Multistory buildings can be modeled as lumpedparameter sprin

Multi-story buildings can be modeled as lumped-parameter spring mass systems. For example, the walls and floors can be modeled as masses m, and the interstory lateral stiffness can modeled as springs Consider the pictured system where m_1 = m, m_2 = m/2, k_1 = k, and h_2 = k/2. Find the natural frequencies and mode shapes for the system.

Solution

Solution:

From given arrangement

both the spring are connected in parallel

therefore equivalent stiffness is given by

1/ke = 1/k1 + 1/k2

=1/k + 2/k

Ke = K/3

total mass for the system is me = m1 + m2

= m + m/2 = 3/2m

Natural frequency is given by w = sqrt (ke/me)

= sqrt ((k/3) /(3/2m))

= sqrt ((2k)/(9m))

As there are two displacements are given therefore exists two DOF for given arrangement

Hence there are 2 modes for the system.