For a two-degree-of-freedom (2DOF) system, the number of independent second-order differential equations is two. With respect to a vector composed of the displacements associated with each degree of freedom, these two differential equations are represented as a single equation. In this vector equation, the coefficients of acceleration, velocity, and displacement vectors are defined as the mass matrix, the damping matrix, and the stiffness matrix, respectively. Next, the method to compute the natural frequencies and the modal vectors, also known as mode shapes, is presented. The number of natural frequencies equals the number of degrees of freedom, which is two. Unlike in a single-degree-of-freedom (SDOF) system, there is a mode shape associated with each natural frequency. Next, free and forced vibration of both undamped and damped 2DOF systems are analyzed. Using these techniques, vibration absorbers are designed next. A vibration absorber consists of a spring, a mass, and a damper, and is attached to an SDOF main system experiencing vibration problems. After the addition of a vibration absorber to an SDOF main system, the complete system has two degrees of freedom. Last, the response is represented as a linear combination of the modal vectors, and it is shown that the response of each mode of vibration is equivalent to the response of an SDOF system.

Mass, Stiffness, and Damping Matrices

Let x 1(t) and x 2(t) be the displacements (linear or angular) associated with two degrees of freedom. Then, displacement x(t), velocity ẋ(t), and acceleration ${\bf\skew2\ddot x}(t)$ vectors are defined as follows:Let f 1(t) and f 2(t) be the forces (or the torques) associated with each degree of freedom. Then, the force vector f(t) is described as follows:The dynamics of a 2DOF system is governed by a set of two coupled second-order differential equations, which is written in the matrix form as follows:where the matrices M, C, and K are known as mass, damping, and stiffness matrices, respectively.