Chapter 2
Free Vibration of Single Degree
of Freedom System
2.1 Introduction
The simplest physical system is one having single degree of freedom. Almost all
practical systems are much more complex than this simple model. However, for
obtaining an approximate idea about vibration characteristics, some of the systems
are at times reduced to that of a single degree of freedom, such as the water tower of
Fig. 2.1.
In some cases such as seismic instruments, this model is sufficient for the necessary study. Single degree of freedom systems may be translational or rotational. The
rotational problems related to the torsional vibration of elastic systems are of considerable importance. In this chapter, we shall deal with free vibrations of the systems
whose motion can be described by a single coordinate [1–10].
2.2 Equation of Motion of Single Degree of Freedom (Sdf)
System
Consider the single bay portal frame of Fig. 2.2. The masses of the columns are
considered to be small in comparison with the mass of the girder, and they are
neglected in the analysis. Further, the girder is assumed to be infinitely rigid, so that
the stiffness of the system is provided only by columns. The entire motion may be
determined once the mass centre of the girder is known. When the frame vibrates in
the horizontal direction, the forces associated with the motion are the inertia forces
connected with the mass of the body, the restoring force provided by spring elements
(the spring constant can be determined from the properties of the column and the
boundary conditions), the damping force provided by the damper and the external
force.
A machine mounted on a spring is shown in Fig. 2.3. The damping of the system
is indicated by the dashpot. When the machine undergoes vertical motion, the forces
© 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_2
15
Free Vibration of Single Degree
of Freedom System
2.1 Introduction
The simplest physical system is one having single degree of freedom. Almost all
practical systems are much more complex than this simple model. However, for
obtaining an approximate idea about vibration characteristics, some of the systems
are at times reduced to that of a single degree of freedom, such as the water tower of
Fig. 2.1.
In some cases such as seismic instruments, this model is sufficient for the necessary study. Single degree of freedom systems may be translational or rotational. The
rotational problems related to the torsional vibration of elastic systems are of considerable importance. In this chapter, we shall deal with free vibrations of the systems
whose motion can be described by a single coordinate [1–10].
2.2 Equation of Motion of Single Degree of Freedom (Sdf)
System
Consider the single bay portal frame of Fig. 2.2. The masses of the columns are
considered to be small in comparison with the mass of the girder, and they are
neglected in the analysis. Further, the girder is assumed to be infinitely rigid, so that
the stiffness of the system is provided only by columns. The entire motion may be
determined once the mass centre of the girder is known. When the frame vibrates in
the horizontal direction, the forces associated with the motion are the inertia forces
connected with the mass of the body, the restoring force provided by spring elements
(the spring constant can be determined from the properties of the column and the
boundary conditions), the damping force provided by the damper and the external
force.
A machine mounted on a spring is shown in Fig. 2.3. The damping of the system
is indicated by the dashpot. When the machine undergoes vertical motion, the forces
© 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_2
15