12
2 Single Degree of Freedom (SDOF) Systems
In the y-system of coordinates, the solution satisfies an inhomogeneous equation
which consists of the sum of the general solution of the homogeneous equation
y h (t) and a particular solution of the inhomogeneous equation y p (t), i.e.,
y(t) = y h (t) + y p (t) = e
−ζ ω t
A cos(ω d t) + B sin(ω d t)
+ a
since y p (t) = a satisfies the equation of motion so that it is a particular solution.
The initial conditions y(0) = x 0 + a and ˙
y(0) = ˙
x 0 give x 0 + a = A + a and
−ζ ω A + B ω d = ˙
x 0 so that A = x 0 , B =
˙
x 0 + ζ ω x 0
/ω d and
y(t) = e
−ζ ω t
x 0 cos(ω d t) +
˙
x 0 + ζ ω x 0
ω d
sin(ω d t)
+ a
which is consistent with the free vibration solution x(t) since y(t) = x(t) + a.
This simple example illustrates clearly that equations of motions are meaningless
if system of coordinates is not specified and that the nature of the solution depends
on the system of coordinates. In this example, the oscillator is in free and forced
vibration in the x-coordinate and the y-coordinate.
The following two subsections construct general homogeneous solutions for
damping ratios ζ > 1, ζ = 1, and ζ < 1 (Sect. 2.4.1) and provide details on these
solutions for damping ratios ζ < 1 (Sect. 2.4.2). Sections 2.4.3 and 2.4.4 construct
particular inhomogeneous solutions for simple and arbitrary forcing functions. The
construction of particular solutions for arbitrary forcing functions uses the Duhamel
integral. The responses of undamped and damped SDOF systems to harmonic forces
are examined in Sects. 2.4.5 and 2.4.6.
2.4.1 General Homogeneous Solution
The general form of the homogeneous solution is exp
λ t
. The parameter λ is
determined by requiring that this function satisfies the homogeneous version of
Eq. 2.8, which gives (see Appendix B)
λ
2 e
λ t
+ 2 ζ ω λ e
λ t
+ ω
2 e
λ t
= 0 or
λ
2
+ 2 ζ ω λ + ω
2
e
λ t
= 0, t ≥ 0.
Since the equality has to hold at all times and the exponential exp
λ t
is not zero in
finite times for bounded λ, the above condition is satisfied only if λ 2 +2 ζ ω λ+ω 2 =
0. This is second degree polynomial in λ whose roots can be real or complex. The
general solution of the homogenous equation has the form
x h (t) =
c 1 e λ 1 t + c 2 e λ 2 t , if λ 1 = λ 2
c 1 e λ 1 t + c 2 t e λ 1 t , if λ 1 = λ 2 ,
(2.17)
2 Single Degree of Freedom (SDOF) Systems
In the y-system of coordinates, the solution satisfies an inhomogeneous equation
which consists of the sum of the general solution of the homogeneous equation
y h (t) and a particular solution of the inhomogeneous equation y p (t), i.e.,
y(t) = y h (t) + y p (t) = e
−ζ ω t
A cos(ω d t) + B sin(ω d t)
+ a
since y p (t) = a satisfies the equation of motion so that it is a particular solution.
The initial conditions y(0) = x 0 + a and ˙
y(0) = ˙
x 0 give x 0 + a = A + a and
−ζ ω A + B ω d = ˙
x 0 so that A = x 0 , B =
˙
x 0 + ζ ω x 0
/ω d and
y(t) = e
−ζ ω t
x 0 cos(ω d t) +
˙
x 0 + ζ ω x 0
ω d
sin(ω d t)
+ a
which is consistent with the free vibration solution x(t) since y(t) = x(t) + a.
This simple example illustrates clearly that equations of motions are meaningless
if system of coordinates is not specified and that the nature of the solution depends
on the system of coordinates. In this example, the oscillator is in free and forced
vibration in the x-coordinate and the y-coordinate.
The following two subsections construct general homogeneous solutions for
damping ratios ζ > 1, ζ = 1, and ζ < 1 (Sect. 2.4.1) and provide details on these
solutions for damping ratios ζ < 1 (Sect. 2.4.2). Sections 2.4.3 and 2.4.4 construct
particular inhomogeneous solutions for simple and arbitrary forcing functions. The
construction of particular solutions for arbitrary forcing functions uses the Duhamel
integral. The responses of undamped and damped SDOF systems to harmonic forces
are examined in Sects. 2.4.5 and 2.4.6.
2.4.1 General Homogeneous Solution
The general form of the homogeneous solution is exp
λ t
. The parameter λ is
determined by requiring that this function satisfies the homogeneous version of
Eq. 2.8, which gives (see Appendix B)
λ
2 e
λ t
+ 2 ζ ω λ e
λ t
+ ω
2 e
λ t
= 0 or
λ
2
+ 2 ζ ω λ + ω
2
e
λ t
= 0, t ≥ 0.
Since the equality has to hold at all times and the exponential exp
λ t
is not zero in
finite times for bounded λ, the above condition is satisfied only if λ 2 +2 ζ ω λ+ω 2 =
0. This is second degree polynomial in λ whose roots can be real or complex. The
general solution of the homogenous equation has the form
x h (t) =
c 1 e λ 1 t + c 2 e λ 2 t , if λ 1 = λ 2
c 1 e λ 1 t + c 2 t e λ 1 t , if λ 1 = λ 2 ,
(2.17)