# In a four mass six spring vibration, how is the kinetic energy represented

This is from Hobson, Riley, Bence Mathematical Methods, p 322. A spring system is described as follows (they are floating in air like molecules):

The equilibrium positions of four equal masses M of a square with sides 2L are $R_n=\pm L_i\pm L_j$ and displacements from equilibrium are $q_n=x_ni+y_nj$. According to the text,

"The coordinates for the system are thus x1, y1, x2, . . . , y4 and the kinetic energy of matrix A is given trivially by $MI_8$ where $I_8$ is the 8x8 identity".

What does that mean? The velocity doesn't even appear. How does that relate to energy?

## 2 Answers

An engineer would call it the "mass matrix" not the "kinetic energy matrix". The KE is given by $\frac 1 2 \mathbf{v}^T \mathbf{M} \mathbf{v}$ where $\mathbf{v}$ is the vector of the velocity components $\dot x_1, \dots, \dot x_4, \dot y_1, \dots, \dot y_4$ and $\mathbf{M} = M \mathbf{I}_8$ - i.e. an $8\times8$ diagonal matrix with all the diagonal terms equal to $M$.

"Kinetic energy matrix" seems a silly name IMO, because as you said it doesn't fully represent the kinetic energy of the system.

The "stiffness matrix" (or whatever the mathematicians who wrote your book call it!) can similarly be written as an $8\times8$ matrix $\mathbf{K}$, though it's not so simple as the mass matrix. The potential energy stored in the springs is then given by $\frac 1 2 \mathbf{x}^T \mathbf{K} \mathbf{x}$

Reading a textbook or web page on matrix methods for modelling multi-degree-of-freedom (MDOF) systems, written for engineers or physicists rather than for mathematicians, might help to understand the basic ideas.

The origin of the mass matrix lies in a change of coordinates from Cartesian to generalized coordinates.

The kinetic energy of a system of $N$ particles in terms of Cartesian coordinates is $$K=\frac 12\sum_{a=1}^Nm_a\dot{\vec r}_a\cdot\dot{\vec r}_a.$$ If the system is scleronomic, i.e., the relation between Cartesian and generalized coordinates do not involve time explicitly, $\vec r_i=\vec r_i(q_1,\ldots,q_n)$, then by the chain rule $$K=\frac 12\sum_{a=1}^Nm_a\sum_{i=1}^n\frac{\partial \vec r_a}{\partial q_i}\dot q_i\cdot\sum_{j=1}^n\frac{\partial \vec r_a}{\partial q_j}\dot q_j.$$ By rearranging terms this can be written as $$K=\frac12\sum_{i,j}A_{ij}\dot q_i\dot q_j=\frac 12\dot q^TA\dot q,$$ where $A_{ij}\equiv \sum_am_a\frac{\partial \vec r_a}{\partial q_i}\cdot\frac{\partial\vec r_a}{\partial q_j}$ are the components of the mass matrix.

Sometimes it is safer calculate $A$ by using its definition. In simpler systems however it is better to write the kinetic energy explicitly in terms of the generalized coordinates and then compare with the quadratic form $\frac 12\dot q^TA\dot q$ to obtain $A$.