DSpace at VNU: Simultaneous resonances involving two mode shapes of parametrically-excited rectangular plates

DSpace at VNU: Simultaneous resonances involving two mode shapes of parametrically-excited rectangular plates tài liệu,...

Simultaneous resonances involving two mode shapes of

parametrically-excited rectangular plates

Department of Engineering Mechanics, Ho Chi Minh City University of Technology, 268 Ly Thuong Kiet Street, District 10,

Hochiminh City, Vietnam

a r t i c l e i n f o

Article history:

Received 20 November 2012

Received in revised form

21 March 2013

Accepted 10 April 2013

Handling Editor: L.N Virgin

Available online 21 May 2013

a b s t r a c t

It is known that, for multi-degree-of-freedom systems under time-dependent excitation, the combination of an internal resonance with an external resonance will give rise to simultaneous resonances, and these resonances are characterized by the fact that the system in question can resonate simultaneously in more than one normal mode while only one resonant mode is directly excited by the excitation This work deals with the problem of the occurrence of simultaneous resonances in a parametrically-excited and simply-supported rectangular plate The analysis is based on the dynamic analog of von Karman's large-deflection theory, and the governing equations are satisfied using the orthogonality properties of the assumed functions The nonlinear temporal response of the damped system is determined by the first-order generalized asymptotic method The solution for simply supported plates indicates the possibility of principal parametric resonances and simultaneous resonances Simultaneous resonances involving two modes

of vibration are presented in this paper, and it is shown for the first time that only one possibility out of two cases can really occur This phenomenon is verified by experimental results and observations

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1 Introduction

A typical example in regard to the dynamic instability of structures is the case of a thin and flat rectangular plate acted upon by an in-plane load of the form n(t)¼n0+nt cosλt, where cosλt is a harmonic function of time Under such circumstances, the plate may become laterally unstable over certain regions of the (n0, nt,λ) parameter space Thus, apart from the forced in-plane vibrations, transverse vibrations may be induced in the plate and the plate is said to be dynamically (or parametrically) unstable

It is well known that when the natural frequencies of this system are distinct, and in the absence of internal resonances and combination resonances, the periodic in-plane load can excite only one normal mode at a time; and when the plate executes lateral vibration at half the driving frequency, the corresponding resonance is called principal parametric resonance

[10–12,14–16] In contrast with this case of simple parametric resonance, simultaneous resonances and combination resonances may also occur in multi-degree-of-freedom system subjected to parametric excitation such as a plate

It has been shown that when a parametric resonance is excited in the presence of an internal resonance, the coincidence

of these two types of resonances will give rise to simultaneous resonances [14–17,19] These kinds of resonances are characterized by the fact that all resonantly involved modes might exist in the response, even though only one mode is

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Journal of Sound and Vibration

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Journal of Sound and Vibration 332 (2013) 5103–5114

directly excited by the parametric excitation It has been specified that internal resonance is responsible for this phenomenon and, as a consequence, for a significant transfer of energy from the directly excited mode to other modes

of vibration through internal mechanisms

Research in the field of parametric instability of plates has not been as extensive as for columns, but their instability behavior has been considerably clarified The first investigation on a rectangular plate subjected to periodic in-plane loads was performed by Einaudi [1] Subsequent works on this subject, however, have only established the boundaries of instability regions associated mostly to principal parametric resonances and a few to combination resonances Bolotin[2]

was the first to investigate the nonlinear problem of parametric response of a rectangular plate Schmidt[3]presented only

a few qualitatively symbolic results in his study on the nonlinear parametric vibrations of sandwich plates Recently, Sassi and Ostiguy[4–6]studied the effects of initial geometric imperfections on the interaction between forced and parametric or combination resonances More recently, chaotic dynamic stability of plates in large deflection was investigated by Sun and Zhang[7], and by Yeh et al.[8], while Wu and Shih[9]considered the dynamic instability of rectangular plate with an edge crack Works with significant results on nonlinear response corresponding to principal parametric resonances of parametrically-excited rectangular plates can be found in Refs [10–17] Research on nonlinear response related to combination resonances of parametrically-excited rectangular plates was performed by the present author and results can only be found in Refs.[15–18] Similarly, nonlinear response corresponding to simultaneous resonances involving two mode shapes of parametrically-excited rectangular plates can only be found in[14,15,17,19,20] Recently, theoretical results concerning the occurrence of simultaneous resonances involving three mode shapes of parametrically-excited rectangular plates was presented by the author for the first time [21] The first experimental studies on plates were conducted by Somerset and Evan-Iwanowski[22,23], and they pertained mainly to the large amplitude, nonlinear parametric response of simply-supported square plates The most significant experimental results on combination and simultaneous resonances of parametrically-excited rectangular plates were obtained by Nguyen[15,17,19]

