Asynchronous parametric excitation, total instability and its occurrence in engineering structures

Journal of Sound and Vibration 428 (2018) 1–12

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Asynchronous parametric excitation, total instability and its occurrence in engineering structures Artem Karev a,* , Daniel Hochlenert b , Peter Hagedorn a a b

Dynamics & Vibrations Group, fnb, TU Darmstadt, Dolivostr. 15, 64293 Darmstadt, Germany Mechatronics and Machine Dynamics, MMD, TU Berlin, Einsteinufer 5, 10587 Berlin, Germany

article info

abstract

Article history: Received 20 October 2017 Revised 27 April 2018 Accepted 2 May 2018 Available online XXX Handling Editor: Ivana Kovacic

In mechanical engineering systems self-excited and parametrically excited vibrations are in general unwanted and sometimes dangerous. There are many systems exhibiting such vibrations which up to this day cannot be completely avoided, such as brake squeal, the galloping vibrations of overhead transmission lines, the ground resonance in helicopters and others. In general, problems of parametric excitation are studied for the case in which all the timeperiodic terms are synchronous. In this case the stability behavior is well understood. However, if the time-periodic terms are asynchronous, an “atypical” behavior may occur: The linear system may then be unstable for all frequencies of the parametric excitation, and not only in the neighborhood of certain discrete frequencies (total instability). Until recently it was believed that such “atypical” behavior would not appear in mechanical systems. The present paper discusses some recent insights and results obtained for linear and nonlinear systems with asynchronous parametric excitation. The method of normal forms is used to prove total instability and to calculate limit cycles of a generalized nonlinear system. Further, a mechanical example of a minimal disk brake model featuring such out of phase parametric excitation is presented. The example outlines the importance of the observed effects from the engineering point of view, since similar terms are also expected in the equations of motion of disk brakes with disks with ventilation channels and most likely also in other physical systems. © 2018 Elsevier Ltd. All rights reserved.

Keywords: Parametric excitation Asynchronous parametric excitation Total instability Stability Nonlinear system

1. Introduction Parametrically excited systems, i.e. systems with time-varying parameters, have been studied since the times of H ILL and MATHIEU, often in the context of celestial mechanics, and then later in mechanical, civil and electrical engineering problems. The MATHIEU equation is the simplest one-dimensional example and is usually presented in most vibration courses. The simplest example of a mechanical system described by the MATHIEU differential equation is the mathematical pendulum with a vertically oscillating suspension point, being moved harmonically [1]. When analyzing stability behavior of parametrically excited systems, systems of higher dimensions – of at least two – are of particular interest. In a one-dimensional case only single parametric resonances involving just one eigenfrequency are possible. Whereas having at least two degrees of freedom, there are various possibilities of combination resonances involving more than one eigenfrequency as well as further phenomena [2,3]. For such systems the equations of motion can be written in the general

* Corresponding author. E-mail addresses: [email protected] (A. Karev), [email protected] (D. Hochlenert), [email protected] (P. Hagedorn).

https://doi.org/10.1016/j.jsv.2018.05.003 0022-460X/© 2018 Elsevier Ltd. All rights reserved.

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A. Karev et al. / Journal of Sound and Vibration 428 (2018) 1–12

form:

𝐌(t )𝐪̈ + (𝐃(t ) + 𝐆(t ))𝐪̇ + (𝐊(t ) + 𝐍(t ))𝐪 = 𝐟 (t , 𝐪, 𝐪) ̇ ,

(1)

with

𝐌 = 𝐌T > 0,

(2a)

𝐃 = 𝐃T ≥ 0,

(2b)

𝐆 = −𝐆T ,

(2c)

𝐊 = 𝐊T > 0 or 𝐊 = 𝐊T ≥ 0,

(2d)

𝐍 = −𝐍T ,

(2e)

where all these matrices may be time dependent. The nonlinear terms and also the (non-parametric) forcing terms are ̇ on the right hand side of Eq. (1). The different matrices have different physical origins lumped into the function 𝐟 (t, 𝐪, 𝐪) and in engineering systems usually are known with different precision. M and K follow from the geometry and from material parameters, and usually can be determined to a high degree of precision. The damping matrix D is in general small and not well defined. It may have different physical origins. The gyroscopic matrix G follows from the kinematics and is therefore usually well defined. The circulatory matrix N is always associated with an energy source, as the corresponding coordinate proportional terms are non-conservative and the associated force field is non-potential. In many cases of selfexcited vibrations in mechanical systems this matrix is due to the contact forces between sliding solid bodies with COULOMB friction. Examples of engineering systems with time-periodic coefficients in the equations of motion are the automotive disk brakes and wind turbines. In disk brakes, ventilation channels lead to sector symmetry in the disk, which, in turn, leads to varying system’s properties depending on the disk’s angular position with respect to the brake pads. In horizontal-axis wind turbines, the periodic coefficients are due to the gravity forces acting on rotating elastic blades [4]. In some special applications, parametric excitation can also be deliberately introduced in order to benefit from the parametric anti-resonance increasing the system’s dissipation properties. A technical example utilizing anti-resonance action of parametric excitation is the flexible rotor in journal bearings with adjustable geometry, as treated by Dohnal et al. [5]. In this case time-periodicity is due to varying bearing geometry, i.e., bearing forces. Most of the literature dealing with parametric excitation treat synchronous excitation only [3,6–8]. There are only few publications dealing with more general cases. The simplest example of differential equations with asynchronous excitation was first given about 70 years ago in 1940 by CESARI [9]: q̈ 1 (t ) + 𝜔21 q1 (t ) = 𝜀 cos(Ωt ) q2 (t ), q̈ 2 (t ) + 𝜔22 q2 (t ) = 𝜀 cos(Ωt + 𝜓 ) q1 (t ).

