ki ли Mi 00 f.(+) 1+x, kz M₂ → f₂(+) 00 1-7x2 A system is modeled by a pair of masses m₁, m2 connected to each other by a spring with constant k₂ and a damper with constant c2, as shown in the figure. m₁ is connected to an inertial reference (i.e., the motionless vertical wall at left) by a spring k₁ and a damper c₁, as shown in the figure. External force fi(t) acts directly on m₁, and external force f2(t) acts directly on m2. The inertial position of m₁ is x1(t), and the inertial position of m2 is x2(t) Use the following numerical values: m₁ = 2, m2 = 1, k₁ = 20, k₂ = 10, c₁ = 3, c₂ = 1 a. Draw free-body diagrams of the two masses, and use F = ma to show that the equations of motion of the two masses are given by -k1x1(t) c₁₁(t) + k₁(x2(t) − x1(t)) + 2(x2(t) − x1(t)) + ƒ₁(t) = m₁₁(t) -k2(x2(t)x1(t)) - c2(x2(t) - 1(t)) + f2(t) = m22(t) b. Convert the differential equations from (a) into a state-space model, and create it in Matlab. There are two outputs of the model: y2(t) is the position of m2, and y₁(t) is the force exerted by spring k₂ on m₁, defined (+) to the right. Define fi and f2 as two separate inputs. c. Use Matlab function initial to find the outputs of the system if m₁ is held motionless at x1 = 2 and then released at t = 0 (assume that the system is in static equilibrium prior to t = 0, with f₁ = 2k1, f2 = 0; then at t = 0, force fi is suddenly removed; note that y₁(0) = 0, y2(0) = 2 in this configuration) d. Use Matlab function eig to find the settling time, natural frequencies (if any) and corresponding damping ratios (if any) of the system e. Compare the results of (d) with the plot of y2(t) from (c); does the homogeneous output appear to match the calculated system properties? f. Use function step to find the step response of both outputs to both inputs. Note that there are four results (two outputs for each of the two inputs); plot them on four separate sets of axes. (a) Show that the steady-state unit step response outputs are correct (note: use statics to determine the correct values, and check that your outputs match them) (b) Use the plotted outputs to estimate their maximum overshoot and peak time for all four step responses

ki ли Mi 00 f.(+) 1+x, kz M₂ → f₂(+) 00 1-7x2 A system is modeled by a pair of masses m₁, m2 connected to each other by a spring with constant k₂ and a damper with constant c2, as shown in the figure. m₁ is connected to an inertial reference (i.e., the motionless vertical wall at left) by a spring k₁ and a damper c₁, as shown in the figure. External force fi(t) acts directly on m₁, and external force f2(t) acts directly on m2. The inertial position of m₁ is x1(t), and the inertial position of m2 is x2(t) Use the following numerical values: m₁ = 2, m2 = 1, k₁ = 20, k₂ = 10, c₁ = 3, c₂ = 1 a. Draw free-body diagrams of the two masses, and use F = ma to show that the equations of motion of the two masses are given by -k1x1(t) c₁₁(t) + k₁(x2(t) − x1(t)) + 2(x2(t) − x1(t)) + ƒ₁(t) = m₁₁(t) -k2(x2(t)x1(t)) - c2(x2(t) - 1(t)) + f2(t) = m22(t) b. Convert the differential equations from (a) into a state-space model, and create it in Matlab. There are two outputs of the model: y2(t) is the position of m2, and y₁(t) is the force exerted by spring k₂ on m₁, defined (+) to the right. Define fi and f2 as two separate inputs. c. Use Matlab function initial to find the outputs of the system if m₁ is held motionless at x1 = 2 and then released at t = 0 (assume that the system is in static equilibrium prior to t = 0, with f₁ = 2k1, f2 = 0; then at t = 0, force fi is suddenly removed; note that y₁(0) = 0, y2(0) = 2 in this configuration) d. Use Matlab function eig to find the settling time, natural frequencies (if any) and corresponding damping ratios (if any) of the system e. Compare the results of (d) with the plot of y2(t) from (c); does the homogeneous output appear to match the calculated system properties? f. Use function step to find the step response of both outputs to both inputs. Note that there are four results (two outputs for each of the two inputs); plot them on four separate sets of axes. (a) Show that the steady-state unit step response outputs are correct (note: use statics to determine the correct values, and check that your outputs match them) (b) Use the plotted outputs to estimate their maximum overshoot and peak time for all four step responses

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter11: Topics From Analytic Geometry

Section: Chapter Questions

Problem 10RE

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