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26th International Conference ENGINEERING MECHANICS 2020 Svratka, Czech Republic, November 24 - 25, 2020 Using Harmonic Balance Method for Solving Frequency Response of Systems with Nonlinear Elastic Foundation Filip Zaoral, Petr Ferfecki
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Page 1: Using Harmonic Balance Method for Solving Frequency ...

26th International Conference ENGINEERING MECHANICS 2020

Svratka, Czech Republic, November 24 - 25, 2020

Using Harmonic Balance Method for Solving Frequency Response of Systems with Nonlinear Elastic FoundationFilip Zaoral, Petr Ferfecki

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Introduction

โ–ช In practice, modeling of mechanical systems supported on an elastic foundation is a common problem in theconstruction industry, mining, rail transport, etc.

โ–ช A mathematical description of behavior of the elastic foundation usually causes the computational model of amechanical system, although otherwise linear, to become nonlinear

โ–ช To obtain the frequency response of such system, a necessity arises to use one of the continuation methods

โ–ช Description of the frequency response then requires repeated solutions of the steady-state component ofvibration response for varying excitation frequency

โ–ช However, direct integration of nonlinear motion equations includes a solution of the transient state as well,which in practice generally leads to a large number of integration steps and very long solution times

โ–ช A possible way-out is the application of the harmonic balance method, which allows for the determination ofsteady-state response component directly, provided it has a periodic or quasi-periodic time course

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โ–ฒ Source: www.daibau.rs/cene/temelji_i_temeljna_ploca โ–ฒ Source: www.rail-fastener.comโ–ฒ Source: www.justmovingaround.com/2016/07/06

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Harmonic Balance Method

โ–ผ Equation of motion of the system in time domain:

๐Œ แˆท๐ฑ + ๐ แˆถ๐ฑ + ๐Š๐ฑ = ๐ฉ๐‘  + ๐ฉโ„Ž ๐‘ก โˆ’ ๐ซ ๐ฑ, ๐‘ก ; ๐ฉโ„Ž ๐‘ก = ๐ฉโ„Ž๐‘Žcos ๐œ”๐‘ก + ๐œ“

๐€ ๐œ” ๐ฎ = ๐ช โˆ’ ๐› ๐ฎ ; ๐€ ๐œ” = diag ๐Š,๐€1, โ€ฆ , ๐€๐‘˜ , โ€ฆ , ๐€๐‘๐น ; ๐€๐‘— =๐Šโˆ’ ๐‘˜๐œ” 2๐Œ ๐‘˜๐œ”๐

โˆ’๐‘˜๐œ”๐ ๐Š โˆ’ ๐‘˜๐œ” 2๐Œ;

โ–ผ Equation of motion of the system in frequency domain:

๐ฎ = ๐œ0 ๐œ1 ๐ฌ1 โ‹ฏ ๐œ๐‘˜ ๐ฌ๐‘˜ โ‹ฏ ๐œ๐‘๐น ๐ฌ๐‘๐น T; ๐ช = ๐ฉ๐‘  sin ๐œ“ ๐ฉโ„Ž๐‘Ž cos ๐œ“ ๐ฉโ„Ž๐‘Ž ๐ŸŽ โ‹ฏ ๐ŸŽ T

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๐ฑ ๐‘ก =๐œ02+

๐‘˜=1

๐‘๐น๐œ๐‘˜cos ๐‘˜๐œ”๐‘ก + ๐ฌ๐‘˜sin ๐‘˜๐œ”๐‘ก ;

โ–ช Where ๐Œ, ๐, and ๐Š are a mass, damping and stiffness matrix, respectively, ๐ฑ, แˆถ๐ฑ, and แˆท๐ฑ are a displacement,velocity and acceleration vector, respectively, ๐ฉ๐‘  is a vector of static loading, ๐ฉโ„Ž is a vector of harmonicexcitation, ๐ซ is a vector of nonlinear foundation forces, ๐ฉโ„Ž๐‘Ž is a vector of amplitudes of harmonic excitation,๐œ” is an angular excitation frequency, ๐œ“ is a phase shift, and ๐‘ก is the time

โ–ช Where ๐€ is a dynamic stiffness matrix, vector ๐ฎ contains vectors of the Fourier coefficients ๐œ0, ๐œ๐‘˜, and ๐ฌ๐‘˜ ofabsolute, cosine and sine terms, respectively, ๐ช and ๐› are a vector of linear and nonlinear forces in frequencydomain, respectively, and ๐‘๐น is the number of harmonic terms of the turncated Fourier series

Solution is assumed in the form of the turncated Fourier series.

