The Two Ways a Two-Story Building Sways

Model a two-story building as two floors on springy columns. It has two natural ways to sway, each with its own frequency, and both come from a fourth-degree equation in ω that's secretly a quadratic in ω².

How it works

Equations reducible to quadratic form

In one sentence

An equation is reducible to quadratic form when a substitution such as $u = x^2$ turns it into $au^2 + bu + c = 0$ . Here $u = \omega^2$ , and each nonnegative solution $u$ gives the two solutions $\omega = \pm\sqrt{u}$ of the original quartic.

General formula

{\htmlData{sym=0}{m_1}}{\htmlData{sym=1}{m_2}}\,{\htmlData{sym=3}{\omega}}^4 - {\htmlData{sym=2}{k}}({\htmlData{sym=0}{m_1}} + 2{\htmlData{sym=1}{m_2}})\,{\htmlData{sym=3}{\omega}}^2 + {\htmlData{sym=2}{k}}^2 = 0 \;\xrightarrow{\;{\htmlData{sym=4}{u}}\,=\,{\htmlData{sym=3}{\omega}}^2\;}\; {\htmlData{sym=0}{m_1}}{\htmlData{sym=1}{m_2}}\,{\htmlData{sym=4}{u}}^2 - {\htmlData{sym=2}{k}}({\htmlData{sym=0}{m_1}} + 2{\htmlData{sym=1}{m_2}})\,{\htmlData{sym=4}{u}} + {\htmlData{sym=2}{k}}^2 = 0

The quartic only cares about ω², so its graph is symmetric: solve once for ω², then take square roots.

What the symbols mean

$m_1$
First-floor mass
The mass of the lower floor.
$m_2$
Top-floor mass
The mass of the upper floor.
$k$
Column stiffness
How hard the columns of each story push back per meter of sway.
$\omega$
Natural frequency
How fast the building sways in one of its modes, in radians per second.
$u$
Substitute variable
ω², which turns the quartic into a quadratic.

Treat each floor as a mass and the columns between floors as springs. Newton's second law for the two floors gives two linked equations.
A natural frequency $\omega$ is one at which both floors can swing in step forever; it must solve $m_1m_2\,\omega^4 - k(m_1 + 2m_2)\,\omega^2 + k^2 = 0$ .
Only even powers of $\omega$ appear, so substitute $u = \omega^2$ : the quartic becomes a quadratic in $u$ , solved with the quadratic formula.
Each positive $u$ gives $\omega = \pm\sqrt{u}$ , so the quartic has four roots mirrored across zero. The two positive ones are the building's sway frequencies.

Learn the math

Equations Reducible to Quadratic FormAlgebra · Lesson 14, Solving Equations and Inequalities