Comparing Measured and Computed Nonlinear Frequency Responses to Calibrate Nonlinear System Models

Abstract

Many systems of interest contain nonlinearities that are difficult to accurately model from first principles, so it would be preferable to characterize the system experimentally. For many nonlinear systems, it is now possible to measure frequency response curves with stepped sine testing and to compute frequency response curves with numerical continuation. Nonlinear frequency response curves are very sensitive to the system model and the nonlinearities and they provide a lot of insight into the response of the system to a variety of inputs. This paper explores the feasibility of a nonlinear model updating approach based on nonlinear frequency response and the experimental and analytical tools that are needed. For the experiment, a cantilever beam with an unknown nonlinearity is driven with a harmonic force at various frequencies. The steady-state response is measured and processed with the fast Fourier transform to obtain the frequency response curve. Some subtle yet important details regarding how this is implemented are discussed. An analytical model is also constructed and its frequency response computed using a recently developed technique. The measured and simulated frequencies are then compared and used to tune the analytical model.

Appendix A: Ritz-Galerkin Discrete Model

A Galerkin approach was used to create a finite-order model of the experimental structure. Assuming that the beam behaves linear-elastically, mode shapes corresponding to transverse bending motion were used as shape functions to construct the Ritz-Galerkin representation [21]. The displacement of the beam at a position x was approximated as

$$ y\left( {x,t} \right) = \sum\limits_{{r = 1}}^{{{N_m}}} \it Psi {}_r(x){q_r}(t) $$

(A.1)

where ψ _r (x) is the rth Euler-Bernoulli beam mode shape for a cantilever, q _r (t) is the rth generalized coordinate, and N _m is the number of modes used. The system’s undamped equations of motion are provided in the following equation, where the coordinates are the amplitudes of the basis functions.

$$ \rho {A_b}L\left[ M \right]\ddot{q} + \frac{{{E_b}I}}{{{L^3}}}\left[ K \right]q = Q = \sum {{f_{{ext}}}{\it Psi_r}({x_f})} $$

(A.2)

For the generalized coordinates, a time derivative is denoted with an over-dot (e.g. the generalized acceleration vector is $\ddot{q}$). Modal damping was added to the system by performing an eigenvector analysis on the linear system and then using,

$$ \left[ C \right] = {(\rho {A_b}L)^2}\left[ M \right]\,\left[ {{\phi_b}} \right]\,\left[ {diag\left( {2{\zeta_r}{\omega_r}} \right)} \right]\,{\left[ {{\phi_b}} \right]^{\text{T}}}\,\left[ M \right] $$

(A.3)

where [φ _b ] is a matrix containing the eigenvectors in the columns, ω _r is the rth circular natural frequency, and ζ _r is the rth desired damping ratio. The generalized force vector Q is a sum of the product between all external forces and the value of the shape functions at the point where the force is applied, x _f . Therefore, Q includes the applied or external forces, F _ext in Fig. 21.5, as well as the nonlinear restoring force due to the spring [21]. The beam provides linear stiffness at the tip due to its flexural rigidity, so the discrete spring’s stiffness was chosen to be purely nonlinear

$$ {k_{{nl}}} = {k_3}y(L)^2 $$

(A.4)

where k ₃ is a stiffness constant associated with the nonlinear spring. The physical restoring force due to the spring is then equal to ${f_{{sp}}} = {k_3}y{(L)^3}$. The generalized force vector then has components corresponding to the nonlinear spring located at x = L and the externally applied force located at x = x _f .

$$ \left\{ Q \right\} = {k_3}y(L)^3\left\{ {\begin{array}{*{20}{c}} {{\psi_1}(L)} \\ \vdots \\ {{\psi_N}(L)} \\ \end{array} } \right\} + {A_{{ext}}}\sin \left( {{\omega_T}t} \right)\left\{ {\begin{array}{*{20}{c}} {{\psi_1}({x_f})} \\ \vdots \\ {{\psi_N}({x_f})} \\ \end{array} } \right\} $$

(A.5)

A _ext is the amplitude and ${\omega_T}$ the frequency of the external forcing term that produces the limit cycle.

After using the Ritz-Galerkin method to form the discrete beam model and to account for the nonlinear applied force of the spring, the equations of motion were transformed back into physical coordinates using the relationship in (A.1). The differential equations of motion can then be arranged in state space format.

$$ \begin{gathered} \left\{ {\begin{array}{*{20}{c}} {\dot{y}} \\ {\ddot{y}} \\ \end{array} } \right\} = \left\{ {\begin{array}{*{20}{c}} {\dot{y}} \\ { - {{\left[ {{M_p}} \right]}^{{ - 1}}}\left( {\left[ {{C_p}} \right]\dot{y} + \left[ {{K_p}} \right]y + \left\{ F \right\}} \right)} \\ \end{array} } \right\} \hfill \\ \left[ {{M_p}} \right] = \rho {A_b}L{\left[ \Psi \right]^{{ - {\text{T}}}}}\left[ M \right]{\left[ \Psi \right]^{{ - 1}}},\quad \left[ {{C_p}} \right] = {\left[ \Psi \right]^{{ - {\text{T}}}}}\left[ C \right]{\left[ \Psi \right]^{{ - 1}}}, \hfill \\ \left[ {{K_p}} \right] = \frac{{{E_b}I}}{{{L^3}}}{\left[ \Psi \right]^{{ - {\text{T}}}}}\left[ K \right]{\left[ \Psi \right]^{{ - 1}}},\quad \left\{ F \right\} = {\left[ \Psi \right]^{{ - {\text{T}}}}}\left\{ Q \right\} \hfill \\ \end{gathered} $$

(A.6)

The matrix $\left[ \Psi \right]$has the numerical values of the mode vectors for specific position coordinates on the beam. Then, $\left[ \Psi \right]$can contain shape functions evaluated at the nodal degrees of freedom on the beam. In this study the number of mode shapes used in the expansion and the number of degrees of measurement points (shown in Fig. 21.5) was N = N _m  = 3. The nodes were located at the center and tip of the beam as shown in Fig. 21.5.