Analysis and synthesis of modal and non-modal self-excited oscillations in a class of mechanical systems with nonlinear velocity feedback

Journal of Sound and Vibration 334 (2015) 296–318

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Analysis and synthesis of modal and non-modal self-excited oscillations in a class of mechanical systems with nonlinear velocity feedback Anindya Malas, S. Chatterjee n Department of Mechanical Engineering, Indian Institute of Engineering Science and Technology, Shibpur, P.O. Botanic Garden, Howrah 711103, West Bengal, India

a r t i c l e in f o

abstract

Article history: Received 7 March 2014 Received in revised form 3 September 2014 Accepted 8 September 2014 Handling Editor: L.G. Tham Available online 2 October 2014

Many devices and processes utilize self-excited oscillations either as the working principle or as the performance enhancer. The present paper investigates a nonlinear velocity feedback control method for generating artificial self-excited oscillations in a class of two degrees-of-freedom mechanical systems. The paper derives the conditions of existence and stability of the modal oscillations and numerically corroborates this. It also proposes a methodology to design the control system for inducing natural oscillations in one of the desired modes. Other than modal oscillations, the system can also be designed to oscillate in a non-modal state with the desired frequency and amplitude-ratio within a specific range (depending on the system). Numerical simulations confirm the analytical results. Further analysis shows that for each frequency of excitation, the control cost is minimum when the system operates at an optimal amplitude-ratio. & 2014 Elsevier Ltd. All rights reserved.

1. Introduction Many machines and processes utilize artificially generated self-excited vibration; a few to mention are vibratory machines [1–3], vibratory material transportation [4], rotary drilling [5], ultrasonically assisted cutting [6], atomic force microscopy (AFM) [7–9], pick and place robots [10], pipe-crawling robots [11], self-excited biped mechanisms [12,13], flutter wing mechanism (hypothetical model of insect wing) [14], bio-sensors [15], strain and stress transducers [16,17], mass sensing [18], mass measurements in microgravity conditions [19], electrostatic field sensing [20] etc. Babitsky [10] observes that vibration and percussion machines, control of the displacement of materials and parts, mixing, separation, grinding, batching and many other processes may be efficiently intensified by self-excited vibration. Thus, the analysis and design of the control system for generating artificial self-excited oscillation in a given mechanical system with predefined amplitude and frequency is an important research problem. Self-excited oscillations in mechanical systems are generally attributed to some-form of nonlinear state-dependent forces. The simplest mathematical models used to explain the phenomenology of self-excited oscillations are the van der Pol and Rayleigh oscillators [21]. In these systems, self-excited oscillations are driven by the motive forces that (mathematically) comprise a negative, linear dissipative part leading to the oscillatory destabilization (the Hopf bifurcation) of the static equilibrium and a nonlinear part that limits the growth of this oscillatory instability. As a result, a new dynamic equilibrium n

Corresponding author. E-mail address: [email protected] (S. Chatterjee).

http://dx.doi.org/10.1016/j.jsv.2014.09.011 0022-460X/& 2014 Elsevier Ltd. All rights reserved.

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(limit cycle) emerges and the system continues to vibrate autonomously. Thus, in general, a nonlinear active feedback is necessary for inducing self-excited oscillation in an otherwise passive mechanical system. The feedback force must destabilize the static equilibrium, allowing for the growth of oscillation as well as arrest the growth of the instability far away from the static equilibrium. Many previous researchers have studied the efficacy of the van der Pol type or relay feedback control for self-exciting any preselected natural modes of oscillations of a vibratory system and applied the techniques in many mechanical or micromechanical systems [5–10,14,19,22–25]. Aguilar et al. [25] propose a two-relay feedback control law for generating self-excited oscillation with any desired frequency and amplitude. Ono and coworkers [12,13,26] have utilized self-excited oscillation at the anti-resonance frequency in robotic devices by introducing (through feedback) asymmetry in the stiffness matrix and nonlinear damping. Kurita and co-workers [1–3] have discussed the applications of different control laws for developing self-excited oscillations in vibratory machines. Hideomi et al. [27] have used a Voice Coil Motor (VCM) for generating self-excited vibration in a mechanical system, thereby circumventing the need of any vibration sensors. In this method, the self-excited vibration is produced by the positive feedback control based on the velocity estimated from the voltage induced in the VCM and the inward current of the VCM. Lee and White [28] have fabricated a self-excited micro-cantilever transducer using bulk-micromachining technology. The transducer consists of a feedback electrode and a drive electrode which, when connected by an amplifier, result in selfexcited oscillation of the cantilever in the acoustic range of frequencies. Zook et al. [17] build a device using an integral photodiode that converts optical power into an electrostatic force to produce self-oscillation in a clamped–clamped silicon microbeam. The device can be used in strain sensing. Chatterjee and Malas [29] investigate the efficacy of a stiffness switching control for artificially inducing self-excited oscillation in mechanical and mircromechanical systems. Chatterjee proposes a nonlinear time-delayed feedback control [30] to generate self-excited oscillation in single and two degrees-of-freedom mechanical systems. Recently, photothermal self-excitation principles are utilized to drive selective mechanical modes of micro- and nanomechanical resonators [31,32]. Pulsed Digital Oscillator, another form of self-excited oscillators, can also be used for exciting the desired mode of natural vibration in MEMS oscillators [33]. Periodic and rhythmic motions of animal organs involved during locomotion (manifested in many forms like walking, crawling, swimming, running, flying, etc.) are believed to be self-excited oscillations of the natural modes of vibration of the mechanical structure of the animal limbs interacting with the environment and controlled by neuronal circuits called the Central Pattern Generator (CPG) [34–39]. The design of the control systems regulating the complex periodic movements of many robotic devices are inspired by this fact. Various neural network based control techniques are proposed in the literature to generate self-excited oscillations in mechanical systems [40–42]. Such controllers mimic the CPG as nonlinear oscillators that when coupled to a mechanical structure can induce self-excited motion in it. Recently, a more simplified version of the controller has been proposed to excite natural oscillations in mechanical systems and the controller can be designed to excite only a particular natural oscillatory mode of interest [43]. A large number of previous works clearly establish the significance of the research towards the generation and control of self-excited oscillations in mechanical and micromechanical systems either in the laboratory or in-situ conditions. This paper investigates the efficacy of a nonlinear control method, based on the velocity feedback, to excite self-oscillations in a class of two degrees-of-freedom mechanical systems. Analysis shows that the proposed control method can induce modal (oscillation in one of the natural modes of vibration) as well as non-modal (oscillation condition is different from the natural modes of vibrations in terms of the frequency and amplitude-ratio) self-excited oscillations in mechanical systems. The paper also discusses a simple control design procedure, based on the describing function (DF) method, to achieve the preset values of the frequency and amplitudes of vibration of a known mechanical system. The optimal operating conditions corresponding to the minimal control cost are also obtained. 2. Mathematical model The mechanical system considered for the present study is a simple two degrees-of-freedom mass-spring-damper chain as shown in Fig. 1. The system represents a mass-spring chain oscillating in a lubricated, linear guideway. The interfacial friction between the masses and the guideway is represented by linear viscous friction forces. The basic motivation behind selecting such a simple system is that the efficacy of the control can be easily demonstrated in the laboratory (the same system is studied in [22]). Two feedback control forces Fc1 and Fc2 are applied on the first and second mass, respectively, to generate selfexcited oscillation in the system. With reference to Fig. 1, one can write the governing equations of motion of the system as m1 x001 ðtÞ þ c1 x01 ðtÞ þ ðk1 þ k2 Þx1 ðtÞ  k2 x2 ðtÞ ¼ F c1

(1a)

m2 x002 ðtÞ þ c2 x02 ðtÞ þ k2 x2 ðtÞ k2 x1 ðtÞ ¼ F c2 ;

(1b)

and

where x1 and x2 denote the displacements of the masses m1 and m2, respectively, and the prime denotes differentiation with respect to time t.

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Fig. 1. Mathematical model of a two degrees-of-freedom mechanical system with control.

