本文研究的主題是液壓缸及氣壓致動器運動之阻尼係數的估算，由本文研究提出的計算阻尼係數的方法共有三種，包括逆解模式、差分模式與應用拉氏轉換法的模式，配合實驗數據，推導得到計算阻尼係數的公式。
本文研究的氣壓缸運動時，出現黏滯滑動現象，與本文研究的液壓缸運動不同；本文提出逆解模式與差分模式，計算液壓缸運動之阻尼係數，而本文的氣壓缸運動時，除了逆解模式與差分模式外，再提出應用拉氏轉換法的模式，配合實驗數據，推導得到計算阻尼係數的公式。本文的氣壓馬達轉動也應用逆解模式與差分模式以及應用拉氏轉換法的模式來計算氣壓馬達轉動之阻尼係數。
本文研究的主要目的是提出一逆解模式，對於本文所探討的液壓缸運動實驗與氣壓缸運動實驗中，將量測得到的實驗活塞位移與實驗力，利用本文所提出的逆解模式，疊代計算得到的活塞位移，如果滿足位移誤差函數的收斂法則時，再配合Tikhonov正則化法進行疊代計算的最佳化，並採用Levenberg-Marquardt法的觀念，調整平滑因子及進行靈敏度分析，可以同時計算得到合理的阻尼係數及活塞速度。
而差分模式在液壓缸運動實驗與氣壓缸運動實驗中，當活塞速度不為零時，估算得到的阻尼係數可以為合理的有限值，與逆解模式計算得到的阻尼係數相比較，其值差異不大。但是當活塞速度接近零或等於零時，估算得到的阻尼係數卻是很大或是無窮大。而逆解模式估算的阻尼係數係經由疊代計算，滿足位移收斂法則與疊代計算最佳化後，同時估算得到阻尼係數與活塞速度。故即使氣壓缸活塞發生黏滯現象，逆解模式仍然可以估算得到可接受的阻尼係數。
在氣壓馬達轉動實驗中，將量測得到的實驗角速度與實驗力矩，利用本文所提出的逆解模式，疊代計算得到的角速度，如果滿足角速度誤差函數的收斂法則時，同樣地進行疊代計算的最佳化及靈敏度分析，也可以同時計算得到合理的阻尼係數及角位移。同樣地，差分模式在氣壓馬達轉動實驗中，分別計算得到合理的阻尼係數及角位移，比較兩種模式計算得到的阻尼係數及角位移，其值差異都不大。
此外本文再應用拉氏轉換法配合實驗數據，分別推導得到氣壓缸運動與氣壓馬達轉動的阻尼係數的計算公式，經由疊代計算得到的氣壓缸運動的阻尼係數與氣壓馬達轉動的阻尼係數。由分析結果可知，由逆解模式、差分模式與拉氏轉換法推導計算得到的阻尼係數，這三種阻尼係數的值其差異為可接受，且拉氏轉換法推導計算得到之結果不受時間步距大小的影響，然而拉氏轉換法無法直接計算得到氣壓缸運動的活塞速度或氣壓馬達轉動的角位移。
當氣壓缸活塞的位移不變，即發生黏滯現象，此時的活塞速度為零時，應用拉氏轉換法推導得到阻尼係數的計算公式，仍然可以進行疊代，計算得到合理的阻尼係數。又應用拉氏轉換法推導計算得到的阻尼係數具有唯一解，與逆解模式疊代計算得到的阻尼係數比較，兩者差異不大，可以證明本文提出的逆解模式疊代計算得到的阻尼係數是合理的。
本文提出之逆解模式的優點是經由疊代計算可以得到液壓缸運動實驗、氣壓缸運動實驗與氣壓馬達轉動實驗之阻尼係數，同時也得到液壓缸運動實驗、氣壓缸運動實驗之活塞速度或氣壓馬達轉動實驗的角位移。綜合分析結果，本文所提出的逆解模式對於液壓缸運動實驗、氣壓缸運動實驗與氣壓馬達轉動實驗均可合理計算得到其阻尼係數。

The subject of this study includes the estimation of damping coefficients for hydraulic cylinder and pneumatic actuator motion. The paper studies three methods, including the proposed inverse model, finite difference model and the application of Laplace transform method to deduce the equations for calculation of damping coefficients.
