Nonlinear damping in nonlinear vibrations of ...

Journal of the Mechanics and Physics of Solids 118 (2018) 275–292

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Nonlinear damping in nonlinear vibrations of rectangular plates: Derivation from viscoelasticity and experimental validation Marco Amabili Department of Mechanical Engineering, McGill University, 817 Sherbrooke Street West, Montreal H3A 0C3, Canada

a r t i c l e

i n f o

Article history: Received 22 April 2018 Revised 1 June 2018 Accepted 2 June 2018 Available online 4 June 2018 Keywords: Plates Nonlinear damping Nonlinear vibrations Viscoelasticity Standard linear solid Rectangular plate

a b s t r a c t Even if still little known, the most significant nonlinear effect during nonlinear vibrations of continuous systems is the increase of damping with the vibration amplitude. The literature on nonlinear vibrations of beams, shells and plates is huge, but almost entirely dedicated to model the nonlinear stiffness and completely neglecting any damping nonlinearity. Experiments presented in this study show a damping increase of six times with the vibration amplitude. Based on this evidence, the nonlinear damanaping of rectangular plates is derived assuming the material to be viscoelastic, and the constitutive relationship to be governed by the standard linear solid model. The material model is then introduced into a geometrically nonlinear plate theory, carefully considering that the retardation time is a function of the vibration mode shape, exactly as its natural frequency. Then, the equations of motion describing the nonlinear vibrations of rectangular plates are derived by Lagrange equations. Numerical results, obtained by continuation and collocation method, are very successfully compared to experimental results on nonlinear vibrations of a rectangular stainless steel plate, validating the proposed approach. Geometric imperfections, in-plane inertia and multi-harmonic vibration response are included in the plate model. © 2018 Elsevier Ltd. All rights reserved.

1. Introduction Even if still little known, the most significant nonlinear effect during nonlinear vibrations of continuous systems is the increase of damping with the vibration amplitude. The literature on nonlinear vibrations of beams, shells and plates is huge, but almost entirely dedicated to model the nonlinear stiffness and completely neglecting any damping nonlinearity. Vibration of viscoelastic plates has received attention (Aboudi et al., 1990; Xia and Lukasiewicz, 1994; 1995; Sun and Zhang, 2001; Rossihkin and Shitikova, 2006; Bilasse et al., 2011; Boutyour et al., 2006; Mahmoudkhani and Haddadpour, 2013; Mahmoudkhani et al., 2014; Balkan and Mecitog˘ lu, 2014; Amabili, 2016; Balasubramanian et al., 2017), but little has been studied about nonlinear vibration regime and even less about nonlinear damping. In particular, a nonlinear damping was derived by using the Kelvin-Voigt viscoelastic model in Amabili (2016) and Balasubramanian et al. (2017). However, this type of model is not capable of reproduce experimental results. On the other hand, the literature on nonlinear vibrations of elastic (i.e., non viscoelastic) rectangular plates is really voluminous. Some examples and review studies are just briefly listed here (Chia, 1980; 1988; Sathyamoorthy, 1987; Amabili and Païdoussis, 2003; Alijani and Amabili, 2014; Ribeiro and Petyt, 1999a; 1999b; Amabili, 20 04; 20 06; 20 08; Alijani and Amabili, 2014). In all these studies, there is no one that has de-

E-mail address: [email protected] https://doi.org/10.1016/j.jmps.2018.06.004 0022-5096/© 2018 Elsevier Ltd. All rights reserved.

276

M. Amabili / Journal of the Mechanics and Physics of Solids 118 (2018) 275–292

rived a theory for nonlinear vibrations of plates which introduces a realistic nonlinear damping model capable of reproduce the experimental results available in the literature (Balasubramanian et al., 2017; Alijani et al., 2016; Amabili et al., 2016; Davidovikj et al., 2017). Phenomenological nonlinear damping has been introduced, in most of cases for a single degree of freedom systems (Ravindra and Mallik, 1994; Trueba et al., 20 0 0; Lifshitz and Cross, 20 08; Eichler et al., 2011; Zaitsev et al., 2012; Gottlieb and Habib, 2012; Elliot et al., 2015). There is a significant disagreement about the nonlinear damping term to be used to reproduce experiments. Terms of the type x2 x˙ (Eichler et al., 2011; Zaitsev et al., 2012; Gottlieb and Habib, 2012) or x˙ 3 Elliot et al. (2015) have been introduced, where x is the oscillation displacement of the system and the dot superscript represents the time derivative. Geometrically nonlinear damping has been obtained as a resultant of two mechanical dampers acting on the same point, but relatively inclined by an angle, in Jeong et al. (2013). The increased complication of the nonlinear dynamics introduced by nonlinear dampers and its stability has been investigated in Andersen et al. (2012), while nonlinear energy sinks have been introduced in Sapsis et al. (2012) with the purpose to subtract mechanical energy from an excited vibrating system. Description of nonlinear damping based on phonons and scattering for nanosystems has been proposed in Croy et al. (2012); Atalaya et al. (2016); Guttinger et al. (2017) and De et al. (2017). Nonlinear damping has been recently derived for a singledegree-of-freedom system based on the application of geometrically nonlinear stiffness to viscoelasticity (standard linear solid model or fractional solid model) in Amabili (2018); Amabili (2018); the approach has been validated through nonlinear vibration experimental results on continua (beam, plates and shallow shells) modeled as single degree of freedom systems. These studies justify the introduction of a nonlinear damping term of the type x2 x˙ . Experiments on a stainless steel rectangular plate presented in this study show a damping increase of six times with the vibration amplitude. Based on this evidence, the nonlinear damping of rectangular plates is derived assuming the material to be viscoelastic, and the constitutive relationship to be governed by the standard linear solid model. The material model is then introduced into a geometrically nonlinear plate theory, carefully considering that the retardation time is a function of the vibration mode, exactly as its natural frequency. Then, the equations of motion describing the nonlinear vibrations of rectangular plates are derived by Lagrange equations. Numerical results, obtained by continuation and collocation method, are very successfully compared to experimental results on nonlinear vibrations of a rectangular stainless steel plate, validating the proposed approach. Geometric imperfections, in-plane inertia and multi-harmonic vibration response are included in the plate model. 2. Constitutive equation for viscoelastic material The constitutive equation of the standard linear solid viscoelastic material is given by the following ordinary differential equation (Lakes, 2009; Amabili, 2018)

