Nonlinear Modal Decomposition Using Normal Form Transformations

7/30/2019 Nonlinear Modal Decomposition Using Normal Form Transformations

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Nonlinear modal decomposition using normal form

transformations

Simon A. Neild, Andrea Cammarano and David J. Wagg

Department of Mechanical Engineering

University of Bristol

Queens Building, University Walk,

Bristol, UK, BS8 1TR

ABSRACT

In this paper we discuss a technique for decomposing multi-degree-of-freedom weakly nonlinearsystems into a simpler form. This type of decomposition technique is an established cornerstoneof linear modal analysis. Extending this type of technique to nonlinear multi-degree-of-freedomsystems has been an important area of research in recent years. The key result in this work is that atheoretical transformation process is used to reveal both the linear and nonlinear system resonances.For each resonance, the parameters which characterise the backbone curves and higher harmoniccomponents of the response, can be obtained. The underlying mathematical technique is based ona near identity normal form transformation for systems of equations written in second-order form.This is a natural approach for structural dynamics where the governing equations of motion arewritten in this form as standard practice. The example is a system with cubic nonlinearities, andshows how the transformed equations can be used to obtain a time independent representation ofthe system response. It is shown that when the natural frequencies are close to an integer multiple

of each other, the backbone curve bifurcates. Examples of the predicted responses are compared totime-stepping simulations to demonstrate the accuracy of the technique.

Keywords: normal form; resonances; backbone curves; nonlinear; multi-degree-of-freedom

Introduction

The concept of normal modes of vibration is fundamental to the analysis of linear multi degree-of-freedom systems, with each mode relating to a physical configuration of the system and a naturalfrequency. For linear systems, the governing equations of motion can be decomposed into a modalmodel by applying a linear modal transform. This results in a modal model of the system consisting

of a series of independent oscillators that capture the complete system behaviour via superpositionof the modal responses. The ability to create a modal modal from a structural model is key to theexperimental identification of dynamic properties known as modal testing [1] and model updating[2]. While a linear model can capture the key characteristics of many systems, nonlinear b ehaviour isencountered in a large number of applications, such as structures that exhibit large deflections resultingin geometric nonlinearities.

The idea of using a modal approach to analysing the behaviour of nonlinear systems has manychallenges. The concept of nonlinear normal modes was introduced in the 1960s by Rosenberg [3] andmany researchers have pursued this idea see for example [4, 5, 6] and references therein.

In this paper we consider a different approach, in that rather than looking to define modes, weattempt to decompose the nonlinear system into its simplest (aka normal) form. In the context of

multi degree-of-freedom nonlinear structural systems, simplest form means elimination of as manyAddress all correspondence to this author [email protected].


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cross-coupling and nonlinear terms as possible. In fact the method we use will find this simplified formand also includes expressions for the modeshapes as a by product of the process as well.

The method of normal forms has a long history (see [7] for example), and involves applying trans-formations to the governing equations of motion with the aim of finding a simplified form. Detaileddiscussions on the theory of normal forms are given in [8, 9] among others. Of particular interest hereis the application of normal forms to find periodic steady-state system response solutions, as consid-

ered by Jezequel and Lamarque [10] and Nayfeh [11]. Potential advantages of using normal forms overother perturbation techniques for this type of problem include the ease in which it can be extendedto consider multi-degree-of-freedom systems and non-autonomous systems [12] and its suitability foranalysis using symbolic manipulation programs [13]. Relating to nonlinear normal modes, Jezequeland Lamarque [10] considered a multi-mode system using normal forms. Touze et al. [14] linked thenear-identity transform, which provides an asymptotic non-linear change of coordinates for the system,to the normal modes of the system.

The normal form transformation is normally applied to dynamic equations expressed in their state-space, or first-order differential equation, form. Recently a method of applying the normal form tech-nique directly to second-order nonlinear oscillators has been reported [15, 16]. As linear modal analysisis based on second-order differential equation representation of the equations of motion, this second-

order normal form technique has potential to provide a more natural nonlinear extension to the linearproblem. Here, we use the second-order normal form technique to find the backbone curves for a twodegree-of-freedom system.

We present results for a two degree-of-freedom system with cubic spring nonlinearities. The back-bone curves are found using the second-order normal form technique and it is shown that when thenatural frequencies are close to an integer multiple of each other, one of the backbone curves bifur-cates. The curves are obtained by considering the initial conditions which results in one of four types ofmodeshape for the system. Examples of the resulting motion from a selection of the initial conditionsare also shown. These predicted responses are compared to time-stepping simulations to demonstratethe accuracy of the technique.

