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Generalized Modal Coordinates

So, is there a way to extend modal analysis to nonlinear systems? There is, and addressing that question is the purpose of this website. The method discussed on this site applies to linear and nonlinear systems and reduces directly to classical modal analysis for linear systems. I refer to this approach as Generalized Modal Analysis or GMA.

For nonlinear systems, the definition of modal analysis has to be generalized. First we drop the restriction that the coordinate function be a linear function - any function which decouples the system of ODEs will be allowed. Second, we relax the restriction on the form of the decoupled ODE - any decoupled ODE will be allowed. The key to calculating modes for a nonlinear system is to express a generalized mode in terms of the chain rule rather than using eigenvectors for the transformation.

First Order Form

In order to apply the chain rule to a system of ODEs, it is useful to write the system in first order form, where the term on the left hand side of the equation is a lone single derivative, and the term on the right contains only state variables:

${dx_1}/{dt} = f_1(x_1, x_2, ..., x_n)$
${dx_2}/{dt} = f_2(x_1, x_2, ..., x_n)$
...
${dx_n}/{dt} = f_n(x_1, x_2, ..., x_n)$

Example System

As an example, the same linear system will be used as on the previous page so that we can verify that we get the same linear modes. However, we will also see that there are other generalized modes associated with a linear system. The linear system in first order form is:

${dx_1}/{dt} = -2 x_1 + x_2$
${dx_2}/{dt} = x_1 - 2 x_2$

Vector Field Form

A closely related form that is useful for expressing the chain rule is vector field form, where $∂/{∂ x_i}$ is used to represent a coordinate vector for coordinate $x_i$. This might seem like a strange notation for a basis vector if you have done multivariate calculus. However, there is a good reason for this notation. This is a case where the notation helps you know how to apply a transformation of coordinates correctly. The system of ODEs represented in vector field form is:

(1)    $V  =  {dx_1}/{dt} ∂/{∂ x_1} + {dx_2}/{dt} ∂/{∂ x_2} + ... + {dx_n}/{dt} ∂/{∂ x_n}$
(2)    $V  =  f_1 ∂/{∂ x_1} + f_2 ∂/{∂ x_2} + ... + f_n ∂/{∂ x_n}$

Note that when the ODE vector field is expressed in form (1), it kind of resembles the chain rule.

Example Vector Field

The example system in vector field form is:

$V = (-2 x_1 + x_2) {∂}/{∂ x_1} + (x_1 - 2 x_2) {∂}/{∂ x_2}$

Coordinate Transformation (The Chain Rule)

If a new coordinate function $q(x_1, x_2, ..., x_n)$ is defined, ${dq}/{dt}$ is calculated by:

(3)    ${dq}/{dt} = V(q) = {dx_1}/{dt} {∂ q}/{∂ x_1} + {dx_2}/{dt} {∂ q}/{∂ x_2} + ... + {dx_n}/{dt} {∂ q}/{∂ x_n}  =  f_1 {∂ q}/{∂ x_1} + f_2 {∂ q}/{∂ x_2} + ... + f_n {∂ q}/{∂ x_n}$

Supposed we defined a complete set of new coordinates $q_1, q_2, ... , q_n$, where these coordinate are expressed as functions of $x_1, x_2, ..., x_n$, and would like to represent the vector field in terms of the new coordinate system. The new vector field is calculated as follows:

(3)    $W = V(q_1) {∂}/{∂ q_1} + V(q_2) {∂}/{∂ q_2} + ... + V(q_n) {∂}/{∂ q_n} = {dq_1}/{dt} {∂}/{∂ q_1} + {dq_2}/{dt} {∂}/{∂ q_2} + ... + {dq_n}/{dt} {∂}/{∂ q_n}$

Combining the definition of $V(q_i)$ from (3) with (4), you can see that this is the chain rule, and the the vector field notation make the transformation of a system of ODEs from one coordinate system more obvious. For this reason, it is a preferred notation for generalizing modal analysis to nonlinear systems.

Example Transformations

Lets try a couple of random examples to show how you calculate ${dq}/{dt}$ for a function $q$. Note that the coordinate examples chosen below are NOT modal coordinates.

Generalized Modal Analysis

For generalized modal analysis the coordinate transformation from (3) is combined with the decoupling condition ${dq}/{dt} = h(q)$. A coordinate function satifying this equation is a generalized modal coordinate and results in a decoupled ODE which can be solved independent of any other choice of coordinate for the system of ODEs.

${dq}/{dt} = V(q) = f_1 {∂ q}/{∂ x_1} + f_2 {∂ q}/{∂ x_2} + ... + f_n {∂ q}/{∂ x_n} = h(q)$
$f_1 {∂ q}/{∂ x_1} + f_2 {∂ q}/{∂ x_2} + ... + f_n {∂ q}/{∂ x_n} = h(q)$

Verify the Linear Modes

On the Linear Modal Analysis page, the linear modes of the example system were calculated. Applying the chain rule gives: