IMPROVED ANALYSIS OF A PROPPED CANTILEVER UNDER LATERAL VIBRATION

Continuous systems are sometimes analysed as lumped masses connected by massless elements. This reduces the structure’s degree of freedom and therefore simplifies the analysis. However this over simplification introduces an error in the analysis and the results are therefore approximate. In this work sections of the vibrating beam were isolated and the equations of the forces causing vibration obtained using the Hamilton’s principle. These forces were applied to the nodes of an equivalent lumped mass beam and the stiffness modification needed for it to behave as a continuous beam obtained. The beam’s stiffness was modified using a set of stiffness modification factors to . It was observed that by applying these factors in the dynamic analysis of the beam using the Lagrange’s equation, we obtain the exact values of the fundamental frequency irrespective of the way the mass of the beam was lumped. From this work we observed that in order to obtain an accurate dynamic response from a lumped mass beam there is need to modify the stiffness composition of the system and no linear modification of the stiffness distribution of lumped mass beams can cause them to be dynamically equivalent to the continuous beams. This is so because the values of the modification factors obtained for each beam segment were not equal. The stiffness modification factors were obtained for elements at different sections of the beam

Dimensionless wave numbers evolution of a three spans simply supported beam when the intermediate supports are moving along the whole beam

Continuous beams simply supported with several intermediate supports are very common in engineering achievements everywhere. The paper shows the evolution of the dimensionless wave number in 3D format, respectively of the eigenfrequencies for a continuous beam with three openings when the intermediate supports take any position inside the beam. The frequency equation for calculating the dimensionless wave number is presented and the modal function is given with an example for the case where the eigenfrequency has the maximum value at fist vibration mode.

Physical-mathematical model for determination of variation of internal forces in the simplified analysis of a beam

Springs are often taught in subjects of physics such as statics and solid mechanics belonging to civil engineering programs and mechanical engineering. This knowledge can be applied successfully in the modeling of structures and the consequent development of structural analysis. This paper presents the results of an investigation on physic-mathematical models which uses springs to replace complex connective conditions attempting to simplify the structural analysis process. The work focuses on the analysis of beams supported upon masonry walls, applying variations to the span lengths, sections and loads on them and considering realistic variations of the stiffness conditions required in the supports to meet the demands that these variations impose. For this, continuous beams with two spans with three types of section that are supported by walls that support different levels of restriction from different heights of the building of which they are part are modeled. It is concluded that there is an important influence of the slenderness of the beams and the degree of confinement of the supports upon the precision of the simplified model.