Civil Engineering Reference
In-Depth Information
When depicting an impedance or mobility in a frequency diagram one usually uses
a logarithmic scale and the modulus of the actual quantity is given in decibels (dB) with
an agreed reference value. In the impedance diagram, shown in
Figure 2.5
, a reference
value of 1N⋅s/m is used and the solid lines show the impedance for lumped mass and
spring elements. Using such diagrams for plotting the impedance of composite and more
complex systems one will immediately observe in which part of the frequency range we
may characterize the systems behaviour in terms of mass, a spring or damping. (What
would the corresponding mobility diagram look like?)
2.4
TRANSFER FUNCTIONS. SIMPLE MASS-SPRING SYSTEMS
The simple mass-spring system is the classical example for illustrating transfer functions
in a physical system, a system also called a linear harmonic oscillator. We shall model
the system using three concentrated (lumped) elements, a mass, a spring and a damper.
The system is assumed to have one degree of freedom only, a transverse movement as
shown in
Figure 2.6
. The justification for using such a simple system is twofold: it
exhibits most of the phenomena found in more complex systems and furthermore, it
gives us the opportunity to define and clarify the concepts and quantities to be used later
on. We shall either use the displacement or the velocity as the response quantity, the
latter when dealing with the impedance of the system.
As seen from the figure we assume that an outside force
F
drives the mass. The
first task will be to calculate the movement of the mass as a function of frequency,
second, we shall calculate the transmitted force
F'
to the base or foundation. The latter
task is fundamental to the subject area of
vibration isolation.
F
x(t)
m
c
k
F´
Figure 2.6
Simple mass-spring system (harmonic oscillator).
2.4.1
Free oscillations (vibrations)
We shall start by assuming that the system in one way or other is displaced from its
equilibrium position and thereafter left to move freely. Without any outside forces
operating, i.e.
F
equal to zero, the sum of inertial forces
F
m
, spring forces
F
k
and viscous
damping forces
F
c
will be equal to zero:
FFF
+
+=
0,
(2.11)
mc
k
giving