The control forces should ideally be nonlinear, but must possess linear state-depended terms that can destabilize the static equilibrium of the system. The nonlinear terms must be dissipative in nature, such that a limit cycle oscillation is reached in the steady state. In the present paper, the linear feedback terms are considered to introduce controllable offdiagonal terms in the damping matrix. The nonlinear control forces considered here are given by F c1 ¼ g 1 x02 þ n1 x03 1

(2a)

F c2 ¼ g 2 x01 þ n2 x03 2;

(2b)

and

where g i 's are the linear control parameters and the nonlinear control parameters are ni o 0; i ¼ 1; 2. It is worth mentioning here that a similar but restricted version of the above control is studied in [14] where g 1 ¼ g 2 and the nonlinear damping terms are of van der Pol type. Clearly, the control law studied in the present paper is the generalization of that used in [14] and the nonlinear damping term is of the Rayleigh type. Eqs. (1) and (2) can be recast in the following non-dimensional form: " # " # " # ( ) 1 0   f c1 h1 0   1 þ kr  kr   y€ þ y_ þ y ¼ ; (3a) 0 μ f c2 0 h2 kr kr where fyg ¼ fy1 y2 gT is the non-dimensional displacement vector. The quantities are y1 ¼ x1 =x0 , y2 ¼ x2 =x0 , μ ¼ m2 =m1 , h1 ¼ c1 =m1 ω0 , h2 ¼ c2 =m1 ω0 and kr ¼ k2 =k1 , where ffiffiffiffiffiffiffiffiffiffiffiffiffiffi pnon-dimensional ω0 ¼ k1 =m1 and x0 is some reference length quantity. The non-dimensional control forces are expressed as f c1 ¼ γ 1 y_ 2 þ β 1 y_ 31 and f c2 ¼ γ 2 y_ 1 þ β 2 y_ 32 ;

(3b)

where γ 1 ¼ g 1 =m1 ω0 , γ 2 ¼ g 2 =m1 ω0 , β ω0 =m1 o0 and β2 ¼ ω0 =m1 o 0. The ‘dot’ denotes the differentiation with respect to the non-dimensional time τ ¼ ω0 t. 2 1 ¼ n1 x 0

n2 x20

3. Linear stability analysis As the primary requirement for inducing self-excited oscillation in the system is to destabilize the static equilibrium, the linear feedback parameters γ i for i¼1, 2 must be appropriately chosen outside the stability boundaries of the static equilibrium in the parameter space. To this end, the governing equations of motion given by Eqs. (3) are linearized around the trivial equilibrium as " # " # " #   h1 1 0    γ1   1 þ kr  kr   0 €y þ _y þ y ¼ : (4) 0 μ  γ2 h2  kr kr 0 The corresponding characteristic equation of the linearized system is b4 s4 þ b3 s3 þ b2 s2 þ b1 s þ b0 ¼ 0; where s is the standard Laplace variable and the coefficients of the polynomial are b4 ¼ μ;

b3 ¼ ðh1 μ þ h2 Þ;

b2 ¼ ðμ þkr μ þ h1 h2 þ kr  γ 1 γ 2 Þ;

b1 ¼ ðh2 þkr h2 þ kr h1  kr γ 1  kr γ 2 Þ;

b0 ¼ kr :

(5)

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299

Fig. 2. General structure of the stability boundaries.

Fig. 3. (a) Variations of the critical frequencies along the stability boundaries: μ¼0.8, kr ¼ 0.9, h1 ¼ 0.07, and h2 ¼ 0.05. (b) Variations of the critical frequencies with γ1 along the stability boundaries: μ¼ 0.8, kr ¼ 0.9, h1 ¼0.07, and h2 ¼0.05.

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On the stability boundary, some of the system poles are on the imaginary axis. Thus, putting s ¼ jωc (where ωc is the critical frequency) in Eq. (5) and separating the real and imaginary parts, one obtains b3 ω3c þ b1 ωc ¼ 0

(6)

and b4 ω4c  b2 ω2c þ b0 ¼ 0: (7) pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi Clearly, Eq. (6) has two solutions of ωc : ωc ¼ 0 and ωc ¼ b1 =b3 . The nontrivial solution ωc ¼ b1 =b3 corresponds to the flutter boundary where a pair of complex conjugate roots of the characteristic equation crosses the imaginary pffiffiffiffiffiffiffiffiffiffiffiffiffi axis and thus, can give rise to oscillation. The equation of the flutter boundary is obtained after substituting ωc ¼ b1 =b3 in Eq. (7) as f ðγ 1 ; γ 2 Þ ¼ e1 γ 1 þ e2 γ 2 þ e3 γ 21 þe4 γ 22 þ e5 γ 1 γ 2 þ e6 γ 21 γ 2 þ e7 γ 22 γ 1 þ e8 ¼ 0;

(8)

where ei for i¼1–8 are functions of the mechanical system parameters Clearly, Eq. (8) represents a cubic curve, which has one or many asymptotes. Fig. 2 delineates the general structure of the flutter boundaries (the Hopf bifurcation lines) of the static equilibrium in γ 2 vs: γ 1 plane. Apparently, there are two boundary lines; the lower flutter boundary has vertical and horizontal asymptotes at γ 1 ¼ κ and γ 2 ¼ κ (where κ ¼ μkr =ðh1 μ þ h2 Þ), respectively in the parameter plane. The upper flutter line is asymptotic to another line defined by γ 1 þ γ 2 ¼ h2 =kr þðh1 þ h2 Þ. The trivial solution ωc ¼ 0 corresponds to the divergence boundary where at least one root of the characteristic equation crosses the imaginary axis along the real axis and thus, cannot give rise to any oscillation. One can easily see from Eq. (6) that ωc ¼ 0 is obtained on the boundary given by b1 ¼ 0;

(9a)

which is equivalent to a line (in the parameter plane) defined as

γ 1 þ γ 2 ¼ h2 =kr þ ðh1 þ h2 Þ

(9b)

It is observed from Fig. 2 that the upper flutter boundary is asymptotic to the line defined by Eq. 9(b); thus, divergence is absent in the present system. It is evident from Eq. pffiffiffiffiffiffiffiffiffiffiffiffi ffi (8) that for each chosen value of γ 1 , there are two values of γ 2 and hence two values of the critical frequency ωc ¼ b1 =b3 , provided b1 4 0. However, it can be shown that one value of b1 is negative for γ 1 4 κ and thus, there exists only one value of γ 2 and ωc . The variations of the critical frequencies (low-frequency – ωl and high-frequency – ωh) along the flutter lines (for some selected values of the mechanical system parameters) are plotted in Fig. 3(a). The minimum and maximum values of ωh and ωl are observed (from Fig. 3(b)) to be very close to the second and the first modal frequencies, respectively. With reference to Figs. 2 and 3, henceforth the upper and lower flutter boundaries will be called as the low and high-frequency flutter lines, respectively. Thus, it can be concluded that the proposed control law can generate modal as well as low (lower than the first natural frequency) and high-frequency (higher than the second natural frequency) oscillations in the system provided the control parameters are appropriately selected outside the region of stability of the static equilibrium.

4. Nonlinear analysis In the present section, an averaging method is used to calculate the amplitudes and frequency of self-excited oscillation generated in the system when the linear feedback parameters are suitably chosen as discussed above. First a general analysis is presented assuming that the self-excited oscillation can take place at any arbitrary frequency. Then the analysis corresponding to the modal oscillations is considered as a special case.