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Alternating Frequency-Time Scheme

๐ฎ ๐ฑ โ†’ ๐ซ ๐ฑ, ๐‘ก ๐› ๐ฎDFTDFTโˆ’1

๐› = ๐“+ ๐œ” ๐ซ๐ฑ = ๐“ ๐œ” ๐ฎ

โ–ผ Obtaining actual iteration of nonlinear forces using the alternating frequency-time scheme:

โ–ผ Linear operator of the inverse Fourier transform (DFT) has the form:

๐“ ๐œ” = 0,5 โˆ™ ๐Ÿ ๐ญ๐‘,1 ๐ญ๐‘ ,1 โ‹ฏ ๐ญ๐‘,๐‘˜ ๐ญ๐‘ ,๐‘˜ โ‹ฏ ๐ญ๐‘,๐‘๐น ๐ญ๐‘ ,๐‘๐น ;

๐ญ๐‘,๐‘˜ = cos ๐‘˜๐œ”๐‘ก1 โ‹ฏ cos ๐‘˜๐œ”๐‘ก๐‘š โ‹ฏ cos ๐‘˜๐œ”๐‘ก๐‘T; ๐ญ๐‘ ,๐‘˜ = sin ๐‘˜๐œ”๐‘ก1 โ‹ฏ sin ๐‘˜๐œ”๐‘ก๐‘š โ‹ฏ sin ๐‘˜๐œ”๐‘ก๐‘

T

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โ–ช Where ๐“ is a linear operator of the inverse Fourier transform and ๐“+ , being the Moore-Penrosepseudoinverse of ๐“, is a linear operator of the direct Fourier transform

โ–ช Where ๐Ÿ is a vector of ones and ๐‘ is a number of collocation points over a period of vibration

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Arc Length Continuation Method

โ–ผ Calculation of initial guess (predictor) from arc length ๐‘Ž and two last solutions ๐ฎ๐‘–โˆ’2, ๐œ”๐‘–โˆ’2 and ๐ฎ๐‘–โˆ’1, ๐œ”๐‘–โˆ’1:

๐ฎ0๐‘– = ๐ฎ๐‘–โˆ’1 + ๐‘Ž

๐ฎ๐‘–โˆ’1 โˆ’ ๐ฎ๐‘–โˆ’2

๐ฎ๐‘–โˆ’1 โˆ’ ๐ฎ๐‘–โˆ’2 T ๐ฎ๐‘–โˆ’1 โˆ’ ๐ฎ๐‘–โˆ’2 + ๐œ”๐‘–โˆ’1 โˆ’๐œ”๐‘–โˆ’2 2;

โ–ผ Corrections of displacements ฮด๐ฎji and excitation frequency ๐›ฟ๐œ”j

i are in j-th iteration coupled by equation:

โ–ผ Where vectors ๐ฑ1 and ๐ฑ2 are obtained from solution of a set of equations:

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ฮด๐ฎ๐‘—๐‘– = ๐ฑ1 โˆ’ ๐›ฟ๐œ”๐‘—

๐‘–๐ฑ2 ( 1 )

๐€ ๐œ”๐‘–โˆ’1 โˆ’๐œ•๐› ๐ฎ๐‘–โˆ’1

๐œ•๐ฎ๐ฑ1 = ๐€ ๐œ”๐‘–โˆ’1 ๐ฎ๐‘–โˆ’1 โˆ’ ๐› ๐ฎ๐‘–โˆ’1 โˆ’ ๐ช; ๐€ ๐œ”๐‘–โˆ’1 โˆ’

๐œ•๐› ๐ฎ๐‘–โˆ’1

๐œ•๐ฎ๐ฑ2 =

๐œ•๐€ ๐œ”๐‘–โˆ’1

๐œ•๐œ”๐ฎ๐‘–โˆ’1

๐œ”0๐‘– = ๐œ”๐‘–โˆ’1 + ๐‘Ž

๐œ”๐‘–โˆ’1 โˆ’ ๐œ”๐‘–โˆ’2

๐ฎ๐‘–โˆ’1 โˆ’ ๐ฎ๐‘–โˆ’2 T ๐ฎ๐‘–โˆ’1 โˆ’ ๐ฎ๐‘–โˆ’2 + ๐œ”๐‘–โˆ’1 โˆ’ ๐œ”๐‘–โˆ’2 2

โ–ช Here, ๐œ•๐› ๐ฎ๐‘–โˆ’1 /๐œ•๐ฎ is a matrix of partial derivatives of the vector of nonlinear forces in frequency domain ๐›

with respect to displacement vector ๐ฎ and ๐œ•๐€ ๐œ”๐‘–โˆ’1 /๐œ•๐œ” is a vector of partial derivatives of the dynamic

stiffness matrix ๐€ with respect to angular excitation frequency ๐œ”

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Arc Length Continuation Method

โ–ผ Correction of excitation frequency ๐›ฟ๐œ”๐‘—๐‘– is one of the roots ๐›ฟ๐œ”1 and ๐›ฟ๐œ”2 of quadratic equation:

๐‘Ž1ฮด๐œ”2 + ๐‘Ž2ฮด๐œ” + ๐‘Ž3 = 0;

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๐ฎ๐‘—๐‘– + ๐ฑ1 โˆ’ ๐›ฟ๐œ”1๐ฑ2 โˆ’ ๐ฎ๐‘–โˆ’1

