Coriolis dynamic vibration absorber is a device working in nonlinear zone. In stochastic design of this device, the Monte Carlo simulation requires large computation time. A simplified model of the system is built to retain the most important nonlinear term, the Coriolis damping of the dynamic vibration absorber. Applying the full equivalent linearization technique to the simplified model is inaccurate to describe the nonlinear behavior. This paper proposes a combination of partial stochastic linearization and KarhunenLoeve expansion to solve the problem. The numerical demonstration of a ropeway gondola induced by wind load is presented. A design example based on the partial linearization supports the advantage of the proposed approach.
Dynamic vibration absorber (DVA) or tuned mass damper composed of mass, spring, and damper is widely used to suppress vibration. The theory of linear DVA is well developed in literature [
In our recent study [
The novelty of this paper is the proposal of partial linearization technique combined with KL expansion. This approach is much more accurate than the full linearization technique while the time consumed is much shorter than the Monte Carlo simulation. The structure of paper is as follows. Firstly, in Section
In Figure
Pendulum structure with a Coriolis dynamic vibration absorber.
Assume that the pendulum structure is subjected to a random external force
The following parameters are introduced:
in which
The stochastic differential equations (
(i) The effect of the variation of the arm length is ignored, such as
The approximations are appropriate because, in the practical application, the DVA normalized displacement
(ii) The nonlinearity of the trigonometric functions is retained up to the second order, such as
These approximations produce acceptable errors (<5%) for the angle up to 30 degrees.
Equations (
In the next sections, the linearization techniques are discussed in the simplified equation (
The simplest equivalent linearization technique is based on the assumption of Gaussian distributed responses [
Any nonlinear vector function
Applying (
Substituting (
An important remark is drawn from (
The full linearization technique linearizes the system too much that hides some important nonlinearities. The partial linearization is considered instead. The idea has been studied in [
Setting the derivative of
Substituting the replacements (
The first equation of (
The KL expansion is well known in literature. The orthonormality relation of the KL expansion is presented in the Appendix. Let us discuss the details of the implementation of KL expansion to the linearization techniques presented in Section
The relations (
The solutions of full linearized equation (
The mean values are calculated as
The iterative procedure of the implementation of KL expansion to full linearization technique is shown in Figure
Flowchart of implementation of KL expansion to full linearization technique.
Because (
Substituting (
Equation (
This rearrangement yields the following form of DVA response:
By using the useful relations (
The iterative procedure of the implementation of KL expansion to partial linearization technique is shown in Figure
Flowchart of implementation of KL expansion to partial linearization technique.
This section will verify the accuracy of the proposed approach through an example of a ropeway gondola subjected to wind load. The values for parameters are selected to model a middlesize gondola as pendulum mass
The wind velocity contains two parts, the mean velocity and the fluctuating velocity. The fluctuating velocity
The covariance function of
The KL expansion (
In the numerical calculation, the time span
(a) 100 first eigenvalues and (b) 4 first eigenfunctions.
Because there are no exact solutions of nonlinear systems (
Comparing with the Monte Carlo simulation, the partial linearization procedure presented in Figure
The comparisons of the time histories are shown in Figures
RMS values versus normalized time
RMS values versus normalized time
RMS values versus normalized time
RMS values versus normalized time
RMS values versus normalized time
RMS values versus normalized time
The results yield the following conclusions:
In all cases, the partial linearization technique gives the solutions agreeing with the solutions of Monte Carlo simulation. In some cases (Figures
The DVA natural frequency is chosen near the resonant frequency
In comparison with the Monte Carlo simulation, the time required to process the procedure in Figure
Sweeping parameters.
Parameter  Interval (from–to)  Number of points  Scale 


1.8–2.2  20  Linearly equally spaced points 

0.01–1  20  Logarithmically equally spaced points 
The total data size of
Performance index
It can be seen that the optimal value of the DVA frequency ratio
This paper considers the stochastic design of the pendulum structures attached with the dynamic vibration absorber using Coriolis force. The standard Gaussian equivalent linearization method applying to a simplified model fails to describe the system behaviors. Meanwhile, a partial linearization technique realized by the KarhunenLoeve expansion works well in this simplified model. The proposed combination method is checked by the Monte Carlo simulations. The usefulness of the presented approach is demonstrated by a numerical example of a ropeway gondola induced by wind and by a design example of the dynamic vibration absorber.
Consider a continuous stochastic process
The discrete KL expansion of
The following orthonormality relation can be obtained [
However, in this paper, we need to derive the orthonormality relation of higher orders of
If we choose
At last, the eigenvalue problem (
The orthonormality relations (
The author declares that there are no conflicts of interest regarding the publication of this paper.
This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant no. “107.012015.35.”