Stability Analysis of Newmark's Method for Non-Singular Linear Systems

[traianus]

Newmark's method deals with second order differential equation in the time domain.
The equation is of the type

M*xddot + C*xdot + K*x = F(t) (1)

Now suppose to have a GENERIC system:

A*xddot + B*xdot + D*x = G(t) (2)

where A,B,C are nonsingular matrices. A is NOT necessarily positive definite as the mass matrix in equation (1). If I apply Newmark's method in equation (2), would I have stability? What are the conditions to have stability in equation (2)? Do you know good references? I applied equation (2) to my case with success, but I like to know what is the mathematics behind it. Thank you

 

[AlephZero]

Yes, it is well behaved in some cases when M is not positive definite.

There are lots of ways to formulate the Newmark method but one way (assuming beta = 1/2 and gamma = 1/4) involves solving the equations

(M + Ch/2 + Kh^2/4)x_{n+1} = something complicated (and not relevant to the issue).

at each time step.

One way to consider stability is to think how the eigenvalues and vectors of the left hand side matrix relate to the the eigenpairs of M, K, and C. For example, if K is positive definite and M is positive semidefinite (i.e. some degrees of freedom have no mass) and C represents Rayleigh damping (C = aK + bM) then the equation system will be positive definite, and this case works fine in practice.

To think about stability for a nonlinear problem, you need to consider M C and K as the linearised (small perturbations, "tangent stiffness", whatever terminology you prefer) matrices at the current state of the system. Dumping all the nonlinearities into the right hand side doesn't make stability problems go away!

For example if the structure buckles, and K is non-positive-definite, that can cause problems for large values of h, if Kh^2/4 dominates M in the equation above.

One formal approach to stability analysis is to do an eigensolution of the LHS matrix, then transform the problem into the modal coordinates so you have N uncoupled scalar equations rather than an NxN matrix equation. However physical insight into what the matrices mean is often quicker than flogging through the maths.

My "bedtime background reading" on this type of problem is T J R Hughes, "The Finite Element Method: Linear statics and dynamic FE Analysis", Prentice-Hall, 1987. Of course there may good modern texts as well - but I'm still learning from re-reading that one occasionally!

 

[traianus]

Thank you! Do you have a reference paper in which it is stated that Newmark can be applied (with stability!) on a general second order problem?

 

[AlephZero]

A general second order problem isn't always stable anyway, so if Newmark gave always gave stable answers it would be wrong!

As a trivial example, take a 1-dof problem with A = D = 1 and B < 0 in your equation. In other words a mass on a spring, with "negative damping". Give it any non-zero initial conditions you like, apply no external force, and it will go exponentially to infinity (which it should do).

Most likely the physics of your problem puts some constraints on what form the matrices can have (constraints like the matrices must be symmetric, no negative eigenvalues, etc) and what forms the solutions can have (e.g. is the system's energy conserved or it is dissipated). You need to use all the information you have to draw useful conclusions.

 

[traianus]

My problem is a bit more complex . I have a structural system which interacts with the aerodynamics. Then, each structural matrix is "contaminated" by an aerodynamic matrix. Sometimes there can be a stable system and sometimes there can be un unstable system, depending on the geometry. But Your aswer is enough for me: if I have a physically stable system, Newmark's method works just fine.

 

[mina1363]

Hi,
I know this thread goes a while back but I'm trying to write a code for a linear system of spring elements using the Newmark's method of integration. The system is undamped and forced and the mass and stiffness matrices are non-singular . I was wondering if you found a good reference in the end?
I've written a code but my displacements keep increasing with time.
Thanks!