Appendix G
Forced Oscillations
G.1 Solution of Equation
¨
x + ω
2
0 x =
F (t)
m
,
(G.1)
where ω 0 is the free oscillations frequency. Equation G.1 can be integrated in a
general form for an arbitrary external force F (t). This is easily done by rewriting
the equation as
d
dt
( ˙
x + iω 0 x) − iω 0 ( ˙
x + iω 0 x) =
1
m
F (t)
or
dξ
dt
− iω 0 ξ =
F (t)
m
(G.2)
where
ξ = ˙
x + iω 0 x
(G.3)
is a complex quantity. Equation G.2 is of the first order. Its solution when the righthand side is replaced by zero is ξ = Aexp(iω 0 t) with constant A. We seek a solution
of the inhomogeneous equation in the form ξ = A(t)exp(iω 0 t), obtaining for the
function A(t) the equation ˙
A = F (t)exp(−iω 0 t)/m. Integration gives the solution
of Eq. G.3:
ξ = exp(−iω 0 t)
t
0
1
m
F (t)exp(−iω 0 t)dt + ξ 0
(G.4)
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
J. A. Fornés, Principles of Brownian and Molecular Motors, Springer Series in
Biophysics 21, https://doi.org/10.1007/978-3-030-64957-9
181
Forced Oscillations
G.1 Solution of Equation
¨
x + ω
2
0 x =
F (t)
m
,
(G.1)
where ω 0 is the free oscillations frequency. Equation G.1 can be integrated in a
general form for an arbitrary external force F (t). This is easily done by rewriting
the equation as
d
dt
( ˙
x + iω 0 x) − iω 0 ( ˙
x + iω 0 x) =
1
m
F (t)
or
dξ
dt
− iω 0 ξ =
F (t)
m
(G.2)
where
ξ = ˙
x + iω 0 x
(G.3)
is a complex quantity. Equation G.2 is of the first order. Its solution when the righthand side is replaced by zero is ξ = Aexp(iω 0 t) with constant A. We seek a solution
of the inhomogeneous equation in the form ξ = A(t)exp(iω 0 t), obtaining for the
function A(t) the equation ˙
A = F (t)exp(−iω 0 t)/m. Integration gives the solution
of Eq. G.3:
ξ = exp(−iω 0 t)
t
0
1
m
F (t)exp(−iω 0 t)dt + ξ 0
(G.4)
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
J. A. Fornés, Principles of Brownian and Molecular Motors, Springer Series in
Biophysics 21, https://doi.org/10.1007/978-3-030-64957-9
181