(3)

According to CESARI [9], a phase shift 𝜓 = 𝜋 /2 between time-periodic coupling terms of the two-dimensional system (3) leads to instability of the trivial solution for all frequencies of excitation Ω (total instability or “atypical” behavior). More general systems were analyzed in the 70’s and 80’s: SCHMIEG [10] analyzed a system with out of phase parametric excitation in stiffness, while E ICHER [11,12] investigated a much more general case of simultaneous out of phase excitation in mass, damping and in stiffness terms. These papers confirmed and extended earlier findings by CESARI [9] concerning the phenomenon of total instability: it was found out that any infinitesimally small phase shift 𝜓 ≠ z𝜋 , z ∈ ℤ, leads to the same effect. While the phase relations investigated by E ICHER [11,12] are of the most general shape, neither circulatory nor gyroscopic terms in the unperturbed constant system are considered. Also no mechanical example featuring such behavior is given. The present paper gives an overview of the typical and “atypical” stability behavior of parametrically excited systems. The method of normal forms is used to analyze a nonlinear model with asynchronous excitation in order to outline the effects of total instability, in particular the simultaneous occurrence of equally strong instability regions at sum and at difference combination resonance frequencies. Further, a mechanical example featuring parametric excitation with various phase relations is introduced. Throughout the paper the Lyapunov’s definition for the stability of a solution of the equations of motion is used. 2. Parametric resonance, typical cases In this section we review some of the typical known cases of parametric resonance, which are for example well described in the excellent survey paper by METTLER [3] and also, e.g., in Refs. [6] and [7]. For the time being, we will assume that the matrices D and G are equal to zero. In the simple case of the MATHIEU equation q̈ (t ) + 𝜔2 [1 + 𝜀 cos(Ωt )]q(t ) = 0

(4)

A. Karev et al. / Journal of Sound and Vibration 428 (2018) 1–12

3

Fig. 1. MATHIEU equation with 𝜔 = 1: stability map; parameter values in the colored regions correspond to instability.

Fig. 2. Stability maps in Ω 𝜀-plane with 𝜔1 = 1, 𝜔2 = nation resonance.

√

3. Part (a): Single resonances and sum combination resonance. Part (b): Single resonances and difference combi-

the regions of instability in the parameter space of 𝜀 and Ω emanate from the critical circular frequencies

Ωcrit =

2𝜔 , p

p ∈ ℕ,

(5)

as depicted in the well-known stability map of Fig. 1 (for simplicity only two of the infinitely many instability regions are shown). Graphs such as this one are easily obtained by assessing stability of the trivial solution for a set of parameter values in the Ω 𝜀-plane. This is done by numerically computing the monodromy matrix P, which is defined through the fundamental matrix 2𝜋 is the period of the parametric excitation [13]. The trivial solution solution X(t) of the system as X (t + T) = X(t) P, where T = Ω of the system is unstable if the magnitude of at least one of the eigenvalues 𝜇i of P is larger than one. All the stability graphs presented in this paper were calculated numerically in this manner, and the corresponding parameter values are given in the respective figures. Next, consider the 2 dof system

𝐪( ̈ t) + [𝐊0 + 𝜀𝐂 cos(Ωt)]𝐪(t) = 0,

(6)

where without loss of generality we set

(

𝜔21

𝐊0 =

0

)

0

(7)

𝜔22

and assume a parametric excitation in the stiffness with

(

𝐂=

0

−1

−1

0

)

.

(8)

Here, the instability regions in the Ω 𝜀-plane originate from the critical excitation frequencies

Ωcrit,single = Ωcrit,combi =

2𝜔i , 2p

i = 1, 2,

𝜔1 + 𝜔2 2p − 1

, p ∈ ℕ,

(9a) (9b)

as shown in Fig. 2a. If, on the other hand, we assume a parametric excitation in the circulatory terms such as in

(

𝐂=

)

0

1

−1

0

,

(10)

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A. Karev et al. / Journal of Sound and Vibration 428 (2018) 1–12

Fig. 3. Stability maps in Ω |𝜇 i |max -plane with 𝜔1 = 1, 𝜔2 = combination resonance.