Modal decomposition for a nonlinear system

The normal forms technique as defined by [19] involves applying a series of transformations resultingin an equation of motion using the following steps:

apply a linear transform to decouple the linear terms,

apply a forcing transformation,

apply a nonlinear near-identity transform.

The aim of the forcing and the near-identity transforms is to remove the non-resonant terms for each

mode. The non-resonant terms are defined as those in the equation of motion that result in harmonicsof the natural frequency (in the case of the unforced system) or of the dominant response frequency (inthe case of forced systems). The forcing transformation targets the non-resonant forcing terms, andthe near-identity transform removes the nonlinear non-resonant terms. Note that the response due tothe non-resonant terms is not lost, instead it is captured in the transform equations. Transformingthese non-resonant terms terms out of the equations of motion, for, say, the nth mode, allows the useof a trial solution of the form Un cos(rnt n) to solve the equation exactly, thereby removing theneed for a harmonic balance type approximation.

Full details of the method can be found in [19, 16], here we consider a two mass oscillator withthree springs, one connecting each mass to ground and one between the masses. The springs connectingthe masses to ground have a force-deflection relationship F = k + 3 and the spring between the

masses has the relationship F = k2 + 23

, where is the length change of the spring. Linearviscous dampers are also connected between the masses and ground (damping c) and between themasses (damping c2). The configuration of the system is such that the resulting system has symmetric


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mass, damping and stiffness matrices. Single frequency forcing P1 cos(t) and P2 cos(t) is applied tothe two masses. The resulting equation of motion is

m 00 m

x1x2

+

c + c2 c2c2 c + c2

x1x2

+ (1)

k + k2 k2

k2

k + k2

x1

x2 +

x31

+ 2(x1 x2)3

x

3

2 + 2

(x2

x1

)

3 = 12 P1 P1

P2

P2 r

where r = {rp, rm}T = {eit, eit}T is used to represent the harmonic forcing.

Applying the linear modal transform x = q, where is a matrix of eigenvectors of M1K resultsin

q + q +Nq(q) = Pqr, (2)

where,

=

1 11 1

, =

k/m 0

0 (k + 2k2)/m

,

Nq(q) =

m q31 + 3q1q22

3q2

1q2 + q3

2+ 21n1 0

0 22n2 q,

Pq =1

4m

P1 + P2 P1 + P2P1 P2 P1 P2

. (3)

Also = 1 + (82/), the linearized natural frequencies n1 = k/m and n2 = (k + 2k2)/m anddamping relationships 21n1 = c/m and 22n2 = (c + 2c2)/m.

The forcing transform q = v+[e]r is now applied and considering each mode in turn, this transformremoves forcing terms that are away from resonance while retaining those that are near-resonant. Inthis paper we will not consider forced equations of motion in detail, and as a result we write q = vand Pq = 0. For the case where q = v, the resulting equation of motion is

v + v + Nv(v, v, r) = Pvr (4)

where Pv = Pq = 0. The resulting nonlinear term, without damping, can written as

Nv(v) = Nq(v) =

m

v31

+ 3v1v22

3v21

v2 + v32

(5)

Now the equation of motion is in a form in which the near-identity nonlinear transform can beapplied. This transform is written as

v = u + h(u, u, r) (6)

The aim of the transform is to simplify the dynamic equation, Eqn. (4), into a form that can be solvedexactly using a single frequency trial solution for each mode. The resulting transformed (i.e. simplified)normal form equation is expressed as

u + u + Nu(u, u, r) = Pur (7)

where currently Nu, and the transform h, are the unknown.To keep track of the relative sizes of terms we introduce , a small parameter, which can be viewed

as a bookkeeping aid. Using this bookkeeping notation, the nonlinear vectors and the transform vectorsare expressed as a power series in

Nv(v, v, r) = nv1(v, v, r) + 2nv2(v, v, r) +

Nu(u, u, r) = nu1(u, u, r) + 2nu2(u, u, r) + (8)

h(u, u, r) = h1(u, u, r) + 2h2(u, u, r) +

Note that there are no 0 terms as Nv, Nu and h are all small compared to the linear terms.


