Principal Parametric Resonance Research Articles

Nonlinear deterministic dynamic analysis of a GPL micropipe conveying pulsating laminar flow

In several industries, the handling of fluid-filled micropipes is common. New-brand structures are increasingly being manufactured using advanced materials. This article tackles one of the crucial dynamic analyses that an advanced fluid-filled micropipe encounters. The dynamics of a graphene nanoplatelets-reinforced micropipe (GPL micropipe) under pulsating laminar flow for the first time is examined. The Euler-Bernoulli beam model follows the modified couple stress theory (MCST) and von-Karman’s nonlinear strain to formulate the problem. The impression of the flow profile exerted by a real flow as a consequence of fluid viscosity, is taken into account. The plug flow model results are compared to those regarding real flow consideration. Using the method of multiple scales (MMS), we determine the instability area and the steady state response of the GPL micropipe subjected to the principal parametric resonance of one of its modes due to pulsating laminar flow by applying this method to the discretized couple nonlinear gyroscopic governing equations. The Floquet theory treats the linear equations and fourth-order Runge–Kutta (RK4) method attacks nonlinear ones to validate MMS findings. It is confirmed that the 0.3 % addition of the GPL to matrix phase significantly enhances its advanced attributes, making it a highly effective material. The GPL reinforcement phase in the FGO pattern increases the critical flow velocity that exhibits the micropipe to static instability by 37.98 % , and critical flow stimulation amplitude fraction by 152.19 % , while declines instability area bandwidth by 36.95 % , and steady state response by 67.17 % relative to a non-reinforced micropipe. A denser flow degrades the corresponding values by 18.40 % , 42.59 % , 41.67 % , and 227.28 % in comparison to a lighter fluid. The declared findings provide crucial insights into the key design elements affecting significantly the functioning of fluid-filled GPL micropipes system, and regulate future research and expand openings for industry applications.

Nonlinear dynamic analysis of a spring coupled double beam based piezoelectric energy harvester under parametric base excitation

This paper proposes a base excited, spring coupled double cantilever beam with attached tip mass based piezoelectric energy harvester. The dimensions of the beams are so chosen that the system experiences 1:1 internal resonance. The governing coupled electro-mechanical equations of motion of the system are obtained by using the Lagrange principle which is reduced to the temporal form by using generalized Galerkin’s method. The governing temporal equations of motion are in the form of that of a parametrically excited coupled system with geometric and inertial nonlinearities. These nonlinear equations are solved using the method of multiple scales and the results are compared with those solved by numerical method (4th order Runge Kutta method) for principal parametric resonance conditions. Parametric studies are conducted by varying the tip masses of the beams, coupled spring stiffness, load resistance, external excitation frequency and excitation amplitude. Results show that by introducing spring coupling, the system operational frequency range increases significantly and also energy transfer process takes place due to the internal resonance. Significant enhancement of voltage and power are observed in the present work in comparison to the earlier studied equivalent single beam-based energy harvester.

Effect of Angular Speed Variations on the Nonlinear Vibrations of a Rotational Spring-Mass System

A rotating spring-mass system is considered using polar coordinates. The system contains a cubic nonlinear spring with damping. The angular velocity harmonically fluctuates about a mean velocity. The dimensionless equations of motion are derived first. The velocity fluctuations resulted in a direct and parametric forcing terms. Approximate analytical solutions are sought using the Method of Multiple Scales, a perturbation technique. The primary resonance and the principal parametric resonance cases are investigated. The amplitude and frequency modulation equations are derived for each case. By considering the steady state solutions, the frequency response relations are derived. The bifurcation points are discussed for the problems. It is found that speed fluctuations may have substantial effects on the dynamics of the problem and the fluctuation frequency and amplitude can be adjusted as passive control parameters to maintain the desired responses.

Effect of nonlinear inertia and Duffing-type nonlinearity on the dynamic behaviour of nonlinear parametrically excited cantilever beams with tip mass

The dynamic response of parametrically excited cantilever beams with a tip mass (PECBM) is investigated in this paper. Employing the Method of Multiple Scales (MMS) and the Method of Varying Amplitudes (MVA), the frequency-response equations are obtained and the stability of the response is analysed. MVA results show that, for frequency ranges around the principal parametric resonance, the term containing a linear combination of nonlinear inertia and Duffing-type nonlinearity in the frequency response equation can tend to zero, resulting in an exponential growth of the vibrations. These results are supported by numerical simulations obtained by direct integration (DI) of the equation of motion, while the MMS fails to predict the critical frequency at which this phenomenon occurs. Furthermore, based on the MVA, a criterion for determining the hardening and softening characteristics of PECBM is developed, showing that both Duffing nonlinearity and nonlinear inertia terms contribute to determining this characteristic. To verify the results, experimental measurements of the response of a PECBM are presented, showing good agreement with analytical and numerical results. It is shown that the mass added at the cantilever tip can change the characteristics of the system's dynamics, enhancing the softening behaviour of the PECBM.

