本研究探討了在單自由度動力方程式中考慮系統參數不確定性的問題，其中包括材料特性的變異、製造公差和測量誤差等多種不確定性因素，將系統參數假設為隨機變數，準確地估計和分析系統參數的不確定性對於確保系統性能和安全至關重要。
在這項研究中，列出了兩種方法來處理系統參數不確定性，分別是參數擾動法和正交函數展開法，參數擾動法通過假設擾動參數來模擬參數的隨機變動，並利用數值分析方法求解動力方程式以獲得系統的位移反應，正交函數展開法則是將系統的參數空間展開成正交函數的級數，將參數估計轉化為正交函數的系數求解問題。
為了比較這兩種方法的效果，我們在兩個具體的案例中使用數值積分法的計算結果做為比較基準，並與另外兩種方法進行比較，結果顯示這兩種方法在考慮系統參數不確定性方面都取得了良好的效果，能夠提供準確的參數估計和敏感性分析結果，在某些情況下，正交函數展開法表現出更好的性能，尤其是在隨機變數空間較大且不確定性較高時。
總結而言，本研究比較了兩種方法來處理單自由度動力方程式中的系統參數不確定性問題，並比較了它們的效果，這些方法對於確定系統參數估計和敏感性分析具有重要的實際價值，並能夠提供更準確和可靠的工程解決方案，在具體應用時，可以根據問題的特點選擇適合的方法，以獲得最佳的結果。

This study investigates the problem of considering the uncertainty of system
parameters in single-degree-of-freedom dynamic equations, which includes various
uncertainties such as material property variability, manufacturing tolerances, and
measurement errors, etc. Assuming these system parameters as random variables,
accurate estimation and analysis of the uncertainty of system parameters are essential
to ensure system performance and safety.
In this study, two methods are listed to deal with the uncertainty of system
parameters, namely the parameter perturbation method and the orthogonal function
expansion method. The parameter perturbation method simulates the random variation
of parameters by assuming the perturbation parameters and solves the dynamic equation
to obtain the displacement response of the system using numerical analysis. The
orthogonal function spreading method solves the problem by spreading the system
parameter space into a hierarchy of orthogonal functions and converting the parameter
estimates into coefficients of orthogonal functions.
In order to compare the effectiveness of these two methods, we use the results of
the numerical integration method as a benchmark in two specific cases and compare
them with the other two methods. In some cases, the orthogonal function spreading
method shows better performance, especially when the random variable space is large
and the uncertainty is high.
In conclusion, this study compares two methods to deal with the uncertainty of
system parameters in single-degree-of-freedom dynamical equations and compares
their effects. These methods have important practical value in determining system
parameter estimation and sensitivity analysis, and can provide more accurate and
reliable engineering solutions. In specific applications, suitable methods can be selected
according to the characteristics of the problem to obtain the best results.

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