Chapter 3
Forced Vibration of Single Degree
of Freedom System
3.1 Introduction
In the previous chapter, solutions have been obtained for the differential equation
of free vibration of single degree of freedom system (SDF) and also, determination
of free vibration characteristics of SDF systems by energy methods. In this chapter,
attention is directed towards the study of forced vibration of SDF systems and some
of its applications. The chapter is begun with the study of harmonic loading. Towards
the end of the chapter, the earthquake response analysis of structures has been dealt
with.
3.2 Response of Damped Systems to Harmonic Loading
Consider a single degree of freedom damped system acted upon by force F 0 sin ω t.
The amplitude of this force is F 0 and angular frequency ω. The system has a spring
of constant k and a damper having coefficient c.
For the dynamical equilibrium of the system for which the freebody diagram is
shown in Fig. 3.1, the equation of motion is
m ¨
x + c ˙
x + kx = F 0 sin ωt
(3.1)
Equation (3.1) is recast as
¨
x + 2n ˙
x + p
2 x =
F 0
m
sin ωt
(3.2)
where n = c/2 m and p
2
= k/m.
General solution of Eq. (3.2) consists of two parts, the complementary function,
which is the solution of homogeneous equation and the particular integral. Thus,
© 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_3
59
Forced Vibration of Single Degree
of Freedom System
3.1 Introduction
In the previous chapter, solutions have been obtained for the differential equation
of free vibration of single degree of freedom system (SDF) and also, determination
of free vibration characteristics of SDF systems by energy methods. In this chapter,
attention is directed towards the study of forced vibration of SDF systems and some
of its applications. The chapter is begun with the study of harmonic loading. Towards
the end of the chapter, the earthquake response analysis of structures has been dealt
with.
3.2 Response of Damped Systems to Harmonic Loading
Consider a single degree of freedom damped system acted upon by force F 0 sin ω t.
The amplitude of this force is F 0 and angular frequency ω. The system has a spring
of constant k and a damper having coefficient c.
For the dynamical equilibrium of the system for which the freebody diagram is
shown in Fig. 3.1, the equation of motion is
m ¨
x + c ˙
x + kx = F 0 sin ωt
(3.1)
Equation (3.1) is recast as
¨
x + 2n ˙
x + p
2 x =
F 0
m
sin ωt
(3.2)
where n = c/2 m and p
2
= k/m.
General solution of Eq. (3.2) consists of two parts, the complementary function,
which is the solution of homogeneous equation and the particular integral. Thus,
© 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_3
59