Question

The building is modeled as a single degree of freedom spring-mass system where the building mass is lumped atop of two beams used to model the walls of the building in bending. Assume the ground motion is modeled as having amplitude of 0.1 m at a frequency of 8 rad/s. Approximate the building mass by 500 kg and the stiffness of each wall by 2x10^6 N/m. Compute the magnitude of the deflection of the top of the
building.

Answer 1

As here given building is single degree of freedom spring-mass system

So, we can write its equation of motion as

$m \ddot{x}(t)+2kx(t)=A cos \omega t$

Now, here m=mass of building as given as 500kg,A= amplitude=0.1, and frequency= 8 rad/s

k=stiffnes=$2\times 10^6 N/m$

Now, we will calculate natural frequency as

$\omega_n= \sqrt{\frac{2k}{m}}=\sqrt\frac{2\times 10^6}{500}=63.24$

and frequency ratio

$r=\frac{\omega}{\omega_n}=\frac{8}{63.24}=0.126$

Now, for the maximum deflection of top of building we know that

$\delta= A\frac{1}{1-r^2}=0.1\frac{1}{1-.126^2}=0.101 m$

So, the magnitude of deflection of top= 0.101 m