SSMtool Manual

SSMtool_manual

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Introduction
- Download ssmtool
- Installation
How to use ssmtool
Final remarks

SSMtool: A short guide

S. Ponsioen & G. Haller

Nonlinear Dynamics Group

Institute for Mechanical Systems

SSMtool: A short guide

S. Ponsioen ·G. Haller

March 1, 2019

S. Ponsioen ( ) ·G. Haller

Institute for Mechanical Systems, ETH Zürich, Leonhardstrasse 21, 8092, Zürich, Switzerland

E-mail: stenp@ethz.ch

SSMtool: A short guide 3

Contents

1 Introduction.................................................. 4

1.1 Download ssmtool ........................................... 5

1.2 Installation ............................................... 5

2 How to use ssmtool.............................................. 6

2.1 Loading a mechanical system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Selecting the spectral subspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 SSMcomputation............................................ 9

2.4 Parameterization/Backbone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.5 Integrate trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.6 Invarianceerror............................................. 14

2.7 Plotoptions............................................... 14

3 Finalremarks................................................. 15

4 S. Ponsioen, G. Haller

1 Introduction

ssmtool is a matlab-based computational tool for computing two-dimensional spectral sub-

manifolds (SSMs) in nonlinear mechanical systems with arbitrary degrees of freedom. ssmtool is

intended for researchers and students who are interested in extracting key-information, such as

reduced-order models or backbone curves from a nonlinear, dissipative mechanical system with

a potentially large number of degrees of freedom. The software achieves this without using any

numerical integration or numerical continuation techniques, purely based on a reduction to SSMs.

SSMs where ﬁrst introduced by Haller and Ponsioen [1] as the smoothest nonlinear contin-

uations of modal subspaces of the linearized system, providing mathematically exact reduced

models for the nonlinear system (cf. Ponsioen et al. [2, 3], Breunung and Haller [4] and Jain

and Haller [5]). More speciﬁcally, let us consider n-degree-of-freedom, autonomous mechanical

systems of the form

M¨

y+C˙

y+Ky +g(y,˙

y) = 0,g(y,˙

y) = O|y|2,|y| | ˙

y|,|˙

y|2,(1.1)

where y∈Rnis the generalized position vector; M=MT∈Rn×nis the positive deﬁnite mass

matrix; C=CT∈Rn×nis the damping matrix; K=KT∈Rn×nis the stiﬀness matrix and

g(y,˙

y)denotes all the nonlinear terms in the system. These nonlinearities are assumed to be

analytic for simplicity. Additionally, we require that the trivial ﬁxed point of system (1.1) is

asymptotically stable.

Under appropriate non-resonance conditions, we can construct a two-dimensional autonomous

SSM, W(E), over a chosen spectral subspace Eas an embedding of a two-dimensional open set U

into the full phase space of the system via a mapping W(z). This mapping is approximated in a

neighborhood of the origin using a Taylor expansion in the parameterization coordinates z∈C2.

Additionally, there exists a polynomial function R(z)representing the reduced dynamics on the

SSM via

z=R(z).(1.2)

By introducing a change to polar coordinates, eq. (1.2) can be rewritten as

˙ρ=a(ρ), Ω =˙

θ=b(ρ),(1.3)

where we refer to section 5.2 of Ponsioen et al. [2] for a detailed explanation of this derivation.

In other words, using SSM theory we are able to reduce the n-degree-of-freedom system (1.1)

to the two ordinary diﬀerential equations in (1.3). Additionally, as can be seen in eq. (1.3), the ˙

θ-

equation provides a direct relationship between the amplitude ρand the instantaneous frequency

Ω. This relationship is generally called the backbone curve, corresponding to a given mode. We

illustrate this concept in Figure 1.1.

ssmtool is designed to construct the Taylor approximations of the autonomous SSM W(z)

and the reduced dynamics R(z)in an automated fashion. The graphical user interface (GUI)

of ssmtool allows the user to import/deﬁne their nonlinear mechanical system and compute the

SSM over an arbitrarily chosen spectral subspace under appropriate non-resonance conditions.

The software then computes the reduced dynamics up to any predicted order of accuracy and

determines the associated backbone curves from the reduced dynamics on the SSM without any

further user input.

