Algebraic Harmonic Balance Method

This package provides a Python 3.10 implementation of the algebraic harmonic balance method (algebraic HBM) as proposed by [1].

Installation

The following commands install algebraic_hbm from the Python Package Index. You will need a working installation of Python and pip.

pip install algebraic-hbm

Usage

Evaluation

This example shows how to use the algebraic HBM framework for a softening Duffing oscillator that generates the executable functions

$F_n$

and

$\mathrm D F_n=(\frac{\mathrm d F_n}{\mathrm d \mathbf c}, \frac{\mathrm d F_n}{\mathrm d a})_n$

(see Theoretic background). This example can also be found here.

from algebraic_hbm import ODE_2nd_Order_Poly_Coeffs, softening_Duffing, Algebraic_HBM

Calling import softening_Duffing importes an instance of the ODE_2nd_Order_Poly_Coeffs-class that represents classical softening Duffing oscillator

$$r(t,x) = x''(t) + 0.4 x'(t) - 0.4 x^3(t) - 0.3 \cos(\Omega t) = 0$$

This can also be achieved by instead calling

softening_Duffing = ODE_2nd_Order_Poly_Coeffs(mass=1, damping=.4, stiffness=1, excitation=(0,.3), monomials={3: -.4})

Then, initialize the algebraic HBM for the softening Duffing oscillator and ansatz order

$n$

.

HBM = Algebraic_HBM(ODE=softening_Duffing, order=n)

Now generate the multivariate polynomials that define the algebraic equation system

$F_n$

as obtained from the algebraic HBM.

HBM.generate_multivariate_polynomials()

Finally, compile the polynomials into excecutable functions

$F_n$

and

$\mathrm DF_n$

.

F, DF = HBM.compile()

The functions

$(\mathbf c_n, a) \mapsto F_n(\mathbf c_n, a)$

and

(\mathbf c_n, a) \mapsto \mathrm D F_n(\mathbf c_n, a)

may now be used to perform a bifurcation analysis of the softening Duffing oscillator. For example, computing the frequency response of the softening Duffing oscillator for different initial guesses as done here [3].

Coefficient matrix for Macaulay framework

This example shows how to build the coefficient matrix of the algebraic representation (again, see Theoretic background) that can be used in conjunction with the Macaulay matrix framework [4]. This example can also be found here.

Most of the steps are as in the above example, but instead of compiling executable functions we request the coefficient matrix at a given excitation frequency

\Omega = 3.14

by invoking HBM.get_monomial_coefficient_matrix.

from algebraic_hbm import softening_Duffing, Algebraic_HBM

n, a = 1, 3.14
HBM = Algebraic_HBM(ODE=softening_Duffing, order=n)
HBM.generate_multivariate_polynomials()
A = HBM.get_monomial_coefficient_matrix(a)

Theoretic background

We are considering second order ordinary differential equations (ODEs) with polynomial coefficients in the state

x : \mathbb T \subset \mathbb R \to \mathbb R

, that is ODEs of the form

r(t,x;u) = \rho x''(t) + \delta x'(t) + \sum_{i=1}^q \alpha_i x^i(t) - u(t) = 0 \,, \quad u(t) = \hat u \cos(\Omega t) \,.

The idea of the HBM is to yield approximations

x_n(t) = c_0 + \sum_{i=1}^n c_{2i-1} \cos(i \Omega t) + c_{2i} \sin(i \Omega t)

of stationary periodic solutions

x

of the ODE. Given a excitation frequency

\Omega

, the algebraic HBM of order

n

yields a system of multivariate polynomials

R_i

,

i=0,1,\ldots,2n

, in the variables

c_0,c_1,\ldots,c_{2n}

that solve the algebraic system

F_n(\mathbf c; \Omega) = [R_i(\mathbf c; \Omega)]_{i=0}^{2n} = 0

where

\mathbf c = [c_0,c_1,\ldots,c_{2n}] \in \mathbb R^{2n+1}

. A solution

\mathbf c \leftrightarrow x_n

of

F_n(\mathbf c; \Omega) = 0

is also a solution of the (original) HBM defining system of integral equations

\langle r(x_n), \phi_j\rangle = \frac{1}{T} \int_0^T r(t,x_n(t)) \phi_j(t) \, \mathrm d t \,, \quad j = 0,1,\ldots,2n \,,

with basis functions

\phi_0(t) = 1

,

\phi_{2i-1}(t) = \cos(i \Omega t)

and

\phi_{2i}(t) = \sin(i \Omega t)

,

i=1,\ldots,n

. Note that building and evaluating

F_n

does not require the computation of integrals as e.g. in the classical or Alternating Frequency-Time HBM [2].

References

1. Hannes Dänschel and Lukas Lentz. "An Algebraic Representation of the Harmonic Balance Method for Ordinary Differential Equations with Polynomial Coefficients". Manuscript PDF: /algebraic_hbm.pdf
2. Malte Krack and Johann Gross. "Harmonic Balance for Nonlinear Vibration Problems". Springer, 2019. isbn: 978-3-030-14022-9. DOI: 10.1007/978-3-030-14023-6
3. Hannes Dänschel, Lukas Lentz, and Utz von Wagner. "Error Measures and Solution Artifacts of the Harmonic Balance Method on the Example of the Softening Duffing Oscillator". In: Journal of Theoretical and Applied Mechanics 62.2 (Apr. 2024), pp. 435–455. DOI: 10.15632/jtam-pl/186718
4. Philippe Dreesen, Kim Batselier, and Bart De Moor. "Back to the Roots: Polynomial System Solving, Linear Algebra, Systems Theory". In: IFAC Proceedings Volumes 45.16 (2012), pp. 1203–1208. issn: 1474-6670. DOI: 10.3182/20120711-3-BE-2027.00217