T๐ฎ๐‘—๐‘– โˆ’ ๐ฎ๐‘–โˆ’1 ; ๐ฎ๐‘—

๐‘– + ๐ฑ1 โˆ’ ๐›ฟ๐œ”2๐ฑ2 โˆ’ ๐ฎ๐‘–โˆ’1T๐ฎ๐‘—๐‘– โˆ’ ๐ฎ๐‘–โˆ’1

๐‘Ž2 = 2 ๐ฎ๐‘–โˆ’1 โˆ’ ๐ฎ๐‘—๐‘– โˆ’ ๐ฑ1

T๐ฑ2 + ๐œ”๐‘—

๐‘– โˆ’๐œ”๐‘–โˆ’1 ;

๐‘Ž1 = ๐ฑ2T๐ฑ2 + 1;

๐‘Ž3 = ๐ฎ๐‘—๐‘– โˆ’ ๐ฎ๐‘–โˆ’1 + ๐ฑ1

T๐ฎ๐‘—๐‘– โˆ’ ๐ฎ๐‘–โˆ’1 + ๐ฑ1 + ๐œ”๐‘—

๐‘– โˆ’ ๐œ”๐‘–โˆ’1 2โˆ’ ๐‘Ž2

โ–ผ The root that represents forward sense of continuation can be identified from the larger of the two products:

โ–ผ Finally, after calculating the correction of displacements ฮด๐ฎ๐‘—๐‘– from equation ( 1 ) (see previous slide), all that

remains in current iteration is to update the vector of displacements and angular excitation frequency:

๐ฎ๐‘—+1๐‘– = ๐ฎ๐‘—

๐‘– + ฮด๐ฎ๐‘—๐‘–; ๐œ”๐‘—+1

๐‘– = ๐œ”๐‘—๐‘– + ๐›ฟ๐œ”๐‘—

๐‘–

โ–ช The iterative process ends when suitable convergence criteria are met

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Test Problem

โ–ผ Finite element model of the test problem:

โ–ผ Physical parameters of the test problem:

r(x) = ฮบ1x + ฮบ3x3

p = pacos(ฯ‰t) โ€“ pmParameter Symbol Value UnitYoungโ€™s modulus ๐ธ 69 โˆ™ 1010 PaPoissonโ€™s number ๐œ‡ 0.33 โ€“Mass density ๐œŒ 2 700 kg/m3

Mass proportional damping coefficient ๐›ผ 100 1/s

Stiffness proportional damping coefficient ๐›ฝ 5 โˆ™ 10โˆ’5 s

Constant pressure ๐‘๐‘  5 โˆ™ 106 PaAmplitude of distributed excitation pressure ๐‘๐‘Ž 1 โˆ™ 106 Pa

Mean value of distributed excitation pressure ๐‘๐‘š 5 โˆ™ 106 Pa

Linear stiffness coefficient of foundation ฮบ1 1 โˆ™ 109 N/m3

Cubic stiffness coefficient of foundation ฮบ3 5 โˆ™ 1015 N/m5

Number of elements(hexahedral) 2 232Number of nodes 3 965Number of degrees of freedom 11 895

โ–ผ Basic finite element mesh parameters:

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โ–ช A console, discretized by 3D finite elements,excited by a pulsating pressure load and mountedon a bilateral nonlinear elastic support, objective isto calculate frequency response of the system

ps

ps

p = pacos(ฯ‰t) + pmps

A

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Results of the test problem

โ–ผ Frequency response of the test problem at node A (see previous slide):

B

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Results of the test problem

โ–ผ Vibration response for excitation frequency ๐‘“ = 500 Hz (at point B, see previous slide):

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Results of the test problem

โ–ผ Frequency spectrum of the foundation forces for excitation frequency ๐‘“ = 500 Hz:

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Harmonic term number k [-]

๐ซ ๐ฑ, ๐‘ก =๐ซ02+

๐‘˜=1

๐‘๐น๐ซ๐‘˜cos ๐‘˜๐œ”๐‘ก + ๐œ“๐‘˜

No

rm o

fF

ouri

er c

oef

fici

ents

vec

tor

|rk| [

-]

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Conclusion

โ–ช The frequency response of a console, discretized by three-dimensional finite elements, excited by a pulsatingpressure load and mounted on a bilateral nonlinear elastic support, was investigated

โ–ช The arc length continuation method was used to calculate the frequency response

โ–ช The harmonic balance method was used to solve the steady-state vibration response at each increment ofcontinuation

โ–ช For 4 harmonic terms of the Fourier series, no significant change in the shape of the frequency responsecurve was observed for the excitation frequency values from 300 Hz above, compared to the case with 8harmonic terms

โ–ช The frequency spectrum of nonlinear reaction forces of the elastic foundation was also investigated

โ–ช Already for 8 harmonic members of the Fourier series, a significantly decreasing trend of the superharmoniccomponents of the reaction forces has been observed from the 4th term above

11 / 12

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VSB - Technical University of OstravaIT4Innovations National Supercomputing Center

Parallel algorithms research labStudentskรก 6231/1B

708 00 Ostrava-PorubaCzech Republic

www.it4i.cz

Thank You for your attention.

26th International Conference ENGINEERING MECHANICS 2020

Svratka, Czech Republic, November 24 - 25, 2020

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