應用分佈質量及集中質量轉移矩陣法於攜帶各種集中元素樑的自由振動分析

Free Vibration Analysis of Beams Carrying Concentrated Elements Using CTMM and LTMM

The purpose of this paper is to extend the continuous-mass transfer matrix method (CTMM) and the lumped-mass transfer matrix method (LTMM) to determine the natural frequencies and the associated mode shapes of the uniform or non-uniform beams carrying any number of point masses, translational springs and/or rotational springs with various classical or non-classical boundary conditions. To this end, a continuous beam is subdivided into several (massive or massless) beam segments with any two adjacent beam segments connected by a node, and then each kind of concentrated elements (including a point mass, a translational spring together with a rotational spring) is attached to each node (including each end of the beam). In such way, one may easily establish the mathematical model of a uniform or non-uniform beam with various (classical or non-classical) boundary conditions by only adjusting the cross-sectional area and length of each beam segment, and the stiffness constants of each translational spring and rotational spring. It is evident that, for any node without attachments, the associated physical quantity for each kind of concentrated elements is equal to zero at that node. 本文之主旨在於使用延伸的分佈質量轉移矩陣法（CTMM）及集中質量轉移矩陣法（LTMM），來求解一均勻樑或不均勻樑攜帶多個集中質量、線性彈簧及螺旋彈簧時，在各種傳統及非傳統邊界條件下之自然頻率與振態。為達上述目標，吾人將一連續樑細分成數（十）根（有質量或無質量）的段樑，每相鄰的兩根段樑並以一節點連接之，然後，再將上述各種集中元素附著於各個節點上（樑的每一端點也算一節點）。因此，吾人只須調整各根段樑的剖面積與長度，以及附著於各個節點上的各種集中元素之大小，便可輕易建立一攜帶多個集中質量、線性彈簧及螺旋彈簧之均勻樑或不均勻樑的數學模型。根據此數學模型，吾人便可進行該樑的自由振動分析。當然，若有任何節點未附帶任一集中元素，則位於該節點之各種集中元素之大小須等於零。