A planetary gear train has four gears and one carrier. Angular velocities of the gears are $${\omega _1},{\omega _2},{\omega _3},$$ and $${\omega _4}$$ respectively. The carrier rotates with angular velocity $${\omega _5}.$$

For $${{\omega _1} = 60}$$ rpm clockwise $$(CW)$$ when looked from the left, what is the angular velocity of the carrier and its direction so that Gear $$4$$ rotates in counter clockwise $$(CCW)$$ direction at twice the angular velocity of Gear $$1$$ when looked from the left?

If $${C_f}$$ is the coefficient of speed fluctuation of a flywheel then the ratio of $${\omega _{\max }}/{\omega _{\min }}$$ will be

The differential equation governing the vibrating system is

A machine of $$250$$ kg mass is supported on springs of total stiffness $$100$$ kN/m. Machine has an unbalanced rotating force of $$350$$ N at speed of $$3600$$ rpm. Assuming a damping factor of $$0.15,$$ the value of transmissibility ratio is