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Using the method described in [19, 16] 2 and higher terms are assumed to be zero, and thecoefficients of the 1 terms can be written as

nv1(u, u, r) = [nv]u(up,um, r)

nu1(u, u, r) = [nu]u(up,um, r) (9)

h1(u, u, r) = [h]u(up,um, r)

where the [] matrices are N L where N is the number of degrees-of-freedom and L is the length ofu which contains all the nonlinear terms. The expressions in (9) can be used to obtain a relationshipfor the coefficients of the form

[h] = [nv] [nu] (10)

where the (n, )th element in [h], hn,l, is related to the corresponding element in [h], hn,l, using

hn, =

(mp mm) + N

n=1

(snp snm)rn

2 2rn

hn,

= n,hn,. (11)

Here we use the matrix [] in which the (n, )th element is n, see [15] or [16] for full details of thisderivation.

For the near-identity transform, v u, the nonlinear term is rewritten in terms ofu, the substi-tution u = up + um is made and nonlinear term is written in matrix form

Nv(up + um) = [nv]u, (12)

using Eqns. (9) and (8), where for each element un in u we have

un = unp + unm : unp =Un2

eineirnt, unm =Un2

eineirnt. (13)

Using these relationships for our current example we obtain

u

=

u31p

u21pu1m

u1pu21m

u31m

u1pu22p

u1pu2pu2mu1pu

22m

u1mu22p

u1mu2pu2mu1mu

22m

u21pu2pu21pu2m

u1pu1mu2pu1pu1mu2m

u21mu2p

u21mu2m

u32p

u22pu2m

u2pu22m

u32m

, [nu]T

=

m

1 03 03 01 03 06 03 03 06 03 0

0 30 30 60 60 30 30 0 30 30

[]T

= 2

r1

8 0 0 8

4(n2 + n) 0

4(n2 n) 4(n2 n)

0 4(n2 + n)

4(1 + n) 4(1 n) 0 0 4(1 n) 4(1 + n) 8n2

0 0 8n2

(14)

where r2 = nr1 and, in [], a dash has been used where the corresponding value in [nv] is zero andhence the value in [] is of no importance. Note that as ri ni the linear natural frequencies mustbe approximately related by n2 nn1.


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We are now in a position to select both the transform terms and the nonlinear terms in the dynamicequation for u. Considering element (n, ), if possible we wish to remove the nonlinear term from thedynamic equation and place it in the transform equation. However this is not possible for all terms.From Eqn. (11) it can be seen that hn, can be large compared to hn, if n, is close to zero. Also fromEqn. (10), it can be seen that [h] is of a similar size to [nu] and [nv]. Hence for the terms where n,is close to zero, this would suggest that if the element in the nonlinear term in matrix [ nv] is placed in

the transform matrix [h], the transform can not be said to be near-identity. This results in two optionsfor the (n, ) element, that satisfy Eqn. (10) and ensure the near-identity condition it met. These arethe preferred option in which the term is removed from the equation of motion

Preferred Option: nu,n, = 0, hn, = nv,n,/n, (15)

and, to ensure that the transform is near-identity, the near-resonant option in which the nonlinear termis kept in the equation of motion

Near-resonant Option: nu,n, = nv,n,, hn, = 0 (16)

Using these transforms the equation of motion can be solved, for each mode in turn, using a trial

solution in the form un =Un2 e

in

eirnt

+Un2 e

in

eirnt

(for the nth

mode) without the need for aharmonic balance approximation. Information regarding the harmonics of the response are capturedin the transform equation

v = u + [h]u + (17)

This equation can also be used to identify the nonlinear normal modes of a system.From [] it can be seen that regardless of the value of n the following terms are resonant; [1, 2],

[1, 3], [1, 6], [1, 9], [2, 13], [2, 14], [2, 18] and [2, 19]. In addition if n = 1 the terms [1, 7], [1, 8], [2, 12] and[2, 15] are resonant. For the resonant terms we set the relevant terms in [ nu] to equal those in [nv] andset the corresponding terms in [h], the transform matrix, to equal zero. Provided n = 1, this results inthe following normal form for the system

u1 + 2n1u1 + 3m

u1pu1mu1 + 6m

u2pu2mu1 = 0 (18)

u2 + 2n2u2 +

3

mu2pu2mu2 +

6

mu1pu1mu2 = 0 (19)

where uip + uim = ui has been used.