SSMtool: A short guide 5

Observed forced response

under external forcing with

frequency .

Fig. 1.1: Illustration of the backbone curve construction using the parameterized autonomous

SSM. For each ﬁxed radius ρ0on the SSM, we can identify an instantaneous frequency Ω(ρ0)

from the reduced system on the SSM, which, subsequently, will represent a point on the damped

backbone curve.

1.1 Download ssmtool

This software is made public for research use only. It may be modiﬁed and redistributed under

the terms of the GNU General Public License. ssmtool can be downloaded from the Haller Group

software page at the ETH Zürich:

http://www.georgehaller.com

ssmtool requires matlab R2016b or newer. This code may be improved and is subject to change.

1.2 Installation

To install ssmtool, unzip the source code into a local directory. In matlab, change the current

matlab folder to the ssmtool folder.

6 S. Ponsioen, G. Haller

2 How to use ssmtool

After installing ssmtool, type “SSM” in the matlab command window to open the graphical

user interface. The graphical user interface shown in Figure 2.1 is displayed.

Fig. 2.1: Overview of the graphical user interface of ssmtool.

2.1 Loading a mechanical system

To load an autonomous mechanical system into ssmtool, there is a total of three options. The

ﬁrst option is to load a predeﬁned mechanical system from the drop-down list in the top-left

corner of the GUI, as shown in Figure 2.2 (a). The systems listed here have been analyzed, using

ssmtool, in the work of Ponsioen et al. [2] and consist out of a two-degree-of-freedom nonlinear

mass-spring-damper system with a near-inner resonance and a near-outer resonance respectively,

and, additionally, a discretized clamped nonlinear Timoshenko beam having a 32-dimensional

phase space. A graphical illustration of both systems is shown in Figure 2.3.

SSMtool: A short guide 7

(a) (b)

Fig. 2.2: (a) You can load a predeﬁned mechanical system from the drop-down list or load a

custom mechanical system from a .mat ﬁle. (b) Additionally, you can manually enter a mechanical

system in the predeﬁned input windows.

The second option is to deﬁne your mechanical system in the predeﬁned input windows as

shown in Figure 2.2 (b). Here, you are asked to specify the position and velocity vectors, then

indicate if the system is conservative or non-conservative, specify the mass, damping and stiﬀness

matrices, the nonlinear force vector and the scaling factor of the transformation matrix containing

the eigenvectors of the linearized system at the trivial ﬁxed point. The manually entered system

can be saved to a .mat ﬁle by using the Save button and loaded back into ssmtool for future

use by using the Browse ﬁle button.

The third and last option available is to load a custom mechanical system from a .mat ﬁle

containing the variables listed in Table 2.1, where the "conservative" variable is either 0or 1to

indicate if the system is non-conservative or conservative, respectively. The "scaling" variable,

mentioned above, can be used to scale the eigenvectors of the system and can be any positive

value. The .mat ﬁle can be loaded by using the Browse ﬁle button shown in Figure 2.2 (a).

8 S. Ponsioen, G. Haller

ϕy

nonlinear Timoshenko beam

zϕy

Fig. 2.3: Graphical illustration of the predeﬁned mechanical systems in ssmtool. You can load

the modiﬁed Shaw-Pierre example as explained in section 7.1 and 7.2 of Ponsioen et al. [2] or

load the discretized clamped nonlinear Timoshenko beam from section 7.3 of [2].

Table 2.1: Required variables and variable names to load a .mat ﬁle into ssmtool.

Variable name Class

Mass matrix Mn×ndouble

Damping matrix Cn×ndouble

Stiﬀness matrix Kn×ndouble

Nonlinear force vector fn×1sym

Position vector xn×1sym

Velocity vector xd n×1sym

Conservative constant conservative integer or double

Scaling constant scaling double

2.2 Selecting the spectral subspace

After specifying an autonomous mechanical system, press Analyze to extract the eigenvalues

of the system (see Figure 2.4 (a)). Select a two-dimensional spectral subspace E, by selecting a

pair of complex conjugate eigenvalues and pressing the Select button, as shown in Figure 2.4

(b). ssmtool will calculate the spectral quotient σ(E), which indicates the minimum order of the

Taylor expansion needed to be able to the capture the unique SSM. Additionally, ssmtool will

check if the outer non-resonance conditions (equation (9) in Ponsioen et al. [2]) are satisﬁed in

order to guarantee that the SSM of interest exists (see Figure 2.4 (b)). In case of a lightly damped

spectral subspace E,ssmtool will check for near inner-resonances (see section 5.2 in Ponsioen et

al. [2]), up to the order dictated by the spectral quotient.