4.1. Analysis for oscillations at any arbitrary frequency First, Eq. (3) is transformed from the physical coordinates {y} to the quasi-normal coordinates {z} using the following similarity transformation:   y ¼ Pfzg; (10) " # 1 1 2 1 þ k  ω r i wherepfffiffiffiffiffiffiffiffiffiffiffiffiffi zg ¼ fffiz1 z2 gT and P ¼ and ωi as the ith natural frequency normalized with respect to ρ1 ρ2 with ρi ¼ kr ω0 ¼ k1 =m1 . The quasi-normal form, thus obtained, is written as " 2 # " # " #( )  ρ2 1=μ ω1 0 c11 c12   u1   1 1 _ €z þ z þ fzg ¼  ; (11) 0 ω22 ρ1  ρ2 c21 c22 ρ1  ρ2 ρ1 1=μ u2

A. Malas, S. Chatterjee / Journal of Sound and Vibration 334 (2015) 296–318

where

"

c11 c12 c21 c22

#

301

"    # h1 ρ2  h2 ρ1 =μ h1 ρ2  h2 ρ2 =μ     ; ¼ h1 ρ1 þ h2 ρ2 =μ  h1 ρ1 þh2 ρ1 =μ

u1 ¼ γ 1 ðρ1 z_ 1 þ ρ2 z_ 2 Þ þ β1 ðz_ 1 þ z_ 2 Þ3 and u2 ¼ γ 2 ðz_ 1 þ z_ 2 Þ þ β2 ðρ1 z_ 1 þ ρ2 z_ 2 Þ3 : Eq. (11) are finally recast as z€ 1 þ ω2 z1 ¼ εf^ 1 ðz1 ; z2 ; z_ 1 ; z_ 2 Þ

(12)

z€ 2 þ ω2 z2 ¼ εf^ 2 ðz1 ; z2 ; z_ 1 ; z_ 2 Þ;

(13)

and

where ω is the frequency of oscillation of the system and ε is a small book-keeping parameter. Thus, f^ 1 ðz1 ; z2 ; z_ 1 ; z_ 2 Þ ¼

1

ρ1  ρ2

ρ2 ~ 1 u~ u þ ρ1  ρ2 1 μðρ1  ρ2 Þ 2

(14)

ρ1 ~ 1 u~ ; u  ρ1  ρ2 1 μðρ1  ρ2 Þ 2

(15)

~ z  ðc~ 11 z_ 1 þ c~ 12 z_ 2 Þ þ Ω 1 1

and f^ 2 ðz1 ; z2 ; z_ 1 ; z_ 2 Þ ¼

1

ρ1  ρ2

~ z þ ðc~ 21 z_ 1 þ c~ 22 z_ 2 Þ þ Ω 2 1

~ , ðω2  ω2 Þ ¼ εΩ ~ , u ¼ εu~ and u ¼ εu~ . where c11 ¼ εc~ 11 , c12 ¼ εc~ 12 , c21 ¼ εc~ 21 , c22 ¼ εc~ 22 , ðω2  ω21 Þ ¼ εΩ 1 2 1 1 2 2 2 The solutions of Eqs. (12) and (13) are sought in the following form: zi ¼ r i cos ðωτ þ θi Þ;

z_ i ¼  r i ω sin ðωτ þ θi Þ;

i ¼ 1; 2;

(16)

where the amplitude r i and the phase θi of the oscillation are assumed to be slowly varying functions of τ and satisfy the following equations:

ωr_ i ¼ ε sin ðψ i ÞF i ðψ 1 ; ψ 2 Þ

(17)

ωr i θ_ i ¼ ε cos ðψ i ÞF i ðψ 1 ; ψ 2 Þ;

(18)

and

where ψ i ¼ ωτ þ θi and F i ðψ 1 ; ψ 2 Þ ¼ 

(

) 2 1 ~ r cos ðψ Þ þð 1Þi ρ u~  1 u~ :  ω ∑ c~ ik r k sin ðψ k Þ þ ðρ1  ρ2 ÞΩ i i i 3i 1 ρ1  ρ2 μ 2 k¼1

Eqs. (17) and (18) can be approximated (in an average sense) as Z ε 2π =ω r_ i ¼ F i sin ðωτ þ θi Þdτ 2π 0

(19)

and

ε r i θ_ i ¼ 2π

Z 2π =ω 0

F i cos ðωτ þ θi Þdτ:

(20)

Integrating and simplifying Eqs. (19) and (20), gives amplitude and phase equations as r_ 1 ¼ G1 ðr 1 ; r 2 ; ω; ϕÞ; ;

(21)

r_ 2 ¼ G2 ðr 1 ; r 2 ; ω; ϕÞ;

(22)

θ_ 1 ¼ G3 ðr1 ; r2 ; ω; ϕÞ;

(23)

θ_ 2 ¼ G4 ðr1 ; r2 ; ω; ϕÞ;

(24)

and

where ϕ ¼ ðθ1  θ2 Þ and the expressions of G1, G2, G3 and G4 are given in Appendix A. The fixed points of Eqs. (21)–(24), are obtained by solving the following four nonlinear algebraic equations: Gi ðr 1 ; r 2 ; ω; ϕÞ ¼ 0;

i ¼ 1–4

(25)

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The stability of these steady-state solutions can be ascertained by the eigenvalues of the Jacobian matrix of the right-hand sides of Eqs. (21)–(24) evaluated at the corresponding equilibrium points. If the real parts of all eigenvalues are negative, then the corresponding solution is stable, otherwise unstable. Obviously, the solutions of Eq. (25) correspond to the amplitudes (r1 and r2), frequency (ω) and phase difference (ϕ) of the oscillations of the quasi-normal coordinates. Using the transformation (10) once again, the amplitudes (Ai) and phase difference (θ) of the physical coordinates are obtained as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (26a) A1 ¼ r 21 þ r 22 þ 2r 1 r 2 cos ϕ; A2 ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ρ21 r 21 þ ρ22 r22 þ 2ρ1 ρ2 r1 r 2 cos ϕ

(26b)

and

θ ¼ cos  1



 1  ρ1 r21 þ ρ2 r22 þ ðρ1 þ ρ2 Þr1 r2 cos ϕ : A1 A2

(26c)

4.2. Analysis for modal oscillations When the system oscillates at one of the natural frequencies while maintaining the corresponding amplitude-ratio, it is possible to simplify the above analysis by replacing ω ¼ ω1 and ω ¼ ω2 in Eqs. (12) and (13), respectively. Accordingly, ~ ¼ 0; i ¼ 1; 2 in Eqs. (14) and (15). Under these circumstances, one seeks solutions of Eqs. (12) and (13) as Ω i zi ¼ r i cos ðωi τ þ θi Þ;

z_ i ¼ r i ωi sin ðωi τ þ θi Þ;

i ¼ 1; 2

(27)

Proceeding in a similar fashion as discussed in Section 4.1, one finally obtains the amplitude and phase equations as r_ 1 ¼

 r1 p þp1 r 21 þp2 r 22 ; 2 0

(28)

r_ 2 ¼

 r2 q þ q1 r 21 þ q2 r 22 ; 2 0

(29)

θ_ 1 ¼ 0

(30)

θ_ 2 ¼ 0;

(31)

and

where pi and qi for i¼0–2 are defined in Appendix B. The steady-state solutions of Eqs. (28) and (29) can be obtained by solving the following nonlinear algebraic equations: p0 r 1 þ p1 r 31 þ p2 r 1 r 22 ¼ 0

(32)

q0 r 2 þ q1 r 21 r 2 þq2 r 32 ¼ 0:

(33)

and

Four different steady-state solutions are possible as listed below: 1. 2. 3. 4.

0, ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 2 ¼ 0 (static equilibrium). r1 ¼ p r 1 ¼  p0 =p , r 2 ¼ 0 (oscillation in the first mode). p1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r1 ¼ p 0, ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 2 ¼ q0 =q2 (oscillation in the second mode). pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 1 ¼ ðq0 p2 p0 q2 Þ=ðp1 q2  p2 q1 Þ, r 2 ¼ ðp0 q1  p1 q0 Þ=ðp1 q2 p2 q1 Þ (mixed-mode oscillation; both the modes are present in the oscillation).

The existence and stability of the modal and mixed-mode oscillations depend on the system parameters (see Appendix B for details). It is shown in Appendix B that the low and high-frequency flutter lines can be approximated, to the limits of small values of the linear feedback parameters γ i ; i ¼ 1; 2, by the two lines: ðγ 1 þ γ 2 Þ ¼ ðμh1 jρ2 j þ h2 ρ1 Þ and ðγ 1 þ γ 2 Þ ¼ ð  μh1 ρ1  h2 jρ2 jÞ, respectively. Accordingly, the following observations are made: 1. 2. 3. 4.

The static equilibrium is stable if ð  μh1 ρ1  h2 jρ2 jÞ oðγ 1 þ γ 2 Þ o ðμh1 jρ2 j þh2 ρ1 Þ. The first mode is excited if ðγ 1 þ γ 2 Þ 4 ðμh1 jρ2 j þh2 ρ1 Þ. The second mode is excited if ðγ 1 þ γ 2 Þ o ð  μh1 ρ1  h2 jρ2 jÞ. Mixed-mode oscillation does not exist.