σ ε ε σ + τr d = E ε + τr d + E 1 τr d . dt

dt

(1)

dt

where σ is the second Piola-Kirchhoff stress, ɛ the Green’s strain, E1 and E the stiffness moduli of the two springs of the mechanically equivalent model in Fig. 1(a), t is time, τr = η/E1 is the retardation time constant, measured in seconds, which is a characteristic of the viscoelastic material, and η is the viscosity coefficient of the dashpot, measured in (N s m−2 ). In order to solve the differential Eq. (1), it is possible to use the harmonic balance method. Here the harmonic balance is introduced just to show a simple solution of the problem. Then, the harmonic balance solution is replaced by a more general formulation. Therefore, for the present purpose, it is sufficient to introduce the zero-order term (constant) and first harmonic, since addition harmonics do not alter the solution on the zero and first-order terms being Eq. (1) linear. In this way, the strain and the stress in Eq. (1) can be written as

ε (t ) = ε0 + ε1 sin(ω t ) + ...,

(2)

σ (t ) = σ0 + σ1s sin(ω t ) + σ1c cos(ω t ) + ...,

(3)

where ω is the vibration frequency and ɛ0 , ɛ1 , σ 0 , σ 1s , σ 1c are coefficients to be determined. Substituting Eqs. (2) and (3) into (1), it gives

σ0 + σ1s sin(ω t ) + σ1c cos(ω t ) + τr ω σ1s cos(ω t ) − τr ω σ1c sin(ω t ) = E ε0 + E ε1 sin(ω t ) + E τr ω ε1 cos(ω t ) + E1 τr ω ε1 cos(ω t ) .

(4)

The zero-order terms give

σ0 = E ε 0 ,

(5)

while the first-order terms are obtained as

σ1 s =

1 + τr2 ω2 E + τr2 ω2 E1 1+τ

2 r

ω

2

ε1 = E ε1 +

τr2 ω2 E1 ε1 , 1 + τr2 ω2

(6)


277

σ

E1 E

ε

η

σ (a)

σ

E

η

σ (b) Fig. 1. Material models. (a) Standard linear solid (Zener) model; (b) Kelvin–Voigt model.

σ1 c =

τr ω

1 + τr2 ω2

E1 ε1 .

(7)

Eqs. (5) and (6) can be combined into

σ0 + σ1s sin(ω t ) = E ε0 + E ε1 sin(ω t ) +

τr2 ω2

1 + τr2 ω2

E1 ε1 sin(ω t ) = E ε +

τr2 ω2

1 + τr2 ω2

E1 (ε − ε0 ).

(8)

Eq. (7) can be rewritten as

σ1c cos(ω t ) =

τr

1 + τr2 ω2

E1 ε˙ ,

(9)

where the superimposed dot indicates the partial time derivative. While Eq. (8) represents the stress-strain law in the case of dynamic load with a stress in-phase with the strain, Eq. (9) introduces a stress that has a π /2 phase shift with the strain,

278


y

h

b

x a Fig. 2. Rectangular plate and coordinate system.

giving damping. The in-phase elastic stress is referred as σ E ; the viscoelastic stress at π /2 of phase is labeled σ V . The sum of Eqs. (8) and (9) is the particular solution of the constitutive Eq. (1) in the case of harmonic functions, independently of the harmonic balance method, as it can be directly verified. In the case of lightly damped systems (this is not valid for rubber and biological materials), τr2 ω2 1 and it can be neglected in Eqs. (8) and (9). Under this assumption, Eqs. (8) and (9) become

σE = σ0 + σ1s sin(ω t ) = E ε ,

(10)

σV = σ1c cos(ω t ) = τr E1 ε˙ .

(11)

Therefore, in the case of lightly damped systems, the stress-strain uniaxial constitutive law takes the usual form (10) and the damping is represented by Eq. (11), which is the viscous part of the constitutive law. It is convenient to introduce the following relationship between the retardation time constant and the damping ratio ζ

τr,i =

2 ζi

ωi

,

(12)

where ωi is the natural frequency and ζ i the damping ratio of the ith natural mode of the system. Eq. (12) shows that the retardation time constant is a function of the mode shape considered, exactly as the damping ratio. The assumption that (τ r ω)2 1 has the double effect to eliminate the material stiffening with the frequency (which is negligible for many metals) and to remove the frequency dependency of the damping ratio (which is again accurate for metals, in which case the vibration response is controlled by damping only around the resonance peak). This simplifies the standard linear solid material model (also known as Zener model) transforming it practically in a Kelvin-Voigt model, which is shown in Fig. 1(b). 3. Potential energy of the plate Fig. 2 shows a rectangular plate with coordinate system (O; x, y, z), having the origin O at one corner made of isotropic and viscoelastic material responding to the standard linear model. The plate is assumed to be thin, so shear deformation and rotary inertia can be neglected. The displacements of an arbitrary point of coordinates (x, y) on the middle surface of the plate are denoted by u, v and w, in the x, y and out-of-plane (z) directions, respectively. Initial geometric imperfections of the rectangular plate associated with zero initial tension are denoted by out-of-plane displacement w0 ; only out-of-plane initial imperfections are considered. For thin isotropic plates, there is no linear coupling between in-plane stretching and transverse bending and shear deformation can be neglected. The variation of the potential energy UP of a plate, neglecting the transverse normal stress σ z under Kirchhoff’s hypotheses (i.e. neglecting shear deformation), is given by Amabili (2016)

δ UP =

0

a

b h/2 0

−h/2

(σx δ εx + σy δ εy + τxy δ γxy ) dx dy dz = δ UE + δ WN ,

(13)


279

where h is the plate thickness, a and b are the in-plane dimensions in x and y directions, respectively, δ is the functional derivative, σ x and σ y are the normal stresses in x and y directions, ɛx and ɛy are the corresponding strains, τ xy is the in-plane shear stress, γ xy is the correspond strain, and

δ UE =

a

0

δ WN =

−h/2

0 a

0

b h/2

b h/2 0

−h/2

σxE δ εx + σyE δ εy + τxyE δ γxy dx dy dz,

(14)

σxV δ εx + σyV δ εy + τxyV δ γxy dx dy dz.

(15)

Therefore, the variation of the potential energy of the plate has been divided in the variation of the classical potential elastic energy, δ UE , and the component δ WN , which corresponds the variation of the virtual work done by the nonconservative viscous stresses (forces). Eqs. (10) and (11) represents the stress-strain uniaxial constitutive law and they have to be applied to a plate. The generalization of Eq. (10) to the plate is carried out first. In the case of plane stress approximation for the plate, i.e. σz = σx z = σy z = 0, which is equivalent to assume a classical plate theory, the elastic stresses σ xE , σ yE and τ xyE are related to the strains for homogeneous and isotropic material by Amabili (2008)

σxE =

E (εx + ν εy ), 1 − ν2

(16a)

σyE =

E ( ε y + ν ε x ), 1 − ν2

(16b)

τxyE =

E γxy , 2 (1 + ν )

(16c)

where E is the Young’s modulus and ν is the Poisson’s ratio. Making use of Eq. (11), the viscoelastic stresses σ xV , σ yV and τ xyV are related to the strains by

σxV =

τr E 1 (ε˙ x + ν ε˙ y ), 1 − ν2

(17a)

