Coupled motion and the matrices

Two floors, two coordinates

Take a two-storey shear building. Floor 1 has mass m₁ and sits on columns of total lateral stiffness k₁. Floor 2 has mass m₂ and sits on columns of stiffness k₂ connecting it to floor 1.

Let u₁ and u₂ be the lateral displacements of the two floors, both measured from the ground.

Write equilibrium for each floor separately

Floor 2 is connected only to floor 1. The columns between them are stretched by the RELATIVE displacement (u₂ − u₁), so:

m₂ü₂ + k₂(u₂ − u₁) = F₂(t)

Floor 1 is connected to the ground below and floor 2 above. The lower columns are deformed by u₁; the upper columns pull back with the force they are carrying, which is k₂(u₂ − u₁) acting in the opposite direction:

m₁ü₁ + k₁u₁ − k₂(u₂ − u₁) = F₁(t)

That is the whole derivation. Two free bodies, two equilibrium equations, and the only subtlety is remembering that a storey's columns respond to the RELATIVE movement of its two ends.

Collect into matrices

Rearranging and writing in matrix form:

[m₁ 0 ] {ü₁} [k₁+k₂ −k₂] {u₁} {F₁} [0 m₂] {ü₂} + [ −k₂ k₂] {u₂} = {F₂}

or simply Mü + Ku = F.

What every entry means

The mass matrix is diagonal. Accelerating floor 1 produces no inertia force on floor 2 — the masses are separate lumps and the inertia of one has nothing to do with the other. This is the lumped-mass assumption of Module 2, and here it is exactly true if the mass really is concentrated at the floors.

The stiffness matrix is not diagonal, and its entries have a precise meaning:

K[i][j] is the force required at i to hold every other coordinate at zero while coordinate j is moved by one unit.

Read that way, each entry can be checked by inspection:

• K[0][0] = k₁ + k₂. Move floor 1 by one unit while holding floor 2. Both the lower columns and the upper columns are deformed, so both resist.
• K[1][1] = k₂. Move floor 2 while holding floor 1. Only the upper columns deform.
• K[0][1] = K[1][0] = −k₂. Move floor 2 by one unit while holding floor 1. The upper columns pull floor 1 upwards in the direction of the movement, so a force of −k₂ must be applied at floor 1 to hold it still.

The symmetry K[0][1] = K[1][0] is not a coincidence. Betti's reciprocal theorem guarantees it, and a stiffness matrix that comes out unsymmetric has an assembly error in it.

Coupling

Two coordinates are coupled when the equation for one contains the other.

Here the coupling is in the stiffness matrix, through the −k₂ terms. Physically: floor 1 cannot move without deforming the columns that hold floor 2, so it cannot move without pushing floor 2.

This is static coupling. A system can also have inertial coupling, where the mass matrix has off-diagonal terms — a consistent-mass beam element, or a floor whose centre of mass is offset from its centre of stiffness. Module 10 meets both.

If both matrices were diagonal, the two equations would be entirely independent and there would be no interesting problem: two separate SDOF systems that happen to be drawn near each other. Modal analysis is the search for a set of coordinates in which that is true.