Question

Consider a machine of mass 70 kg mounted to ground through an isolation system of total stiffness 30,000 N/m, with a measured damping ratio of 0.2. The machine produces a harmonic force of 450 N at 13 rad/s during steady-state operating conditions. Determine

Answer 1

Complete Question

Consider a machine of mass 70 kg mounted to ground through an isolation system of total stiffness 30,000 N/m, with a measured damping ratio of 0.2. The machine produces a harmonic force of 450 N at 13 rad/s during steady-state operating conditions. Determine

(a) the amplitude of motion of the machine,  

(b) the phase angle of the motion,  

(c) the transmissibility ratio,  

(d) the maximum dynamic force transmitted to the floor, and  

(e) the maximum velocity of the machine.

Answer:

a)  $X=0.0272m$

b)  $\phi=22.5 \textdegree$

c)  $T_r=1.57$

d)  $F=706.5N$

e)  $V_{max}=0.35m/s$

Explanation:

From the question we are told that:

Mass $M=70kg$

Total Stiffness $\mu=30000$

Damping Ratio $r=0.2$

Force $F=450N$

Angular velocity $\omega =13rad/s$

Generally the equation for vibration in an isolated system is mathematically given by

 $\omega_n=\sqrt{\frac{k}{m}}$

 $\omega_n=\sqrt{\frac{30000}{70}}$

 $\omega_n=20.7rad/s$

a)

Generally the equation for Machine Amplitude is mathematically given by

$X=\frac{F_O/m}{(\omega_n^2-\omega^2)^2-(2*r*\omega)*\omega_n*\omega^2)^{1/2}}$

$X=\frac{450}{70}}{(20.7^2-(137^2)^2-(2*0,2*(20.7(13)))^2)^{1/2}$

$X=0.0272m$

b)

Generally the equation for Phase Angle is mathematically given by

$\phi=tan^{-1}\frac{2*r*\omega_n*\omega}{\omega_n^2*\omega^2}$

$\phi=tan^{-1}\frac{2*0.2*20.7*13}{\20.7^2*13^2}$

$\phi=22.5 \textdegree$

c)

Generally the equation for transmissibility ratio is mathematically given by

$T_r=\sqrt{\frac{1+(2r\beta)^2}{(1-r^2)^2+(2*\beta*r)^2}}$

Where

$\beta=Ratio\ of\ angular\ velocity$

$\beta=\frac{13}{20.7}\\\beta=0.638$

Therefore

$T_r=\sqrt{\frac{1+(2*(0.2)(0.638))^2}{(1-(0.2)^2)^2+(2*0.2*0.638)^2}}$

$T_r=1.57$

d)

Generally the equation for Maximum dynamic force transmitted to the floor is mathematically given by

 $F=(T_r)*F_o$

 $F=(1.57)*450$

 $F=706.5N$

e)

Generally the equation for Maximum Velocity of Machine is mathematically given by

 $V_{max}=\omega*x$

 $V_{max}=13*0.0272$

 $V_{max}=0.35m/s$