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Fig. 4. Linear feedback parameter values chosen for numerical analysis: μ¼0.8, kr ¼0.9, h1 ¼0.07, and h2 ¼ 0.05.

Table 1 Numerical results: β1 ¼  0:3; β2 ¼ 0. Point

P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12

γ1 ; γ2

0.2, 0.2 0, 0.5 0.5, 0  0.1,  0.1 0,  0.25  0.25, 0  2, 1 1,  2  1, 2 1,  0.8  1.5,  1.5 1.5, 1.5

Analytical results

Results of numerical simulations

Amplitude (A1, A2)

Frequency (ω)

Phase (rad)

Amplitude (A1, A2)

Frequency (ω)

2.14, 3.54 2.51, 4.3 2.51, 4.03 0.31, 0.24 0.39, 0.32 0.34, 0.3 0.35, 0.26 0.35, 0.49 4.71, 8.24 1.1, 1.67 10.54, 191.28 11.55, 202.35

0.66 0.65 0.65 1.61 1.61 1.61 2.31 2.31 0.46 0.55 1.11 1.01

0.09 0.29 0.06 3.03 2.79 3.07 1.91 1.8 0.76 0.99 2.63 0.51

2.28, 3.6 2.72, 4.36 2.71, 4.12 0.31, 0.24 0.39, 0.33 0.39, 0.3 0.34, 0.26 0.35, 0.49 4.83, 8.0 1, 1.6 12.95, 227.6 13.22, 194.18

0.66 0.66 0.65 1.61 1.61 1.57 2.2 2.32 0.46 0.54 1.08 1.01

Remarks

1st mode

2nd mode

High-frequency non-modal Low-frequency non-modal Non-modal oscillation near the first anti-resonance frequency with large amplitude-ratio

5. Numerical results and discussions In this section, numerical solutions of Eq. (25) are obtained for some specific cases and compared with the results of direct numerical simulations of Eqs. (3). As discussed earlier, the linear feedback parameters are so chosen that the static equilibrium is unstable and the resulting motion of the system is oscillatory in nature. Finite-amplitude self-oscillation is then excited due to the nonlinear damping terms present in the feedback force. For the numerical analysis, the parameters of the mechanical systems are kept fixed at the values mentioned in Fig. 4. The undamped natural frequencies of the system, for these selected parameter values, are 0.659 and 1.6096 with the corresponding amplitude-ratios (second mass amplitude: first 1.6286 and 0.7675, respectively. The two anti-resonance frequencies of the system are pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffimass amplitude) kr =μ ¼1.0607 and kr þ 1 ¼1.3784 (see Appendix C for details). The linear feedback parameters are chosen from outside the stable region as depicted in Fig. 4. Apparently, some parameter values are chosen close to the stability boundaries and some are far away. The analytical results (as discussed in Section 4.1) are presented in Tables 1–3 and compared with the corresponding results obtained from direct numerical simulations, carried out in MATLAB SIMULINK, for various combinations of the linear and nonlinear feedback parameter values. 5.1. Modal and non-modal oscillations For small values of the linear feedback parameters selected close to the low-frequency flutter line (points P1, P2 and P3 in Fig. 4), the system oscillates with the characteristics very similar to the first natural mode. On the other hand, the system can be made to oscillate with the characteristics very similar to the second natural mode by selecting small values of the linear

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Table 2 Numerical results: β1 ¼ 0; β2 ¼  0:3. Point

P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12

γ1 ; γ2

0.2, 0.2 0, 0.5 0.5, 0  0.1,  0.1 0,  0.25  0.25, 0  2, 1 1,  2  1, 2 1,  0.8  1.5,  1.5 1.5, 1.5

Analytical results

Results of numerical simulations

Amplitude (A1, A2)

Frequency (ω)

Phase (rad)

Amplitude (A1, A2)

Frequency (ω)

0.82, 1.31 0.94, 1.5 1, 1.54 0.52, 0.4 0.63, 0.5 0.7, 0.5 0.47, 0.34 0.25, 0.35 1.85, 3.1 0.47, 0.71 127.2, 7.72 121.27, 7.9

0.66 0.66 0.66 1.61 1.62 1.62 2.31 2.31 0.46 0.56 1.41 1.34

0.11 0.03 0.32 3.13 3 2.9 1.81 2 0.5 0.52 2.76 0.44

0.82, 1.33 0.94, 1.6 1.01, 1.6 0.52, 0.4 0.63, 0.5 0.7, 0.5 0.47, 0.35 0.25, 0.35 1.88, 3.1 0.46, 0.71 146.5, 9.34 118.73, 9.05

0.66 0.66 0.66 1.6 1.61 1.61 2.32 2.32 0.47 0.54 1.4 1.34

Remarks

1st Mode

2nd Mode

High-frequency non-modal Low-frequency non-modal Non-modal oscillation near the second anti-resonance frequency with large amplitude-ratio

Table 3 Numerical results: β1 ¼  0:3; β2 ¼  0:3. Point

P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12

γ1 ; γ2

0.2, 0.2 0, 0.5 0.5, 0  0.1,  0.1 0,  0.25  0.25, 0  2, 1 1,  2  1, 2 1,  0.8  1.5,  1.5 1.5, 1.5

Analytical results

Results of numerical simulations

Amplitude (A1, A2)

Frequency (ω)

Phase (rad)

Amplitude (A1, A2)

Frequency (ω)

0.76, 1.23 0.88, 1.45 0.93, 1.44 0.26, 0.2 0.33, 0.27 0.34, 0.25 0.28, 0.21 0.2, 0.28 1.72, 2.9 0.43, 0.65 1.59, 1.57 3.08, 3.53

0.66 0.65 0.65 1.61 1.62 1.62 2.31 2.31 0.46 0.55 1.6 0.74

0.09 0.06 0.28 3.07 2.85 3.02 1.87 1.86 0.52 0.52 3.05 0.3

0.76, 1.24 0.88, 1.48 0.93, 1.47 0.26, 0.2 0.33, 0.27 0.34, 0.26 0.28, 0.21 0.2, 0.28 1.77, 2.93 0.43, 0.65 1.7, 1.71 3.58, 4.71

0.66 0.65 0.65 1.61 1.61 1.61 2.32 2.31 0.46 0.54 1.53 0.62

Remarks

1st Mode

2nd Mode

High-frequency non-modal Low-frequency non-modal

feedback parameters close to the high-frequency flutter line (points P4, P5 and P6 in Fig. 4). These results are consistent with the analysis presented in Section 4.2. High-frequency non-modal oscillations can be generated by selecting the parameter values close to the high-frequency flutter line, but far away from the origin of the parameter plane (points P7 and P8 in Fig. 4). Similarly, low-frequency nonmodal oscillations are obtained for the parameter values close to the low-frequency flutter line, but far away from the origin of the parameter plane (points P9 and P10 in Fig. 4). Non-modal oscillations with very large or small amplitude-ratios (A2/A1) and the frequency of oscillation close to one of the anti-resonance frequencies can be generated by selecting large values of the linear feedback parameters in the lower left and upper right corner regions of the parameter plane (points P11 and P12 in Fig. 4) when only one of the system masses is nonlinearly damped (Tables 1 and 2). The anti-resonance frequency depends on which mass is nonlinearly damped. The oscillation frequency is close to the first anti-resonance frequency (1.0607 for the present system) when the first mass is nonlinearly damped. On the other hand, the frequency of oscillation is close to the second anti-resonance frequency (1.3784 for the present system) when the second mass is nonlinearly damped. The contour plots of the frequency of oscillations presented in Figs. 5 and 6 give a more comprehensive numerical evidence of the above observations. 5.2. Simultaneous variations of the linear feedback parameters Figs. 7 and 8 depict the variations of the frequency and amplitude-ratio of the oscillation with the linear feedback parameters varied along the line: γ 1 ¼ γ 2 ¼ γ . It is observed from these figures that for small values (absolute) of γ, the oscillation takes place at one of the natural modes. However, the oscillation frequency is gradually approaching one of the anti-resonance frequencies for very large values of γ. When the first mass is nonlinearly damped (i.e., β2 ¼ 0), the oscillation

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Fig. 5. Frequency contour plots: β1 ¼  0.3, β2 ¼0, μ¼ 0.8, kr ¼ 0.9, h1 ¼ 0.07, and h2 ¼ 0.05.