SSMtool: A short guide 9

(a) (b) (c)

Fig. 2.4: (a) Analyzing the spectrum of the linearized system at the trivial ﬁxed point. (b)

Selecting a two-dimensional spectral subspace E. (c) Specifying the order of the Taylor expansions

and choosing the desired output coordinates.

2.3 SSM computation

You are now required to specify the order of the Taylor expansion and select if the output

coordinates of the SSM should be in modal, physical or complex coordinates. The "Check for

higher-order internal resonances" checkbox is there to specify if ssmtool should check for near

internal-resonances up to the order of expansion instead of the order dictated by the spectral

quotient (see Figure 2.4 (c)).

After pressing the Compute SSM button for the ﬁrst time, ssmtool will try to start-

up the local parallel matlab pool and identify the number of cores available, which will be

indicated by the message window shown in Figure 2.5 (a). A small portion of the autonomous

SSM computations can be run in parallel, however, it is not required to run the computations in

parallel.

After this stage, ssmtool will proceed with constructing the SSM and the reduced dynamics.

The progress is graphically displayed with the loading bar as shown in Figure 2.5 (b). When the

SSM computation is complete, ssmtool will output a lower-dimensional projection of the SSM

in modal or physical coordinates, depending on the coordinate selection (see Figure 2.6). If you

selected the complex coordinates option, no ﬁgure is displayed.

10 S. Ponsioen, G. Haller

(a) (b)

Fig. 2.5: (a) ssmtool will try to start the local parallel matlab pool during the ﬁrst SSM compu-

tation and will automatically detect the available cores. If matlab fails to start the local pool,

the SSM computation will be continued in series. (b) The progress of the SSM computation is

shown in the loading bar.

(a)

Fig. 2.6: Lower-dimensional projection of the computed SSM in modal or physical coordinates.

SSMtool: A short guide 11

2.4 Parameterization/Backbone

The constructed SSM expressions and the reduced dynamics expressions can be inspected

by pressing the Show button under the header "Parameterization/Backbone". A new window

will be opened, as shown in Figure 2.7. The mapping W(z)and the reduced dynamics R(z)

being displayed are both stored in a matlab function ﬁle under the names SSM_function.m and

R_function.m.

Fig. 2.7: The expressions for the autonomous SSM, W(z), and the reduced dynamics R(z)are

displayed. The reduced dynamics has automatically been transformed to polar coordinates.

As explained in the introduction of this guide and in section 5.2 of Ponsioen et al. [2], the

backbone curve can be extracted from the Ω(ρ) = ˙

θ(ρ)equation of the reduced system. By

pressing the Plot button, ssmtool will compute the backbone curve in the physical coordinate

system, see Figure 2.8. The output can be speciﬁed to be either the maximum absolute value of

one of the physical coordinates of the system, or to be an average over all positional coordinates

as deﬁned in eq. (48) of [2].

12 S. Ponsioen, G. Haller

Fig. 2.8: Extracted backbone curve by ssmtool from the ˙

θ(ρ)reduced equation.

2.5 Integrate trajectories

By pressing the Open Panel button under the "Integrate trajectories" header, a new window

will open up, shown in Figure 2.9 (a). Here, you can integrate the reduced system for a given

initial position (ρ0, θ0)and specify the total integration time tend and the maximum stepsize

used by the matlab ordinary-diﬀerential-equation solver. The result is mapped back into the full

phase space using the mapping W(z)as shown in Figure 2.10 (a). To compare the result of the

reduced system with the full system, you can integrate a trajectory of the full system with the

same initial condition of the reduced system on the SSM. This can be accomplished by checking

the "Integrate full system" box and subsequently pressing the Integrate button. The results are

shown in Figure 2.10 (b). As an extra feature, it is also possible to integrate the full system from

an initial condition that is not necessarily on the SSM by simply modifying the initial position

values as shown in Figure 2.9 (b).