The occurrence of simultaneous resonances involving two spatial forms of vibration of a parametrically-excited rectangular plate is dealt in this paper and covers an existing gap in our understanding of the dynamic buckling of structures The simply-supported rectangular plate under investigation is acted upon by periodic in-plane forces uniformly distributed along two opposite edges; the other two edges are stress-free The analysis is based on the dynamic analog of von Karman's large-deflection theory and the governing equations are satisfied using the orthogonality properties of the assumed functions The first-order generalized asymptotic method is used to solve the temporal equations of motion, and attention is focused on principal parametric resonances and an internal resonance involving two modes of vibration The coincidence of this type of internal resonance with corresponding principal parametric resonances will give rise to two cases

of simultaneous resonances, in which only one case is resonantly possible, the other case is impossible The reason for the latter case is explained for the first time This explanation is also based on experimental results obtained[15]

2 Statement of the problem

The mechanical system under investigation is a rectangular plate, simply supported along its edges (in-plane movable edges) and subjected to the combined action of both static and dynamic compressive forces uniformly distributed along two opposite edges The two vertical edges are stress free The geometry of the plate, the load configuration and the coordinate system are shown inFig 1 The x–y plane is selected in the middle plane of the undeformed plate The plate is assumed to be thin, initially flat, of uniform thickness, and the plate material is elastic, homogeneous and isotropic

Restricting the problem to the relatively low frequency range where the plate oscillations are predominantly flexural, the effect of transverse shear deformations as well as in-plane and rotatory inertia forces can be neglected From these restrictions, the plate theory used in the analysis may be described as the dynamic analog of von Karman's large-deflection theory The dimensionless differential equations governing the nonlinear flexural vibrations of the plate can be written as

R4F;XXXXþ 2R2

F;XXYYþ F;YYYY¼ R2

½W2

R4W;XXXXþ 2R2W;XXYYþ W;YYYY¼ ζ½R2ðF;YYW;XX−2F;XYW;XYþ F;XXW;YYÞ−R4W;TT; (2)

in which a comma denotes partial differentiation with respect to the corresponding coordinates, R¼b⧸a is the plate aspect ratio,ζ¼12(1−ν2) whereν is the Poisson ratio, and where

X¼ x=a; Y ¼ y=b; W ¼ w=h; F ¼ f =Eh2

; T ¼ t½Eh2

In Eq.(3), w(x, y, t) is the lateral displacement and f(x, y, t) the Airy stress function, h denotes the plate thickness,ρ the density per unit volume, E the Young modulus, and t the time The nonlinearity arising in the problem under consideration

is due to large amplitudes generating membrane forces

The non-dimensional membrane forces NX, NYand NXY, arising from a combination of the dimensionless in-plane loading

NYðTÞ ¼ ða2=Eh3

with the large amplitude lateral deflection, are related to the non-dimensional force function by

ðN ; N ; N Þ ¼ ðF;YY=R2

ðNX; NY; NXYÞ ¼ ða2

=Eh3

in which nx, nyand nxyare membrane forces In Eq.(4), NY0is the dimensionless constant component of the in-plane force,

NYTis the dimensionless amplitude of the harmonic in-plane loading, andΛ is the dimensionless excitation frequency The boundary conditions are related to both the stress function, F, and the lateral displacement, W The stress conditions may be expressed in the dimensionless form as

The supporting conditions for a simply-supported rectangular plate are written as

W¼ R2

W¼ W;YYþ νR2

The problem consists in determining the functions F and W which satisfy the governing equations, together with the boundary conditions

3 Method of solution

An approximate solution of the governing Eqs.(1)and(2)is sought in the form of double series in terms of separate space and time variables The non-dimensional force function is expressed as

FðX; Y; TÞ ¼ ∑

n

FmnðTÞXmðXÞYnðYÞ−12X2NYðTÞ (8) and the dimensionless lateral displacement as

Fig 1 Plate and load configuration.

