From 1e16cae61e30c65a1ba4fb140d53be5c601f2d9f Mon Sep 17 00:00:00 2001 From: Dane Sabo Date: Thu, 13 Mar 2025 16:05:45 -0400 Subject: [PATCH] me 2046 hw4 complete! --- ME_2046/HW4/hw4.py | 76 +++++++++++++++++- .../Project/ME2046 - Project Description.pdf | Bin 0 -> 74913 bytes 2 files changed, 74 insertions(+), 2 deletions(-) create mode 100644 ME_2046/Project/ME2046 - Project Description.pdf diff --git a/ME_2046/HW4/hw4.py b/ME_2046/HW4/hw4.py index aeb1b60..f516618 100644 --- a/ME_2046/HW4/hw4.py +++ b/ME_2046/HW4/hw4.py @@ -33,13 +33,13 @@ sm.pprint(g_t.expand()) # make z substitution subs -G_z = ( +G_z = (1 - z**-1) * ( K_T * T * RADIUS * RHO / B * (z / (z - 1) ** 2) - J * K_T * RADIUS * RHO / B**2 + J * K_T * RADIUS * RHO / B**2 * (z / (z - sm.exp(-B / J * T))) ) -sm.pprint(G_z.simplify()) +sm.pprint(G_z) # part c was on the board print("part c") @@ -54,3 +54,75 @@ sm.pprint(F) sm.pprint(G) sm.pprint(A) sm.pprint(B) + +########################################################a +print("PROBLEM 4:") +print("bilinear transform") +T = 0.2 # s + +G_s = 10 / (s + 10) +G_z = G_s.subs(s, 2 * (z - 1) / T * (z + 1)) +sm.pprint(G_z) +sm.pprint(G_z.simplify()) + +print("bilinear with warping") +omega_star = 10 # rad/s +G_z_warped = G_s.subs(s, (omega_star / np.tan(omega_star * T / 2)) * (z - 1) / (z + 1)) +sm.pprint(G_z_warped) +sm.pprint(G_z_warped.simplify()) + +# Create a frequency vector from 0 to 10 rad/s (avoid exactly 0 to prevent log(0) issues) +omega = np.linspace(5, 15, 500) + +# Continuous-time frequency response (evaluate at s = j*omega) +G_s_response = 10 / (10 + 1j * omega) + +# For the discrete transfer functions, substitute z = exp(j*omega*T) +z_vals = np.exp(1j * omega * T) + +# Convert symbolic expressions into functions that accept numpy arrays +G_z_func = sm.lambdify(z, G_z, "numpy") +G_z_warped_func = sm.lambdify(z, G_z_warped, "numpy") + +G_z_response = G_z_func(z_vals) +G_z_warped_response = G_z_warped_func(z_vals) + + +# Helper function to compute magnitude (in dB) and phase (in degrees) +def compute_mag_phase(response): + mag = 20 * np.log10(np.abs(response)) + phase = np.angle(response, deg=True) + return mag, phase + + +# Compute responses +mag_s, phase_s = compute_mag_phase(G_s_response) +mag_z, phase_z = compute_mag_phase(G_z_response) +mag_z_warped, phase_z_warped = compute_mag_phase(G_z_warped_response) + +# Plot the Bode plots +plt.figure(figsize=(10, 8)) + +# Magnitude plot +plt.subplot(2, 1, 1) +plt.semilogx(omega, mag_s, label="Continuous: G(s)") +plt.semilogx(omega, mag_z, label="Bilinear: G(z)") +plt.semilogx(omega, mag_z_warped, label="Warped: G(z)_warped") +plt.title("Bode Magnitude Plot") +plt.ylabel("Magnitude (dB)") +plt.legend() +plt.grid(True) + +# Phase plot +plt.subplot(2, 1, 2) +plt.semilogx(omega, phase_s, label="Continuous: G(s)") +plt.semilogx(omega, phase_z, label="Bilinear: G(z)") +plt.semilogx(omega, phase_z_warped, label="Warped: G(z)_warped") +plt.title("Bode Phase Plot") +plt.xlabel("Frequency (rad/s)") +plt.ylabel("Phase (degrees)") +plt.legend() +plt.grid(True) + +plt.tight_layout() +plt.show() diff --git a/ME_2046/Project/ME2046 - Project Description.pdf b/ME_2046/Project/ME2046 - Project Description.pdf new file mode 100644 index 0000000000000000000000000000000000000000..2400c426e699f762fbc06c4ede4e83be2195c182 GIT binary patch literal 74913 zcmb@tWpo_Nk~J!_n38+8J3w^YRihh*?-W z0~|k()`rdi5rDCs34oA624HLEY);6*#mqv;&kyb7>Bn2dYiy;Z~Oj)=D;Hbl=xHUT$0NeHtt>e6@SttMQv_ zFz9{e7X}zr?Y0&X?6zE8K8L;De|p}}Hn{j~EmeM@*X?Sb8*kD}@odxl*|TJ)zo_5k zayNmXcf4&lXxOqLv$s4}=fZq;;?PvH_4~2ysl=jTt7#sVAJLtdn*wKB!i1&yyyP+y zONuCOEXO26I@&`hl$A@!#SL23j z$MFt-@L^8=wkDJm=g|l=uD1-hr0b6hiXY>pp7|2|g?SgMo~Tf5hYhH^4%LuiT?(FH z&Yl%&etINA)mKNz3~Mo&7XFi{usK9cqZV!*lQXOxSk9$`_5E}YvJJ;;8?W&kl?!~k z%eEX-3eYW#d|bCxle|>ko}ro^65H4oEgq%I7GGdYNEt2g0tzlw?CS3G4y>dcwW0(LWH7^zd((r6mTK zjtHHFZZSfenb9SH25@E_Onx~VhbxQ?g*NF)&5i zu3104mM`SlTb1it=Ig54g*IQT+HAnzfBZIkmwx|RiXl;@XbNY}L&zcjuF}$hH=b-D 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