  By introducing four fundamental types of disorders into a two-dimensional
triangular lattice separately, we determine the role of each type of disorder
in the vibration of the resulting mass-spring networks. We are concerned mainly
with the origin of the boson peak and the connection between the boson peak and
the transverse Ioffe-Regel limit. For all types of disorders, we observe the
emergence of the boson peak and Ioffe-Regel limits. With increasing disorder,
the boson peak frequency $\omega_{BP}$, transverse Ioffe-Regel frequency
$\omega_{IR}^T$, and longitudinal Ioffe-Regel frequency $\omega_{IR}^L$ all
decrease. We find that there are two ways for the boson peak to form:
developing from and coexisting with (but remaining independent of) the
transverse van Hove singularity without and with local coordination number
fluctuation. In the presence of a single type of disorder, $\omega_{IR}^T\ge
\omega_{BP}$, and $\omega_{IR}^T\approx \omega_{BP}$ only when the disorder is
sufficiently strong and causes spatial fluctuation of the local coordination
number. Moreover, if there is no positional disorder, $\omega_{IR}^T\approx
\omega_{IR}^L$. Therefore, the argument that the boson peak is equivalent to
the transverse Ioffe-Regel limit is not general. Our results suggest that both
local coordination number and positional disorder are necessary for the
argument to hold, which is actually the case for most disordered solids such as
marginally jammed solids and structural glasses. We further combine two types
of disorders to cause disorder in both the local coordination number and
lattice site position. The density of vibrational states of the resulting
networks resembles that of marginally jammed solids well. However, the relation
between the boson peak and the transverse Ioffe-Regel limit is still indefinite
and condition-dependent.
