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Research Papers: Gas Turbines: Structures and Dynamics

# Forced Response Prediction of Constrained and Unconstrained Structures Coupled Through Frictional Contacts

[+] Author and Article Information
Ender Cigeroglu1

Department of Mechanical Engineering, The Ohio State University, Columbus, OH 43210ender@metu.edu.tr

Ning An, Chia-Hsiang Menq

Department of Mechanical Engineering, The Ohio State University, Columbus, OH 43210

1

Corresponding author. Present address: Middle East Technical University, Ankara, Turkey.

J. Eng. Gas Turbines Power 131(2), 022505 (Dec 23, 2008) (11 pages) doi:10.1115/1.2940356 History: Received December 24, 2007; Revised March 17, 2008; Published December 23, 2008

## Abstract

In this paper, a forced response prediction method for the analysis of constrained and unconstrained structures coupled through frictional contacts is presented. This type of frictional contact problem arises in vibration damping of turbine blades, in which dampers and blades constitute the unconstrained and constrained structures, respectively. The model of the unconstrained/free structure includes six rigid body modes and several elastic modes, the number of which depends on the excitation frequency. In other words, the motion of the free structure is not artificially constrained. When modeling the contact surfaces between the constrained and free structure, discrete contact points along with contact stiffnesses are distributed on the friction interfaces. At each contact point, contact stiffness is determined and employed in order to take into account the effects of higher frequency modes that are omitted in the dynamic analysis. Depending on the normal force acting on the contact interfaces, quasistatic contact analysis is initially employed to determine the contact area as well as the initial preload or gap at each contact point due to the normal load. A friction model is employed to determine the three-dimensional nonlinear contact forces, and the relationship between the contact forces and the relative motion is utilized by the harmonic balance method. As the relative motion is expressed as a modal superposition, the unknown variables, and thus the resulting nonlinear algebraic equations in the harmonic balance method, are in proportion to the number of modes employed. Therefore the number of contact points used is irrelevant. The developed method is applied to a bladed-disk system with wedge dampers where the dampers constitute the unconstrained structure, and the effects of normal load on the rigid body motion of the damper are investigated. It is shown that the effect of rotational motion is significant, particularly for the in-phase vibration modes. Moreover, the effect of partial slip in the forced response analysis and the effect of the number of harmonics employed by the harmonic balance method are examined. Finally, the prediction for a test case is compared with the test data to verify the developed method.

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## Figures

Figure 1

Contact model for the wedge damper

Figure 2

Figure 3

Wedge damper contact planes and coordinate systems

Figure 4

Schematic view for the bounded configuration

Figure 5

Finite element models for (a) blade and (b) wedge dampers

Figure 6

Tracking plot for the first mode

Figure 7

Tracking plot for the second mode

Figure 8

Optimal and frequency shift curves: first mode

Figure 9

Optimal and frequency shift curves: second mode

Figure 10

Tracking plots: (a) third and seventh (b) modes

Figure 11

Optimum and frequency shift curves: (a) third and seventh (b) mode

Figure 12

Effect of normal load on the rigid body motion of damper (bias component)

Figure 13

Effect of normal load on the rigid body motion of damper (vibratory component)

Figure 14

Effect of rotational modes for normal load; (a) 100, (b) 200, (c) 1000, and (d) 10,000; ——, translational and rotational modes; ----------, translational modes

Figure 15

Effect of the number of contact points for normal load; (a) 100, (b) 200, (c) 1000, and (d) 25,000

Figure 16

Contact status of sample points on (a) left (b) right contact planes: (red) stick, (blue) slip, and (green) separation). (Color version of this figure available online only.)

Figure 17

Schematic for the blade to ground damper

Figure 18

Tracking plots, (⋯⋯⋯), stuck; single-harmonic: (— — —) 1.0e6, (-------) 5.0e5, (⋅–⋅–⋅–⋅–) 2.5e5, (⋅–⋯–⋯) 1.0e5, (– — – —) 5.0e4, (– – – – –) 1.0e4; multiharmonic: (⋯▽⋯) 1.0e6, (⋯◇⋯) 5.0e5, (⋯◻⋯) 2.5e5, (⋯△⋯) 1.0e5, (⋯○⋯) 5.0e4, and (- – -◻- – -) 1.0e4

Figure 19

Schematic view of the test case

Figure 20

Finite element model for the test case

Figure 21

Frequency shift curve

Figure 22

Predicted normalized optimal curve and test data

## Errata

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