σyV =

τr E 1 (ε˙ y + ν ε˙ x ), 1 − ν2

(17b)

τxyV =

τr E 1 γ˙ xy . 2 (1 + ν )

(17c)

Eqs. (17a–c) have been generalized for the plate from the viscous part of the uniaxial constitutive law (11) taking into account the hypothesis of plane stress introduced in Eqs. (16a–c). According to Eq. (12), the retardation time constant is a function of the mode shape and is indicated here for sake of brevity as τ r , without indicating the mode shape dependence explicitly. It is assumed that the deformation of the plate during vibrations is discretized by a base of the low-frequency natural modes of vibrations. A generic strain, which is a quadratic nonlinear function of the generalized modal coordinates qi , can be written as

ε=

N

ai qi (t ) fi (x )gi (y )+

i=1

N N

ai j qi (t ) q j (t ) fi (x ) f j (x )gi (y )g j (y ),

(18)

2ai j q˙ i (t ) q j (t ) fi (x ) f j (x )gi (y )g j (y ),

(19)

i=1 j=1

and its time derivative

ε˙ =

N

ai q˙ i (t ) fi (x )gi (y )+

i=1

N N i=1 j=1

where ai and aij are coefficients of linear and quadratic terms that depend on the transverse coordinate z,fi (x)gi (y) is the shape of the ith mode and N is the number of degrees of freedom used to model the plate. The symmetry of the coefficients aij has been used in Eq. (19), where quadratic terms appear due to geometric nonlinearity. A generic viscoelastic contribution to the stress can be written as

σV

N N N E1 ˙ = τ b q ( t ) f ( x ) g ( y ) + τr,i j 2bi j q˙ i (t ) q j (t ) fi (x ) f j (x )gi (y )g j (y ) , r,i i i i i 1 − ν2 i=1

(20)

i=1 j=1

where bi and bij are coefficients of linear and quadratic terms that depend on the transverse coordinate z, different from those in Eqs. (18) and (19) in order to accommodate Eq. (17), and τ r, i and τ r, ij are the retardation time constants, function of the vibration shape i and i,j, respectively. The retardation time constants τ r, i and τ r, ij depends on the functions in the space coordinates, since damping and dissipation depend on the vibration shapes. While τ r, i is associated to the “linear” shape

280


function fi (x)gi (y), τ r, ij is associated to the “quadratic” shape function fi (x)fj (x)gi (y)gj (y), which is clearly quite different. It is important to note that the retardation time constants are not a function of the generalized modal coordinates. The virtual work done by the non-conservative viscous stresses of the plate has the following form:

E1 WN = 1 − ν2

×

0

N

a

b h/2

−h/2

0

N

a˜i qi (t ) fi (x )gi (y )+

i=1

E1 WN = 1 − ν2

+ +

k=1 l=1 N N

τr,kl 2b˜ kl q˙ k (t ) ql (t ) fk (x ) fl (x )gk (y )gl (y )

a˜i j qi (t ) q j (t ) fi (x ) f j (x )gi (y )g j (y ) dx dydz,

(21)

i=1 j=1

which gives

+

τr,k b˜ k q˙ k (t ) fk (x )gk (y )+

k=1

N N

a

0

b h/2 −h/2

0

N N N i=1 j=1 k=1 N N N

N N

τr,k cik qi (t ) q˙ k (t ) fi (x ) fk (x )gi (y )gk (y )

i=1 k=1

τr,k cijk qi (t ) q j (t ) q˙ k (t ) fi (x ) f j (x ) fk (x )gi (y ) g j (y )gk (y ) (22)

τr,kl dikl qi (t ) qk (t ) q˙ l (t ) fi (x ) fk (x ) fl (x )gi (y ) gk (y )gl (y )

i=1 k=1 l=1 N N N N

τr,kl cijkl qi (t ) q j (t ) qk (t )q˙ l (t ) fi (x ) f j (x ) fk (x ) fl (x )gi (y ) g j (y )gk (y )gl (y ) dx dydz ,

i=1 j=1 k=1 l=1

where cij , cijk , dijk and cijkl are coefficients of quadratic, cubic, cubic again and quartic terms, respectively; they depend on the transverse coordinate z. Therefore, the quadratic coefficients are linked to τ r, k , the cubic to both τ r, k and τ r, kl , and the quartic to τ r, kl only, for k = 1,…N and l = 1,…N. In Eq. (21) the tilde has been superimposed to the coefficients in order to represent Eq. (15), which is more general than the integral of the product of Eqs. (18) and (20). Eqs. (21) and (22) show the structure of Eq. (15), which is the virtual work done by the non-conservative viscous stresses. The von Kármán nonlinear strain-displacement relationships are used so that shear deformation and rotary inertia are neglected. This is accurate for thin plates. The strain components ɛx , ɛy and γ xy at an arbitrary point of the plate are related to the middle surface strains ɛx, 0 , ɛy, 0 , γ xy, 0 and to the changes in the curvature and torsion of the middle surface kx , ky and kxy by the following three relationships

εx = εx,0 + z kx ,

εy = εy,0 + z ky ,

γxy = γxy,0 + z kxy ,

(23a-c)

where z is the distance of the arbitrary point of the plate from the middle surface. By inserting Eqs. (23a-c) into (15) and (17a–c), the virtual work done by the non-conservative viscous stresses of the plate is given by

WN =

1−ν τr εx,0 ε˙ x,0 + εy,0 ε˙ y,0 + ν εx,0 ε˙ y,0 + ν ε˙ x,0 εy,0 + γxy,0 γ˙ xy,0 dx dy 2 0 0 a b 3 E1 h 1−ν ˙ ˙ ˙ ˙ + τ kxy k˙ xy dx dy + O(h4 ), r kx kx + ky ky + ν kx ky + ν kx ky + 2 12 1 − ν 2 0 0 E1 h 1 − ν2

a

b

(24)

where ɛi, 0 , with i = x, y, and γ xy, 0 are the middle surface strains, ki , with i = x, y, xy, are the changes in curvature and torsion, and O(h4 ) is a higher-order term in h that is usually neglected. In Eq. (24), the different expressions for the retardation time constants τ r, i and τ r, ij must be used for the different shape functions. The elastic potential energy of the plate is given by

UE =

1−ν 2 εx,2 0 + εy,2 0 + 2ν εx,0 εy,0 + γxy,0 dx dy 2 0 0 a b 3 1 Eh 1−ν 2 + k2x + k2y + 2ν kx ky + kxy dx dy + O h4 , 2 2 12 1 − ν 2 0 0 1 Eh 2 1 − ν2

a

b

(25)

The first term in Eq. (25) is the membrane (also referred to as stretching) energy and the second one is the bending energy. Eq. (24) can be obtained from Eq. (25)

WN =

τr E1 ∂ UE , E ∂t

(26)

keeping in mind that τ r must be inside the double integral operator and takes different values τ r, i and τ r, ij according to the shape functions, so that different coefficients are obtained for the quadratic, cubic and quartic terms. Also, since the product τ r E1 appears in Eqs. (24) and (26), it is possible to set E1 = E without giving any restriction to the method.


