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
Introduction
โช In practice, modeling of mechanical systems supported on an elastic foundation is a common problem in theconstruction industry, mining, rail transport, etc.
โช A mathematical description of behavior of the elastic foundation usually causes the computational model of amechanical system, although otherwise linear, to become nonlinear
โช To obtain the frequency response of such system, a necessity arises to use one of the continuation methods
โช Description of the frequency response then requires repeated solutions of the steady-state component ofvibration response for varying excitation frequency
โช 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
โช 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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โฒ Source: www.daibau.rs/cene/temelji_i_temeljna_ploca โฒ Source: www.rail-fastener.comโฒ Source: www.justmovingaround.com/2016/07/06
Harmonic Balance Method
โผ Equation of motion of the system in time domain:
๐ แท๐ฑ + ๐ แถ๐ฑ + ๐๐ฑ = ๐ฉ๐ + ๐ฉโ ๐ก โ ๐ซ ๐ฑ, ๐ก ; ๐ฉโ ๐ก = ๐ฉโ๐cos ๐๐ก + ๐
๐ ๐ ๐ฎ = ๐ช โ ๐ ๐ฎ ; ๐ ๐ = diag ๐,๐1, โฆ , ๐๐ , โฆ , ๐๐๐น ; ๐๐ =๐โ ๐๐ 2๐ ๐๐๐
โ๐๐๐ ๐ โ ๐๐ 2๐;
โผ Equation of motion of the system in frequency domain:
๐ฎ = ๐0 ๐1 ๐ฌ1 โฏ ๐๐ ๐ฌ๐ โฏ ๐๐๐น ๐ฌ๐๐น T; ๐ช = ๐ฉ๐ sin ๐ ๐ฉโ๐ cos ๐ ๐ฉโ๐ ๐ โฏ ๐ T
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๐ฑ ๐ก =๐02+
๐=1
๐๐น๐๐cos ๐๐๐ก + ๐ฌ๐sin ๐๐๐ก ;
โช Where ๐, ๐, and ๐ are a mass, damping and stiffness matrix, respectively, ๐ฑ, แถ๐ฑ, and แท๐ฑ are a displacement,velocity and acceleration vector, respectively, ๐ฉ๐ is a vector of static loading, ๐ฉโ is a vector of harmonicexcitation, ๐ซ is a vector of nonlinear foundation forces, ๐ฉโ๐ is a vector of amplitudes of harmonic excitation,๐ is an angular excitation frequency, ๐ is a phase shift, and ๐ก is the time
โช Where ๐ is a dynamic stiffness matrix, vector ๐ฎ contains vectors of the Fourier coefficients ๐0, ๐๐, and ๐ฌ๐ ofabsolute, cosine and sine terms, respectively, ๐ช and ๐ are a vector of linear and nonlinear forces in frequencydomain, respectively, and ๐๐น is the number of harmonic terms of the turncated Fourier series
Solution is assumed in the form of the turncated Fourier series.
Alternating Frequency-Time Scheme
๐ฎ ๐ฑ โ ๐ซ ๐ฑ, ๐ก ๐ ๐ฎDFTDFTโ1
๐ = ๐+ ๐ ๐ซ๐ฑ = ๐ ๐ ๐ฎ
โผ Obtaining actual iteration of nonlinear forces using the alternating frequency-time scheme:
โผ Linear operator of the inverse Fourier transform (DFT) has the form:
๐ ๐ = 0,5 โ ๐ ๐ญ๐,1 ๐ญ๐ ,1 โฏ ๐ญ๐,๐ ๐ญ๐ ,๐ โฏ ๐ญ๐,๐๐น ๐ญ๐ ,๐๐น ;
๐ญ๐,๐ = cos ๐๐๐ก1 โฏ cos ๐๐๐ก๐ โฏ cos ๐๐๐ก๐T; ๐ญ๐ ,๐ = sin ๐๐๐ก1 โฏ sin ๐๐๐ก๐ โฏ sin ๐๐๐ก๐
T
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โช 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
โช Where ๐ is a vector of ones and ๐ is a number of collocation points over a period of vibration
Arc Length Continuation Method
โผ Calculation of initial guess (predictor) from arc length ๐ and two last solutions ๐ฎ๐โ2, ๐๐โ2 and ๐ฎ๐โ1, ๐๐โ1:
๐ฎ0๐ = ๐ฎ๐โ1 + ๐
๐ฎ๐โ1 โ ๐ฎ๐โ2
๐ฎ๐โ1 โ ๐ฎ๐โ2 T ๐ฎ๐โ1 โ ๐ฎ๐โ2 + ๐๐โ1 โ๐๐โ2 2;
โผ Corrections of displacements ฮด๐ฎji and excitation frequency ๐ฟ๐j
i are in j-th iteration coupled by equation:
โผ Where vectors ๐ฑ1 and ๐ฑ2 are obtained from solution of a set of equations:
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ฮด๐ฎ๐๐ = ๐ฑ1 โ ๐ฟ๐๐
๐๐ฑ2 ( 1 )
๐ ๐๐โ1 โ๐๐ ๐ฎ๐โ1
๐๐ฎ๐ฑ1 = ๐ ๐๐โ1 ๐ฎ๐โ1 โ ๐ ๐ฎ๐โ1 โ ๐ช; ๐ ๐๐โ1 โ