Fig. 6. Frequency contour plot: β1 ¼0, β2 ¼  0.3, μ¼0.8, kr ¼ 0.9, h1 ¼0.07, and h2 ¼0.05.

Fig. 7. Variations of the frequency and amplitude-ratio with γ 1 ¼ γ 2 ¼ γ when the first mass is nonlinearly damped: β1 ¼  0.3, β2 ¼ 0, μ¼ 0.8, kr ¼ 0.9, and h1 ¼ 0.07, h2 ¼ 0.05. Gray-shaded regions signify stable static equilibrium.

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Fig. 8. Variations of the frequency and amplitude-ratio with γ 1 ¼ γ 2 ¼ γ when the second mass is nonlinearly damped: β1 ¼ 0, β2 ¼  0.3, μ¼ 0.8, kr ¼ 0.9, h1 ¼0.07, and h2 ¼ 0.05. Gray-shaded regions signify stable static equilibrium.

Fig. 9. Variations of the frequency of oscillation and amplitude-ratio with γ1 when the first mass is actuated: β1 ¼  0.3, β2 ¼0, μ¼0.8, kr ¼ 0.9, h1 ¼ 0.07, and h2 ¼0.05. Gray-shaded regions signify stable static equilibrium.

pffiffiffiffiffiffiffiffiffiffi approaches the non-modal state with the frequency of oscillation close to the first anti-resonance frequency kr =μ (1.06 in the present numerical example) and large amplitude-ratio A2 =A1 with the increasing value of γ (Fig. 7). On the other hand, when the second mass is nonlinearly damped (β1 ¼ 0), the oscillation approaches the frequency of oscillation close to the pffiffiffiffiffiffiffiffiffiffiffiffi second anti-resonance frequency kr þ 1 (1.37 for the present numerical example) and small amplitude-ratio A2 =A1 with the increasing value of γ (Fig. 8).

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5.3. Under-actuated control The under-actuated control configurations, i.e., when only one of the masses is actuated, is practically more appealing. For γ2 ¼ β2 ¼ 0 (i.e., when only the first mass is actuated), the variations of the frequency of oscillation and amplitude-ratio with γ1 are shown in Fig. 9(a) and (b) (note that both the frequency and amplitude ratio characterize modal oscillations). It is observed from these plots that the first natural mode can be generated for positive values of γ1 very close to the stability boundary. However, for higher positive values of γ1, the oscillation drifts considerably from the first natural mode. On the other hand, the second natural mode is generated for any negative value of γ1 outside the stability boundary (even when the value is far away from the stability boundary). For γ1 ¼ β1 ¼0, (i.e., the second mass is actuated) the variations of the frequency of oscillation and the amplitude-ratio with γ2 are shown in Fig. 10(a) and (b). It is observed from these plots that the second natural mode can be generated for negative value of γ2 very close to the stability boundary. However, for higher negative values of γ2, the oscillation drifts considerably from the second natural mode. On the other hand, the first natural mode is generated for any positive value of γ2 outside the stability boundary (even when the value is far away from the stability boundary). 5.4. Effects of nonlinear feedback parameters The effects of nonlinear feedback parameters β1 and β2 on the dynamic characteristics of the system are tabulated in Table 4. It is observed that one can regulate the amplitude of oscillation of the system by varying the nonlinear feedback parameters. However, the frequency of oscillation remains more or less independent of β1 and β2. The nonlinear terms being completely dissipative and velocity dependent can only affect the amplitude not the frequency. 5.5. Qualitative/quantitative summary Though the above results are specific to only the selected values of the mechanical system parameters, the results remain qualitatively the same for any other parameter values. Thus, one may infer that the proposed control method can generate all types of self-excited motions with almost any frequency, like modal, low, high and intermediate frequency non-modal oscillations. Fig. 11 depicts a graphical summary of the overall qualitative characteristics of the system. In general, it is observed that the analytical results match closely with numerical results when the parameter values are selected close to the stability boundaries. The frequency of oscillation obtained from the averaging analysis matches well with the simulation results. However, for the parameter values chosen far from the stability boundaries, the analytically obtained amplitude-ratio shows poor numerical matching with that obtained from simulations. This is because of the fact

Fig. 10. Variations of the frequency of oscillation and amplitude-ratio with γ2, when the second mass is actuated: β1 ¼ 0, β2 ¼  0.3, μ¼ 0.8, kr ¼ 0.9, h1 ¼ 0.07, and h2 ¼ 0.05. Gray-shaded regions signify stable static equilibrium.

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Table 4 Effects of nonlinear feedback parameters. γ1 ; γ2

β2

β1  0.1

0

 0.3

 0.5

 0.7

ω

A1

A2

A1

A2

A1

A2

A1

A2

A1

A2

0.1, 0.1 (1st mode)

0  0.1  0.3  0.5  0.7

– 0.74 0.42 0.33 0.28

– 1.2 0.69 0.54 0.45

2 0.69 0.41 0.32 0.27

3.18 1.12 0.68 0.53 0.45

1.14 0.61 0.4 0.32 0.27

1.84 1 0.65 0.51 0.44

0.88 0.56 0.38 0.31 0.26

1.42 0.91 0.62 0.5 0.43

0.75 0.52 0.37 0.3 0.26

1.2 0.84 0.6 0.49 0.42

0.66

 0.1,  0.1 (2nd mode)

0  0.1  0.3  0.5  0.7

– 0.9 0.52 0.4 0.34

– 0.67 0.4 0.31 0.26

0.53 0.45 0.37 0.32 0.28

0.42 0.35 0.28 0.25 0.22

0.31 0.29 0.26 0.24 0.23

0.24 0.23 0.2 0.19 0.17

0.24 0.22 0.22 0.2 0.19

0.19 0.18 0.17 0.16 0.15

0.2 0.2 0.19 0.18 0.17

0.16 0.15 0.15 0.14 0.13

1.61

 2, 1 (high-frequency non-modal)

0  0.1  0.3  0.5  0.7

– 0.82 0.47 0.37 0.31

– 0.6 0.35 0.27 0.23

0.61 0.49 0.37 0.31 0.28

0.45 0.36 0.27 0.23 0.2

0.35 0.32 0.28 0.25 0.23

0.26 0.24 0.21 0.19 0.17

0.27 0.26 0.23 0.22 0.2

0.2 0.19 0.17 0.16 0.15

0.23 0.22 0.21 0.2 0.18

0.17 0.16 0.15 0.14 0.13

2.31

 1, 2 (low-frequency non-modal)

0  0.1  0.3  0.5  0.7

– 3.25 1.88 1.46 1.23

– 5.38 3.11 2.41 2.03

8.37 3.07 1.84 1.44 1.22

13.8 5.08 3.05 2.38 2.02

4.83 2.75 1.77 1.41 1.2

7.97 4.54 2.93 2.33 1.99

3.74 2.5 1.71 1.37 1.18

6.17 4.13 2.82 2.27 1.95

3.16 2.3 1.64 1.34 1.16

5.22 3.8 2.71 2.22 1.92

0.46

– 205.6 118.7 92 77.7

– 15.7 9 7 5.9

22.9 336.3 13.2 194.2 10.2 150.4 8.7 127.1 1.01 Non-modal oscillation near the anti-resonance frequency will occur when any one 1.34 of the two nonlinearities is zero.