where Fmn and Wpqare undetermined functions of the dimensionless time T, and where Xm, Yn, Φp, andΨq are beam eigenfunctions given by

XmðXÞ ¼ coshαmX−cosαmX− coshαm−cosαm

sinhαm−sinαm

YnðYÞ ¼ coshαnY−cosαnY− coshαn−cosαn

sinhαn−sinαn

in which the coefficientsαiare obtained from the transcendental equations

These beam functions satisfy the relevant boundary conditions

Applying the approach of generalized double Fourier series[24]to the governing equations, using the orthogonality properties of the assumed functions, together with the indicial notation, leads to

∑

n

Amnij FmnðTÞ ¼ ∑

s

€

WuvðTÞ þ ω2

uvWuvðTÞ− πv

R

 2

NYðTÞWuvðTÞ þ ∑

s

Gmnrs

in which Amnij ; Bpqrs

ij and Gmnrsuv are coefficient matrices, and

ω2

uv¼πζ4 u4þ2u2v2

R2 þv4

R4

(14)

is the vibration frequency of the unloaded plate In Eq (13), the dots denote differentiation with respect to the non-dimensional time T

Generally, Eq.(12)can be solved for the time-dependent stress coefficients Fmn(T) in terms of Wuv(T) coefficients and substituting into Eq.(13), leads to

€

Wuvþ ω2

uvWuv− πvR 2NYWuvþ ∑

s

where

½Mklpqrs

uv  ¼ ½Gmnkl

uv ½Amn

ij −1½Bpqrs

It is known that the rectangular plate buckles in such a way that there can be several half-waves in the direction of compression but only one wave in the perpendicular direction Hence, omitting all indices associated with the half-wave spatial mode in the unloaded direction, and introducing linear (viscous) damping lead to a system of nonlinear ordinary differential equations for the time functions as follows:

€

Wmþ 2CmW_mþ Ω2

mð1−2μmcosθÞWmþ ∑

k

MijkmWiWjWk¼ 0; m ¼ 1; 2; 3; ⋯ (17) where Cmrepresents the coefficient of viscous damping,Ωm¼ωm[1−NY0/Nm]1/2is the free vibration circular frequency of a rectangular plate loaded by the constant component NY0of the in-plane force while Nmrepresents the static critical load according to linear theory, _θðTÞ ¼ Λ is the instantaneous frequency of excitation, and μm¼NYT/2(Nm−NY0) is the load (excitation) parameter, in which NYT is the dimensionless amplitude of the harmonic in-plane loading as previously mentioned

By taking the first three terms in the expansion for the lateral displacement, the continuous system is reduced to a three-degree-of-freedom system and we get the following set of temporal equations of motion:

€

W1þ Ω2

W1¼ −2C1W_1þ 2μ1Ω2

cosθW1−ðΓ11W3þ Γ12W2W2þ Γ13W2W3þ Γ14W1W2

þΓ15W1W2

þ Γ16W3

þ Γ17W2W3þ Γ18W2W2

þ Γ19W3

þ Γ110W1W2W3Þ;

€

W2þ Ω2W2¼ −2C2W_2þ 2μ2Ω2cosθW2−ð⋯Γ2⋯Þ;

€

W3þ Ω2W3¼ −2C3W_3þ 2μ3Ω2cosθW3−ð⋯Γ3⋯Þ (18a–c)

in whichΓm1throughΓm10are the coefficients of the nonlinear (cubic) terms, and are defined as follows:

Γm1¼ M111

Γm4¼ M122

Γ ¼ M223þ M232þ M322; Γ ¼ M233þ M323þ M332; Γ ¼ M333;

Γm10¼ M123

Eq (18) constitutes the final form assumed by the equations of motion They represent a system of second-order nonlinear differential equations with periodic coefficients, which may be considered as extensions of the standard Mathieu-Hill equation

4 Solution of the temporal equations of motion

Mathematical techniques for solving nonlinear problems are relatively limited, and approximate methods are generally used The method of asymptotic expansion in powers of a small parameter, ε, developed by Mitropolskii [25] and generalized by Agrawal and Evan-Iwanowski [26,10], is an effective tool for studying nonlinear vibrating systems with slowly varying parameters In the present analysis, this method is used to solve the equations of motion

