(2) Using the superposition of ramp responses (see response form in notes of Fig. 8.2), derive...

 

 

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(2) Using the superposition of ramp responses (see response form in notes of Fig. 8.2), derive the response expression (a single equation associated with step functions) of a viscously damped SDOF system to the triangular pulse shown in Fig. 8.2. Assume zero initial conditions (see the hints below the figure). Plot the response (for 0 - 3 s) for the following system parameters: i. m = 10 kg ii. iii. iv. V. k = 4000 N/m c = 20 kg/s Fo= 100 N T = 1s Fo +F(t) 712 T T Figure 8.2: Triangular pulse for Problem 2. Hint/Note i: Response of a viscously damped SDOF system (with 0 < < 1) to a ramp excitation of the form Framp(t) = Fot u(t), where u(t) is the Heaviside (unit step function), is: Fo XRamp (t) = - 23 Wn 23 n +e-funt (22 cos(wat) + 2 = 1 sin(wat))](t) - Wd (2) Using the superposition of ramp responses (see response form in notes of Fig. 8.2), derive the response expression (a single equation associated with step functions) of a viscously damped SDOF system to the triangular pulse shown in Fig. 8.2. Assume zero initial conditions (see the hints below the figure). Plot the response (for 0 - 3 s) for the following system parameters: i. m = 10 kg ii. iii. iv. V. k = 4000 N/m c = 20 kg/s Fo= 100 N T = 1s Fo +F(t) 712 T T Figure 8.2: Triangular pulse for Problem 2. Hint/Note i: Response of a viscously damped SDOF system (with 0 < < 1) to a ramp excitation of the form Framp(t) = Fot u(t), where u(t) is the Heaviside (unit step function), is: Fo XRamp (t) = - 23 Wn 23 n +e-funt (22 cos(wat) + 2 = 1 sin(wat))](t) - Wd