Thermal Effects on the Nonlinear Forced Responses of Hinged-Clamped Beam with Multimodal interaction

It has widely been shown that a beam with dissymmetric boundary conditions, precisely a Hinged-Clamped beam, presents a rich dynamic due to interaction between its substructures [1]-[9]. If one takes the first three substructures of a nonlinear Hinged-Clamped beam, it can be seen that their frequencies are naturally commensurable. The second mode frequency is three times the frequency of the first mode : ω2 = 3ω1. The frequency of the third mode is given by a linear combination of the frequencies of the first and the second mode, namely : ARTICLE INFO ABSTRACT


Introduction
It has widely been shown that a beam with dissymmetric boundary conditions, precisely a Hinged-Clamped beam, presents a rich dynamic due to interaction between its substructures [1]- [9]. If one takes the first three substructures of a nonlinear Hinged-Clamped beam, it can be seen that their frequencies are naturally commensurable. The second mode frequency is three times the frequency of the first mode : 2 = 3 1 . The frequency of the third mode is given by a linear combination of the frequencies of the first and the second mode, namely :

Article history
Received 04 August 2021 Revised 16 August 2021 Accepted 18 August 2021 Nonlinear analysis of a forced geometrically nonlinear Hinged-Clamped beam involving three modes interactions with internal resonance and submitted to thermal and mechanical loadings is investigated. Based on the extended Hamilton's principle, the PDEs governing the thermoelastic vibration of planar motion is derived. Galerkin's orthogonalization method is used to reduce the governing PDEs of motion into a set of nonlinear non-autonomous ordinary differential equations. The system is solved for the three modes involved by the use of the multiple scales method for amplitudes and phases. For steady states responses, the Newton-Raphson shooting technique is used to solve the three systems of six parametric nonlinear algebraic equations obtained. Results are presented in terms of influences of temperature variations on the response amplitudes of different substructures when each of the modes is excited. It is observed for all substructures and independent of the mode excited a shift within the frequency axis of the temperature influenced amplitude response curves on either side of the temperature free-response curve. Moreover, it is found that thermal loads diversely influence the interacting substructures. Depending on the directly excited mode, higher oscillation amplitudes are found in some substructures under negative temperature difference, while it is observed in others under positive temperature change and in some others for temperature free-response curves.
To the best of our knowledge, minimal studies are available on the influence of temperature change on the substructures of a Hinged-clamped beam involving internal resonance, whereas several studies have shown that a moderate temperature change can be of great influence on structural responses [10]- [12]. Xia et al. [12] stated that changes in the vibration characteristics of the structure due to damage might be smaller than changes in ones due to variations in temperature. Moreover, it is widely known that structures are permanently exposed to the thermal field from accidental or environmental origins that induce temperature variation of their members [10]- [24]. Yaobing et al. [13] have paid attention to the temperature effects on the nonlinear vibration behavior of Euler-Bernoulli beams with different boundary conditions. Effects of uniform temperature rise on the structural responses have been investigated through the authors limited their study to singlemode analysis for the sake of simplicity. They showed that the enhancement of the constraint conditions from Hinged-Hinged, Hinged-Fixed to Fixed-Fixed reduces the temperature effects on the vibration behaviors and that the hardening behavior tends to increase for positive temperature change and to decrease for a negative one. Warminska et al. [16] studied the nonlinear dynamics of a reduced multimodal Timoshenko beam subjected to thermal and mechanical loadings. They considered the first three modes of a simply supported beam at both ends and investigated the influence of temperature on primary resonances around the frequencies. Meanwhile, several studies [25]- [29] have shown that although there are commensurable linear natural frequencies such as 3 = 1 + 2 2 (where = 2 2 ) there is no internal resonance in the case of Hinged-Hinged beam because of the vanishing of the coupling coefficients leading to internal resonance, unlike the cases of Hinged-Clamped and Clamped-Clamped.
The research contribution is to deal with the nonlinear analysis of a forced geometrically nonlinear Hinged-Clamped beam with three modes interactions and submitted to thermal and mechanical loadings. The goal is to investigate the effect of temperature change on the steadystate responses of different modes when the excitation frequency is near the frequency of a given mode. It is observed for all substructures and independent of the mode excited a shift within the frequency axis of the temperature influenced amplitude response curves on either side of the temperature free-response curve. The shift depends on the magnitude and sign of the temperature difference. Oscillation amplitudes of the substructures are remarkably influenced by thermal effects and diversely respond to temperature change depending on the directly excited mode, the sign, and magnitude of temperature difference. For some substructures, higher oscillation amplitudes are observed under negative temperature difference, while it is observed for others in the presence of positive temperature change and for temperature free-response curves for some others.
The structure of the paper is organized as follows: In section 2, the governing equations of the beam for thermoelastic vibration are derived using the extended Hamilton's principle. The PDE of the planar motion is reduced to a coupled nonlinear ordinary differential equations by means of modal projection and then by using the Galerkin orthogonalization method. The system is solved for amplitudes and phases of the three modes using the multiple scales method. The steady-state responses are numerically obtained for different temperature changes in section 3. Some numerical results and discussions are presented in section 4 to illustrate the influences of temperature variations on the response amplitude of the substructures when a given mode is excited. Some concluding remarks are drawn in section 5. The thermal stress-strain relation is given as follows: The total strain in the beam is obtained by summing up the thermal and mechanical strains, that is: The extended Hamilton's principle reads: Where is the work done by external forces, T is the kinetic energy, and U is the elastic energy.
Hamilton's principle leads to the following PDEs of motion governing the thermoelastic behavior of the beam under-considered loadings.  Where the mechanical load ( , ) has been assumed harmonic with spatial distribution ( ) in the direction, and Ω is the excitation frequency.
Introducing the quasistatic assumptions, the acceleration and velocity terms in the direction are neglected [13]. This leads to: One obtains the average strain of the system: Substituting (7) into (5), one obtains the following nonlinear partial differential equations of the planar motion.