Assuming that the actual mechanical system is weakly nonlinear, the damping, the excitation, and the nonlinearity can

be expressed in terms of the above-mentioned small parameter, and that the instantaneous frequency of excitation and the load parameter vary slowly with time Then, the generalized system of temporal equations of motion (17) can be rewritten

in the following asymptotic form:

€

Wmþ Ω2

mWm¼ ε½2μmΩ2

mcosθWm−2CmW_m−∑

k

Mijk

mWiWjWk; m ¼ 1; 2; 3: (20)

Confining ourselves to the first order of approximation inε, we seek a solution for the system of Eq.(20)in the following form:

whereτ¼εT represents the “slowing” time, and where amandφmare functions of time defined by the system of differential equations

dam=dT ¼ _am¼ εAm

dφm=dT ¼ _φm¼ ΩmðτÞ þ εBm

Functions A1m(τ,θ, am,φm) and B1m(τ,θ, am,φm) are selected in such a way that Eq.(21)will, after replacing amandφmby the functions defined in Eqs.(22)and(23), represent a solution of the set of Eq.(20)

Following the general scheme of constructing asymptotic solutions for Eq.(18)and performing numerous transforma-tions and manipulatransforma-tions, we arrive finally at a system of equatransforma-tions describing the nonstationary response of the discretized system By integrating this system of equations, amplitudes amand phase anglesφmcan be obtained as functions of time

5 Stationary response

For simply-supported rectangular plates excited parametrically, the results of the investigations conducted by the present author [15,16] indicate, besides the possibility of principal parametric resonances, the presence of internal resonances The stationary response associated with the assumed spatial forms of vibration of our system may be calculated

as a special case of the nonstationary motions in the resonant regime described previously

It is known that governing equations with cubic nonlinearities are associated with many physical systems The presence

of these nonlinear terms has an important influence upon the behavior of the system, especially under a condition of internal resonance An internal resonance is possible when two or more natural frequencies are commensurable or almost commensurable

∑

i

where mi are positive or negative integers When an internal resonance coincides with a parametric resonance, the combination of the two types gives rise to simultaneous resonances This kind of resonances is characterized by the fact that the system in question vibrates simultaneously in more than one normal mode and at different frequencies, although only one of the modes is directly excited by the parametric excitation

In the absence of internal resonances, the parametric excitation can excite only one mode at a time In this case, principal parametric resonance occurs when the excitation frequency is approximately equal to twice the natural frequency associated with a particular mode of vibration, that is, Λ≈2Ωm Stationary values for principal parametric response associated with various spatial modes of vibration are given by

am¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

4Ωm

3Mm Λ−2Ωm7

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2μmΩ2 m

Λ

−ð2CmÞ2

s 2

4

3 5

v u

where only positive real values for the amplitude are admitted The“7” sign upon the inner radical indicates the possibility

of two solutions; the larger solution is stable and attainable by the real system, while the lower is unstable and physically

not realizable Hence, the most important condition for this resonant case is that the stable solution must be higher than its unstable solution When the unstable solution coincides with– or is larger than – its stable solution, the parametrically excited system becomes stable; this condition is then related to the non-resonant case

In the presence of internal resonances, as mentioned above, the coincidence between an internal resonance and a principal parametric resonance will give rise to simultaneous resonances Of all possible internal resonances associated with

a flat rectangular plate we will, for convenience, consider only an internal resonance of the type 3Ω1≅Ω3 Consequently, the two following cases of simultaneous resonances will be investigated: (1)Λ ¼ 2Ω1and 3Ω1≅Ω3, and (2)Λ ¼ 2Ω3andΩ3≅3Ω1 Case 1 Λ ¼ 2Ω1AND 3Ω1≅Ω3

It is supposed that the principal parametric resonanceΛ ¼ 2Ω1and the internal resonance 3Ω1≅Ω3occur simultaneously Then, performing numerous transformations and manipulations of the asymptotic solutions, we arrive finally at a system of equations describing the stationary response for this case as follows:

−C1a1þ1Λμ1Ω2a1sinψ þ Γ12

4ðΩ3−Ω1Þa2a3sinψ′ ¼ 0; (26a) Λ−2Ω1−3Γ11

4Ω1

a2−Γ14

2Ω1

a2þ2Λμ1Ω2

cosψ− Γ12

2ðΩ3−Ω1Þa1a3cosψ′ ¼ 0; (26b)