This study on the pneumatic cylinder motion with stick-slip is different from the hydraulic cylinder motion. The proposed inverse model and finite difference model are used to calculate the damping coefficients of hydraulic cylinder motion. Beside this two models, this paper proposes the application of Laplace transform method to calculate the damping coefficients of pneumatic actuator motion.
The main purpose of this study is to propose an inverse model. In the hydraulic cylinder motion experiment and the pneumatic cylinder motion experiment investigated by this paper, the measured experimental piston displacement and experimental force are employed to find out the piston displacement by using inverse model and iteration calculation proposed by this paper. If the piston displacement satisfies the convergence principle of the error function of displacement, it can carry out the optimization of iteration calculation by using Tikhonov regularization method. Then the sensitivity analysis is made and the smoothing parameter is modified by using the concept of Levenberg-Marquardt method. In addition, reasonable damping coefficients and piston velocity can be calculated at the same time.
For the use of difference model in hydraulic cylinder motion experiment and pneumatic cylinder motion experiment, when the piston velocity is not zero, the estimated damping coefficient can be a reasonable limited value. Comparing it to the damping coefficient calculated by inverse model, the difference is not significant. However, when the piston velocity is close to zero or equal to zero, the estimated damping coefficient will be very great or infinitely great. The damping coefficient estimated by inverse model has gone through iteration calculation. After the damping coefficient satisfies the convergence principle of displacement and goes through the optimization of iteration calculation, the damping coefficient and piston velocity can be estimated at the same time. Therefore, even though there is sticking phenomenon occurred to the piston of pneumatic cylinder, an acceptable damping coefficient could still be estimated by inverse model.
In the air motor rotation experiment, the measured experimental angular velocity and experimental torque can be employed to calculate the angular velocity by using the inverse model and iteration calculation proposed by this paper. If it satisfies the convergence principle of the error function of angular velocity, the optimization of iteration calculation and sensitivity analysis can likewise be conducted. Meanwhile, reasonable damping coefficient and angular displacement can also be calculated. Similarly, by using difference model in the air motor rotation experiment, reasonable damping coefficient and angular displacement can be calculated respectively. Comparing the damping coefficients and angular displacements calculated by these two models, the differences in between are not great.
Besides, the paper applies Laplace transform method to match with the experimental data for deducting a formula for calculation of the damping coefficients of hydraulic cylinder motion and air motor rotation. After iteration calculation, the damping coefficient of hydraulic cylinder motion and the damping coefficient of air motor rotation are obtained. As known from the analytic results, the difference among the three damping coefficients calculated by inverse model, difference model and the deduction of Laplace transform method is acceptable. Besides, the result calculated by the deduction of Laplace transform method is not affected by the magnitude of time step. Nevertheless, Laplace transform method cannot directly calculate the piston velocity of hydraulic cylinder motion or the angular displacement of air motor rotation.
When the displacement of the piston of hydraulic cylinder remains unchanged, sticking phenomenon would occur immediately. During this time, the piston velocity is zero. The formula for calculation of damping coefficient deduced from Laplace transform method can still undergo iteration calculation to acquire a reasonable damping coefficient. In addition, the damping coefficient calculated by the deduction of Laplace transform method is the sole solution. The difference between such solution and the damping coefficient acquired by the iteration calculation of inverse model is not great. Thus, it is proved that the damping coefficient acquired by the iteration calculation of inverse model is reasonable.
The use of inverse model proposed by this paper has the advantage that after it has gone through iteration calculation, the damping coefficients for hydraulic cylinder motion experiment, pneumatic cylinder motion experiment, and air motor rotation experiment can be obtained. At the same time, the piston velocities for hydraulic cylinder motion experiment and pneumatic cylinder motion experiment, or the angular displacement for air motor rotation experiment can be acquired. Synthesizing the above analytic results, the use of inverse model, as proposed by this paper, can reasonably calculate the damping coefficients for hydraulic cylinder motion experiment, pneumatic cylinder motion experiment, and air motor rotation experiment.

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