√

3. Part (a): Single resonances and sum combination resonance. Part (b): Single resonances and difference

we obtain the stability diagram of Fig. 2b, with the critical excitation frequencies

Ωcrit,single =

2𝜔i , 2p

Ωcrit,combi =

|𝜔1 − 𝜔2 | , 2p − 1

i = 1, 2,

(11a)

p ∈ ℕ.

(11b)

Instead of the critical sum frequencies, we now have the critical difference frequencies. Finally, if we consider the system with parametric excitation as a linear combination of symmetric and skew-symmetric terms

(

𝐂 = 𝛼1

0

) −1

−1

0

(

+ 𝛼2

)

0

1

−1

0

,

(12)

the kind of combination resonance depends on 𝛼 i : there are sum combination resonances for 𝛼 1 > 𝛼 2 , difference combination resonances for 𝛼 1 < 𝛼 2 , and no parametric instability at all for 𝛼 1 = 𝛼 2 . In other words, as defined by METTLER [3], if the product of the anti-diagonal excitation terms is positive (negative), then there are sum (difference) combination resonances. However, there is no linear combination of symmetric and skew-symmetric excitation leading to simultaneous appearance of sum and difference combination resonances. With view to the next section, we also represent the behavior of the systems corresponding to Fig. 2 in an alternative fashion for a fixed value of 𝜀: Fig. 3. In this figure, the largest absolute value of the (in general complex) eigenvalues of the monodromy matrix |𝜇i |max is depicted as a function of the excitation frequency Ω. The stability behavior in parametrically excited systems discussed so far will be termed “typical stability behavior”. 3. “Atypical” parametric resonance, total instability Around 1940 CESARI [9] studied the differential system

𝐪( ̈ t) + [𝐊0 + 𝜀𝐂(t)]𝐪(t) = 0

(13a)

with

(

𝐊0 =

𝜔21 0

0

𝜔22

)

( and

𝐂(t ) =

0

− sin Ωt

− cos Ωt

0

)

.

(13b)

The trivial solution of this system is generically unstable, i.e. it is unstable for all values of Ω ≠ 0 and 𝜀 ≠ 0. More precisely, in a rather involved proof, C ESARI showed that in an arbitrarily small neighborhood of any given set of values of these parameters, there are infinitely many parameter values for which the system is unstable in the sense of LYAPUNOV. Although the original proof given by CESARI is rather complex, numerical investigations of the problem easily show that, in fact, the trivial solution is unstable for arbitrary values of the mentioned parameters. This kind of parametric resonance is called by CESARI [9] “total instability”. E ICHER [12] explained total instability as infinitely wide open sum and difference resonance tongues in the Ω 𝜀plane. Recently, a very simple proof of the total instability was also given by A. STEINDL [14], using normal form theory. Of course, the resonance diagrams as in Figs. 1 and 2 do not give any information in this case, but a plot of the type of Fig. 3a or 3b makes sense, see Fig. 4. Apart from the distinct parametric resonance areas in the vicinity of the sum and difference frequencies, Fig. 4 shows total instability of the trivial solution, since |𝜇i |max is always larger than one. In the rest of this section we will try to understand what could possibly be the reason for this “atypical” behavior.

A. Karev et al. / Journal of Sound and Vibration 428 (2018) 1–12

Fig. 4. CESARI equation with 𝜔1 = 1, 𝜔2 =

5

√

3: total instability.

3.1. Simultaneous sum and difference combination resonances Following the introduced scheme of splitting the excitation terms, the matrix C(t) of the original CESARI system (13) is written as

⎛ 0 1 𝐂(t ) = 𝐊1 (t ) + 𝐍1 (t ) = √ ⎜ 𝜋 ⎜ 2 ⎝− sin(Ωt + ) 4

𝜋 ⎞ )

⎛ 0 1 ⎜ 4 ⎟ +√ 𝜋 ⎟ ⎜ 2 ⎝sin(Ωt − ) ⎠ 4

− sin(Ωt + 0

− sin(Ωt − 0

𝜋 ⎞ )

4 ⎟,

(14)

⎟ ⎠

⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟

⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟

symmetric

skew−symmetric

splitting the parametric excitation in a symmetric part K1 (t) and a skew-symmetric part N1 (t). This highlights that CESARI ’s system contains simultaneous symmetric and skew-symmetric as well as out of phase parametric excitations. Considering systems containing either the symmetric or the skew-symmetric part

𝐪( ̈ t) + [𝐊0 + 𝜀𝐊1 (t)]𝐪(t) = 0,

(15a)

𝐪( ̈ t) + [𝐊0 + 𝜀𝐍1 (t)]𝐪(t) = 0,

(15b)

typical parametric resonance behavior can be observed: single and sum combination resonances for Eq. (15a) and single and difference combination resonances for Eq. (15b). However, as already mentioned, the combination of the symmetric and the skew-symmetric parts as in Eq. (14) here leads to a totally different behavior with both distinct sum and difference combination resonances superimposed by total instability, in the sense that even outside of the usual resonance areas |𝜇i |max is larger than one. Compared to the system in Eq. (12) showing the typical behavior, there is only one major difference: being in phase for 𝜋 themselves, the matrices K1 (t) and N1 (t) are now phase shifted with respect to each other by . Applying METTLER ’ S condition 2 for the presence of combination resonances, it is clear that the sign of the product built by the anti-diagonal excitation terms is not constant, but changes multiple times over one period of excitation. This explains the simultaneous presence of the sum and difference combination resonances. 3.2. Normal form analysis of a nonlinear system The original C ESARI system (13) is a special case of

