Consider a mechanical mass-spring-damper system that is critically damped. For this system with a damping factor...

  

Transcribed Image Text:

Consider a mechanical mass-spring-damper system that is critically damped. For this system with a damping factor of <= 1 the governing equation has a solution that takes the form: x= [xo + (vo+wnxo)t]e-wnt An experiment is conducted to measure the system response and produces the results shown in Figure 1 and Table 1. Due to experimental constraints in the measurement system, it is not possible to measure the initial displacement x, and the initial velocity vo at time t = 0 s. Using the supplied data using any appropriate combination of analytical/mathematical/numerical techniques determine the quantitative values of the initial displacement xo, the initial velocity vo, and the natural angular velocity @ such that when your values of these three parameters when substituted into the corresponding formula for the velocity v = dx produce results for the predicted velocity to an accuracy of ± 2.5% for the time interval 0.25 ≤t/[s] ≤ 1.25. In your investigation you must show that your values actually correctly predict the correct value of the velocity at the different known times. The formula for checking the accuracy is: de accuracy 100 x- In the above formula the (true value) is the value of the velocity at the various times from Table 1, and the predicted value is the value you work out with your calculated values of xo.o. @n [(predicted value)-(true value)] (true value) displacement x(t)/[m] -0.1 -02 -0.3 -0.4 -0.5 -0.6 G -0.7 -0.8 velocity v(1)/m.s 02 08 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 02 B Time t/[s] 0.2500 0.36111 0.47222 0.58333 0.69444 0.80556 0.91667 1.02778 1.13889 1.25000 04 0.4 Graphs of System Response 0.6 0.6 0.8 time [s] 0.8 time [s] 1 Figure 1 Graph of system response displacement and velocity -0.50974 -0.43171 -0.35673 -0.28941 -0.23147 -0.18304 -0.14341 12 12 Table 1 Tabular data of system response displacement and velocity Displacement x/[m] -0.62196 -0.62892 -0.58133 14 0.64464 0.56475 0.47801 0.39481 0.32019 1.4 Velocity v/m.s -0.44145 0.24131 0.57057 0.69242 0.69838 Consider a mechanical mass-spring-damper system that is critically damped. For this system with a damping factor of <= 1 the governing equation has a solution that takes the form: x= [xo + (vo+wnxo)t]e-wnt An experiment is conducted to measure the system response and produces the results shown in Figure 1 and Table 1. Due to experimental constraints in the measurement system, it is not possible to measure the initial displacement x, and the initial velocity vo at time t = 0 s. Using the supplied data using any appropriate combination of analytical/mathematical/numerical techniques determine the quantitative values of the initial displacement xo, the initial velocity vo, and the natural angular velocity @ such that when your values of these three parameters when substituted into the corresponding formula for the velocity v = dx produce results for the predicted velocity to an accuracy of ± 2.5% for the time interval 0.25 ≤t/[s] ≤ 1.25. In your investigation you must show that your values actually correctly predict the correct value of the velocity at the different known times. The formula for checking the accuracy is: de accuracy 100 x- In the above formula the (true value) is the value of the velocity at the various times from Table 1, and the predicted value is the value you work out with your calculated values of xo.o. @n [(predicted value)-(true value)] (true value) displacement x(t)/[m] -0.1 -02 -0.3 -0.4 -0.5 -0.6 G -0.7 -0.8 velocity v(1)/m.s 02 08 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 02 B Time t/[s] 0.2500 0.36111 0.47222 0.58333 0.69444 0.80556 0.91667 1.02778 1.13889 1.25000 04 0.4 Graphs of System Response 0.6 0.6 0.8 time [s] 0.8 time [s] 1 Figure 1 Graph of system response displacement and velocity -0.50974 -0.43171 -0.35673 -0.28941 -0.23147 -0.18304 -0.14341 12 12 Table 1 Tabular data of system response displacement and velocity Displacement x/[m] -0.62196 -0.62892 -0.58133 14 0.64464 0.56475 0.47801 0.39481 0.32019 1.4 Velocity v/m.s -0.44145 0.24131 0.57057 0.69242 0.69838

Answer rating: 100% (QA)

To determine the values of the initial displacement xo initial velocity vo and natural angular veloc... View the full answer