One Degree of Freedom

Time response is the reaction of a vibrating system to nonzero initial conditions and/or time forcing function, usually transient excitations. Therefore, time response may also be called transient response. The response of one DOF system to such conditions will be studied in this chapter. Consider a one degree of freedom (DOF) vibrating system. The equation of motion of the system is a forced linear second-order differential equation along with a set of given initial conditions. The coefficients of mass m, stuffiness k, damping c, are assumed constant. When there is no forcing term, there is no excitation, and the equation of motion is called homogeneous, otherwise it is nonhomogeneous. The solution of the general equation of motion is made up of two parts: the homogeneous solution, and the particular solution. The homogeneous solution is for the no force case which is also called a free vibration and its solution is also called the free vibration response. The nonhomogeneous case is called forced vibration and its solution is called forced vibration response. We rewrite the equation of motion of free vibrations based on only two parameters; the natural frequency and the damping ratio, instead of three parameters m, k, c. In this way, the results of analysis are general and can be applied on all similar systems. The time response of the free vibration is based on the eigenvalues characteristic parameters of the equations of motion. Depending on the value of the damping ratio, we will have five types of time response for free vibrations. The most practical ones with applications in mechanical vibrations are: 1. Underdamped: The system shows vibrations with a decaying amplitude. 2. Critically damped: The system approaches its equilibrium in fastest time with no vibrations. 3. Overdamped: The system approaches its equilibrium asymptotically and shows no vibrations. The particular solution of forced equation of motion is the special solution of the equation associated to the forcing function. If the force function is a continuous function of time and is a combination of the following functions: (1) a constant, (2) a polynomial function (3) an exponential function, (4) a harmonic function, then, the particular solution of the linear equation of motion has the same form as the forcing term with unknown coefficients. The coefficients of particular solution are calculated by substituting the solution in the differential equation and solving the resultant algebraic equations to satisfy the differential equation. The coefficients will be functions of the differential equation known coefficients. The coefficients of homogenous solution are functions of the initial conditions. The initial conditions must be applied after the exact particular solution and the general homogenous solution are found and added together to have the general solution. When the forcing function disappears after a while, the system is under transient excitation. The particular part of the solution associated to the transient forcing function will be determine by using the convolution integral, which will be studied in this chapter. If it is not possible to represent the transient force by a mathematical function, we model the force by piecewise linear connected functions and determine the solution step by step. Then convolution integral and unit impulse force method determines the response. The forcing function is approximated by a piecewise linear function connecting n selected points. In the piecewise linear interpolation, the variation of the force in any time interval is linear. The response of the system in a time interval will be found by calculating the response of the system to the applied linear force function during the interval. The solution in the next time interval would be similar by replacing the initial conditions from the end conditions of the previous step. This method makes a set of recurrence relations to determine the response of the vibrating system to any arbitrary transient forcing function.