(

)( )

1

0

q̈ 1

0

1

q̈ 2

(

+

𝛿

0

0

𝛿

)( ) q̇ 1 q̇ 2

[(

+

𝜔21 0

0

)

𝜔22

(

+𝜀

)] ( )

0

cos(Ωt − 𝜓 )

q1

cos Ωt

0

q2

( )

+𝛾

q̇ 31 q̇ 32

( )

+𝜅

q31 q32

=0

(16)

for 𝜓 = 𝜋 /2 and 𝛾 = 𝜅 = 0. We will now study (16) by applying normal form theory. The general background on normal form theory can be found in Refs. [15,16] for example. Details of the following analysis are based on [17,18]. In a first step (16) is written as a time-autonomous first order system of the form u̇ 1 = u2 , u̇ 2 = −𝜔21 u1 − u7 u2 −

(17a) 1 𝜀(u5 e−j𝜓 + u6 ej𝜓 )u3 − 𝛾 u32 − 𝜅 u31 , 2

u̇ 3 = u4 , u̇ 4 = −𝜔22 u3 − u7 u4 − u̇ 5 = jΩu5 ,

(17b) (17c)

1 𝜀(u5 + u6 )u1 − 𝛾 u34 − 𝜅 u33 , 2

(17d) (17e)

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A. Karev et al. / Journal of Sound and Vibration 428 (2018) 1–12

u̇ 6 = −jΩu6 ,

(17f)

u̇ 7 = 0

(17g)

with u1 = q1 , u2 = q̇ 1 , u3 = q2 , u4 = q̇ 2 and u5 = e

jΩt

, u6

= e−jΩt , u

7

= 𝛿 . In matrix notation (17) reads

𝐮̇ = 𝐀𝐮 + 𝐟 (𝐮),

(18)

where 𝐟 contains nonlinear (in the present case quadratic and cubic) terms in the generalized coordinates. The linear part of (17) can be decoupled by a modal transformation u = Rx resulting in

𝐱̇ = 𝚲𝐱 + 𝐑−1 𝐟 (𝐑𝐱)

(19a)

= 𝚲𝐱 + 𝐟2 (𝐱) + 𝐟3 (𝐱) + …

(19b)

= 𝐟 (𝐱)

(19c)

with the diagonal matrix

𝚲 = diag(𝜆1 , … , 𝜆n ) = diag(j𝜔1 , −j𝜔1 , j𝜔2 , −j𝜔2 , jΩ, −jΩ, 0)

(20)

and fi (x) containing nonlinear terms of ith order. The basic idea of the normal form transformation is to eliminate as many nonlinear terms of (19) as possible by the nearidentity transformation

𝐱 = 𝐠(𝐲) = 𝐲 + 𝐠2 (𝐲) + 𝐠3 (𝐲) + …

(21)

yielding the system in normal form

𝐲̇ = 𝐡(𝐲) = 𝚲𝐲 + 𝐡2 (𝐲) + 𝐡3 (𝐲) + … ,

(22)

where gi (y) and hi (y) contain nonlinear terms of ith order. Inserting (21) and (22) in (19c) yields

𝜕 𝐠(𝐲) 𝐡(𝐲) = 𝐟 (𝐠(𝐲)). 𝜕𝐲

(23)

This partial differential equation can be solved for the coefficients of gi (y) and hi (y) under the condition that as many nonlinear m m m terms of h(y) as possible are eliminated. In doing so, all monomials y1 1 y2 2 · · · yn n (in the present case n = 7) can be eliminated except the ones for which the resonance condition

𝜆j = m1 𝜆1 + m2 𝜆2 + · · · + mn 𝜆n , j = 1, … , n

(24)

is fulfilled, where mj are natural numbers. In the present case, the resonance condition reads

{±𝜔1 , ±𝜔2 , ±Ω, 0} = m1 𝜔1 − m2 𝜔1 + m3 𝜔2 − m4 𝜔2 + m5 Ω − m6 Ω + m7 0.

(25)

Obviously, the resonance condition covers simple resonances as well as combination resonances. For system (17) the normal form reads ẏ 1 = j𝜔1 y1 + h12 (𝐲) + h13 (𝐲) + … ,

(26a)

ẏ 2 = −j𝜔1 y1 + h22 (𝐲) + h23 (𝐲) + … ,

(26b)

ẏ 3 = j𝜔2 y3 + h32 (𝐲) + h33 (𝐲) + … ,

(26c)

ẏ 4 = −j𝜔2 y4 + h42 (𝐲) + h43 (𝐲) + … ,

(26d)

ẏ 5 = jΩy5 ,

(26e)

ẏ 6 = −jΩy6 ,

(26f)

ẏ 7 = 0.