281

4. Strain-displacement relationships and discretization According to von Kármán’s theory, the middle surface strain-displacement relationships and changes in the curvature and torsion are given by Amabili (2008)

2 ∂u 1 ∂ w ∂ w ∂ w0 εx,0 = + + , ∂x 2 ∂x ∂x ∂x εy,0 =

(27a)

2 ∂v 1 ∂w ∂ w ∂ w0 + + , ∂y 2 ∂y ∂y ∂y

(27b)

γxy,0 =

∂ u ∂ v ∂ w ∂ w ∂ w ∂ w0 ∂ w0 ∂ w + + + + , ∂y ∂ x ∂x ∂y ∂x ∂y ∂x ∂y

(27c)

kx = −

∂ 2w , ∂ x2

(27d)

ky = −

∂ 2w , ∂ y2

(27e)

kxy = −2

∂ 2w , ∂ x∂ y

(27f)

where u, v and w are displacements of a generic point of the middle plane in x, y and z directions, respectively. The displacement w0 indicates the initial geometric imperfection, assumed associated to zero initial stress, in z direction. The kinetic energy TP of a rectangular plate, by neglecting rotary inertia, is given by Amabili (2008)

TP =

1 ρP h 2

a 0

b

0

(u˙ 2 + v˙ 2 + w˙ 2 ) dx dy,

(28)

where ρ P is the mass density of the plate. In Eq. (28) the overdot denotes the time derivative. The boundary conditions for the simply supported (in-plane) movable plate with additional rotational distributed springs at the edges are (Amabili, 2008)

v = w = Nx = 0,

Mx = kt ∂ w/∂ x

at x = 0, a,

(29a-d)

u = w = Ny = 0,

My = kt ∂ w/∂ y at y = 0, b.

(30a-d)

where Nx , Ny are the in-plane force resultants, Mx , My are the moment resultants per unit length that are expressed as (Amabili, 2008)

Nx Mx Ny My

=

h 0

=

h 0

σx

1 z, z d

(31)

σy

1 z. z d

(32)

Moreover, in Eqs. (29d) and (30d), kt represents an elastic rotational constraint (N/rad) at the panel edges, distributed along their length, giving rotational constrains from zero rotational stiffness for kt = 0 (simply supported movable edges) to completely rotational constrained boundaries for kt → ∞, corresponding to ∂ w/∂ x = ∂ w/∂ y = 0. In order to reduce the system to finite dimensions, the middle surface displacements u, v and w are expanded by using trial functions, which satisfy identically the geometric boundary conditions (29a-c; 30a-c) (Amabili, 2008) ˜ ˜ M N

u(x, y, t ) =

um,n (t ) cos(m π x/a ) sin(n π y/b),

(33a)

vm,n (t ) sin(m π x/a ) cos(n π y/b),

(33b)

m=1 n=1 ˜ ˜ M N

v(x, y, t ) =

m=1 n=1

w(x, y, t ) =

ˆ M Nˆ m=1 n=1

wm,n (t ) sin (m π x/a ) sin(n π y/b),

(33c)

282


where m and n are the numbers of half-waves in x and y directions, respectively, and t is the time; um,n (t), vm,n (t) and wm,n (t) ˜ and N ˜ indicate the terms necessary in the expansion of are the generalized coordinates that are unknown functions of t. M ˆ and N ˆ , respectively, which indicate the terms in the expansion the in-plane displacements and are generally larger than M of w. Initial geometric imperfections of the rectangular plate are considered only in z direction. They are associated with zero initial stress. Under the hypothesis of w0 = 0 at the edges (perfectly flat edges), the imperfection w0 is expanded in a double Fourier sine series M¯ N¯

w0 (x, y ) =

Am,n sin (m π x/a ) sin(n π y/b),

(34)

m=1 n=1

where Am, n are the amplitudes of the imperfection contributions; N¯ and M¯ are integers indicating the number of terms in the expansion. It should be noted that natural boundary conditions (e.g. Nx = 0) are not identically satisfied by expressions (33a-c). Only geometric boundary conditions must be satisfied by using a variational, energy based method. However, if natural boundary conditions are also satisfied, a convergence of the solution can be achieved with a smaller number of terms (superconvergence), which is convenient in building smaller models for nonlinear dynamics. Therefore, additional terms can be added to the in-plane displacements u and v after eliminating the null terms at the panel edges as follows (Amabili, 2008)

ˆ ˆ ˆ M N M Nˆ 1 s uˆ = − (mπ /a ) wm,n (t ) sin(nπ y/b) w (t ) sin(kπ y/b) sin( (m + s )π x/a ) 2 m + s s,k m=1 n=1

k=1 s=1

+ wm,n (t ) sin(nπ y/b)

ˆ M Nˆ i=1 j=1

i Ai, j sin( jπ y/b) sin( (m + i )π x/a ) , m+i

(35a)

ˆ ˆ M Nˆ M Nˆ 1 k vˆ = − (nπ /b) wm,n (t ) sin(mπ x/a ) w (t ) sin(sπ x/a ) sin( (n + k )π y/b) 2 n + k s,k m=1 n=1

k=1 s=1

+ wm,n (t ) sin(mπ x/a )

ˆ M Nˆ i=1 j=1

j Ai, j sin(iπ x/a ) sin( (n + j )π y/b) . n+ j

(35b)

Moreover, the additional energy stored by the elastic distributed rotational springs at the panel edges must be added to the one previously obtained. It must be observed that, since the virtual work done by the non-conservative viscous stresses in the proposed model is related to the potential elastic energy through Eq. (26), the rotational springs have corresponding distributed damping elements in parallel, which simulate the dissipated energy at the boundary. The energy contribution associated to the springs and dampers in parallel is given by the following two contributions, the elastic one and the virtual work done by the non-conservative viscous forces (Balasubramanian et al., 2017)

UKE

WKV

1 = 2

b

kt 0

∂w ∂x

2

+

x=0

∂w ∂x

x=a

2

1 dy + 2

⎧ ⎨

a

kt 0

⎩

∂w ∂y

2

+

y=0

∂w ∂y

2 ⎫ ⎬

y=b

2 2 ∂w ∂w + dy ∂ x x=0 ∂ x x=a 0 ⎧ 2 2 ⎫ ⎨ ⎬ 1 a τr E1 kt ∂ ∂w ∂w + + x, 2 0 E ∂t⎩ ∂ y y=0 ∂ y y=b ⎭ d

1 = 2

b

τr E1 kt ∂ E ∂t

⎭

dx,

(36a)

(36b)

where kt is the rotational stiffness of the distributed spring and Eq. (36b) has been obtained from 36a) by using expression ((26). For simplicity, it can be assumed E1 = E. In Eq. (36b), the retardation time constant assumes different values τ r, i according to each shape function; all shape functions are “linear” in this case, so that nonlinear damping is not introduced by these rotational springs and the corresponding dampers. 5. Equations of motion The virtual work W done by an external harmonic concentrated force f˜ of z constant direction is

W =

a 0

b 0

f˜ δ (y − y˜ ) δ (x − x˜) cos(ω t )w dx dy = f˜ cos(ω t )(w )x=x˜, y=y˜ .