๐๐ ๐ฎ๐โ1
๐๐ฎ๐ฑ2 =
๐๐ ๐๐โ1
๐๐๐ฎ๐โ1
๐0๐ = ๐๐โ1 + ๐
๐๐โ1 โ ๐๐โ2
๐ฎ๐โ1 โ ๐ฎ๐โ2 T ๐ฎ๐โ1 โ ๐ฎ๐โ2 + ๐๐โ1 โ ๐๐โ2 2
โช Here, ๐๐ ๐ฎ๐โ1 /๐๐ฎ is a matrix of partial derivatives of the vector of nonlinear forces in frequency domain ๐
with respect to displacement vector ๐ฎ and ๐๐ ๐๐โ1 /๐๐ is a vector of partial derivatives of the dynamic
stiffness matrix ๐ with respect to angular excitation frequency ๐
Arc Length Continuation Method
โผ Correction of excitation frequency ๐ฟ๐๐๐ is one of the roots ๐ฟ๐1 and ๐ฟ๐2 of quadratic equation:
๐1ฮด๐2 + ๐2ฮด๐ + ๐3 = 0;
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๐ฎ๐๐ + ๐ฑ1 โ ๐ฟ๐1๐ฑ2 โ ๐ฎ๐โ1
T๐ฎ๐๐ โ ๐ฎ๐โ1 ; ๐ฎ๐
๐ + ๐ฑ1 โ ๐ฟ๐2๐ฑ2 โ ๐ฎ๐โ1T๐ฎ๐๐ โ ๐ฎ๐โ1
๐2 = 2 ๐ฎ๐โ1 โ ๐ฎ๐๐ โ ๐ฑ1
T๐ฑ2 + ๐๐
๐ โ๐๐โ1 ;
๐1 = ๐ฑ2T๐ฑ2 + 1;
๐3 = ๐ฎ๐๐ โ ๐ฎ๐โ1 + ๐ฑ1
T๐ฎ๐๐ โ ๐ฎ๐โ1 + ๐ฑ1 + ๐๐
๐ โ ๐๐โ1 2โ ๐2
โผ The root that represents forward sense of continuation can be identified from the larger of the two products:
โผ Finally, after calculating the correction of displacements ฮด๐ฎ๐๐ from equation ( 1 ) (see previous slide), all that
remains in current iteration is to update the vector of displacements and angular excitation frequency:
๐ฎ๐+1๐ = ๐ฎ๐
๐ + ฮด๐ฎ๐๐; ๐๐+1
๐ = ๐๐๐ + ๐ฟ๐๐
๐
โช The iterative process ends when suitable convergence criteria are met
Test Problem
โผ Finite element model of the test problem:
โผ Physical parameters of the test problem:
r(x) = ฮบ1x + ฮบ3x3
p = pacos(ฯt) โ pmParameter Symbol Value UnitYoungโs modulus ๐ธ 69 โ 1010 PaPoissonโs number ๐ 0.33 โMass density ๐ 2 700 kg/m3
Mass proportional damping coefficient ๐ผ 100 1/s
Stiffness proportional damping coefficient ๐ฝ 5 โ 10โ5 s
Constant pressure ๐๐ 5 โ 106 PaAmplitude of distributed excitation pressure ๐๐ 1 โ 106 Pa
Mean value of distributed excitation pressure ๐๐ 5 โ 106 Pa
Linear stiffness coefficient of foundation ฮบ1 1 โ 109 N/m3
Cubic stiffness coefficient of foundation ฮบ3 5 โ 1015 N/m5
Number of elements(hexahedral) 2 232Number of nodes 3 965Number of degrees of freedom 11 895
โผ Basic finite element mesh parameters:
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โช 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(ฯt) + pmps
A
Results of the test problem
โผ Frequency response of the test problem at node A (see previous slide):
B
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Results of the test problem
โผ Vibration response for excitation frequency ๐ = 500 Hz (at point B, see previous slide):
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Results of the test problem
โผ Frequency spectrum of the foundation forces for excitation frequency ๐ = 500 Hz:
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Harmonic term number k [-]
๐ซ ๐ฑ, ๐ก =๐ซ02+
๐=1
๐๐น๐ซ๐cos ๐๐๐ก + ๐๐
No
rm o
fF
ouri
er c
oef
fici
ents
vec
tor
|rk| [
-]
Conclusion
โช 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
โช The arc length continuation method was used to calculate the frequency response
โช The harmonic balance method was used to solve the steady-state vibration response at each increment ofcontinuation
โช 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
โช The frequency spectrum of nonlinear reaction forces of the elastic foundation was also investigated
โช 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
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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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