Excitation and response of dynamic systems - Building Acoustics - page 44

Stiffness-controlled range, ω << ω 0
In this case, we have ω 2 << k/m = ω 0 2 and | Z m | ≈ k/ ω. The spring stiffness is the
determining factor for the impedance. At very low frequencies the conditions are the
same as when the force is a simple static load. The displacement is proportional to the
force and in phase with it. Assuming a frequency independent force the velocity
amplitude will increase with frequency and arrive at its maximum amplitude in the
damping-controlled range.

Damping-controlled range , ω ≈ ω 0
Here we find | Z m | ≈ c , telling us that it is damping only that controls the amplitude at
resonance. The velocity and force are in phase whereas the displacement and force are
90° out of phase.

Mass-controlled range, ω >> ω 0
Above the resonance frequency the mass will start to be the controlling factor, | Z m | ≈ ω m .
Again assuming a frequency independent force the velocity will be inversely
proportional to frequency and there will be a 90° phase difference between force and
velocity. The acceleration will be constant and in phase with the force. (How will the
displacement behave?)
Most people will be more familiar with a description showing a resonance to be a
maximum value and not a minimum as shown here. We shall therefore use the mobility
as the descriptive quantity when giving further examples.

2.4.3 Transmitted force to the foundation (base)

Using the equations above we are now in a position to calculate the force transmitted to
the foundation or base, i.e. the ratio of the transmitted force and the applied force. This is
of great interest in the field of vibration isolation of rotating machinery exhibiting
unbalanced forces. The task here is to design an elastic supporting system reducing the
transmission of forces to the foundation and thereby prevent harmful vibrations to be
transmitted to the environment. We shall again use the simple model depicted in Figure
2.6 where the machine is modelled as a lumped mass and is placed on an elastic element,
a simple spring. This model is, for several reasons, not a very realistic one especially due
to the assumption of a foundation of infinite stiffness. If that is the case in practice, we
should not need the isolation! We shall however, treat the more general case later.

From the figure we obtain for the force F' transmitted to the base

j v

k

⎛ ⎞ ⎛
′ =⋅− + = −

⎠

Fk vvc j .

(2.30)

⎜ ⎟ ⎜

ω ω

⎝ ⎠ ⎝

Using the expression for the velocity v from Equation (2.28) we find

k

c − j j ω t

Fe ⋅ k

⎛ ⎞

ω 0

F ′ = c − j = ⋅ .

(2.31)

⎜ ⎟

Z ω ⎛ k ⎞

⎝ ⎠

m

c +⋅ j ω m −

⎜ ⎟

ω

⎝ ⎠

We shall introduce the fundamental frequency ω 0 and the damping ratio ζ arriving at