(37)


283

where ω is the excitation frequency, t is the time, δ is the Dirac delta function, x˜ and y˜ give the position of the point of application of the excitation force. The following notation is introduced for sake of brevity

˜ orM ˆ and n = 1, . . . N ˜ orN ˆ q = {um,n , vm,n , wm,n }T , m = 1, . . . M

(38)

The generic element of the time-dependent vector of the generalized coordinates q is referred to as qj ; the dimension of q is N, which is the number of degrees of freedom used in the mode expansion. The generalized forces Qj are obtained by differentiation of: (i) the virtual work done by external forces and, (ii) the virtual work done by the non-conservative forces with negative sign

Qj =

∂W ∂ − WN + WkV , ∂ qj ∂ qj

where

∂W = ∂ qj

if q j = um,n ,

0 f˜ cos (ω t )

(39)

vm,n ; or wm,n with m, n even,

if q j = wm,n with m, n odd,

and WN + WkV represents the work done by the non-conservative forces. The second term on the right-hand side of Eq. (39) represents the dissipative generalized force. The Lagrange equations of motion are (Amabili, 2008)

d dt

∂ TP ∂ q˙ j

+

∂ UE + UkE = Q j , ∂ qj

j = 1, . . . N.

(40)

The first element in Eq. (40) is given by

d dt

∂ TP ∂ q˙ j

= ρP h (a b/4 ) q¨ j ,

(41)

which shows that no inertial coupling among the Lagrange equations exists for the plate with the mode expansion used. The stiffness term giving linear, quadratic and cubic nonlinearities can be written in the form N N N ∂ UE + UkE = k j,i qi + k j,i,k qi qk + k j,i,k,l qi qk ql , ∂ qj i=1 i,k=1 i,k,l=1

(42)

where coefficients k have very long expressions that involve also geometric imperfections; these expressions are handled by computer program for algebraic manipulation and are not reported here for sake of space. The generalized dissipation forces giving linear, quadratic and cubic nonlinearities in damping, are given by N N N ∂ WN + WkV = g j,i q˙ i + g j,i,k qk q˙ i + g j,i,k,l qk ql q˙ i , ∂ qj i=1 i,k=1 i,k,l=1

(43)

where coefficients g have also long expressions. However, differently from the coefficients k in Eq. (42), they must be obtained from experimental data since the retardation time constant τ r is a function of the mode shape. Eq. (40) can be written in the following matrix form

Mq¨ + [G + G2 (q ) + G3 (q, q )]q˙ + [K + K2 (q ) + K3 (q, q )]q = f0 cos (ωt ),

(44)

where M is the diagonal mass matrix of dimension N × N; G is the linear damping matrix with elements gj,i , the matrix G2 gives the quadratic nonlinear damping terms, the matrix G3 denotes the cubic nonlinear damping terms; K is the linear stiffness matrix with elements kj,i , K2 gives the quadratic nonlinear stiffness terms, K3 denotes the cubic nonlinear stiffness terms and f0 is the vector representing the projection of the concentrated harmonic force on the generalized coordinates. In particular, the generic elements k2 j, i and k3 j, i of the matrices K2 and K3 , respectively, are given by

k2 j, i (q ) =

N

k j,i,k qk ,

k3 j, i (q, q ) =

k=1

N

k j,i,k,l qk ql

(45a,b)

k,l=1

The generic elements g2 j, i and g3 j, i of the matrices G2 and G3 , respectively, are given by

g2 j, i (q ) =

N k=1

g j,i,k qk ,

g3 j, i (q, q ) =

N

g j,i,k,l qk ql .

(46a,b)

k,l=1

In order to obtain the equations of motion in a suitable form for implementation in computational routine for numerical integration, the system (44) is pre-multiplied by the inverse of mass matrix and then is written in the state-space form as

284


follows

q˙ = y,

·

y˙ = − M−1 G + M−1 G2 (q ) + M−1 G3 (q, q ) q

− M−1 K + M−1 N2 (q ) + M−1 N3 (q, q ) q + M−1 f0 cos (ω t ),

(47)

where y is the vector of the generalized velocities. 6. Identification of damping An accurate model of damping is a particularly challenging goal. In general, it is necessary to identify experimentally the linear and nonlinear damping terms in Eq. (47). In fact, damping coefficients must be measured experimentally. It is known that linear damping ratios vary significantly with the vibration mode shape in a way that is not linked to the natural frequency. This means that they are not just determined by the material and boundary conditions. They are also linked to the shape of the deformation. Therefore, they cannot be calculated with accuracy, but only measured, at least with the present knowledge. The matrix M−1 G in Eq. (47), associated to the linear damping, is assumed to be given by

⎡

2ω1,1 ζ1,1 .. M−1 G = ⎣ . 0

... .. . ...

0 .. .

⎤

⎦.

(48)

2ωm,n ζm,n

In Eq. (48), ω1,1 is the natural frequency of mode (m = 1, n = 1), being m and n the number of axial half-waves of the mode shape in x and y direction, respectively, and ζ 1, 1 is the corresponding modal damping ratio. Matrix (48) is assumed to be diagonal in order to use modal damping, which strongly reduces the number of parameters to be identified. In particular, N damping ratios and N natural frequencies are used in Eq. (48). It can be already very challenging to obtain all these parameters by experimental modal analysis, especially if N is not a very small number. In the present study, the generalized coordinates qi are not exactly modal coordinates; for example, there is a coupling between two in-plane coordinates and the corresponding transverse coordinate. However, in this case, the same modal damping coefficient can be used for the three generalized coordinates associated to the same natural mode. Therefore, the assumption of diagonal linear damping in Eq. (48) seems reasonable. If a base of functions that are not natural modes of vibration is used in Eq. (33) to discretize the system, then the damping matrix (48) would result non-diagonal. In that case, Eq. (47) can be replaced by the use of the modal matrix to diagonalize the equations of motion, which would result again in a diagonal linear damping matrix in the case of proportional damping. On the other hand, M−1 G2 (q ) and M−1 G3 (q, q ) are associated to quadratic and cubic nonlinear damping, respectively. The nonlinear damping terms associated to generalized coordinates with small amplitude can be neglected. Therefore, only nonlinear terms in the transverse displacement coordinates wm, n (t) can be retained. In particular, in the case of vibrations in the frequency neighborhood of the fundamental mode (m = 1, n = 1), w1, 1 (t) has the largest amplitude and terms containing this coordinate are the most significant for nonlinear damping. In the case of experimental identification of nonlinear damping coefficients, it is essential to keep the number of parameters as small as possible in order to develop an efficient algorithm. Identification of a large set of nonlinear damping coefficients lead to failure. In the case that no internal resonances are active and the nonlinear dynamics is dominated by a single nonlinear mode, experimental evidence suggests to set the nonlinear quadratic damping to zero (Amabili, 2018; Amabili, 2018), i.e.