(26g)

Therefore, the solutions of the differential equations (26e) and (26f) can be easily calculated and substituted in (26a) to (26d). We will now investigate (3.2) in detail for 𝜔1 ≠ 𝜔2 in the case of non-resonant parametric excitation1 and in the case of the combination resonances Ω ≈ 𝜔1 + 𝜔2 and Ω ≈ 𝜔2 − 𝜔1 . 1 Non-resonant parametric excitation means Ω ≠ m1 𝜔1 − m2 𝜔1 + m3 𝜔2 − m4 𝜔2 for any mi (i = 1, …4) and has to be distinguished from the notion (non)resonant in the sense of (24) or (25) in the context of normal form theory.

A. Karev et al. / Journal of Sound and Vibration 428 (2018) 1–12

7

Fig. 5. Critical 𝜖 -values for the instability of the trivial solution; solid: numerical integration of system (16), dotted: normal form (NF) in the non-resonant case, Eq. (27). √ Parameter values: 𝜔1 = 1, 𝜔2 = 3, 𝜓 = 𝜋 /2, 𝛾 = 0, 𝜅 = 0.

3.2.1. Non-resonant parametric excitation First, in order to point out the total instability of the trivial solution of the generalized CESARI system (16), normal form is derived explicitly for the non-resonant parametric excitation. In this case the normal form (3.2) can be written as

[

]

1 𝜀2 Ω sin 𝜓 3 ṙ 1 = − 𝛿 + r1 − 𝛾𝜔21 r13 + … , 2 8 2[Ω2 − (𝜔1 + 𝜔2 )2 ][Ω2 − (𝜔1 − 𝜔2 )2 ]

[ 1 ṙ 2 = − 𝛿 − 2

(27a)

]

𝜀2 Ω sin 𝜓 3 r2 − 𝛾𝜔22 r23 + … , 8 2[Ω2 − (𝜔1 + 𝜔2 )2 ][Ω2 − (𝜔1 − 𝜔2 )2 ]

(27b)

𝜑̇ 1 = −

𝛿2 3 𝜅 2 + 𝜔1 + r + …, 8 𝜔1 8 𝜔1 1

(27c)

𝜑̇ 2 = −

𝛿2 3 𝜅 2 + 𝜔2 + r + …, 8 𝜔2 8 𝜔2 2

(27d)

where the polar coordinates y1 =

1 𝜔 r ej𝜑1 , 2 1 1

y2 =

1 𝜔 r e−j𝜑1 , 2 1 1

y3 =

1 𝜔 r ej𝜑2 , 2 2 2

y4 =

1 𝜔 r e−j𝜑2 2 2 2

(28)

have been introduced. It can be seen from (27) that for 𝛿 = 0 and 𝜓 ≠ z𝜋 , z ∈ ℕ (note that in CESARI ’s system (13) 𝜓 = 𝜋 /2 holds), the trivial solution is unstable in the sense of LYAPUNOV for all 𝜀 ≠ 0, especially for all non-resonant frequencies Ω of the parametric excitation: as the coefficients of r1 and r2 are equal except for the opposite sign, at least one of them will always be positive. This corresponds to the notion of “atypical” parametric resonance and total instability and is in agreement with results obtained in Ref. [12], where instability regions of resonant parametric excitation are investigated. For 𝛿 = 0 and 𝜓 = z𝜋 the trivial solution is stable for 𝛾 > 0 (positive cubic damping) and all non-resonant frequencies Ω of the parametric excitation. However, if 𝛿 > 0, the linear damping can quickly mask the total instability making the trivial solution stable. Fig. 5 shows the critical values of 𝜖 , above which the trivial solution becomes unstable (the representation is equivalent to the stability maps in Figs. 1–2). It has to be noted, that these values are valid for non-resonant Ω only; in particular, Eqs. (27a)–(27d) are not valid in the vicinity of the combination resonances, which will be treated separately in the next subsection. Nevertheless, Fig. 5 shows that the results of the non-resonant normal form match very well with the numerical integration of the original system (16) over wide regions of Ω. The implications of linear damping in parametrically excited systems have also been extensively studied in the past: for linear and non-linear systems with “typical” excitation [7,19], as well as for linear systems with total instability [12]. Therefore, further analysis of non-linear behavior in this subsection is performed for 𝛿 = 0 in order to avoid interference with other effects and to emphasize the impact of total instability. In the case 𝜓 ≠ z𝜋 the positive cubic damping leads to limit cycle oscillations, since there is a fixed point approximately given by

√ ri =

4 𝜀2 Ω sin 𝜓 , 3 𝛾𝜔2 [Ω2 − (𝜔1 + 𝜔2 )2 ][Ω2 − (𝜔1 − 𝜔2 )2 ] i

i = 1 or 2.