Method
Equations (11) can be solved approximately by Galerkin's method. The deflection is approximated by Where are generalized coordinates and are eigenfunctions of the following eigenvalue problem: where are natural frequencies.
Substituting equation (12) into equations (11), multiplying by , integrating over the length and invoking orthogonality of the eigenfunctions, one obtains a set of ordinary nonlinear differential equations The damping is assumed to be modal. The method of multiple scales can be used to construct a uniformly valid asymptotic expansion. According to this method, we assume that each is a function of different time scales which are defined by = And can be expanded in the form Derivatives with respect to time transform according to Where Substituting equations (15) into equations (14) and equating coefficients of like powers of , we obtain The solution of equation (16) can be written as follows where stands for the complex conjugate of the terms to the left. At this point, the are unknown. These are determined from the elimination of secular terms at the next level of the approximation. Substituting equations (18) into (17) leads to 1 2 In order to eliminate the secular terms from 1 , the must be chosen so that the coefficient of exp( 0 ) is zero. This coefficient will contain when Ω is near as well as the nonlinear terms associated with any combinations of the form The eigenfunction of the eigenvalue problem (13) is And are the roots of tan( ) = tanh ( ).
In the case of = 2, the first three roots and frequencies are  (20). We introduce detuning parameters 12 and 13 as follows: where ISSN 2775-2658  13 = −0.0964. In this study we consider primary resonances. To express the nearness of Ω to , we introduce a detuning parameter 2 as follows: Substituting the resonance conditions (21) and (22) into equations (19), and eliminating the secular terms, we have the following solvability conditions The coefficients and can be found in [6], and 3 is the Kronecker delta. To solve equations (23), we write in the polar form Where and are real. Then we separate the result into its real and imaginary parts and obtain = 1,2,3: Where 1 = 1 (3 12 We consider the case of interest = 1,2,3. These cases are discussed separately. Case I: Ω ≈ 1 ( = 1)
Since we are interested in the steady-state response, we can disregard ( ≥ 4). then equations (25a)-(25f) can be reduced to a set of autonomous ordinary differential equations in amplitudes and phase for = 1,2,3. letting ′ 1 = ′ 2 = ′ 3 = ′ 1 = ′ 2 = ′ 3 = 0, one obtains the following algebraic equations giving the steady-state responses.  Proceeding in a similar way, we obtain the following algebraic equations giving the steadystate responses. The following algebraic equations are obtained when the excitation frequency is near the natural frequency of the third mode

Temperature-free responses
At room temperature, the resolution of systems of equations (27)- (29) leads to the following results in agreement with the work done by authors of papers [5]- [7]. The frequency response curves are presented in figures (2)-(4) for excitation frequency near the frequency of the first mode and in figures (5)-(6) for excitation frequency near the frequency of the second mode. When the excitation frequency is near the frequency of the first mode, the response amplitude is dominated by the first mode (Fig. 3) with a small amplitude for the second mode (Fig. 4) and a very weak amplitude for the third mode (Fig. 5). When the second mode is excited, Fig. 6 shows that there is a range of excitation frequency for which the response is dominated by the first mode (internal resonance). The third mode still presents small amplitude (Fig. 7).