M−1 G2 (q ) = 0,

(49)

since damping increases with the vibration amplitude. This may be obvious for a hardening system like a plate, where cubic nonlinearities largely dominate the stiffness, but it has been observed also for softening systems like shells (Amabili, 2018; Amabili, 2018). Under the same hypothesis of no internal resonances, the cubic nonlinear damping terms can be simplified into

⎡ β1,1 q21 (t ) .. −1 M G3 ( q, q ) = ⎣ . 0

... .. . ...

⎤

0 .. ⎦, . 2 βm,n qN (t )

(50)

making much easier the experimental identification of damping parameters. Neglecting all cross-product nonlinear damping terms, in the case of system representation in modal coordinates (or nearly modal coordinates, like in the present case) and absence of internal resonances is a good approximation of the system behavior, as verified by experiments. This is also intuitive since only one vibration mode has large enough vibration amplitude to activate nonlinear damping, while the others remain in the linear damping regime. In fact, cubic damping is significant only for vibration amplitudes close enough or larger than the plate thickness h (for the case experimentally studied in this paper, nonlinear damping is negligible below the peak vibration amplitude of 0.2 h). This amplitude is reached only by one generalized coordinate for each vibration mode if a careful choice of the discretization functions is carried out, as done in Eq. (33). In that case, a single cubic nonlinear


285

vi v

iv i iii ii Fig. 3. Experimental set up of the stainless steel plate; (i) thick bolted frame; (ii) stainless steel rectangular plate; (iii) laser spot generated by a Polytec Laser Doppler Vibrometer OFV 505; (iv) B&K electrodynamic exciter 4810; (v) stinger; (vi) piezoelectric force transducer B&K 8203. The position of the force transducer has been moved in the experiments presented here.

damping coefficient can be identified at each resonance. It is convenient to introduce the non-dimensional coefficients β˜m,n defined as

β˜m,n =

βm,n h2 . ωm,n

(51)

7. Numerical procedure The equations of motion are obtained by the Mathematica computer software (Wolfram, 2003), which is capable to perform analytical surface integrals of trigonometric functions. A non-dimensionalization of variables is introduced: the frequencies are divided by the natural circular frequency ω1, 1 of the fundamental mode (m = 1, n = 1) investigated, the time is multiplied by ω1, 1 and the vibration amplitudes are divided by the plate thickness h. The resulting 2 × N equations are numerically integrated by using the computer program AUTO (Doedel et al., 2007) for continuation and bifurcation analysis of nonlinear ordinary differential equations by arclength continuation and collocation methods. 8. Experiments The experimental set-up for a AISI 304 stainless steel rectangular plate is shown in Fig. 3. The plate is bolted to an AISI 410 stainless steel frame. The bolted frame ensures that the plate is restrained in transverse direction and along the edge, while the edges could move in the in-plane direction orthogonal to the edge itself, due to the flexibility of the frame, and are rotationally elastically constrained by relatively soft rotational distributed springs. Thus, the boundary conditions are close to those indicated in Eqs. (29a-d) and (30a-d). The plate, with reference to the symbols introduced in Fig. 2, has the following dimensions and materials properties: a = 0.25 m, b = 0.24 m, h = 0.0 0 05 m, Young modulus E = 193 GPa, density ρ = 80 0 0 Kg/m3 and Poisson’s ratio ν = 0.29. Fig. 4 is a contour plot presenting the measured geometric imperfections of the bolted plate obtained by a Matsushita ANR 1282 laser triangulation sensor. The nonlinear vibration experiments have been carried out by increasing and decreasing the excitation frequency in very small steps by using a stepped-sine testing technique in the frequency neighbourhood of the fundamental mode (1, 1), detected at 70.5 Hz, which has one half-wave in both x and y directions. The excitation has been provided by an electrodynamic exciter (B&K type 4810), driven by a power amplifier, via a stinger connected to a piezoelectric miniature force

286


Fig. 4. Measured geometric imperfections in normal direction of the stainless steel plate indicating deviations (in mm) from perfectly flat configuration. Table 1 System parameters in the model of the rectangular plate. h (meters)

ω1, 1 (rad/s)

ω1, 3 (rad/s)

ω3, 1 (rad/s)

ω3, 3 (rad/s)

ζ 1, 1

β˜1,1

0.0 0 05

441.6

1475.6

1567.5

2493.4

0.0023

0.044

transducer (B&K type 8203) glued to the plate. The plate responses have been measured by using a Polytec laser Doppler vibrometer (sensor head OFV-505 and controller OFV-50 0 0) in order to have non-contact displacement and velocity measurements without introduction of sensor mass. The time responses have been acquired by using a SCADAS III front-end connected to a computer. The LMS Test.Lab software has been used for signal processing, data analysis and excitation control. In particular, the MIMO Sweep & Stepped Sine Testing application of the LMS system has been utilized to generate the excitation signal and its closed loop control has been used to maintain constant the amplitude of the harmonic force while the excitation frequency was varied in the neighborhood of the fundamental frequency. Experiments at five different harmonic forces have been carried out: 0.1 N, 0.3 N, 0.5 N, 0.7 N and 0.9 N. The vibrations have been measured at the center of the plate, where the fundamental mode shape presents the largest displacement, and the stepped-sine harmonic excitation has been applied at the point of coordinates (0.25a, 0.75b). The experiments have been performed by spanning the excitation frequency up (and down in a second set of experiments) in 0.005 Hz steps. Every time the frequency has been changed in the stepped-sine tests, 40 periods have been discharged and the following 16 periods have been recorded in order to obtain a steady-state response. 9. Comparison of Numerical and Experimental Results In the numerical simulations, a geometric imperfection with only the first term in Eq. (34) has been used, with amplitude A1, 1 = 0.7 h, and a rotational stiffness kt = 100 N/rad, in order to match the experimental natural frequencies of the stainless steel plate and to be compatible with the imperfection data in Fig. 4. The model used has 22 degrees of freedom, and specifically includes u1,1 , u1,3 , u3,1 , u3,3 , u1,5 , u5,1 , u3,5 , u5,3 , u5,5 , v1,1 , v1,3 , v3,1 , v3,3 , v1,5 , v3,5 , v5,3 , v5,5 , w1,1 , w1,3 , w3,1 , w3,3 as generalized coordinates, which are functions of time, in Eqs. (33a–c). The other system parameters used in the numerical model are given in Table 1. Fig. 5(a–j) present comparisons of numerical and experimental vibration amplitudes (normalized with respect to the plate thickness h) measured at the center of the plate versus the excitation frequency (normalized with respect to the fundamental frequency) in the frequency neighborhood of the fundamental mode (1,1) of the stainless steel plate. Five different harmonic excitations are considered: 0.1 N, 0.3 N, 0.5 N, 0.7 N, 0.9 N. It is observed that the plate displays a hardening type nonlinear behavior with clear jumps and hysteresis. Numerical results from two different models are presented in Fig. 5(a– j): the model with the nonlinear damping introduced in the present study, and a model with a linear viscous damping. Results are filtered in order to present only the first-harmonic component of the vibration (zero-order and higher-orders are removed from both experimental and numerical results). The damping coefficients ζ 1, 1 and β˜1,1 used in the model with nonlinear damping are reported in Table 1. These values have been identified by matching the experimental vibration amplitudes at the lowest and highest excitation levels. It is very important to say that just these two parameters are used in the model (the same damping coefficients are assumed for all modes, since mode (1,1) is largely dominant) and they are