1.5, 1.5 (Non-modal oscillation near 0 the anti-resonance frequency)  0.1  0.3  0.5  0.7

Fig. 11. A graphical summery of the qualitative characteristics of the system. LF – low-frequency oscillation (less than first natural frequency), HF – highfrequency oscillation (higher than second natural frequency), IF – intermediate-frequency oscillation (between the first and second natural frequency), M1 – first mode, and M2 – second mode.

that the analytical method treats the control function as weak. Such an assumption is violated for higher values of feedback parameters. 6. Design methodology From the above analysis, it is evident that the proposed control law can produce a large variety of modal and non-modal self-excited oscillations. This section discusses a design methodology to calculate the control parameters for generating selfexcited oscillation with the preset values of the frequency and amplitudes in a given mechanical system. The slow flow equations are not very useful for the purpose. Rather, the describing function method [44] can be more conveniently employed here for designing the control. Only the main results are presented in this section with the details

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given in Appendix D. Given the other system parameters, the desired frequency (ω) and amplitudes of oscillation (A1 and A2), it is possible to compute the unknown control parameters γ 1 , γ 2 and β1 or β2 from the equations obtained in Appendix D. As discussed in Appendix D, two different control configurations are considered. 6.1. Case I: Only the first mass is nonlinearly damped (β2 ¼0) Substituting the desired values of the frequency (ω) and the amplitude-ratio (r ¼ A2 =A1 ) in Eq. (D.11), one can compute the required values of the control parameter γ 2 as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (34) γ 2 ¼ 7 ½r 2 fðkr  μω2 Þ2 þ h22 ω2 g  k2r =ω2 Substituting the above computed value(s) of γ 2 in the frequency equation (Eq. (D.9)) one obtains

γ1 ¼

fμ2 ω6  v44 ω4  ðv22  h2 γ 2 kr Þω2  kr g ; f  γ 2 μω4 þðγ 2  h2 Þkr ω2 g 2

(35)

where v44 ¼ h2 þ kr μ2 þ μ2 þ 2kr μ and v22 ¼ h2 kr þ h2  kr  2kr μ  kr μ. Finally substituting the above values of γ 1 and γ 2 in Eq. (D.10), gives 2

2

2

2

β1 ¼

2

4u11 ðωÞ 3A21 u12 ðωÞ

:

(36)

6.2. Case II: Only the second mass is nonlinearly damped (β1 ¼0) Substituting the desired values of the frequency (ω) and amplitude-ratio (r ¼ A2 =A1 ) in Eq. (D.14), one can compute the required values of the control parameter γ 1 as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (37) γ 1 ¼ 7 ½fðkr þ1  ω2 Þ2 þ h21 ω2 g=r2  k2r =ω2 : Then one finds from Eq. (D.12)

γ2 ¼

fμω6 d44 ω4  ðd22  h1 γ 1 kr Þω2  kr ð1 þ kr Þg f  γ 1 ω4 þ ðγ 1 kr þ γ 1  h1 kr Þω2 g

(38)

and from Eq. (D.13)

β2 ¼

4u11 ðωÞ 3A22 u22 ðωÞ

;

(39)

where d44 ¼ h1 μ þkr þ 2μ þ 2kr μ and d22 ¼  kr ðkr þ 2Þð1 þ μÞ þ  μ þ h1 kr . 2

2

6.3. Existence of a feasible control solution A feasible control design corresponds to the real values of γ 1 and γ 2 and negative real values of β1 or β2 . However, as evident from Eqs. (34) and (37), for some values of the system parameters or the given values of the frequency and amplitudes, γ 1 , γ 2 or βi may turn out to be imaginary. Even when Eqs. (34) and (37) produce real values, there are two possible real solutions to the problem. However, the feasible solutions must correspond to the negative values of βi . From Eqs. (34), (36), (37) and (39) one obtains the conditions of existence of at least one feasible control solution out of the two cases discussed above as r 2 fðkr  μω2 Þ2 þ h2 ω2 g kr 40 or fðkr þ1  ω2 Þ2 þh1 ω2 g=r 2  kr 4 0

(40a)

u11 ðωÞ 40:

(40b)

2

2

2

2

and

As a numerical example, the mechanical system parameters are assumed to be the same as given in Fig. 3. The region of existence of a feasible control solution is thus computed from Eqs. (40) and shown in Fig. 12. 6.4. Numerical examples It is evident from Fig. 12 that there exists a wide range of frequency and amplitude-ratio, for which one can always find a feasible control solution. To demonstrate the efficacy of the proposed design method, a few numerical examples are considered. For the desired amplitude-ratios and frequencies of oscillations, the corresponding design solutions are listed in Table 5. Examples 1 and 2 correspond to oscillations near the first and second natural modes, respectively. The system equations are numerically simulated with the estimated control parameters and the time-history plots for these two

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Fig. 12. Region of a feasible control solution (white region): μ¼ 0.8, kr ¼ 0.9, h1 ¼0.07, and h2 ¼0.05. Table 5 Design examples. Example no.

Desired oscillation characteristics (ω ; r)

A2

A1

1

0.66, 1.62

1

1.62

2

1.61, 0.76

1.0

0.76

3

0.4, 1

4

4

4

0.4, 5

10

50

5

2.5, 0.8

6

2.5, 5

7

1.05, 60

8

1.36, 0.01

1

0.8

1.25

1

1

5

40

2400

400

4

Estimated control parameters

Results of numerical simulation

γ 1 ,γ 2 β1 ,β2

ω

0.135, 0.102 0,  0.081  0.126,  0.065 0,  0.0713 3.72,  0.87 0,  0.73  0.3, 9.39  0.87, 0  2.15, 1.26  0.025, 0  2.15, 1.27 0,  0.014 0.32,  8.2  0.343, 0 14.9, 3.05  0.68, 0 7.89, 1.56 0,  7.64

0.67

0.999

1.629

1.61

0.999

0.76

0.392

3.6

4.075

A1

A2

0.41

10.584

59.97

2.51

0.788

0.63

2.51

1.23

0.989

2.51

1.072

5.468

1.047 1.36

50.75 425.15

2575 4.804

Fig. 13. Numerically simulated time-history plots with the control parameters corresponding to (a) example 1: close to the first natural mode; (b) example 2: close to the second natural mode. Mechanical system parameters are the same as given in Fig. 12.

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Fig. 14. Numerically simulated time-history plots with the control parameters corresponding to (a) example 3: non-modal low-frequency oscillation; (b) example 5: non-modal high-frequency oscillation. Mechanical system parameters are the same as given in Fig. 12.

Fig. 15. Numerically simulated time-history plots with the control parameters corresponding to (a) and (b) example 7: non-modal oscillation near antiresonant frequency and large amplitude-ratio when the first mass is nonlinearly damped; (c) and (d) example 8: non-modal oscillation near the second anti-resonance frequency when the second mass is nonlinearly damped. Mechanical system parameters are the same as given in Fig. 12.

example cases are shown in Fig. 13. Example 3 corresponds to low-frequency non-modal oscillation and the corresponding simulated time-history plot is shown in Fig. 14(a). Similarly the time-history plot corresponding to high-frequency nonmodal oscillation (example 5) is shown in Fig. 14(b). Fig. 15 shows the time-history plots of non-modal oscillations near antiresonance frequencies with large/small amplitude-ratios corresponding to examples 7 and 8.

6.5. Optimal mode of operation Numerical examples presented above clearly demonstrate that the proposed control system can be designed to induce self-excited oscillations with different modal and non-modal frequencies and amplitude-ratios. However, one should operate the system at optimal conditions corresponding to the minimum control cost. The control cost function can be pertinently considered as the average power (over a single cycle) associated with the control signal. Thus, one defines the control cost function as J ¼ minðJ 1 ; J 2 Þ;

(41)

where J1 (corresponding to β2 ¼ 0) and J2 (corresponding to β1 ¼ 0) are the quadratic control cost functions defined, respectively as J1 ¼

ω 2 ∑ 2π i ¼ 1

Z 2π =ω 0

f ci dτ ¼ 2

ω 2π

Z 2π =ω 0

fðγ 1 y_ 2 þ β1 y_ 31 Þ2 þ ðγ 2 y_ 1 Þ2 gdτ

(41a)

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Fig. 16. Variations of the control cost with amplitude-ratio: kr ¼ 1, μ¼ 0.75, h1 ¼ h2 ¼ 0.001, and A1 ¼ 1.

Fig. 17. Variations of the optimal amplitude-ratio and minimum control cost with operating frequency: _ _ _ minimum control cost, _____ optimum amplitude-ratio, kr ¼1, μ¼0.75, h1 ¼h2 ¼ 0.001, and A1 ¼ 1.

and J2 ¼

ω 2 ∑ 2π i ¼ 1

Z 2π =ω 0

f ci dτ ¼ 2

ω 2π

Z 2π =ω 0

fðγ 1 y_ 2 Þ2 þ ðγ 2 y_ 1 þ β2 y_ 32 Þ2 gdτ:

(41b)

The control cost function is minimized with respect to the amplitude-ratio r ¼ A2 =A1 for the given values of A1 and ω. The variations of the control cost J with the amplitude-ratio are plotted in Fig. 16 for the two modal frequencies. These plots clearly show, as expected, that the minimal control cost corresponds to the modal amplitude-ratios. Similarly, for each frequency of excitation, there exists an optimal amplitude-ratio. The variations of the optimal amplitude-ratio and minimum control cost with the operating frequency are plotted in Fig. 17. As expected, the modal excitations are associated with the minimum control costs; the global minimum being in the first mode.

