在工程上，有許多結構物的動態行為，皆可以一均勻或不均勻樑攜帶一組或多組彈簧-質量系統（spring-mass system）來模擬，因此，有關於這一方面的研究報告很多；但為了將問題簡化，在現有資料中，絕大部分皆未將各組彈簧-質量系統的螺旋彈簧質量納入考慮。 本文之主旨即在探討上述螺旋彈簧的質量，對一攜帶一組或多組彈簧-質量系統的均勻或不均勻樑動態特性的影響。為達上述目的，吾人首先將一組螺旋彈簧（helical spring）以一桿子（rod）取代之，然後推導一典型段樑（typical beam segment）與一桿子的位移函數。接著，吾人又推導任一彈簧-質量系統與段樑連接點的位移與斜率之相容方程式，以及力與彎矩之平衡方程式，並根據整段樑兩端點的邊界條件，推導相關的邊界條件方程式。最後，吾人將此方程式寫成矩陣形式，令其係數行列式等於零，得一頻率方程式並解之，得整個系統的自然頻率；將各個無因次化自然頻率常數，代入上述的矩陣方程式，即得各個相關段樑的積分常數。利用這些積分常數與各段樑的位移函數，吾人即可獲得對應的振態。為驗証本文理論與電腦程式之可靠性，有些本文結果，吾人曾以由傳統有限元素法所得結果比較，所有相關數值皆非常接近，故本文理論與電腦程式的可靠性應可被接受。

In engineering applications, many dynamic behaviors of structures can be modeled with a uniform or non-uniform beam carrying one or multiple spring-mass systems, so there exist many research reports in this aspect. However, in order to simplify the problem, most of the existing papers do not take the mass of helical spring of the spring-mass system into account. The objective of this paper is to study the vibration characteristics of the uniform or non-uniform beam carrying one or multiple spring-mass systems with the mass of each helical spring considered. To the last end, the helical spring is replaced by a rod and the displacement functions of a typical helical spring and a rod are determined, Then, the compatibility equations of displacement and slope, the equilibrium equations of force and moment at each attaching point, and the equations for the boundary conditions of the two ends of the beam, one may obtain the simultaneous equations. Setting the coefficient determinant of the last equations to be equal to zero, one obtains the frequency equation, and solving the last equation one will obtain one of the natural frequencies. Substituting the associated dimensionless natural frequency coefficient into the last simultaneous equations will determine the integration coefficients of each beam segment concerned. Based on these integration coefficients and the associated displacement function of each beam segment, the corresponding mode shape will be obtained. To confirm the reliability of the theory and computer program developed for this paper, some results obtained from the present method are compared with those obtained from the conventional finite element method. Good agreement between the corresponding results confirms the reliability of theory and computer program developed for this paper.

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