(29)

It depends on the sign of the coefficients of linear terms in r1 and r2 in (27), if i = 1 or i = 2 holds. These coefficients are of same absolute value, opposite in sign and change sign at Ω2 = (𝜔1 ± 𝜔2 )2 , i.e. at the typical parametric resonance frequencies. Therefore the frequency of limit cycle oscillation in coordinates of the normal form is 𝜔1 for Ω2 < (𝜔1 − 𝜔2 )2 or Ω2 > (𝜔1 + 𝜔2 )2 and is 𝜔2 for (𝜔1 − 𝜔2 )2 < Ω2 < (𝜔1 + 𝜔2 )2 .

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A. Karev et al. / Journal of Sound and Vibration 428 (2018) 1–12

3.2.2. Combination resonances Ω ≈ 𝜔1 + 𝜔2 and Ω ≈ 𝜔2 − 𝜔1 In the case of the combination resonance Ω ≈ 𝜔1 + 𝜔2 and with the polar coordinates y1 =

1 1 j(𝜑 + Ωt) r e 1 2 , 2 1

y2 =

1 −j(𝜑1 + 12 Ωt) , r e 2 1

y3 =

1 1 j(𝜑 + Ωt) r e 2 2 , 2 2

y4 =

1 −j(𝜑2 + 12 Ωt) r e 2 2

(30)

the normal form can be written as 1 ṙ 1 = − 𝛿 r1 + 𝛿 A1 cos(𝜓 + 𝜑1 + 𝜑2 ) r2 − B sin 𝜓 r1 + C sin(𝜓 + 𝜑1 + 𝜑2 ) r2 + Dr13 + … , 2

(31a)

1 ṙ 2 = − 𝛿 r2 + 𝛿 A2 cos(𝜑1 + 𝜑2 ) r1 + B sin 𝜓 r2 + E sin(𝜑1 + 𝜑2 ) r1 + Dr23 + … , 2

(31b)

𝜑̇ 1 = − 𝜑̇ 2 = −

r r 𝛿2 1 + 𝛿 A3 sin(𝜓 + 𝜑1 + 𝜑2 ) 2 + 𝜔1 − Ω + F cos 𝜓 + Gr12 + C cos(𝜓 + 𝜑1 + 𝜑2 ) 2 + … , 8 𝜔1 r1 2 r1

𝛿2

8 𝜔2

− 𝛿 A4 sin(𝜑1 + 𝜑2 )

r1 r 1 + 𝜔2 − Ω + H cos 𝜓 + Ir22 + E cos(𝜑1 + 𝜑2 ) 1 + … . r2 2 r2

(31c)

(31d)

For the combination resonance Ω ≈ 𝜔2 − 𝜔1 the polar coordinates y1 =

1 −j(𝜑1 + 12 Ωt) , r e 2 1

y2 =

1 1 j(𝜑 + Ωt) r e 1 2 , 2 1

y3 =

1 −j(𝜑2 + 12 Ωt) , r e 2 2

y4 =

1 1 j(𝜑 + Ωt) r e 2 2 2 2

(32)

yield 1 ṙ 1 = − 𝛿 r1 + 𝛿 A1 cos(𝜓 + 𝜑1 + 𝜑2 ) r2 − B sin 𝜓 r1 + C sin(𝜓 + 𝜑1 + 𝜑2 ) r2 + Dr13 + … , 2

(33a)

1 ṙ 2 = − 𝛿 r2 + 𝛿 A2 cos(𝜑1 + 𝜑2 ) r1 + B sin 𝜓 r2 + E sin(𝜑1 + 𝜑2 ) r1 + Dr23 + … , 2

(33b)

𝜑̇ 1 =

r r 𝛿2 1 + 𝛿 A3 sin(𝜓 + 𝜑1 + 𝜑2 ) 2 − 𝜔1 − Ω + F cos 𝜓 + Gr12 + C cos(𝜓 + 𝜑1 + 𝜑2 ) 2 + … , 8 𝜔1 r1 2 r1

𝜑̇ 2 = −

r r 𝛿2 1 − 𝛿 A4 sin(𝜑1 + 𝜑2 ) 1 + 𝜔2 − Ω + H cos 𝜓 + Ir22 + E cos(𝜑1 + 𝜑2 ) 1 + … . 8 𝜔2 r2 2 r2

(33c)

(33d)