7. Conclusions Many devices and processes utilize self-excited oscillations of mechanical or micromechanical structures. This paper investigates the nature of artificial self-excited oscillations induced in a class of two degrees-of-freedom mechanical systems by nonlinear velocity feedback control. It is demonstrated that the proposed control can excite self-oscillations very close to the natural modes of the mechanical system provided the control parameters are chosen appropriately. The existence and stability conditions of the modal oscillations are derived and numerically verified.

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Besides natural oscillations, it is also possible to design the control system to induce high (above the second natural frequency), low (below the first natural frequency) and intermediate-frequency (between the first and second natural frequency including the anti-resonance frequencies) non-modal oscillations. Analytical and numerical results are presented to show that the control system can be designed to make the spring-mass chain oscillate with the desired frequency and amplitudes. However, a feasible control solution may not exist for certain combinations of amplitudes and frequency of oscillations. The conditions of existence of a feasible control solution are also derived. Finally, it is shown that for each selected frequency of excitation, there exists an optimal amplitude-ratio that corresponds to minimal control cost. Ideally the system should be excited near these optimal points. Finally, it must be mentioned that the qualitative essence of the results presented in the paper are not specific to the model considered but are more general in nature and are applicable to many systems of the same class. Appendix A The expressions of G1, G2, G3 and G4 in Eqs. (21)–(24) are given below: "  1 3ω2 ρ2 ðβ1 þ ρ41 β2 Þr 31 G1 ¼ fc11  ðγ 1 þ γ 2 Þρ2 gr 1 þfc12  ðρ2 γ 1 þ ρ1 γ 2 Þρ2 g  r 2 cos ϕ  2ðρ1  ρ2 Þ 4

# cos ð2ϕÞ r 1 r 22 ; þðβ1 þ ρ1 ρ32 β2 Þr 32 cos ϕ þ 3ðβ1 þ ρ31 ρ2 β2 Þr 21 r 2 cos ϕ þ2ðβ1 þ ρ21 ρ22 β2 Þ1 þ 2 G2 ¼

  1 3 ω2 ρ1 fc21 þ ðρ1 γ 1 þ ρ2 γ 2 Þρ1 gr 1 cos ϕ þfc22 þ ðγ 1 þ γ 2 Þ  ρ1 ρ2 gr 2 þ ðβ1 þ ρ31 ρ2 β2 Þr 31 cos ϕ 2ðρ1  ρ2 Þ 4

 cos ð2ϕÞ 2 r 1 r 2 þ3ðβ1 þ ρ1 ρ32 β 2 Þr 1 r 22 cos ϕ ; þðβ1 þ ρ42 β2 Þr 32 þ 2ðβ 1 þ ρ21 ρ22 β2 Þ 1 þ 2

G3 ¼ 

 1 r1 r 2 sin ϕ 3ω2 ρ2  fðβ þ ρ1 ρ32 β2 Þr 32 þðβ1 þ ρ31 ρ2 β2 Þr 21 r 2 ðω2  ω21 Þ þ fc12 ðρ2 γ 1 þ ρ1 γ 2 Þρ2 g 2r 1 ω ðρ1  ρ2 Þ 4ðρ1  ρ2 Þ 1 þ 2ðβ1 þ ρ21 ρ22 β2 Þr 1 r 22 cos ϕg sin ϕ

and G4 ¼ 

 1 r 1 sin ϕ r2 3ω2 ρ1 þðω2  ω22 Þ þ fðβ þ ρ3 ρ β Þr 3 þðβ1 þ ρ42 β2 Þr 32 fc21 þ ðρ1 γ 1 þ ρ2 γ 2 Þρ1 g 2r 2 ðρ1  ρ2 Þ ω 4ðρ1  ρ2 Þ 1 1 2 2 1 þ2ðβ1 þ ρ21 ρ22 β 2 Þr 21 r 2 cos ϕ þ ðβ1 þ ρ1 ρ32 β 2 Þr 1 r 22 g sin ϕ

Appendix B. Existence and stability of modal solutions The parameters associated with Eqs. (28) and (29) are defined hereunder



3ω21 ρ3 β 1 γ c11  ρ1 ρ2 γ 1 þ 2 ; p1 ¼  ρ2 β1 þ 1 2 ; p0 ¼ ðρ1  ρ2 Þ μ 4ðρ1  ρ2 Þ μ



3ω22 ρ1 ρ22 β2 1 γ  ρ2 β 1 þ c22 þ ρ1 ρ2 γ 1  2 ; ; q0 ¼ p2 ¼ ðρ1  ρ2 Þ 2ðρ1  ρ2 Þ μ μ



3ω21 ρ21 ρ2 β2 3ω22 ρ32 β2 ρ1 β 1  ρ1 β1  and q2 ¼ ; q1 ¼ 2ðρ1  ρ2 Þ μ 4ðρ1  ρ2 Þ μ where the parameters c11 and c22 are as defined with Eq. (11). Since ρ2 o 0, β1 o 0 and β2 o 0, it is easy to see that p1 , p2 , q1 and q2 are always negative. It follows from the definitions of ρ1 and ρ2 given with Eq. (10) that ρ1 ρ2 ¼ 1=μ. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The system oscillates in the first mode if r 1 ¼ p0 =p1 is real and positive. Thus, the condition of existence of the first mode is p0 4 0, which can be expressed after simplification as (follows from the definitions given above) (B.1) ðγ 1 þ γ 2 Þ 4ðμh1 jρ2 jþ h2 ρ1 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi On the other hand, the system oscillates in the second mode if r 2 ¼ q0 =q2 , is real and positive. Thus, the condition of existence of the second mode is q0 4 0, which can be expressed after simplification as (follows from the definitions given above) ðγ 1 þ γ 2 Þ o ð  μh1 ρ1  h2 jρ2 jÞ:

(B.2)

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Fig. B.1. Regions of the stable static equilibrium and the existence of the modal oscillations in the γ2 vs. γ1 plane (obtained from averaging analysis presented in Section 4.2). .

Table B.1 ðp1 q2  p2 q1 Þ

p0 and q0

ðq0 p2  p0 q2 Þ

Positive

p0 40, q0 o 0 p0 o 0, q0 4 0 p0 o 0, q0 o 0

Negative

p0 40, q0 o 0 p0 o 0, q0 4 0 p0 o 0, q0 o 0

ðp0 q1  p1 q0 Þ

r1

r2

Positive Negative Negative positive Stable static equilibrium region

Real imaginary

Imaginary real

Positive Negative Negative Positive Stable static equilibrium region

Imaginary Real

Real Imaginary

Thus, the region of stability of the static equilibrium is bounded by the two lines ðγ 1 þ γ 2 Þ ¼ ðμh1 jρ2 j þ h2 ρ1 Þ and ðγ 1 þ γ 2 Þ ¼ ð  μh1 ρ1  h2 jρ2 jÞ in the γ2 vs. γ1 plane as shown in Fig. B.1. One must note that this region is approximate and valid only for small values of γ2 and γ1. Thus, the oscillations in the first and second mode exist if the parameter values are selected outside the stable equilibrium region depicted in Fig. B.1. Considering various possibilities of the signs of the quantities (as listed in TablepB.1) appearing in the expressions pdifferent ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi of the amplitudes of mixed-mode oscillations (r 1 ¼ ðq0 p2 p0 q2 Þ=ðp1 q2  p2 q1 Þ and r 2 ¼ ðp0 q1  p1 q0 Þ=ðp1 q2  p2 q1 Þ), one can show that mixed-mode oscillations are not possible in the present case. Stabilities of the modal solutions are ascertained by the eigenvalues of the Jacobian of the flow of Eqs. (28) and (29) evaluated at the equilibrium points. The expression of the Jacobian is given below "1 J¼

2

p0 þ 3p1 r 21 þp2 r 22

 1 2

q1 r 1 r 2



p2 r 1 r 2 q0 þ q1 r 21 þ 3q2 r 22

# 

(B.3)

For the first mode, the Jacobian reduces to " J¼

#

p0

0

0

1 2p1 ðq0 p1  q1 p0 Þ

:

(B.4)

The first mode is stable if p0 4 0 and ððq0 p1  q1 p0 Þ=p1 Þ o 0. Now, it can be easily shown that the above two conditions are equivalent to the condition of existence of the solution of the first mode, i.e., the condition given in Eq. (B.1). For the second mode the Jacobian (B.3) reduces to " J¼

1 2q2 ðp0 q2  p2 q0 Þ

0

0

 q0

# :

(B.5)

Thus, the conditions of stability of the second mode are q0 4 0 and ðp0 q2  p2 q0 Þ=q2 o0 which can be shown to be equivalent to the condtion of existence of the oscillation in the second mode, i.e., the condition given in Eq. (B.2).