Responses under temperature change
Non-free thermal stress responses are presented in this section. In the presence of temperature change, nonlinear algebraic systems of equations (27)- (29) are solved by the mean of Newton-Raphson shooting technique to investigate the thermal effects on the response amplitude of each of the three modes involved when one of the modes is directly excited. All figures presenting temperature-influenced responses are depicted alongside the corresponding temperature-free response in order to appreciate the temperature effect on these responses.
When the excitation frequency is near the frequency of the first mode, Fig. 8a and Fig. 8b present the effect of temperature on the amplitude response of the first mode respectively for ∆ = ±30 ° and ∆ = ±100 °. It is globally observed a shift in frequency along the frequency axis, to the left for positive temperature difference and to the right for negative one 296 International Journal of Robotics and Control Systems ISSN 2775-2658 Vol. 1, No. 3, 2021, pp. 285-307 in comparison to the temperature-free curve (∆ = 0 °). These results agree with those obtained by authors of papers [10]- [13]. The shift increases with the absolute value of temperature difference. There is no notable change, however, in amplitude between temperature-free response and responses under the thermal influence in this case. Fig. 9a and Fig. 9b present the effect of temperature change on the response amplitude of the second mode for temperature difference ∆ = ±50 ° and ∆ = ±100 ° respectively. Similar observation as previously made on the responses of the first mode in terms of a shift in frequency domain due to temperature change can be made here. However, there is a change in regard to the maximal amplitude reached for different responses. In comparison with the temperature-free response curve, amplitudes are larger for negative temperature difference and lower for positive one, for the discussed case of the second mode responding when the first mode is directly excited. As a quantitative example, figure 8a presents maximal amplitude of about 2.5, 1.5, and 0.8 respectively for ∆ = − 50 ° , ∆ = 0 °, and ∆ = + 50 °. These values turn into around 4.5, 1.5, and 0.6 respectively for ∆ = − 100 ° , ∆ = 0 °, and ∆ = + 100 ° in Fig. 9b. When the excitation frequency is near the frequency of the second mode, Fig. 10a -Fig. 10c show the effect of temperature change on the response amplitude of the first mode, Fig. 11a -Fig. 11c present the same results for the second mode, while Fig. 12a and Fig.  12b depict these results for the third mode.
For the first mode ( Fig. 10a -Fig. 10c), results show the previously described shift in the frequency domain of the response amplitude for negative and positive temperature change on either side of the thermal free-response curve. Nonetheless, there is a change in amplitude for different curves. However, in opposition to the aforementioned case, higher amplitudes are observed for positive temperature difference and smaller for negative one as compared with room temperature response. Qualitative change is also observed between these response amplitude curves, and it can be seen that as positive temperature difference increases, the profile of the response curve gradually loses its resonant profile.
The responses of the second mode of vibration when the same mode is excited in the presence of thermal influence are presented in figures 10a-10c. One notes the shift in frequency caused by the temperature difference on either side of the room temperature response depending on the sign of temperature difference. In addition, from a quantitative point of view, it is observed that the temperature free-response curve presents higher amplitude over the temperature influenced response curves. This situation is quite different from those observed previously. The responses of the third mode ( Fig. 12a and Fig. 12b) reacting to the second mode directly excited in the presence of temperature change globally present the same behavior as described for the response of the second mode, but with weak amplitude.
These results clearly state that temperature influence on the responses of a hingedclamped beam undergoing nonlinear vibration under a periodic mechanical excitation is important. This influence is diversely observed on the different substructures when multimodal interaction is considered. The changes are observed both qualitatively and quantitatively and are not the same on the different substructures when one of the modes is directly excited. These outcomes show that one should pay more attention when designing structural elements involving vibrational behavior under simultaneous actions of mechanical excitation and thermal loads. So far, temperature-free analysis has revealed that when multimodal interaction is involved in a hinged-clamped beam vibration, there is a possibility of internal resonance with modes other than the directly excited mode responding with higher amplitude in some frequency interval due to energy transfer in between the interacting substructures.
Actually, it is shown that the effect of temperature mistakenly evaluated as a flat rate in several structural and civil engineering designs needs to be paid more attention as far as multimodal interaction is involved under the action of mechanical excitation. There is a shift in

Conclusion
The vibrational behavior of a hinged-clamped beam with multimodal interaction under combined actions of periodic mechanical loadings and thermal loads has been investigated. Nonlinear PDEs governing the planar motion of the thermoelastic problem has been established and reduced to a set of nonlinear non-autonomous ODEs. For the sake of steadystate solutions, the ODEs have been reduced to three sets of parametric nonlinear algebraic equations solved by the mean of the Newton-Raphson technique. Results have been presented in terms of temperature effects on the response amplitudes of each of the three considered modes when one of them is directly excited. It came out that thermal effects are of great influence on the responses of such a system. More precisely, the temperature change affects the amplitude response curves by shifting them on either side of the temperature-free response curve in the frequency domain, depending on the sign and magnitude of the temperature difference. In terms of oscillation amplitude, the influence of temperature is diversely observed on the substructures depending on the magnitude and sign of the temperature difference but also on the mode directly excited. It is found that higher oscillation amplitude is observed for negative temperature difference for some substructures, while it is observed for positive one for some others and for temperature free-response curve for others. It should, however, be pointed out that for some substructures, temperature change does not significantly influence the oscillation amplitude. As a consequence, this study should contribute to advising civil and Structural designers that evaluating temperature influence on the structural elements engaged in vibrational behavior at a flat rate can produce inefficient results. The future work will be dedicated to the investigation of the dynamical aspect of the studied system in order to have more insights into the system responses to the combined effects of thermal loads and mechanical excitations in the presence of multimodal interaction.

Conflicts of Interest:
The authors declare no conflict of interest.