Shock and Vibration

Volume 2016, Article ID 5450865, 10 pages

http://dx.doi.org/10.1155/2016/5450865

## On the Shaker Simulation of Wind-Induced Non-Gaussian Random Vibration

^{1}School of Reliability and System Engineering, Beihang University, 37 Xueyuan Road, Haidian District, Beijing 100191, China^{2}Mechanical Engineering, Blekinge Tekniska Högskola, 371 79 Karlskrona, Sweden

Received 29 July 2015; Revised 28 September 2015; Accepted 15 October 2015

Academic Editor: Dumitru I. Caruntu

Copyright © 2016 Fei Xu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Gaussian signal is produced by ordinary random vibration controllers to test the products in the laboratory, while the field data is usually non-Gaussian. Two methodologies are presented in this paper for shaker simulation of wind-induced non-Gaussian vibration. The first methodology synthesizes the non-Gaussian signal offline and replicates it on the shaker in the Time Waveform Replication (TWR) mode. A new synthesis method is used to model the non-Gaussian signal as a Gaussian signal multiplied by an amplitude modulation function (AMF). A case study is presented to show that the synthesized non-Gaussian signal has the same power spectral density (PSD), probability density function (PDF), and loading cycle distribution (LCD) as the field data. The second methodology derives a damage equivalent Gaussian signal from the non-Gaussian signal based on the fatigue damage spectrum (FDS) and the extreme response spectrum (ERS) and reproduces it on the shaker in the closed-loop frequency domain control mode. The PSD level and the duration time of the derived Gaussian signal can be manipulated for accelerated testing purpose. A case study is presented to show that the derived PSD matches the damage potential of the non-Gaussian environment for both fatigue and peak response.

#### 1. Introduction

Random vibration testing is usually used to bring a test item to failure to identify weaknesses in the product or to verify if the product can survive a particular random vibration environment. Historically, random vibration controllers accomplish this goal by producing a power spectral density (PSD) that would expose the test item to the type of vibratory environment that the test item would experience in a real-world setting. However, a random process can be characterized by a PSD only when it is stationary and Gaussian distributed. In practice, non-Gaussian vibration is usually encountered, especially in the road transportation [1, 2]. The difference of the probability density function (PDF) between the measured acceleration and Gaussian distribution may lead to totally different accumulated fatigue damage [3]. MIL-STD810 points out that we should always check if the field tested data is non-Gaussian and the testing hardware and software is appropriate [4]. The Time Waveform Replication (TWR) is frequently referred to as a methodology for non-Gaussian testing [4]. The basic idea of TWR is to reproduce a sequence of instantaneous values of the vibration process. Such a test may be non-Gaussian; however, this is only a replication of one particular measured record, not a simulation of a specified road type. Smallwood presented a zero memory nonlinear function (ZMNL) method to synthesize a given non-Gaussian data [5]. Analytical solution was presented later to extend the original offline method to a method that can be used in the closed-loop frequency domain control mode. Iteration method was presented to address the PSD distortion problem introduced by the nonlinear transformation [6]. A. Steinwolf presented a phase selection method to simulate the road vehicle measured data [7]. In particular, selected phase is transformed from random to deterministic in order to obtain a prescribed kurtosis. PSD and PDF are controlled independently. An analytical relation between kurtosis, amplitude, and phase at specific frequencies was presented later to make this method applicable in a closed-loop control implemented by the shake controller [8–10]. These methods mentioned above are applicable when the non-Gaussianity is caused by significant shocks in vehicle vibration data. For wind-induced vibration data, however, the non-Gaussianity may be caused by other factors, for example, by the running root mean square. Rouillard and Sek presented a novel technique to synthesize non-Gaussian vibrations by generating a sequence of random Gaussian processes of varying RMS levels and durations [11].

Two methodologies are presented in this paper for shaker simulation of wind-induced vibration. The essential idea of the first methodology is to synthesize the non-Gaussian signals offline and replicate them on the shaker in the Time Waveform Replication mode. The non-Gaussian signal is modeled as a Gaussian signal multiplied by an amplitude modulation function (AMF). A two-parameter Weibull distribution is used to create the AMF. A case study is presented to show that the synthesized non-Gaussian signal has the same power spectral density (PSD), probability density function (PDF), and loading cycle distribution (LCD) as the field data. The rainflow cycle counting (RFCC) method [12] and Dirlik method [13] are used for the LCD calculation.

The fatigue damage spectrum (FDS) and the extreme response spectrum (ERS) have been used as a quantitative way to take the fatigue damage and overstress damage into consideration for field data [14–20]. The FDS and ERS are used in the second methodology to derive a damage equivalent Gaussian signal (characterized by a PSD and time duration) from the non-Gaussian signal, so that they can be easily produced on the shaker in the closed-loop frequency domain control mode. Improvement of the derived PSD by taking its spectrum shape into consideration is also presented. The PSD level and the duration time of the derived Gaussian signal can be manipulated for accelerated testing purpose. A case study is presented to show that the derived PSD matches the damage potential of non-Gaussian environment for both fatigue and peak response.

#### 2. Theory

##### 2.1. Non-Gaussian Random Signal

If a Gaussian process has a zero mean value and is ergodic, then the PDF of the instantaneous acceleration values is given by the Gaussian distribution with zero mean: where is standard deviation.

For large durations , the variance is given bywhere is the single-sided PSD.

It shows that random Gaussian processes with zero mean can be completely described by the PSD function. One useful method for establishing how well a random process can be described by the Gaussian distribution is by computing the higher order moments of the process defined as (continuous and discrete form)

When the mean value is zero,where is the skewness and is the kurtosis.

For a truly Gaussian process, the skewness is 0 and the kurtosis is 3. Any deviation from these indicates that the process is non-Gaussian. From the perspective of simulating non-Gaussian vibration, kurtosis is a more important parameter than skewness, because it represents the probability of peak values in time history.

##### 2.2. Response Spectra

All structures experience fatigue as they are repeatedly exposed to adequate levels of stress. The total damage a product experiences in a particular time period can be calculated from field data and plotted for a specific range of frequencies. The FDS is essentially a plot that shows the responses of a number of single degree of freedom (SDOF) systems to a base input acceleration time history. Many SDOF systems that are tuned to a range of natural frequencies are assessed using the same input. The final plot, the FDS, shows the fatigue damage encountered for a particular SDOF system anywhere within the analyzed time, as shown in Figures 1 and 2.