A. Malas, S. Chatterjee / Journal of Sound and Vibration 334 (2015) 296–318

315

Appendix C. Anti-resonance Equations of motion of the system (given by Eq. (3)) without damping and the control forces replaced by external dynamic loads can be written as " # " # ( n) 1 0   1 þ kr kr   f1 y€ þ y ¼ ; (C.1) n 0 μ  kr kr f2 n

n

where f 1 and f 2 are the dynamic loads acting on the first and second mass, respectively. Taking the Laplace Transformations on both sides of Eq. (C.1) and rearranging, yields YðsÞ ¼ HðsÞFn ðsÞ;

(C.2)

where the transfer matrix H(s) is obtained as HðsÞ ¼

" # μs2 þkr kr 1 ; fμs4 þ δs2 þ kr g kr s2 þ 1 þ kr

(C.3)

with δ ¼ kr þ μ þ kr μ. The Laplace-transformed displacement and load vectors are defined, respectively, as ( ) ( n ) Y 1 ðsÞ F 1 ðsÞ n YðsÞ ¼ and F ðsÞ ¼ : Y 2 ðsÞ F n2 ðsÞ Clearly, when the system is excited by a harmonic load of frequency Ω, one obtains after substituting s ¼ jΩ in Eq. (C.3) " # 2  μΩ þkr kr 1 HðΩÞ ¼ : (C.4) 2 4 2  Ω þ 1 þ kr fμΩ  δΩ þkr g kr Thus, when only the first mass is excited, i.e., F n2 ðsÞ ¼ 0, one can write Y 1 ðjΩÞ ¼

kr  μΩ

2

μΩ4  δΩ2 þ kr

! F n1 ðjΩÞ

(C.5a)

and Y 2 ðjΩÞ ¼

kr

μΩ4  δΩ2 þ kr

! F n1 ðjΩÞ

(C.5b)

pffiffiffiffiffiffiffiffiffiffi It is apparent from Eq. (C.5) that when the excitation frequency Ω ¼ kr =μ, the first mass does not vibrate at all while the second mass vibrates with some non-trivial finite amplitude. Such a situation is termed as the anti-resonance condition of the system. Following a similar procedure as above, one can easily see that when only the second mass is excited by a harmonic force pffiffiffiffiffiffiffiffiffiffiffiffi of frequency Ω ¼ 1 þkr , i.e., F n1 ðsÞ ¼ 0, the first mass vibrates with a non-trivial finite amplitude, while the second mass remains standstill. Mathematically, the anti-resonance frequencies are identical to the purely imaginary transmission zeros of the transfer matrix. Onepcan (C.3) that the system has two pairs of purely imaginary transmissionp zeros: pffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffifficlearly observe from Eqs. ffiffiffiffiffiffiffiffiffiffi one pair at 7j kr =μ and the other at 7 j 1 þ kr . Thus, the two anti-resonance frequencies of the system are kr =μ and pffiffiffiffiffiffiffiffiffiffiffiffi 1 þ kr . Physically, the anti-resonance conditions are identical to the resonances of the system with an additional restraint created at one of the masses. Appendix D. Describing function analysis Eqs. (3) can be rewritten as y€ 1 þh1 y_ 1 þ ð1 þkr Þy1 kr y2  γ 1 y_ 2 ¼ f 1

(D.1)

μy€ 2 þ h2 y_ 2 þ kr y2  kr y1  γ 2 y_ 1 ¼ f 2 ;

(D.2)

and

and f 2 ¼ β where f 1 ¼ β To simplify the analysis, the nonlinear feedback force is considered to be applied in one of the two masses. Thus, two cases arise – either β2 ¼ 0 (Case I) or β 1 ¼ 0 (Case II). The dynamic displacements of the first and second mass are assumed to be of the forms y1 ¼ A1 sin ðωτÞ and y2 ¼ A2 sin ðωτ þ φÞ, respectively. _3 1 y1

_3 2 y2 .

316

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Fig. D.1. Frequency domain, block diagram representation of the system.

D.1. Case I: Only the first mass is nonlinearly damped (β2 ¼0) Following the formalism of the describing function analysis, the system can be represented in the frequency domain as depicted in Fig. D.1. With reference to Fig. D.1 one obtains L1 ðsÞ ¼

μ

ð s2 þ h2 s þ kr Þ 4 b4 s þ b3 s3 þb2 s2 þ b1 s þ b0

(D.3)

and the describing function N 1 ðA1 ; ωÞ ¼

3 β A2 ω2 s: 4 1 1

(D.4)

From the block diagram shown in Fig. D.1, the characteristic equation of the system is obtained after substituting s ¼ jω as 1  N1 ðA1 ; ωÞL1 ðjωÞ ¼ 0;

(D.5)

u þjv ¼ 0;

(D.6)

u ¼ u11 ðωÞ  f ðA1 Þu12 ðωÞ

(D.7)

v ¼ v11 ðωÞ f ðA1 Þv12 ðωÞ;

(D.8)

which can be finally written as

where

and

with u11 ðωÞ ¼ b4 ω4  b2 ω2 þ b0 ; v12 ðωÞ ¼  μω5 þ kr ω3

u12 ðωÞ ¼  h2 ω4 ; 3 and f ðA1 Þ ¼ β1 A21 : 4

v11 ðωÞ ¼ b3 ω3 þ b1 ω;

Equating the real and the imaginary parts of Eq. (D.6) to zero finally yields the following frequency and amplitude equations v6 ω6 þ v4 ω4 þ v2 ω2 þ v0 ¼ 0

(D.9)

and A¼

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4u11 ðωÞ 3β1 u12 ðωÞ

(D.10)

where v6 ¼  μ2 , v4 ¼ ð  h2 þkr μ2 þ μ2 þ 2kr μ  γ 1 γ 2 μÞ, v2 ¼ γ 1 γ 2 kr h2 kr ðγ 1 þ γ 2 Þ þh2 kr þ h2  kr  2kr μ  kr μ and v0 ¼ kr . Thus, one obtains the amplitude-ratio as vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 kr þ γ 22 ω2 A2 u r¼ ¼t (D.11) 2 A1 ðkr  μω2 Þ2 þ h ω2 2

2

2

2

2

2

2

D.2. Case II: Only the second mass is nonlinearly damped (β1 ¼0) Proceeding in a similar fashion as above, one can express the frequency and amplitude equations as d6 ω6 þ d4 ω4 þ d2 ω2 þ d0 ¼ 0

(D.12)

and sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4u11 ðωÞ ; A2 ¼ 3β2 u22 ðωÞ

(D.13)

A. Malas, S. Chatterjee / Journal of Sound and Vibration 334 (2015) 296–318

317

where d6 ¼  μ;

d4 ¼ ð  h1 μ þ kr þ 2μ þ2kr μ  γ 1 γ 2 Þ 2

d2 ¼ f  kr ðkr þ 2Þð1 þ μÞ þ γ 1 γ 2 ð1 þ kr Þ  μ h1 kr ðγ 1 þ γ 2 Þ þh1 kr g; d0 ¼ kr ð1 þkr Þ and u22 ðωÞ ¼  h1 ω4 : 2

The amplitude-ratio is thus obtained as vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 A2 u ð1 þkr  ω2 Þ2 þ h1 ω2 : ¼t r¼ 2 2 A1 k þ γ ω2 r

(D.14)

1

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