Backbone curves

To understand the modal response of the system using second-order normal forms we will concentrateon the underlying modal dynamics, i.e. looking at the system response when the damping and forcingare zero. Taking the n = 1 case, and making the substitutions uip = (Ui/2)e

rit and uim = (Ui/2)erit,

such that ui = Ui cos(rit), and noting that r2 = nr1 results in the time-independent relationships2r1 +

2

n1 +3

4m(U21 + 2U

2

2 )

U1 = 0, (20)

n22r1 + 2

n2 +3

4m(2U21 + U

2

2 )

U2 = 0. (21)

Note that for the case where n = 1, the [1, 7], [1, 8], [2, 12] and [2, 15] terms will be included in thenormal form equations, which results in time-independent relationships

2r1 + 2n1 +3

4m

(U21 + 3U22 )U1 = 0, (22)

2r1 + 2

n2 +3

4m(3U21 + U

2

2 )

U2 = 0. (23)


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From here, for clarity when deriving the equations we will concentrate on the n = 1 case, howeverthe technique is the same for both. There are two straightforward solutions to Eqns. (20) and (21) andto Eqns. (22) and (23), these are

S1: U2 = 0, 2r1 =

2n1 +

3

4mU21 (24)

S2: U1

= 0, n

2

2

r1 =

2

n2 +

3

4m U

2

2 (25)

These represent two backbone curves for the two degree-of-freedom system with S1 and S2 correspond-ing to a response in the first and second modes respectively.

In addition to these, under certain conditions there are two further solutions in which both U1 andU2 are non-zero. Taking Eqns. (22) and (23), and setting the terms in the square brackets of bothequations to zero gives

2r1 = 2

n1 +3

4m(U21 + 3U

2

2 ) = 2

n2 +3

4m(3U21 + U

2

2 ) (26)

Rearranging this equation, and using = 1 + (82/), gives

U21 = (1 42

)U22 2m3

(2n2 2

n1). (27)

It can be seen that for a real solution to U1 two conditions must be met

42 and U2

2 2m

3( 42)(2n2

2

n1). (28)

This indicates that unless n2 = n1, for this solution to be valid U2 cannot be zero. However, U1 = 0is a valid solution, and in this case comparing Eqns. (25) and (26) we can see that the solution lies onthe S2 backbone curve, and in fact these two solutions branch out of the S2 backbone.

If we eliminate U1 from Eqn. (26) we obtain the response frequency equation

2r1 = 2r2 =

32n1 + 2n22

+3( 2)

mU22 (29)

Taking Eqns. (27) and (29) we can summarise the two further solution as

S3: U1 =

(1 4

2

)U22

2m

3(2n2

2n1),

2r2 =32n1 +

2n2

2+

3( 2)

mU22 ,

S4: U1 =

(1 4

2

)U22

2m

3(2n2

2n1),

2r2 =32n1 + 2n2

2+ 3(

2)m

U22 . (30)

These two solutions indicate that four modal solutions exist despite the system just having two degree-of-freedom.

Modeshapes

Using these backbone solutions we can generate the modeshapes for the system. Here, for the unforced,undamped system we will plot the initial displacement conditions (assuming zero initial velocities) atwhich a mode is observed in the x1x2 plane. Then we consider the resulting modeshapes from theseinitial conditions. The forced response is considered in [22].

Firstly we must consider the near-identity transform, Eqn. (17), to calculate the response in x

x = q = v = (u + [h]u) (31)


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1 0.5 0 0.5 11

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1

initial condition plot X1

X2

1 0.5 0 0.5 11

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1


X2

Figure 1: Initial displacement conditions at which a modal response is observed, assuming the initial velocities are zero(shown as thin lines), for (a) n=1 system and (b) n=3 system. The thick line gives an example of the resultant oscillatorymotion from a point on the S3 solution branch.

Again considering the n = 1 case, using Eqns. (14) and (15) we can write

x1 = (U1 + U2)cos(r1t) +

m

U31

+ 3U21

U2 + 3U1U22

+ U32

322r1 cos(3r1t)

x2 = (U1 U2)cos(r1t) +

m

U31 3U2

1U2 + 3U1U

22 U3

2

322r1cos(3r1t). (32)

For backbone solutions S1 and S2, where U2 = 0 and U1 = 0 respectively, these equations show thatthe corresponding modeshapes (x1 = x2 and x1 = x2 respectively) are independent of amplitude andresponse frequency r1. For solutions S3 and S4 the modeshapes are more complex. For these solutionsthe ratio x2/x1 is a function of amplitude and hence response frequency. The initial displacements thatresult in a modal response may be written as

x1|initial = 2

r1(U1 + U2) + (/32m)(U3

1+ 3U2

1U2 + 3U1U

2

2+ U3

2)

x2|initial = 2r1(U1 U2) + (/32m)(U31 3U21 U2 + 3U1U22 U32 ) (33)

with r1 and the relationship between U1 and U2 are defined by Eqn. (30) (noting r1 = r2 as n = 1).

