# Figure 3.11: Simulation of the forced spring–mass system with different simulation time constants

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Chapter | System Modeling |
---|---|

Figure number | 3.11 |

Figure title | Simulation of the forced spring–mass system with different simulation time constants |

GitHub URL | https://github.com/murrayrm/fbs2e-python/blob/main/figure-3.11-spring mass simulation.py |

Requires | python-control |

**Figure 3.11**: Simulation of the forced spring–mass system with different simulation time constants. The solid line represents the analytical solution. The dashed lines represent the approximate solution via the method of Euler integration, using decreasing step sizes.

# figure-3.11-spring_mass_simulation.py - forced spring–mass simulation approx # RMM, 28 Aug 2021 # # Figure 3.11: Simulation of the forced spring–mass system with # different simulation time constants. The solid line represents the # analytical solution. The dashed lines represent the approximate # solution via the method of Euler integration, using decreasing step # sizes. # import numpy as np import matplotlib.pyplot as plt import control as ct ct.use_fbs_defaults() # Parameters defining the system m = 250 # system mass k = 40 # spring constant b = 60 # damping constant # System matrices A = [[0, 1], [-k/m, -b/m]] B = [0, 1/m] C = [1, 0] sys = ct.ss(A, B, C, 0) # # Discrete time simulation # # This section explores what happens when we discretize the ODE # and convert it to a discrete time simulation. # Af = 20 # forcing amplitude omega = 0.5 # forcing frequency # Sinusoidal forcing function t = np.linspace(0, 100, 1000) u = Af * np.sin(omega * t) # Simulate the system using standard MATLAB routines response = ct.forced_response(sys, t, u) ts, ys = response.time, response.outputs # # Now generate some simulations manually # # Time increments for discrete approximations hvec = [1, 0.5, 0.1] # h must be a multiple of 0.1 max_len = int(t[-1] / min(hvec)) # maximum number of time steps # Create arrays for storing results td = np.zeros((len(hvec), max_len)) # discrete time instants yd = np.zeros((len(hvec), max_len)) # output at discrete time instants # Discrete time simulations maxi = [] # list to store maximum index for iter, h in enumerate(hvec): maxi.append(round(t[-1] / h)) # save maximum index for this h x = np.zeros((2, maxi[-1] + 1)) # create an array to store the state # Compute the discrete time Euler approximation of the dynamics for i in range(maxi[-1]): offset = int(h/0.1 * i) # input offset x[:, i+1] = x[:, i] + h * (sys.A @ x[:, i] + (sys.B * u[offset])[:, 0]) td[iter, i] = (i-1) * h yd[iter, i] = sys.C @ x[:, i] # Plot the results plt.subplot(2, 1, 1) simh = plt.plot( td[0, 0:maxi[0]], yd[0, 0:maxi[0]], 'g+--', td[1, 0:maxi[1]], yd[1, 0:maxi[1]], 'ro--', td[2, 0:maxi[2]], yd[2, 0:maxi[2]], 'b--', markersize=4, linewidth=1 ) analh = plt.plot(ts, ys, 'k-', linewidth=1) plt.xlabel("Time [s]") plt.ylabel("Position $q$ [m]") plt.axis([0, 50, -2, 2]) plt.legend(["$h = %g$" % h for h in hvec] + ["analytical"]), # Save the figure plt.savefig("figure-3.11-spring_mass_simulation.png", bbox_inches='tight')