Multi-degree-of-freedom System Frequency Response Function Curve-fitting

By Tom Irvine

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January 22, 2014

______

Variables

F / Excitation frequency
f r / Natural frequency for mode r
N / Total degrees-of-freedom
/ The steady state displacement at coordinate i due to a harmonic force excitation only at coordinate j
/ Damping ratio for mode r
/ Mass-normalized eigenvector for physical coordinate i and mode number r
/ Excitation frequency (rad/sec)
/ Natural frequency (rad/sec) for mode r

Receptance

The steady-state displacement at coordinate i due to a harmonic force excitation only at coordinate j is

(1)

where

(2)

(3)

Mobility

The steady-state velocity at coordinate i due to a harmonic force excitation only at coordinate j is

(4)

Accelerance

The steady-state acceleration at coordinate i due to a harmonic force excitation only at coordinate j is

(5)

Curve-fitting Equation

Consider that a measured receptance function is available. Estimate the number of modes N within a selected frequency band. Form a trial function

(6)

Generate a set of trial functions by randomly varying the amplitudes , natural frequencies , and modal damping ratios . Initial bounds and estimates may be set for each of these parameters. Subtract each trial function from the measured data to determine which one yields the least residual error. The final chosen function will then yield the modal parameters.

References

  1. T. Irvine, An Introduction to Frequency Response Functions, Vibrationdata, 2000.
  1. T. Irvine, Calculating Transfer Functions from Normal Modes, Revision E, Vibrationdata, 2013.

APPENDIX A

Example

Figure A-1.

A three-degree-of-freedom system is shown in Figure A-1. First, determine the displacement using the full mode set. Then solve for two modes only. Finally solve using mode acceleration with two modes via equation (12). Compare the results at mass 3.

The parameters are

m1 / 0.0895 / lbf sec^2/in
m2 / 0.0887 / lbf sec^2/in
m3 / 0.0770 / lbf sec^2/in
k1 / 1.8522e+04 / lbf/in
k2 / 0.2157e+04 / lbf/in
k3 / 0.2270e+04 / lbf/in
k4 / 1.9429e+04 / lbf/in
k5 / 1.7072e+04 / lbf/in

The damping is 0.05 for all modes.

The mass matrix is

(A-1)

The stiffness matrix is

(A-2)

The natural frequencies are

Hz (A-3)

The mode shapes are

(A-4)

Figure A-2.

The curve-fit will be performed on the H33 frequency response function.

The mass and stiffness parameters were chosen so that the modal frequencies would be closely-spaced as a rigorous test of resolution abilities of the curve-fit method.

Figure A-3.

The H33 complex receptance FRF is shown in Figure A-3.

Figure A-4.

Three natural frequencies are estimated from the close-up view of the complex receptance function.

Figure A-5.

The curve-fitting is performed using Matlab script: mdof_frf_curvefit.m. Excellent agreement was obtained.

A comparison of the analytic values and the numerical experiment curve-fit values are shown in the following tables.

Table A-1. FRF Curve-Fit
Mode / fn(Hz) / Damping Ratio
1 / 73.29 / 0.0452
2 / 78.18 / 0.0526
3 / 86.55 / 0.0488
Table A-2. Natural Frequency Comparison
Mode / Analysis
fn(Hz) / FRF Curve-Fit
fn(Hz) / Difference
1 / 73.6394 / 73.29 / 0.5%
2 / 78.2766 / 78.18 / 0.1%
3 / 86.4757 / 86.55 / -0.1%
Table A-3. Damping Comparison
Mode / Analysis
Damping Ratio / FRF Curve-Fit
Damping Ratio / Difference
1 / 0.05 / 0.0452 / 9.6%
2 / 0.05 / 0.0526 / -5.2%
3 / 0.05 / 0.0488 / 2.4%

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