Simulation Results

Consider the two degree-of-freedom system with parameters; m = 1, k = 1, k2 = 0.005, = 0.4 and2 = 0.05 such that n1 = 1 and n2 = 1.005. Using Eqn. (33), the initial conditions which result ina modal vibration are shown as thin lines in Figure 1(a). The thick line shows the resulting motion ifreleased from a point on the S3 solution branch, calculated using Eqn. (32).

In the previous figure the oscillatory motion is almost an exact straight line, hence the mode shape is

essentially fixed during the oscillation and is just a function of initial displacement conditions. However,for the case where n = 1 this is not the case, the modeshape varies during oscillations if the system isreleased from branches S3 or S4. To see this we consider the case where n = 3 using a system withparameters m = 1, k = 1, k2 = 4.015, = 0.05 and 2 = 10 such that n1 = 1 and n2 = 3.005. Figure1(b) shows the initial conditions which result in a modal response (thin lines). It can be seen that forthis system the bifurcation occurs on the S1 solution. The thick lines show a series of responses forthe system released from the S3 initial conditions. It can be seen that in this case the shape of theresponse varies over the oscillation.

To assess the accuracy of the normal form analysis, Figure 2(a) shows the system response calculatedusing time-stepping simulation for two periods of oscillation (thick dashed line). It can be seen that itagrees well with the normal form prediction. Figure 2(b) shows the corresponding response of x1 and x2

over time, confirming the behaviour is a vibration in unison as defined by [3]. To further highlight thevibration-in-unison behaviour, Figure 3 shows approximately two cycles of time simulation response ofthe system for initial conditions that do not lie on the S1-4 solution curves.


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1 0.5 0 0.5 11

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1


X2

0 2 4 6 8 10 12 141

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1

time (s)

x1,x2

Figure 2: (a) comparison between time-stepping simulation (thick dashed line) over two periods of oscillation and thenormal form predictions for an initial displacement condition lying on the S3 solution and (b) the corresponding responsein x1 (solid line) and x2 (dashed line) over time.

1 0.5 0 0.5 11

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1


X2

Figure 3: Approximately two cycles of time-stepping simulation response for initial conditions that do not lie on theS1-S4 modal lines.

Conclusion

In this paper, we demonstrate how the second-order normal form method can be used to understand thenonlinear modal response of a system. This is done by considering a two degree-of-freedom nonlinear

system with closely matching linear natural frequencies. We show how the system backbone curvescan be found using the second-order normal form technique. These curves are plotted in the x1x2plane as initial displacement conditions (assuming no initial velocities) from which a modal response isobserved. These solutions are more complex than the expressions for the underlying linear system asfor some x1 values there are four x2 values at which a modal response is observed due to a bifurcationin one of the backbone curves. The responses over a cycle of oscillation are also shown for a number ofinitial conditions, highlighting that the mode shape can vary over each cycle of oscillation. The normalform predictions are shown to have high accuracy using time-stepping simulation results.


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[4] Shaw, S.W., Pierre, C., 1991. Non-linear normal modes and invariant manifolds. Journal of Soundand Vibration, 150, pp. 170-173.

[5] Vakakis, A.F. (Ed.), 2002. Normal Modes and Localization in Nonlinear Systems. Kluwer AcademicPublishers, London, UK.

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[19] Wagg, D.J., Neild, S.A., 2009. Nonlinear Vibration with control, Springer-Verlag.

[20] Xin, Z., Neild, S.A., Wagg, D.J., 2011. The selection of the linearized natural frequency for thesecond-order normal form method. Proceedings of the ASME International Design EngineeringTechnical Conferences & Computers and Information in Engineering Conference IDETC/CIE2011, Washington, DC, USA, paper DETC2011-47654.

[21] Doedel, E.J., Paffenroth, R.C., Champneys, A.R., Fairgrieve, T.F., Kuznetsov, Y.A., Oldeman,B.E., Sandstede, B., Wang, X.J., 2007. AUTO-07P: Continuation and bifurcation software forordinary differential equations, URL http://indy. cs. concordia. ca/auto.

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Nonlinear Modal Decomposition Using Normal Form Transformations