−C3a3− Γ31

4ð3Ω1þ Ω3Þa

3Ω1−Ω3þ 9Γ11

8Ω1−Γ32

4Ω3

a2þ 3Γ14

4Ω1−3Γ36

8Ω3

a2−3

Λμ1Ω2cosψ

þ 3Γ12

4ðΩ3−Ω1Þa1a3− Γ31

4ð3Ω1þ Ω3Þ

a3

a3

whereψ¼θ −2φ1is the phase angle associated with the principal parametric resonance involving the first spatial form, and ψ′¼3φ1−φ3 represents the phase angle corresponding to the specified internal resonance The steady-state amplitudes,

a1and a3, and the phase angles,ψ and ψ′, can be obtained by solving Eq.(26)by a numerical technique

It appears from Eq.(26)that there are two possibilities for a nontrivial solution: either a1is nonzero and a3is zero, or both are nonzero The first possibility indicates that the specified internal resonance has no effect on the system response and only the principal parametric resonance involving the first mode may occur For the latter possibility, as the first mode is the only one excited by the parametric excitation, the presence of the third mode in the response is possible only by the transfer of energy from the excitedly first mode to the third mode through internal mechanism As mentioned earlier, the only condition for this resonant case possible is that the stable solution for each mode must be larger than the corresponding unstable solution When the stable solution of any mode coincides with its corresponding unstable solution, the transfer of energy stops, and the first mode continues to be excited by the parametric excitation If any mode showing the stable solution lower than its corresponding unstable solution, then this is a non-resonant case and only the principal parametric resonance of the first mode is possible

Case 2 Λ ¼ 2Ω3ANDΩ3≅3Ω1

As in the previous case, the internal resonance has the same relationship, but this time the third mode is excited parametrically Using the same analysis as before and after a series of calculations, we obtain the following stationary solutions:

−C3a3þΛ1μ3Ω2a3sinψ þ Γ31

Λ−2Ω3−Γ32

2Ω3

a2−3Γ36

4Ω3

a2þ2Λμ3Ω2cosψ− Γ31

2ð3Ω1þ Ω3Þ

a3

a3

−C1a1− Γ12

3Ω1−Ω3þ 9Γ 11

8 Ω 1−Γ 32

4 Ω 3

a2þ 3Γ 14

4 Ω 1−3Γ 36

8 Ω 3

a2−1

Λμ3Ω2cosψ

þ 3Γ12

4ðΩ3−Ω1Þa1a3− Γ31

4ð3Ω1þ Ω3Þ

a3

a3

whereψ¼θ−2φ3is the phase angle corresponding to the principal parametric resonance involving the third mode, and as before,ψ′¼3φ1−φ3is the phase angle associated with the specified internal resonance A numerical method is used to find the solutions of Eq.(27)

As inCase 1, we have two possibilities: either a1is zero and a3is nonzero, or neither is zero The first possibility means that only the principal parametric resonance involving the third mode shape may exist The second possibility indicates the presence of the two spatial forms in the system response and, hence, a transfer of energy from the directly-excited third mode to the first mode through internal resonance As before, the only condition for this resonant case is that the unstable solution of each mode must be lower than its corresponding stable solution When the unstable solution of any mode coincides with its stable solution, the energy transfer stops and only the third mode continues to be excited by the parametric excitation If any of the involving modes having the unstable solution larger than its corresponding stable solution, no transfer of energy is possible and only the third mode is parametrically excited

6 Numerical results

In order to get more insight into the occurrence of simultaneous resonances, numerical evaluation was performed for a thin rectangular plate of aspect ratio R¼1.73 The various values of the plate parameters and material constants used for the numerical calculations are as follows: a¼293 mm, b¼508 mm, h¼1 mm, E¼2.385 GPa, ν¼0.45, and ρ¼1200 kg/m3

In the analysis and numerical calculations, Dcr(¼NYT/Nn) denotes the dynamic component NYT of the periodic in-plane force normalized to the lowest critical load Nnand is called the ratio of dynamic critical loading, Pcr(¼NY0/Nn) designates the static in-plane load NY0 normalized to the lowest critical load Nn and is called the ratio of static critical loading, and

Δ (¼2πCm/Ωm) is the decrement of viscous damping Typical results associated with the two cases of simultaneous resonances analyzed above are shown inFigs 2–6

