A motor is mounted on a relatively lightweight platform that is observed to vibrate excessively at an operating speed of $3,000$ rpm producing a $100$ N excitation force. The motor operating speed is $1$ times the measured natural frequency of the combined motor$/$platform system. The lumped mass of the motor$/$platform system $3$ kg$.$ Otherwise the platform can be considered as a stiffness element.

Design an undamped dynamic vibration absorber to add to the platform, tuned to the motor operating speed, noting that the allowed peak to peak displacement of the absorber is limited to $4$ mm because of geometric and size constraints. In the following the subscript a

signifies the absorber and subscript p

signifies the primary system without the absorber attached.

The spring stiffness of the absorber is

$50000$

N$/$m

Your answer should be accurate to $+/-0\mathrm{.}01.$

The mass of the absorber is

$0\mathrm{.}51$

kg

Your answer should be accurate to $+/-0\mathrm{.}01.$

The mass ratio, defined as $\backslash$mu $=$mam

$,$ is a value of

$0\mathrm{.}168$

Your answer should be accurate to $+/-0\mathrm{.}001.$

The frequency ratio, defined as $\backslash$beta $=\backslash$omega a$\backslash$omega p

$,$ is a value of

$1$

Your answer should be accurate to $+/-0\mathrm{.}001.$

The lower limit of motor speed at which the tuned vibration absorer is considered effective is a value of

Number

rpm$.$

Your answer should be accurate to $+/-1.$