The coefficients Ai , Ai , B, B, C , D, E, F , F , G, H, H, I (i = 1, …, 4) depend on the parameters 𝜔1 , 𝜔2 , 𝜀, 𝜅 , 𝛾 . The fixed points of (31) and (33) can be calculated numerically for a given set of parameters. The results for the combination resonance Ω ≈ 𝜔1 + 𝜔2 and 𝜓 = 0 are identical to those obtained in Ref. [19] by the method of slowly varying phase and amplitude. Figs. 6–7 show the fixed points of Eqs. (31) and (33) for a given set of parameters and 𝜓 = {0, 𝜋 /20, 𝜋 /4, 𝜋 /2}, respectively. These fixed points correspond to periodic solutions of original system (16) with 𝛿 = 0. These periodic solutions can be calculated using the corresponding definition of the polar coordinates (30) and (32) with the coordinate transformation (21). A comparison of the “amplitudes” of the periodic solutions obtained by the normal form theory and a numerical integration of the original equations is shown in Fig. 8. Both results match very well. 4. Example of a mechanical system with asynchronous parametric excitation Until recently is was believed that out of phase parametric excitation, as discussed in the previous sections, is irrelevant for engineering systems [3]. In this section we present a simple mechanical system which exhibits exactly this type of parametric excitation. The starting point is the minimal model of a disk brake as described in Ref. [20], see Fig. 9. The model consists of a rigid disk (inertia tensor Θ, thickness h, radius r) in frictional contact with idealized brake pads (preload N 0 , stiffness k, damping d). The disk is hinged in its center of mass and viscoelastically supported by rotational springs (stiffness kt , damping dt ). In the inertial Cartesian coordinate system defined by the unit vectors ni , i = 1, 2, 3, the disk rotates around the axis n3 at a constant angular velocity Ω and is free to tilt with respect to the n1 n2 plane. All parameter values are set according to [20]. The model is extended by attaching a mass particle mp to the disk in order to introduce time-periodicity. In a real brake the asymmetry of the disk is due to the ventilation channels. This two degrees of freedom model, described in an inertial frame by Eqs. (34)–(35c), contains asynchronous parametric excitation. The equations of motion are

𝐌(t )𝐪̈ + 𝐁(t )𝐪̇ + 𝐂𝐪 = 0 with

𝐌(t ) =

( Θd

0

0

Θd

)

(34)

(

+ Θp

sin2 (Ωt )

− sin(Ωt ) cos(Ωt )

) − sin(Ωt ) cos(Ωt ) cos2 (Ωt )

,

(35a)

A. Karev et al. / Journal of Sound and Vibration 428 (2018) 1–12

9

Fig. 6. Fixed points r1 , r2 near the combination resonance Ω ≈ 𝜔1 + 𝜔2 for 𝜓 = [0, 𝜋 /20, 𝜋 /4, 𝜋 /2], parts (a)–(d). Parameter values: 𝜔1 = 1, 𝜔2 = 𝜖 = 0.2, 𝛿 = 0.

Fig. 7. Fixed points r1 , r2 near the combination resonance Ω ≈ 𝜔2 − 𝜔1 for 𝜓 = [0, 𝜋 /20, 𝜋 /4, 𝜋 /2], parts (a)–(d). Parameter values: 𝜔1 = 1, 𝜔2 = 𝜖 = 0.2, 𝛿 = 0.

1 h2 ⎛ d + 2dr2 + 𝜇N0 𝐁(t ) = ⎜ t 2 Ωr ⎜−2Ω(Θ + Θ ) − 𝜇dhr ⎝ d p

(

⎞

2ΩΘd ⎟

⎟

dt ⎠

+ 2Θp Ω

√

3, 𝛾 = 0.07, 𝜅 = 0.3,

√

3, 𝛾 = 0.07, 𝜅 = 0.3,

)

sin(Ωt ) cos(Ωt )

sin2 (Ωt )

sin2 (Ωt )

− sin(Ωt ) cos(Ωt )

,

(35b)

10

A. Karev et al. / Journal of Sound and Vibration 428 (2018) 1–12

Fig. 8. Comparison of normal form results and numerical integration: q1 for 𝜓 = 𝜋 /2 and 𝜔1 = 1, 𝜔2 =

√

3, 𝛾 = 0.07, 𝛾 = 0.3, 𝜖 = 0.2, 𝛿 = 0.

Fig. 9. Minimal brake model with mass particle.

⎛ k + 2kr2 + N0 h 𝐂=⎜ t ⎜ −𝜇r(kh + 2N ) ⎝ 0

h2 1 ⎞ 𝜇 N0 ⎟= 2 r ⎟ kt + ( 1 + 𝜇 2 ) N0 h⎠

(

c11

c12

c21

c22

)

.

(35c)

After normalizing the mass matrix to M = I, the matrices are split in symmetric and skew-symmetric constant and in timeperiodic parts:

𝐪̈ +

1 4(Θ2d + Θd Θp )

𝐊1 (t ) =

Θp

𝐍1 (t ) =

Θp

(

4

4

[𝐃0 + 𝐆0 + 𝐃1 (t ) + 𝐆1 (t )]𝐪̇ +

1

Θ2d + Θd Θp

2(c11 cos(2Ωt ) + c21 sin(2Ωt ))

(c12 − c21 ) cos(2Ωt ) + (c11 + c22 ) sin(2Ωt ) (

0

−(c12 + c21 ) cos(2Ωt ) − (c22 − c11 ) sin(2Ωt )

[𝐊0 + 𝐍0 + 𝐊1 (t ) + 𝐍1 (t )]𝐪 = 0 with

) (c12 − c21 ) cos(2Ωt ) + (c11 + c22 ) sin(2Ωt )

