Introduction to Vibrations - Maplesoft（介绍振动Maplesoft）.pdf

Introduction to Vibrations Free Response Part 2: Spring-Mass Systems with Damping The equations for the spring-mass model, developed in the previous module (Free Response Part 1), predict that the mass will continue oscillating indefinitely. Through experience we know that this is not the case for most situations. In this tutorial we will introduce the concept of viscous damping to account for decaying motion and study the different types of responses that can result. This module is a continuation of Free Response Part 1: Spring-mass systems. Spring-Mass Model with Viscous Damping To modify the equations of motion to account for decaying motion, an additional term is added that is proportional to the velocity . This term is in the form where is a constant and is called the damping coefficient (or damping constant). This damping corresponds to the type of resistance to motion and energy dissipation that is encountered when a piston with perforations is moved through a cylinder filled with a viscous fluid, for example oil. Air drag at low velocities, internal forces in structures like shafts and springs, etc. can be approximated using this form where the opposing force is directly proportional to the velocity. Returning to the horizontal spring-mass system and adding a damper to it, as shown in Fig. 1, we get the following equation by summing the forces in the x-direction. ... Eq. (1) or ... Eq. (2) Fig. 1: Single-degree-of-freedom with damping This equation can be solved using the same method used to solve the differential equation for the spring-mass system in Part 1. Assuming that the solution has the form , and substituting it into Eq. (2) we get