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, Â
(b) the phase angle of the motion, Â
(c) the transmissibility ratio, Â
(d) the maximum dynamic force transmitted to the floor, and Â
(e) the maximum velocity of the machine.
Answer:
a) Â [tex]X=0.0272m[/tex]
b) Â [tex]\phi=22.5 \textdegree[/tex]
c) Â [tex]T_r=1.57[/tex]
d) Â [tex]F=706.5N[/tex]
e) Â [tex]V_{max}=0.35m/s[/tex]
Explanation:
From the question we are told that:
Mass [tex]M=70kg[/tex]
Total Stiffness [tex]\mu=30000[/tex]
Damping Ratio [tex]r=0.2[/tex]
Force [tex]F=450N[/tex]
Angular velocity [tex]\omega =13rad/s[/tex]
Generally the equation for vibration in an isolated system is mathematically given by
 [tex]\omega_n=\sqrt{\frac{k}{m}}[/tex]
 [tex]\omega_n=\sqrt{\frac{30000}{70}}[/tex]
 [tex]\omega_n=20.7rad/s[/tex]
a)
Generally the equation for Machine Amplitude is mathematically given by
[tex]X=\frac{F_O/m}{(\omega_n^2-\omega^2)^2-(2*r*\omega)*\omega_n*\omega^2)^{1/2}}[/tex]
[tex]X=\frac{450}{70}}{(20.7^2-(137^2)^2-(2*0,2*(20.7(13)))^2)^{1/2}[/tex]
[tex]X=0.0272m[/tex]
b)
Generally the equation for Phase Angle is mathematically given by
[tex]\phi=tan^{-1}\frac{2*r*\omega_n*\omega}{\omega_n^2*\omega^2}[/tex]
[tex]\phi=tan^{-1}\frac{2*0.2*20.7*13}{\20.7^2*13^2}[/tex]
[tex]\phi=22.5 \textdegree[/tex]
c)
Generally the equation for transmissibility ratio is mathematically given by
[tex]T_r=\sqrt{\frac{1+(2r\beta)^2}{(1-r^2)^2+(2*\beta*r)^2}}[/tex]
Where
[tex]\beta=Ratio\ of\ angular\ velocity[/tex]
[tex]\beta=\frac{13}{20.7}\\\beta=0.638[/tex]
Therefore
[tex]T_r=\sqrt{\frac{1+(2*(0.2)(0.638))^2}{(1-(0.2)^2)^2+(2*0.2*0.638)^2}}[/tex]
[tex]T_r=1.57[/tex]
d)
Generally the equation for Maximum dynamic force transmitted to the floor is mathematically given by
 [tex]F=(T_r)*F_o[/tex]
 [tex]F=(1.57)*450[/tex]
 [tex]F=706.5N[/tex]
e)
Generally the equation for Maximum Velocity of Machine is mathematically given by
 [tex]V_{max}=\omega*x[/tex]
 [tex]V_{max}=13*0.0272[/tex]
 [tex]V_{max}=0.35m/s[/tex]