−2(c22 cos(2Ωt ) + c12 sin(2Ωt ))

(36)

,

) (c12 + c21 ) cos(2Ωt ) + (c22 − c11 ) sin(2Ωt ) 0

(37a)

,

(37b)

where expressions for further transformed system matrices are given in Appendix. From the anti-diagonal terms of Eqs. (37a) and (37b) it follows, that the symmetric and the skew-symmetric excitation terms are phase shifted, which may lead to total instability, depending on the damping and also on the values of the other parameters involved. The example does however show that asynchronous parametric excitation can occur in engineering systems, possibly leading to different types of parametric resonances and/or total instability, even if these effects are often masked by damping. Preliminary findings show that even if the total instability is less pronounced or even suppressed by damping, in particular the difference combination resonances may be very important. This is of particular interest if the eigenfrequencies are closely spaced, as is the case in imperfectly balanced rotors or in brake disks with small asymmetry. Possibly the difference combination resonances may help to explain that break squeal occurs only at very low rotational speeds and vanishes at higher speeds. A detailed analysis of the system is beyond the scope of this paper and will be presented elsewhere.

A. Karev et al. / Journal of Sound and Vibration 428 (2018) 1–12

11

5. Conclusion After an introduction (section 1) to periodic, especially harmonic parametric excitation, a short overview on well known “typical” parametric resonances (section 2) was given. In the “typical” case, where the harmonically time dependent coefficients are in phase, distinct sum or difference combination resonances are observed and parametric instability only may occur around these resonance frequencies. For out of phase parametric excitation, “atypical” behavior can be observed, as originally found by CESARI . In this case the trivial solution is unstable for all frequencies of the parametric excitation (total instability) and additionally sum and difference combination resonances occur simultaneously. A generalized CESARI system with linear damping and nonlinearities was studied by normal form theory (section 3) to highlight the “atypical” behavior with an academic example. The results obtained by normal form theory agree very well with a numerical analysis. Up to recently, no practical application or engineering system with this out of phase parametric excitation was presented. However, it turns out, that the minimal model for disk brake squeal, as discussed in Ref. [20], extended by an asymmetry of the brake disk features out of phase parametric excitation which may lead to total instability and both kinds of combination resonances (section 4). Even though total instability can be easily suppressed by damping, the appearance of difference combination resonance may be of great importance, especially if the eigenfrequencies are closely spaced, as is the case in brake disks. Here, difference combination resonances could possibly help to understand the brake squeal appearing at lower rotational speeds only. Generalized criteria for the occurrence of “atypical” behavior as well as its relevance to engineering systems will also be the subject of further studies. Acknowledgements The support through DFG, the German Science Foundation, through grants SCHA 814/26-1 and HA 1060/56-1 is gratefully acknowledged. Thanks also go the ME Department of the University of Canterbury at Christchurch (New Zealand), where the third author did part of this work. Appendix System matrices for Eq. (36): 2 [ ] ⎛𝜇N h + 2(dt + 2dr2 ) 𝐃0 = 2Θd + Θp ⎜ 0 Ωr ⎜ −𝜇dhr ⎝ (

⎞ −𝜇dhr⎟ , ⎟ 2dt ⎠ 8ΩΘd (Θd + Θp ) + 𝜇dhr(2Θd + Θp )

0

𝐆0 =

(A.1)

−8ΩΘd (Θd + Θp ) − 𝜇dhr(2Θd + Θp ) ( ) d11 (t ) d12 (t ) 𝐃1 (t ) = Θp , d21 (t ) d22 (t ) d11 (t ) = 𝜇N0

h2

Ωr

)

0

,

(A.2)

(A.3a)

+ 2(dt + 2dr2 ) cos(2Ωt ) − 2𝜇dhr sin(2Ωt ),

(A.3b)

1 h2 + 2(dt + dr2 )) sin(2Ωt ), d12 (t ) = 𝜇dhr cos(2Ωt ) + ( 𝜇N0 2 Ωr

(A.3c)

d21 (t ) = d12 (t ),

(A.3d)

d22 (t ) = −2dt cos(2Ωt ),

(A.3e)

⎛ 0 ⎜ 𝐆1 (t ) = Θp ⎜ 2 ⎜−𝜇dhr cos(2Ωt ) − ( 1 𝜇N0 h ) sin(2Ωt ) ⎝ 2 Ωr [ ] c11 Θp ⎛ ⎜ 𝐊0 = Θd + 1 ⎜ 2 ⎝ 2 (c12 + c21 ) [ ]( 0 Θd Θp 𝐍0 = + 2 4 −c12 + c21

1 (c + c21 )⎞ ⎟, 2 12 ⎟ c22 ⎠ c12 − c21 0

1 2

𝜇dhr cos(2Ωt) + ( 𝜇N0 0

h2

Ωr

⎞ ) sin(2Ωt )⎟ ⎟, ⎟ ⎠

(A.4)

(A.5)

)

.

(A.6)

12

A. Karev et al. / Journal of Sound and Vibration 428 (2018) 1–12

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