287

Fig. 5. Comparison of numerical (blue lines; continuous line, stable solution; dashed line, unstable solution) and experimental (colored dots) normalized vibration amplitude measured at the center of the plate versus normalized excitation frequency in the frequency neighborhood of the natural frequency of the fundamental mode (1,1) of the stainless steel plate. Five different harmonic excitations are considered: 0.1 N, 0.3 N, 0.5 N, 0.7 N, 0.9 N. Two models are considered: nonlinear damping and linear viscous damping; results are filtered in order to present only the first-harmonic component of the vibration (zero-order and higher-orders are removed). (a) Excitation 0.1 N, nonlinear damping model; (b) excitation 0.1 N, linear viscous damping ζ 1, 1 = 0.0043; (c) excitation 0.3 N, nonlinear damping model; (d) excitation 0.3 N, linear viscous damping ζ 1, 1 = 0.008; (e) excitation 0.5 N, nonlinear damping model; (f) excitation 0.5 N, linear viscous damping ζ 1, 1 = 0.0097; (g) excitation 0.7 N, nonlinear damping model; (h) excitation 0.7 N, linear viscous damping ζ 1, 1 = 0.0121; (i) excitation 0.9 N, nonlinear damping model; (j) excitation 0.9 N, linear viscous damping ζ 1, 1 = 0.0140. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

288

M. Amabili / Journal of the Mechanics and Physics of Solids 118 (2018) 275–292 Table 2 Equivalent damping ratio in case of linear viscous damping for the rectangular plate. Excitation

Damping ratio ζ

0.01 N (linear) 0.1 N 0.3 N 0.5 N 0.7 N 0.9 N

0.0023 0.0043 0.0080 0.0097 0.0121 0.0140

Fig. 6. Nonlinear vibration amplitudes of the fundamental mode (1,1) of the rectangular plate versus excitation frequency for different five harmonic excitation forces; first-order harmonic vibration only measured at the center of the plate. Comparison of experiments (small colored dots) and numerical simulations with nonlinear damping model (blue lines; continuous line, stable solution; dashed line, unstable solution). The contour plot of the mode shape of the fundamental mode is presented on the top-left corner. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

capable to represent all the five excitation levels with great accuracy, as shown in Fig. 5(a, c, e, g, i), for the 0.1 N, 0.3 N, 0.5 N, 0.7 N, 0.9 N levels, respectively. On the other hand, a different damping coefficient ζ must be used at each excitation level in the linear viscous damping model in order to match the experimental results, as shown in Fig. 5(b, d, f, h, j), for the 0.1 N, 0.3 N, 0.5 N, 0.7 N, 0.9 N levels, respectively. The five different damping ratios used for the five levels are also given in Table 2; the value of the damping ratio for very small (linear) vibration amplitude is also reported. Fig. 5(a–j) show that the nonlinear damping model obtained is fully capable, with just two parameters, to reproduce all the obtained experimental results at different excitation levels. Otherwise, it is necessary to adjust the damping ratio at any excitation level if a linear viscous damping is used. In particular, the damping ratio has to be increased six times from the linear value to model the 0.9 N level.


289

Fig. 7. Contribution to the total damping ratio of the linear damping term and of the cubic damping term versus the maximum normalized vibration amplitude (peak amplitude divided by h).

The comparison of the experimental and numerical results, obtained from the nonlinear damping model, is plotted in Fig. 6 combining all the five excitation levels. The agreement of the present model and experiments is remarkable, considering also the difficulty of nonlinear vibration testing. The contribution of the linear damping coefficient and of the cubic damping coefficient (both given in Table 1) to the global damping ratio is presented in Fig. 7 versus the peak normalized vibration amplitude. For peak amplitudes smaller than 0.2 h, the cubic damping term is negligible with respect to the linear damping ratio. The two terms become equal around the peak vibration amplitude 0.6 h. For vibration amplitudes larger than h, the cubic damping dominates. Another interesting aspect of the cubic damping term is the following: the damping coefficient is non-constant in time and varies during the vibration period. In fact, the cubic damping term acting on mode (1,1) is given by

β˜1,1 ω1,1

q1 (t ) h

2

q˙ 1,1 (t ) .

(52)

Eq. (52) shows that the coefficient in front of q˙ 1,1 (t ) is time-dependent and goes with the function (q1 (t))2 ; it means that, in case of harmonic vibration, it has double oscillation frequency than the vibration. For this reason, it is not possible to insert in Eq. (52) the peak value of q1 (t) in order to compare the cubic damping coefficient of mode (1,1) with the corresponding linear damping term, which is given by

2ζ1,1 ω1,1 q˙ 1,1 (t ).

