Civil Insight: A Technical Magazine Volume 3 | Page 29

Civil Insight (2019) 29-36 Civil Insight: A Technical Magazine Effect of Frequency of Harmonic Excitation to the Single-Degree-of-Freedom System in Structural Dynamics 1,* 1 Mahesh Raj Bhatt Lecturer, Department of Civil Engineering, Kathmandu University, Nepal, [email protected] Abstract This article explains about idealization of the single-degree-of-freedom (SDOF) system in structural dynamics to represent real structure by lumping of mass and consideration of equivalent lateral stiffness of the system. Idealized single-degree-of-freedom system was considered to express the generalized equation of the motion and its response towards the externally excited sinusoidal harmonic loading was represented for both undamped and damped systems. Using computer tools, response for the undamped and damped harmonic vibrations was plotted considering steady state with zero initial conditions for the different frequency ratios. Furthermore, for various frequency ratios, the dynamics response factors, namely displacement, velocity and acceleration response factors, were plotted separately for the comparative study. Results show that the damping property ܿ of the system is one of the major factors that govern the response of the system towards external excitations. However, depending upon the magnitude of the frequency ratio  ߚ, stiffness ݇ and mass ݉ may control the vibration response independently. Keywords: Resonant frequency, SDOF system, Response factors, Structural dynamics, Harmonic excitation 1) Introduction Structures can be idealized as a concentrated or lumped mass ݉ supported by a massless structure with stiffness݇ in the lateral direction, which are known as simple structures. For example, a water tank supported by a number of columns may be idealized as the lumped mass concentrated at the top and the equivalent lateral stiffness of the entire columns as the stiffness of the system. External forces like wind, mechanical vibration, and earthquake vibration may cause the system to be disturbed from its static or dynamic initial equilibrium. If the structural system is disturbed by some external force, then ideally it should be disturbed forever. However, practically, it comes to rest, meaning the motion of the structure decays with time. The magnitude of decay may depend upon various factors like type of structure and material used for the construction of the structure. The process by which the vibration steadily diminishes in amplitude is called damping (Chopra, 2007, p. 7). In damping, most of the energy dissipation in the real structures arise from the thermal effect, internal frictions and friction between connections, opening and closing of micro cracks in the concrete, friction between structural and nonstructural elements (Chopra, 2007, p. 13). 1.1) Equation of the motion for dynamic system The number of independent displacements required to define the displaced position of all the masses relative to their original position is called the number of degree of freedom (DOFs) for dynamic analysis (Chopra, 2007). A simple structure can be idealized, as shown in Fig. 1, where mass is concentrated at the roof level, a massless frame that provides stiffness to the system, and a viscous damper that dissipates vibrational *Corresponding Author Email address: [email protected] (Mahesh Raj Bhatt) Submitted on October 13, 2019; Accepted on December 12, 2019 29