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SolSec 2.9 - 2 69 Problems and Solutions Section 2.9(2.87...

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2- 69 Problems and Solutions Section 2.9 (2.87 through 2.93) 2.87*. Compute the response of the system in Figure 2.34 for the case that the damping is linear viscous and the spring is a nonlinear soft spring of the form k ( x ) = kx ! k 1 x 3 and the system is subject to a harmonic excitation of 300 N at a frequency of approximately one third the natural frequency ( ω = ω n /3) and initial conditions of x 0 = 0.01 m and v 0 = 0.1 m/s. The system has a mass of 100 kg, a damping coefficient of 170 kg/s and a linear stiffness coefficient of 2000 N/m. The value of k 1 is taken to be 10000 N/m 3 . Compute the solution and compare it to the linear solution ( k 1 = 0). Which system has the largest magnitude? Solution: The following is a Mathcad simulation. The green is the steady state magnitude of the linear system, which bounds the linear solution, but is exceeded by the nonlinear solution. The nonlinear solution has the largest response.
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2- 70 2.88*. Compute the response of the system in Figure 2.34 for the case that the damping is linear viscous and the spring is a nonlinear hard spring of the form k ( x ) = kx
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