The stationary frequency–response curves are illustrated inFigs 2–4 In these figures, the ordinate amrepresents the steady-state amplitude, corresponding to the involved mode of vibration m, as a function of the plate thickness, while the abscissaλ denotes the exciting frequency (in Hz) The bar over an amplitude means that the amplitude is associated with simultaneous resonances, while the subscript i following the mthmode shape denotes the amplitude is possible due to internal resonance Solid and broken lines in the figures represent the stable and unstable solutions, respectively

Figs 2and3show respectively the amplitudes of the first and third modes (a1and a3) as functions of the excitation frequency (λ, in Hz) InFig 2, the frequency–response curves are associated with the resonances Λ≅2Ω1(principal parametric resonance involving the first spatial form) and 3Ω1≅Ω3(internal resonance involving the first and third mode shapes); hence, the first mode

is parametrically excited and the presence of the third mode is possible only by the transfer of energy from the first mode to the third mode through internal mechanism The result also shows that both stable branches of a1 and a3 are larger than the corresponding unstable branches This means that the energy transfer is possible, and this case of simultaneous resonances can occur The frequency–response curves inFig 3are correspondent to the resonancesΛ≅2Ω3(principal parametric resonance involving the third mode shape) and Ω3≅3Ω1 (the same internal resonance as specified) In this case, the third mode is parametrically excited and its energy is transferred to the first mode through the specified internal resonance However, it can be seen that the unstable branch of a1is over its stable solution; hence, this case of simultaneous resonances cannot occur and only the principal parametric resonance involving the third spatial form is possible

Fig 2 Frequency–response curves associated with the simultaneous resonances Λ≅2Ω 1 (principal parametric) and 3Ω 1 ≅Ω 3 (internal) P cr ¼0.5, D cr ¼0.2, Δ¼0.11.

The interaction between an internal resonance and a principal parametric resonance on the frequency–response curves

is illustrated inFig 4 As can be seen, the parametric response of the first mode occurs whenΛ≅2Ω1 At a certain frequency, however, a small part of energy from the first mode is transferred to the third mode, due to modal coupling between these two modes Consequently, the amplitude of the first mode slightly decreases but remains larger than the one of the third mode After a certain range when both stable and unstable solutions coincide, the energy transfer vanishes; the amplitude of the third mode decays and the steady-state amplitude of the first mode regains its full strength

As explained previously, only the principal parametric resonance involving the third mode is possible, as shown inFig 4 For reference, the theoretical interaction between the internal resonanceΩ3≅3Ω1and the principal parametric resonance Λ≅2Ω3is also illustrated in this figure by dotted curves If this case of simultaneous resonances is possible, i.e., both stable solutions are larger than their unstable solutions, we can see that the system response will be particularly interesting It can

be observed that when the excitation frequency reaches the point where the first mode can be excited through internal resonance, the amplitude of the third mode, which is directly excited by the parametric excitation, drops drastically and becomes less than the amplitude of the first mode which is due to internal resonance This implies that there is a significant transfer of energy from the third mode to the first one As before, when the transfer of energy stops, the amplitude of the first mode disappears and the third mode continues to be excited by the parametric excitation

Fig 3 Frequency–response curves associated with the simultaneous resonances Λ≅2Ω 3 (principal parametric) and 3Ω 1 ≅Ω 3 (internal) P cr ¼0.5, D cr ¼0.2, Δ¼0.11.

Fig 4 Effect of the internal resonance 3Ω 1 −Ω 3 ≅0 on the frequency–response curves corresponding to principal parametric resonances involving the first and third mode shapes P cr ¼0.5, D cr ¼0.2, Δ¼0.11.

The load amplitude–response curves corresponding to the two cases of simultaneous resonances are shown inFigs 5and

6 In the figures, the ordinate am represents again the steady-state amplitude, corresponding to the involved mode of vibration m, as a function of the plate thickness, while the abscissaμmdenotes the load (or excitation) parameter The bar over an amplitude signifies once more that the amplitude is associated with simultaneous resonances, while the subscript i following the mth mode shape denotes the amplitude is possible due to internal resonance Solid and broken lines represent respectively, as before, the stable and unstable solutions

