# Solving for stiffness matrix numerically by a set of measurements

Is it in general possible when you have a mechanical structure (in this case I am referring to the structure in this post) and you want to determine the global stiffness matrix of this structure, to solve for the matrix based on measurements?

More specifically, I have a structure where parts are connected by springs. I apply a known force to one of the parts and measure the displacement. In this way I get $$n$$ force/moment - displacement/rotation pairs. Is it possible to solve for a matrix $$K$$, for

$$F = KX$$,

with $$F =\begin{bmatrix}F_x & F_y & F_z & M_x &M_y&M_z\end{bmatrix}$$ and $$X= \begin{bmatrix}dx & dy & dz & \theta_x &\theta_y&\theta_z\end{bmatrix}$$?

Is this feasible from a mechanical point of view, because e.g. this "stiffness matrix" $$K$$ may not be unique according to the load applied? Are there any constraints or requirements regarding the measurements?

P.S. Applied forces are rather small (within the elastic range of the spring) and the displacement is restricted, so no large deflections should occur. This is why I assumed a linear approximation.

## 1 Answer

This is feasible and can be used to modify a theoretical stiffness matrix calculated by the Finite Element method to match experimental results more accurately. The FE model can then be used to calculate things which would be impractical to measure directly.

The simple approach you suggest is possible but not necessarily the best practical method. It may appear paradoxical that it is more practical to create both the stiffness and mass matrix by measuring the dynamic response of the structure. The reason is that if you excite the structure by an impact at one point and measure the time history of the response at other points (which is straightforward to do using an accelerometer instead of direct measurements of displacement, and a computer to record the real-time response at a sufficiently high sampling rate), you can obtain information about the motion in many different modes of vibration from "one measurement" of the dynamic response.

This can be done effectively on a "lab bench" scale using simple hand-held equipment, but with more sophisticated measuring devices, for example a scanning laser doppler vibrometer, it is possible to measure the response at hundreds of points on a structure "simultaneously" without any physical contact, including measurements under real operating conditions (e.g. at high temperatures, or the behaviour of rotating machinery while it is operating).

Most of this is not covered in a typical first engineering degree. Google for phrases like "experimental modal analysis" "model updating", or "system identification" for more information, both practical and theoretical.

• Thanks for the detailed and fast response! What I actually wanted to do is to simulate the structures in Ansys, add a force and measure the displacement to obtain a dataset. I have to admit I couldn't analytically solve the structure until now.´, would it be enough to use experimental/ simulated data to obtain the final stiffness matrix? – rfn123 May 13 '20 at 19:22