Question: I have two NumPy arrays x_data and y_data. When I try to fit my data using a second order step response model and scipy.optimize.curve_fit with this code: import numpy as np from scipy.optimize import curve_fit import matplotlib.pyplot as plt x_data = np.array([51.5056, 51.5058, 51.5061, 51.5064, 51.5067, 51.5069, 51.5072, 51.5075, 51.5078, 51.5081, 51.5083, 51.5086, 51.5089, 51.5092, 51.5094, 51.5097, 51.51 , 51.5103, 51.5106, 51.5108, 51.5111, 51.5114, 51.5117, 51.5119, 51.5122, 51.5125, 51.5128, 51.5131, 51.5133, 51.5136, 51.5139, 51.5142, 51.5144, 51.5147, 51.515 , 51.5153, 51.5156, 51.5158, 51.5161, 51.5164, 51.5167, 51.5169, 51.5172, 51.5175, 51.5178, 51.5181, 51.5183, 51.5186, 51.5189, 51.5192, 51.5194, 51.5197, 51.52 , 51.5203, 51.5206, 51.5208, 51.5211, 51.5214, 51.5217, 51.5219, 51.5222, 51.5225, 51.5228, 51.5231, 51.5233, 51.5236, 51.5239, 51.5242, 51.5244, 51.5247, 51.525 , 51.5253, 51.5256, 51.5258, 51.5261, 51.5264, 51.5267, 51.5269, 51.5272, 51.5275, 51.5278, 51.5281, 51.5283, 51.5286, 51.5289, 51.5292, 51.5294, 51.5297, 51.53 , 51.5303, 51.5306, 51.5308, 51.5311, 51.5314, 51.5317, 51.5319, 51.5322, 51.5325, 51.5328, 51.5331, 51.5333, 51.5336, 51.5339, 51.5342]) y_data = np.array([2.99 , 2.998, 3.024, 3.036, 3.038, 3.034, 3.03 , 3.025, 3.02 , 3.016, 3.012, 3.006, 3.003, 3. , 2.997, 2.995, 2.993, 2.99 , 2.989, 2.987, 2.986, 2.985, 2.983, 2.983, 2.982, 2.98 , 2.98 , 2.979, 2.978, 2.978, 2.976, 2.977, 2.976, 2.975, 2.975, 2.975, 2.975, 2.974, 2.975, 2.974, 2.974, 2.974, 2.973, 2.974, 2.973, 2.974, 2.973, 2.974, 2.974, 2.974, 2.974, 2.974, 2.974, 2.974, 2.973, 2.974, 2.974, 2.974, 2.974, 2.974, 2.974, 2.974, 2.974, 2.975, 2.974, 2.975, 2.975, 2.976, 2.976, 2.976, 2.976, 2.975, 2.976, 2.976, 2.977, 2.977, 2.976, 2.977, 2.977, 2.977, 2.977, 2.978, 2.978, 2.978, 2.979, 2.978, 2.978, 2.979, 2.979, 2.979, 2.979, 2.979, 2.98 , 2.98 , 2.98 , 2.98 , 2.98 , 2.981, 2.981, 2.981, 2.981, 2.982, 2.981, 2.982]) # Second order function definition def second_order_step_response(t, K, zeta, omega_n): omega_d = omega_n * np.sqrt(1 - zeta**2) phi = np.arccos(zeta) return K * (1 - (1 / np.sqrt(1 - zeta**2)) * np.exp(-zeta * omega_n * t) * np.sin(omega_d * t + phi)) # Initial Guess of parameters K_guess = y_data.max() - y_data.min() zeta_guess = 0.5 # Typically between 0 and 1 for underdamped systems omega_n_guess = 2 * np.pi / (x_data[1] - x_data[0]) # A rough estimate based on data sampling rate # Fit the model with increased maxfev and parameter bounds params, covariance = curve_fit( second_order_step_response, x_data, y_data, p0=[K_guess, zeta_guess, omega_n_guess], maxfev=5000, # Increase max function evaluations bounds=([0, 0, 0], [np.inf, 1, np.inf]) # Bounds for K, zeta, and omega_n ) K_fitted, zeta_fitted, omega_n_fitted = params # Generate data for the fitted model x_fit = np.linspace(min(x_data), max(x_data), 300) # More points for a smoother line y_fit = second_order_step_response(x_fit, K_fitted, zeta_fitted, omega_n_fitted) # Plot plt.figure(figsize=(10, 6)) plt.scatter(x_data, y_data, color='red', label='Original Data') # Original data plt.plot(x_fit, y_fit, 'blue', label='Fitted Model') # Fitted model plt.title('Fitting Second Order System Step Response to Data') plt.xlabel('Time') plt.ylabel('Y') plt.legend() plt.show() # Display fitted parameters print(f"Fitted Parameters: Gain (K) = {K_fitted}, Damping Ratio (zeta) = {zeta_fitted}, Natural Frequency (omega_n) = {omega_n_fitted}")  This is the equation I am using to fit the data  I get the following fit. The K_fitted(gain) and zeta_fitted(dampening coefficient) are within realistic values but omega_n_fitted is way to large.  What is the problem? Edit1: Adding image over underdamped and overdamped systems for context. I am fitting the underdamped system 

Answer: COMMENT : The model equation appears not well compatible with the data. The fitting is far to be correct. A different model equation is considered below. The variable t is repaced by a variable x=ln(t-t0). Of course this isn't significant on physical viewpoint. This is purely mathematical.