(53)

Therefore, Fig. 7 is obtained by evaluating the effect of the nonlinear damping on the forced vibration response and then comparing it to the corresponding curve obtained with equivalent linear viscous damping (see Fig. 5(b, d, f, h, j)). The global nonlinear damping in Fig. 7 is given by the sum of the linear and cubic contributions. Results in time domain are presented for the model with nonlinear damping in Fig. 8(a–d). In particular, the vibration at the maximum amplitude (excitation frequency giving the exact peak of the amplitude in Fig. 5) is presented in Fig. 8(a and b) for the 0.1 N excitation level, and in Fig. 8(c and d) for the 0.9 N level. Fig. 8(a and c) show the numerical results while in Fig. 8(b and d) those results have been filtered in order to remove any vibration frequency different from the excitation (since filtered results are presented in Figs. 5 and 6). Each of Fig. 8(a–d) display five curves. They correspond to the four generalized coordinates in transverse directions (w1,1 , w1,3 , w3,1 , w3,3 ) and the transverse displacement at the center of the plate (wcenter ), which is obtained combining the previous four coordinates. Results show that, while the coordinate w1,1 is basically harmonic with a significant zero-order component due to the presence of quadratic nonlinearities introduced by the geometric imperfection, the other three coordinates present higher vibration frequencies. This is particularly evident for the 0.9 N level in Fig. 8(c), where the third-harmonic becomes significant. The amplitudes of the four generalized coordinates in transverse directions (w1,1 , w1,3 , w3,1 , w3,3 ) and the transverse displacement at the center of the plate, are plotted in Fig. 9(a and b) versus the excitation frequency in the frequency

290


Fig. 8. Time responses (normalized with respect to the period) of the four generalized coordinates in transverse directions (w1,1 , w1,3 , w3,1 , w3,3 ) and transverse displacement at the center of the plate (wcenter ), obtained at the excitation frequency corresponding to the maximum amplitude (peak) in the frequency neighborhood of the fundamental mode (1,1) of the plate. Model with nonlinear damping. (a) Harmonic excitation 0.1 N, non-filtered time response; (b) harmonic excitation 0.1 N, filtered time response leaving only the first harmonic; (c) harmonic excitation 0.9 N, non-filtered time response; (d) harmonic excitation 0.9 N, filtered time response leaving only the first harmonic.

neighborhood of the fundamental mode (1,1) of the plate. The model used is the one with nonlinear damping and results are filtered in order to keep only the first harmonic. Fig. 9(a) presents the 0.1 N excitation level and Fig. 9(b) the 0.9 N level. 10. Conclusions To the author’s knowledge, the present study is the first to derive a nonlinear damping model for a continuous system which is capable to reproduce with accuracy experimental results for geometrically nonlinear vibrations. The material is assumed to be linearly viscoelastic and modeled by the classical standard linear solid. The material model is then introduced into a geometrically nonlinear plate theory, carefully considering that the retardation time is a function of the vibration


291

Fig. 9. Normalized amplitudes of the four generalized coordinates in transverse directions (w1,1 , w1,3 , w3,1 , w3,3 ) and transverse displacement at the center of the plate, versus normalized excitation frequency in the frequency neighborhood of the fundamental mode (1,1) of the plate. Model with nonlinear damping; filtered responses in order to keep only the first harmonic. (a) Harmonic excitation 0.1 N; (b) harmonic excitation 0.9 N.

mode, exactly as its natural frequency. Then, the nonlinear dynamics is derived by Lagrange equations. Only two damping coefficients have been used in the numerical model: the classical linear damping ratio and a single coefficient of cubic nonlinear damping of the type x2 x˙ . The shift of linear versus nonlinear damping regime (e.g. taking the crossing point of the two curves in Fig. 7) seems related to the normalized vibration amplitude (peak vibration amplitude divided by the thickness of the plate) and to the value of the linear damping term. An increase of the damping ratio (e.g., using a material with higher loss factor for the plate) means a rise of the horizontal line representing the linear damping term in Fig. 7. But a shift towards higher values of the cubic damping curve is also very likely to happen. However, cubic damping may increase less than the linear one, as some preliminary experiments on rubber plates seem to indicate (Balasubramanian et al., 2018); but it is too early to make a final statement on this specific point and further studies are necessary. Acknowledgments The author acknowledges the financial support of the NSERC Discovery Grant and the Canada Research Chair program. References Aboudi, J., Cederbaum, G., Elishakoff, I., 1990. Dynamic stability analysis of viscoelastic plates by Lyapunov exponents. J. Sound Vib. 139, 459–467. Xia, Z.Q., Lukasiewicz, S., 1994. Non-linear, free, damped vibrations of sandwich plates. J. Sound Vib. 175, 219–232. Xia, Z.Q., Lukasiewicz, S., 1995. Nonlinear damped vibrations of simply-supported rectangular sandwich plates. Nonlinear Dyn. 8, 417–433. Sun, Y.X., Zhang, S.Y., 2001. Chaotic dynamic analysis of viscoelastic plates. Int. J. Mech. Sci. 43, 1195–1208. Rossihkin, Yu.A., Shitikova, M.V., 2006. Analysis of free non-linear vibrations of a viscoelastic plate under the conditions of different internal resonances. Int. J. Non Linear Mech. 41, 313–325. Bilasse, M., Azrar, L., Daya, E.M., 2011. Complex modes based numerical analysis of viscoelastic sandwich plates vibrations. Comput. Struct. 89, 539–555. Boutyour, E.H., Daya, E.M., Potier-Ferry, M., 2006. A harmonic balance method for the non-linear vibration of viscoelastic shells. C. R. Mecanique 334, 68–73. Mahmoudkhani, S., Haddadpour, H., 2013. Nonlinear vibrations of viscoelastic sandwich plates under narrow-band random excitations. Nonlinear Dyn. 74, 165–188. Mahmoudkhani, S., Haddadpour, H., Navazi, H.M., 2014. The effects of nonlinearities on the vibration of viscoelastic sandwich plates. Int. J. Non Linear Mech. 62, 41–57. Balkan, D., Mecitog˘ lu, Z., 2014. Nonlinear dynamic behavior of viscoelastic sandwich composite plates under non-uniform blast load: Theory and experiment. Int. J. Impact Eng. 72, 85–104. Amabili, M., 2016. Nonlinear vibrations of viscoelastic rectangular plates. J. Sound Vib. 362, 142–156. Balasubramanian, P., Ferrari, G., Amabili, M., Del Prado, Z.J.G.N., 2017. Experimental and theoretical study on large amplitude vibrations of clamped rubber plates. Int. J. Non Linear Mech. 94, 36–45. Chia, C-Y., 1980. Nonlinear Analysis of Plates. McGraw-Hill, New York, USA. Sathyamoorthy, M., 1987. Nonlinear vibration analysis of plates: A review and survey of current developments. Appl. Mech. Rev. 40, 1553–1561. Chia, C-Y., 1988. Geometrically nonlinear behavior of composite plates: A review. Appl. Mech. Rev. 41, 439–451. Amabili, M., Païdoussis, M.P., 2003. Review of studies on geometrically nonlinear vibrations and dynamics of circular cylindrical shells and panels, with and without fluid-structure interaction. Appl. Mech. Rev. 56, 349–381. Alijani, F., Amabili, M., 2014. Non-linear vibrations of shells: A literature review from 2003 to 2013. Int. J. Non Linear Mech. 58, 233–257. Ribeiro, P., Petyt, M., 1999a. Geometrical non-linear, steady-state, forced, periodic vibration of plate, part I: model and convergence study. J. Sound Vib. 226, 955–983.

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