ME321 Kinematics and Dynamics of Machines Steve Lambert Mechanical Engineering, U of Waterloo 12/29/2018

Rotating Unbalance m k c x rt m0 e Fr xr m0 12/29/2018

Rotating Unbalance Equations of Motion: For the unbalanced mass: For the net mass: 12/29/2018

Rotating Unbalance Assume a particular solution of the form: By analogy to earlier solutions: and for r = r/n 12/29/2018

Rotating Unbalance 12/29/2018

Rotating Unbalance Example 6.5: A machine has a rotating unbalance, which results in a maximum steady-state deflection of 1 cm at resonance. Based on a measurement of the free decay of the system, it is estimated that the damping ratio is  = 0.1. The total mass of the system is 100 kg, and it is estimated, from manufacturing considerations, that the magnitude of the mass unbalance is 5 kg. Estimate the effective radius, e of the unbalance, and the amount of mass that would have to be added to the machine to reduce the vibration to 1 mm. 12/29/2018

Base Excitation m k c x(t) y(t) 12/29/2018

Base Excitation Assume a base excitation of the form: This gives the following governing differential equation: Or, in normalized form: 12/29/2018

Base Excitation There are two particular solutions. One due to the force of the damper: And one due to the force of the spring: 12/29/2018

Base Excitation These two solutions have the same frequency, and can be combined as follows: 12/29/2018

Base Excitation Displacement transmissibility: 12/29/2018

Base Excitation The force acting on the mass through the damper and spring is: or: 12/29/2018

Base Excitation The transmitted force can be rewritten as: with: or, in normalized form: 12/29/2018

Base Excitation Force transmissibility: 12/29/2018

Base Excitation Force transmissibility (dashed line) and displacement transmissibility (solid line) for  = 0.05 12/29/2018

Base Excitation Example 6.6: Consider a single degree-of-freedom model of an automobile suspension travelling over a rough road. The road is modeled as providing a base excitation, in m, of The equivalent stiffness of the suspension is k = 4  105 N/m, a damping coefficient, c = 40  103 kg/s, and a mass of 1000 kg. Determine the steady-state amplitude and displacement of the automobile mass. 12/29/2018