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Research Papers

Theoretical and Experimental Identification of Cantilever Beam With Clearances Using Statistical and Subspace-Based Methods

[+] Author and Article Information
Bing Li

Professor
State Key Laboratory for Manufacturing
Systems Engineering,
Xi'an Jiaotong University,
Xi'an 710049, China
e-mail: bli@mail.xjtu.edu.cn

Luofeng Han

State Key Laboratory for Manufacturing
Systems Engineering,
Xi'an Jiaotong University,
Xi'an 710049, China
e-mail: hanluofeng.2008@163.com

Wei Jin

State Key Laboratory for Manufacturing
Systems Engineering,
Xi'an Jiaotong University,
Xi'an 710049, China
e-mail: jw.1108@stu.xjtu.edu.cn

Shuanglu Quan

State Key Laboratory for Manufacturing
Systems Engineering,
Xi'an Jiaotong University,
Xi'an 710049, China
e-mail: qqqyulu@stu.xjtu.edu.cn

1Corresponding author.

Manuscript received May 7, 2014; final manuscript received July 22, 2015; published online October 23, 2015. Assoc. Editor: Corina Sandu.

J. Comput. Nonlinear Dynam 11(3), 031003 (Oct 23, 2015) (17 pages) Paper No: CND-14-1120; doi: 10.1115/1.4031193 History: Received May 07, 2014; Revised July 22, 2015

Clearance turns up in a large number of engineering structures because of the errors during assembling, manufacturing, and wearing. The presence of clearance in engineering structures changes the normal dynamic response and will result in low precision and short lifetime. The clearance parameter identification of such nonlinear system is the prerequisite to control and eliminate the effect of clearance nonlinearity. In this paper, a derivative plot of probability density function (DPPDF) for displacement response is proposed to precisely identify the clearance value of continuous system, and the nonlinear subspace identification (NSI) method is modified to recognize the related contact stiffness based on the frequency response function (FRF) equations of continuous system. The DPPDF method is carried out by analyzing the distribution characteristic of displacement response, and the clearance value is derived through inspecting the probability density function (PDF) plot and the second derivative plot of the PDF. Based on the identified clearance, the clearance nonlinearity is regarded as external force, and the relationship between the dynamic responses and the external forces in frequency domain can be expressed as the form of FRF equations. Based on the FRF equations, the contact stiffness in continuous system is obtained with modified NSI method. This combined identification process is verified by a single-degree-of-freedom (SDOF) system and a cantilever beam system with clearances, and some influence factors of this identification process, including noise, transfer error, and force level, are discussed in detail. In the end, an experiment device with changeable clearance and contact stiffness was designed to conduct identification experiments, and the results show that the proposed methods perform effectively in identifying the clearance parameters.

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References

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Figures

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Fig. 1

Flow chart of DPPDF method

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Fig. 2

Identification process of the clearance parameters

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Fig. 3

SDOF system with clearance

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Fig. 4

Input and output of linear SDOF system: (a) input and (b) output

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Fig. 5

Input and output of nonlinear SDOF system with clearance: (a) input and (b) output

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Fig. 6

PDF plot against x(t)

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Fig. 7

Second derivative plot of the PDF against x(t)

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Fig. 8

Singular value plot of SDOF system with i=50

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Fig. 9

SDOF system estimation: (a) contact stiffness kd and (b) estimation error

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Fig. 10

FRF of SDOF system with clearance: (a) magnitude and (b) phase

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Fig. 11

Cantilever beam system with clearance

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Fig. 12

First four shape functions for a cantilever beam

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Fig. 13

Applied force and displacement response at the clearance location for case 1: (a) applied force and (b) displacement response at the clearance location

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Fig. 14

Displacement response at the clearance locations for case 2: (a) displacement response at the first clearance location and (b) displacement response at the second clearance location

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Fig. 15

Restoring force against displacement for case 1

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Fig. 16

Nonlinear force contributions: the first division

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Fig. 17

Nonlinear force contributions: the fourth division

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Fig. 18

PDF and second derivative of PDF at the clearance location for case 1: (a) PDF plot and (b) second derivative plot of PDF

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Fig. 19

PDF and second derivative of PDF at the first clearance location for case 2: (a) PDF plot and (b) second derivative plot of PDF

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Fig. 20

PDF and second derivative of PDF at the second clearance location for case 2: (a) PDF plot and (b) second derivative plot of PDF

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Fig. 21

Singular value plot of the cantilever beam system with i=50

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Fig. 22

Estimation of the contact stiffness k of case 1: (a) whole plot and (b) detail plot

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Fig. 23

Estimation of the linear FRF H12 of case 1: (a) whole plot and (b) detail plot

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Fig. 24

First clearance estimation of case 2: (a) contact stiffness k1 and (b) estimation error

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Fig. 25

Second clearance estimation of case 2: (a) contact stiffness k2 and (b) estimation error

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Fig. 26

FRF H12 of case 2: (a) magnitude and (b) phase

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Fig. 27

Photograph of the experimental setup: clearance testbed

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Fig. 28

Detail plot of the clearance location

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Fig. 29

Schematic diagram of the complete setup for the experimental test

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Fig. 30

Force and displacement response at the clearance location: (a) exciting force and (b) displacement response

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Fig. 31

PDF and second derivative of PDF at the clearance location: (a) PDF plot and (b) second derivative plot of PDF

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Fig. 32

Singular value plot of the cantilever system with i=50

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Fig. 33

Estimation of the contact stiffness k

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Fig. 34

FRF H12 of the cantilever system with one clearance: (a) magnitude and (b) phase

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Fig. 35

PDF and second derivative of PDF at the first clearance location: (a) PDF plot and (b) second derivative plot of PDF

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Fig. 36

PDF and second derivative of PDF at the second clearance location: (a) PDF plot and (b) second derivative plot of PDF

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Fig. 37

Estimation of the contact stiffness k1

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Fig. 38

Estimation of the contact stiffness k2

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Fig. 39

FRF H12 of the cantilever system with two clearances: (a) magnitude and (b) phase

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