Analysis on Thermo-Electrical Principal Parametric Resonance of an Axially Moving Piezoelectric Thin Plate

In this paper, principal parametric resonance of an axially moving piezoelectric rectangular thin plate under thermal and electric field is investigated. Based on Kirchhoff–Love plate theory and Von Karman theory, the transverse vibration differential equation of a piezoelectric rectangular thin plate under thermal and electric field is derived by using Hamilton’s principle. The dimensionless vibration equation of piezoelectric rectangular thin plate with parametric excitations is discretized by Galerkin’s method. Then, the multiple scales method is applied to derive amplitude-frequency response equation and the stability conditions of the steady-state solution are obtained by Lyapunov stability theory. Numerical method is used to find the influences of specific parameters on the vibration performance and stability of the system. Based on the global bifurcation diagram and corresponding response diagram, the influences of bifurcation control parameters on the nonlinear dynamic characteristics of the system are discussed. Numerical results illustrate that the system amplitude frequency characteristic curve presents soft spring characteristics. There are periodic and chaotic motions with the increase of velocity and the central temperature difference, and the decrease of plate thickness and velocity will result in the decrease of chaotic threshold. The results also show that increase the velocity perturbation amplitude can prolong the chaotic motion.

Multi-harmonic resonance of pipes conveying fluid with pulsating flow

The parametric excitation analysis of pipes conveying pulsating fluid usually focuses on the principal parametric resonance. In this paper, the principal parametric resonance and the combination resonance under parametric excitation are compared for the first time. A dynamic model of a fluid-conveying pipe with elastic torsional constraints at both ends is established. The nonlinear partial differential-integral equation is derived for governing the transverse vibration. The natural frequencies and modes of pipe are obtained. Moreover, the analytical solutions of stable region and nonlinear amplitude-frequency responses for three kinds of parametric resonances are derived. Then, the analytical results are verified numerically. The results show that there is no differential-type combination parametric resonance for pipes supported at both ends. Increasing material damping or torsional stiffness and decreasing mean velocity can increase the stable region of the pipe. However, it is interesting to note that when combination parametric resonance occurs between symmetric and antisymmetric modes, the effect of mean velocity on stability is reduced. In addition, more than one form of resonance occurs in some regions. The sum-type combination resonance is easier to be excited than the principal parametric resonance. Among them, the combination of first-fourth mode is the most sensitive to the torsional boundary, and its response is the largest. In summary, this paper demonstrates that the combination parametric resonance, which has not been researched deeply yet, can be more destructive than the principal parametric resonance.

Parametric resonance of multi-frequency excited MEMS based on homotopy analysis method

The nonlinear dynamics and control of micro electromechanical systems (MEMS) is an important topic at present and homotopy analysis method (HAM) is an effective semi-numerical and semi-analytical method for solving strongly nonlinear problems. In this paper, the parametric resonance of MEMS with multi-frequency excitation is studied by HAM. Firstly, the differential equation of electrostatically driven microbeam is processed by Taylor expansion, and it is transformed into the parametric motion model with multi-frequency excitation. Then, the approximate solution and amplitude–frequency response equation of the system are obtained by HAM, and compared with the numerical solution. Finally, the influence of direct current (DC) and alternating current (AC) on principal parametric resonance and superharmonic resonance is discussed. The results show that HAM is an effective method to analyze the parametric vibration of multi-frequency excitation system, and the amplitude–frequency response curve of microbeam about parametric motion depends on the time scale in high DC and high AC state. This study effectively extends the application of HAM in parametric resonance, which is of great significance to the study of nonlinear vibration of MEMS.