SSMtool: A short guide 13

(a) (b)

Fig. 2.9: (a) The reduced dynamics can be integrated from 0to tend , for a given initial position

(ρ0, θ0). The maximum integration stepsize can be speciﬁed for the matlab ordinary-diﬀerential-

equation solver. (b) Additionally, the full system can be integrated for the same initial position

on the SSM, in order to compare the results of the reduced system with the full system. Here it

is also possible to integrate the full system from an initial condition that is not necessarily on

the SSM.

(a) (b)

Fig. 2.10: (a) The reduced dynamics is integrated for a given initial position. The resulting

reduced trajectory is mapped into the full phase space using the mapping W(z). (b) To compare

the accuracy of the reduced model, the full system can be integrated for the same initial condition

on the SSM, but mapped into the full phase space.

14 S. Ponsioen, G. Haller

2.6 Invariance error

In order to quantify the accuracy of the computed invariant manifolds and the reduced dy-

namics on them, ssmtool can calculate an invariance error measure which compares trajectories

of the full system xi, with trajectories of the reduced system ˜

xi. A total of Ntrajectories from

the full and reduced system are launched from a circle with ﬁxed radius ρ0from the origin, and

integrated until the reduced trajectories cross the inner circle of radius ρε< ρ0, therefore remov-

ing the time dependency. The error measure norm is mathematically formalized in eq. (2.1) (see

section 6 of Ponsioen et al. [2]).

δinv =1

i=1

dist(i)

max

θ∈S1k˜

x(ρ0, θ)k2

,dist(i) = max 



xi|ρ

ρ0−˜

xi|ρ

ρ0



2

.(2.1)

In Figure 2.11 (a) and (b) we show the computed invariance error measures for an O(5)

and O(10) approximation of the SSM. As we increase the order of approximation, and therefore

increase the accuracy of our approximation, the error measure decreases for a ﬁxed radius ρ0.

(a) (b)

Fig. 2.11: (a) Resulting invariance measure error for an O(5) approximation of the SSM and the

reduced dynamics for a ﬁxed radius ρ0= 1. (b) Resulting invariance measure error for an O(10)

approximation of the SSM and the reduced dynamics for the same radius ρ0= 1. As expected

the error decreases for an increasing order of approximation.

2.7 Plot options

You can change the plot coordinates of the lower-dimensional projection of the SSM by using

the "Plot Options" as shown in Figure 2.12. Additionally, the axes limits and the parameterization

domain can be modiﬁed as desired.

SSMtool: A short guide 15

Fig. 2.12: Plot options that can be modiﬁed to change to visualization of the lower-dimensional

projection of the SSM.

3 Final remarks

You are now able to fully take advantage of ssmtool. We would like to note that the future

release of ssmtool will be able to handle the time-dependent periodic forcing case as explained

in Ponsioen et al. [3]. This will allow you to extract forced response curves corresponding to

vibration modes of interest for diﬀerent forcing amplitudes, in an even more numerically eﬃcient

way. Additionally, and due to the use of SSM theory, it will be possible to detect isolated regions

in the forced response curves.

Should you discover any bugs or malfunctions in the GUI, please send an email to stenp@ethz.ch.

References

1. G. Haller and S. Ponsioen. Nonlinear normal modes and spectral submanifolds: existence, uniqueness and

use in model reduction. Nonlinear Dyn., 86(3):1493–1534, 2016.

2. S. Ponsioen, T. Pedergnana, and G. Haller. Automated computation of autonomous spectral submanifolds

for nonlinear modal analysis. Journal of Sound and Vibration, 420:269–295, 2018.

3. S. Ponsioen, T. Pedergnana, and G. Haller. Analytic Prediction of Isolated Forced Response Curves from

Spectral Submanifolds. arXiv preprint arXiv:1812.06664, 2018.

4. T. Breunung and G. Haller. Explicit backbone curves from spectral submanifolds of forced-damped nonlinear

mechanical systems. Proc. R. Soc. A, 474(2213):20180083, 2018.

5. S. Jain, P. Tiso, and G. Haller. Exact nonlinear model reduction for a von Kármán beam: slow-fast decom-

position and spectral submanifolds. J. Sound Vib., 423:195–211, 2018.

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