Flexibility method

Abstract: The new method described in this paper employs an incomplete set of free-boundary normal modes of vibration, augmented with a low-frequency account for the contribution of neglected (residual) modes. The "residual effects" improve the accuracy of forced dynamic response in a manner which is related to the benefit of the mode-acceleration method. The new method adds residual inertial and dissipative effects to the residual flexibility introduced by MacNeal. All effects are derived from the solution of a special statics problem, followed by removal of the contributions of the retained modes. A structural component can then be represented in a stiffness-matrix form for various applications, one of which is modal synthesis. Numerical results of modal analysis for a cantilevered rod show the new method to yield superior accuracy to several other methods (including those of Hurty, Bamford, and MacNeal). All parameters for the new representation can be derived from test; this is not true for most other methods. Required are the free-boundary normal modes and the dynamic flexibility matrix vs frequency for the boundary points. Consequently, any desired mix of analytically derived and experimentally derived parameters can be employed.

Abstract: Nomenclature Lagrange function for the flexibility matrix weighted norm of the errors between the given and the optimal flexibility matrix Lagrange function for the stiffness matrix weighted norm of the errors between the given and the optimal stiffness matrix unity matrix given stiffness matrix mass matrix M» //element of TV //element of N~* Nq general-coordinates vector measured mode shape /th measured_mode shape normalized 7} transpose of [ • ] = optimal flexibility matrix = (/ element of W orthogonal mode shape matrix = (/ element of X optimal stiffness matrix -ij element of Y = matrices of Lagrange multipliers = ij element of 0y and 0W , respectively = matrix of Lagrange multipliers = given flexibility matrix = matrices of Lagrange multipliers = ij element of A^ and A^ , respectively = measured frequency matrix = //element of Q y»(i* w

Abstract: A dynamic model for beams with cross-sectional cracks is discussed. It is shown that a crack can be represented by a consistent, static flexibility matrix. Two different methods for the determination of the flexibility matrix are discussed. If the static stress intensity factors are known, the flexibility matrix can be determined from an integration of these stress intensity factors. Alternatively, static finite element calculations can be used for the determination of the flexibility matrix. Both methods are demonstrated in the present paper. The mathematical model was applied to an edge-cracked cantilevered beam and the eigenfrequencies were determined for different crack lengths and crack positions. These results were compared to experimentally obtained eigenfrequencies. In the experiments, the cracks were modelled by sawing cuts. The theoretical results were, for all crack lengths, in excellent agreement with the experimental data. The dynamic stress intensity factor for a longitudinally vibrating, centrally cracked bar was determined as well. The results compared very well with dynamic finite element calculations. The crack closure effect was experimentally investigated for an edge-cracked beam with a fatigue crack. It was found that the eigenfrequencies decreased, as functions of crack length, at a much slower rate than in the case of an open crack.

Abstract: A formulation for the analysis of pile groups in layered semi-infinite media is presented. The formulation is based on the introduction of a soil flexibility matrix as well as on dynamic stiffness and flexibility matrices of the piles to relate the discretized uniform forces to the corresponding displacements at the pile-soil interface. Pile group analyses revealed that behavior is highly frequency dependent as the result of wave interferences occurring between the various piles in the group. Large values for stiffnesses as well as large magnification factors for the groups essentially follow the low frequency components of the ground motion. The rotational component is negligible for typical dimensions of the foundation.