain the meanings of the symbols used and the condition for
resonance.
A machine of total mass 100 kg is supported by a spring resting on the
floor and its motion is constrained to be in the vertical direction only. The
system is lightly damped with a damping constant of 942 N sm-'. The
machine contains an eccentrically mounted shaft which, when rotating at
an angular frequency w, produces a vertical force on the system of
Fow cos wr, where Fo is a constant. It is found that resonance occurs at
1200 r.p.m. (revolutions per minute) and the amplitude of vibration in the
steady state is then 1 cm. Estimate the amplitude of vibration in the
steady state when the driving frequency is (a) 2400 r.p.m. and (b) very
large. You may assume that gravity has a negligible effect on the motion.
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