Page 57 - Biosystems Engineering

38 Chapter Two

The relation between the time-dependent force f(t), generally called
the forcing function, exerted on mass m and the position of the mass
x(t) with respect to a fixed body can be derived by applying Newton’s
second law:

2
ft() + f t() + f t() = m dx t() (2.7)
s d 2
dt
where f (t) is the force exerted by the spring with spring constant k
s
when the mass has moved from the rest position x to its current posi-
0
tion x(t):
ft () =− k x t () − x ) (2.8)
(
s 0
and f (t) is the force exerted by the viscous damper with damping
d
constant c:

dx t()
ft() =− c (2.9)
d
dt
Combination of Eqs. (2.7) to (2.9), setting x to zero, and moving
0
all terms in x(t) to the right gives us the equation of motion for the
mass–spring–damper system:
2
ft () = m dx t () + c dx t() + kx t() (2.10)
dt 2 dt
This is a second-order differential equation describing the force
f(t) as a function of the position x(t). However, we are interested in the
effect of a force signal f(t) on the position signal x(t). For this purpose,
it would be nice to have a mathematical description that allows us to
consider the input signal, the output signal, and the system as differ-
ent entities. Such a convenient description is available through the
Laplace transform, which is described next.

2.2.3 The Laplace Transform
The Laplace transform F(s) of a time signal f(t) is defined as (Nise 2000)

ft =
[( )] F ( ) = ∫ ∞ − ft − st dt (2.11)
( )e
s
0
where s =σ+ jω.
The time t has been integrated out the time-variable function f(t)
to produce an algebraic function F(s) of the complex variable s, which
is also known as the Laplace variable. The real part of the Laplace
variable Re(s) = σ is related to the boundedness of the signal or the
stability of the system, whereas the imaginary part Im(s) = ω is related