Natural frequencies of a shear frame

The model

The first floor has a mass of 30 t and the roof 20 t. The lower storey has a lateral stiffness of 40,000 kN/m and the upper one 30,000 kN/m. The stiffness matrix is K = [k1 + k2, -k2; -k2, k2] and the mass matrix is diagonal. With two arguments, eigenvals(K, M) solves the generalised problem directly: K must be symmetric and M symmetric positive definite. Each eigenvalue is omega^2 in the units of K over M, so kN/m divided by tonnes comes out per second squared without any conversion by hand.

Frequencies

The natural frequency is f = omega / (2 pi) and the period is its inverse. Results from the engine:

Mode shapes

eigenvecs(K, M) returns one mode per row, in the order of the frequencies, each scaled so that phi^T M phi = 1. The sheet scales them again so the roof moves 1. In the first mode the first floor then moves 0.585 in the same direction: the whole frame sways one way. In the second it moves -1.14: the floors move against each other.

The modes are orthogonal through the mass matrix, so the product phi_1^T M phi_2 is zero; the sheet prints 0 t. That is a useful check that the matrices were entered correctly.

More storeys

Each extra floor adds a row and a column to K and M: K stays tridiagonal, with k_i + k_(i+1) on the diagonal and -k_(i+1) beside it, and M stays diagonal. The same two calls then give all the frequencies and modes. For many storeys, build the matrices in a loop rather than by hand.

What this is, and is not

The sheet source

The sample as it ships (this page adds two lines that pick out the first floor of each mode):

# Shear frame natural frequencies

A two storey shear frame has a lumped mass at each floor and a lateral stiffness per storey.
Free vibration needs K φ = ω² M φ. Given both matrices, `eigenvals(K, M)` solves this
generalized eigenvalue problem directly: K must be symmetric and M symmetric positive definite,
and each eigenvalue is ω² in the units of K/M.

m_1: 30 t
m_2: 20 t
k_1: 40000 kN/m
k_2: 30000 kN/m

K: [k_1 + k_2, -k_2; -k_2, k_2]
M: [m_1, 0 t; 0 t, m_2]

## Frequencies

lambda: eigenvals(K, M)
omega_1: sqrt(lambda.1)
omega_2: sqrt(lambda.2)
f_1: omega_1/(2*π) -> Hz
f_2: omega_2/(2*π) -> Hz
T_1: 1/f_1 -> s

## Mode shapes

Each row of `eigenvecs(K, M)` is one mode, in the order of the frequencies, scaled so that
φᵀ M φ = 1 (its units are t^-0.5). Scaled again so the roof moves 1:

Phi: eigenvecs(K, M)
phi_1: row(Phi, 1)/Phi.(1, 2)
phi_2: row(Phi, 2)/Phi.(2, 2)

The modes are orthogonal through the mass matrix, so this product is zero:

check: dot(phi_1, M*phi_2)

r_1: phi_1.1 | first floor, mode 1
r_2: phi_2.1 | first floor, mode 2