Chapter 5
Vibration of Two Degrees of Freedom
System
5.1 Introduction
Up to the last chapter, we have dealt with systems having only single degree of
freedom. We gradually pass on to the more advanced topics. We embark on this
chapter on systems, which are referred to as two degrees of freedom system. As has
already been explained, if a system requires two independent coordinates to describe
the motion, it is said to have two degrees of freedom.
Two degrees of freedom form the simplest class of systems, referred to multiple
degrees of freedom. The latter category is presented in the next chapter, and treatment
there has been meted out with the help of matrix notations. But for a thorough
understanding of these systems, two degrees of freedom systems have been taken up
separately and treated in an explicit manner. Free vibration of two degrees of freedom
systems has been taken up first, and then their forced vibration characteristics are
dealt with.
5.2 Free Vibration of Undamped Two Degrees of Freedom
Systems
The system shown in Fig. 5.1 has two masses m 1 and m 2 , connected by two springs
having stiffness k 1 and k 2 . It is two degrees of freedom system, as its configuration
is fully described by two displacements, x 1 and x 2 as shown in Fig. 5.1.
Freebody diagrams of the masses m 1 and m 2 are shown in Fig. 5.2. The equations
of motion of the two masses are
m 1 ¨
x 1 + k 1 x 1 + k 2 (x 1 − x 2 ) = 0
m 2 ¨
x 2 + k 2 (x 2 − x 1 ) = 0
(5.1)
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
M. Mukhopadhyay, Structural Dynamics,
https://doi.org/10.1007/978-3-030-69674-0_5
155
Vibration of Two Degrees of Freedom
System
5.1 Introduction
Up to the last chapter, we have dealt with systems having only single degree of
freedom. We gradually pass on to the more advanced topics. We embark on this
chapter on systems, which are referred to as two degrees of freedom system. As has
already been explained, if a system requires two independent coordinates to describe
the motion, it is said to have two degrees of freedom.
Two degrees of freedom form the simplest class of systems, referred to multiple
degrees of freedom. The latter category is presented in the next chapter, and treatment
there has been meted out with the help of matrix notations. But for a thorough
understanding of these systems, two degrees of freedom systems have been taken up
separately and treated in an explicit manner. Free vibration of two degrees of freedom
systems has been taken up first, and then their forced vibration characteristics are
dealt with.
5.2 Free Vibration of Undamped Two Degrees of Freedom
Systems
The system shown in Fig. 5.1 has two masses m 1 and m 2 , connected by two springs
having stiffness k 1 and k 2 . It is two degrees of freedom system, as its configuration
is fully described by two displacements, x 1 and x 2 as shown in Fig. 5.1.
Freebody diagrams of the masses m 1 and m 2 are shown in Fig. 5.2. The equations
of motion of the two masses are
m 1 ¨
x 1 + k 1 x 1 + k 2 (x 1 − x 2 ) = 0
m 2 ¨
x 2 + k 2 (x 2 − x 1 ) = 0
(5.1)
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
M. Mukhopadhyay, Structural Dynamics,
https://doi.org/10.1007/978-3-030-69674-0_5
155