Fig 5(a) shows the amplitude–response curves associated with the resonances Λ≅2Ω1and 3Ω1≅Ω3 Since both unstable solutions are lower than their corresponding stable solutions, this case of simultaneous resonances is totally possible The results illustrate again the domination of the first mode over the third mode The interaction between these two resonances

is presented inFig 5(b) The transfer of energy from the first mode to the third mode closely resembles to the case of frequency response as explained previously

The amplitude–response curves corresponding to the simultaneous resonances Λ≅2Ω3and Ω3≅3Ω1 are illustrated in

Fig 6(a) As shown before, the deflection of the motion is dominated by the first mode, though the third mode is the only one directly excited by the parametric excitation The result shows, however, that the unstable solution of the first mode is much more higher than its corresponding stable solution In practice, this situation signifies a non-resonant case, and only the load–response curves associated with the principal parametric resonance involving the third mode can occur, as shown

in Fig 6(b) In the latter, theoretical results showing the effect of the specified internal resonance on the amplitude– response curves associated with the principal parametric resonance involving the third spatial form of vibration are represented by dotted lines If this case of simultaneous resonances is possible, a very significant transfer of energy from the third mode to the first mode through internal mechanism can be observed

7 Comparison with experimental results

In order to gain further insight into the occurrence of simultaneous resonances, an experimental program[15,17]was undertaken to investigate the effects of internal resonances on the responses of parametrically-excited rectangular plates Experiments were conducted for four different sets of boundary conditions, namely: (1) all edges simply supported; (2) loaded edges simply supported, the others loosely clamped; (3) loaded edges loosely clamped, the others simply supported; (4) all edges loosely clamped The laboratory apparatus, plate specimens, boundary conditions of the specimens, test procedures and recorded data were described in detail in[17], and therefore are not repeated in this paper

Fig 5 (a) Load amplitude–response curves associated with the simultaneous resonances Λ≅2Ω 1 and 3Ω 1 ≅Ω 3 (b) Effect of the specified internal resonance

on the amplitude–response curves associated with the principal parametric resonance involving the first spatial form of vibration λ¼20 Hz, P cr ¼0.5, Δ¼0.11.

For the four test specimens under four different boundary conditions, it was shown in Ref.[17]that various internal resonances were observed Most of the types of internal resonance exhibited by real systems, however, could differ from those generally assumed in the analytical investigations In this case, the response curve associated with the internal resonance mode shape was highly unpredictable and the frequency range of the response was not large

Particularly important are the experimental results, shown inFig 7for the third set of boundary conditions, involving the first and third mode shapes, in which the first mode of vibration is directly excited by the parametric excitation The frequency spectra and mode shape records are also plotted for reference An examination of the frequency spectra reveals that the internal resonance relationship appears exactly as predicted by the theory, and that the natural frequencies associated with the first and third mode are completely commensurable (that is, 3Ω1¼Ω3) The response shape of the third mode appears in the same form as theoretically predicted, and it can be seen that the first mode dominates the response These experimental results show that the theoretical analysis gives an excellent prediction for this particular case of simultaneous resonances In the figure, the dotted curve represents the estimated response curve associated with the principal parametric resonance of the first spatial mode in the absence of internal resonance

For the same plate specimen under the third boundary conditions, experimental results recorded only the frequency– response curves associated with the principal parametric resonance of the third mode, as shown inFig 8by bold lines This means that the specified internal resonance, 3Ω1¼Ω3, has no effect on this principal parametric resonance and, as a consequence, no significant transfer of energy from the directly-excited third mode to the first mode can occur As explained previously, the energy transfer cannot happen since the unstable solution of the first mode is higher than its stable solution Once again, the experimental results ascertain the validity of the analytical results thus obtained

8 Concluding remarks

The present work deals primarily with the problem of the occurrence of simultaneous resonances involving two spatial forms of vibration in a parametrically-excited rectangular plate and covers an existing gap in our understanding of the parametric resonance and dynamic buckling of structures Experimental results are also presented to verify the analytical predictions

The analysis shows that the presence of cubic nonlinearities has an important influence upon the behavior of the system, especially under a condition of internal resonance involving two or more spatial forms of vibration When this type of Fig 6 (a) Load amplitude–response curves associated with the simultaneous resonances Λ≅2Ω 3 and 3Ω 1 ≅Ω 3 (b) Only the principal parametric resonance involving the third spatial form of vibration is possible λ¼60 Hz, P cr ¼0.5, Δ¼0.11.