Principal parametric resonance analysis of a rotating agglomerated nanocomposite beam employing the Chebyshev–Ritz method

The dynamic instability owing to a principal parametric resonance of a rotating agglomerated nanocomposite beam reinforced with randomly oriented carbon nanotubes (a CNTRC beam) in a uniform thermal ambient is explored. The rotation speed fluctuates as a simple harmonic function of time around a mean value. A method on the basis of the Mori–Tanaka’s approach is implemented to determine the effective thermo-elastic constants of the agglomerated CNTRC beam. By considering the excitation frequency of the rotating speed two times the one of longitudinal or flapping natural frequencies, the principal parametric resonance is activated. The Bolotin’s technique and the Floquet theory are implemented to illuminate the instability region of the rotating agglomerated CNTRC beam. The impacts of the agglomeration parameters, and the CNT volume fraction on dynamic stability boundaries are illustrated. The outcomes reveal that for a rotating completely (partially) agglomerated CNTRC beam, the enlargement of the ratio of the inclusions volume with respect to the volume of the nanocomposite (the decrease in the ratio of the volume of the CNT within the inclusions with respect to the total volume of CNTs), reduces the instability region of the first flapping mode. Meanwhile, the required excitation dynamic amplitude for the activation of the dynamic instability improves.

Nonlinear dynamics of parametrically excited cantilever beams with a tip mass considering nonlinear inertia and Duffing-type nonlinearity

The response of a parametrically excited cantilever beam (PECB) with a tip mass is investigated in this paper. The paper is mainly focused on accurate prediction of the response of the system, in particular, its hardening and softening characteristics when linear damping is considered. First, the method of varying amplitudes (MVA) and the method of multiple scales (MMS) are employed. It is shown that both Duffing nonlinearity and nonlinear inertia terms govern the hardening or softening behaviour of a PECB. MVA results show that for frequencies around the principal parametric resonance, the term containing a linear combination of nonlinear inertia and Duffing nonlinearity in the frequency response equation can tend to zero, resulting in an exponential growth of the vibrations, and results are validated by numerical results obtained from direct integration (DI) of the equation of motion, while the MMS fails to predict this critical frequency. A criterion for determining the hardening and softening characteristics of PECBs is developed and presented using the MVA. To verify the results, experimental measurements for a PECB with a tip mass are presented, showing good agreement with analytical and numerical results. Furthermore, it is demonstrated that the mass added at the cantilever tip can change the system characteristics, enhancing the softening behaviour of the PECB. It is shown that, within the frequency range considered, increasing the value of the tip mass decreases the amplitude response of the system and broadens the frequency range in which a stable response can exist.

Nonlinear planar and non-planar vibrations of viscoelastic fluid-conveying pipes with external and internal resonances

In this study, nonlinear vibrations of a viscoelastic fluid-conveying pipe under in-plane and out-of-plane transverse excitations are investigated, with emphasis on external and internal resonances. For this purpose, a three-dimensional pipe model that couples the in-plane and out-of-plane vibrations is developed based on the Euler–Bernoulli beam theory together with the Kelvin–Voigt viscoelasticity. The Galerkin method is performed on the nonlinear governing equations, and the resultant discretized equations are solved via the pseudo-arclength continuation method and a direct integration method. The dynamical responses of the pipe are examined in both sub- and super-critical regions and presented by plotting the frequency–amplitude curves, force–amplitude curves, basins of the attraction, time histories, and phase portraits. A comparison between the simplified two-dimensional model and the present three-dimensional model is presented to showcase the possibility of the in-plane and out-of-plane vibrations in regions of primary and principal parametric resonances, highlighting the modal interaction between the in-plane and out-of-plane transverse modes. In the presence of a two-to-one internal resonance, in addition to typical jumping and hysteresis, the two-peak resonance behavior is observed. The numerical results not only show complex nonlinear dynamics of the fluid–structure interaction system, but also provide hints for safe design and novel application of such structures.

Dynamic analysis of an axially moving underwater pipe conveying pulsating fluid

In this paper, both linear and non-linear dynamics of a slender and uniform pipe conveying pulsating fluid, which is axially moving in an incompressible fluid, are comprehensively studied. The vibration equations of the system are established by considering various factors, including a coordinate conversion system, an “axial added mass coefficient” describing the additional inertia forces caused by the external fluid, the Kelvin–Voigt viscoelastic damping, a kind of non-linear additional axial tension, and the pulsating internal fluid. The vibration equations are discretized by the Galerkin procedure and solved by the Runge–Kutta approach, and the validity of the solution procedure is carefully checked. After that, the linear and non-linear responses of the system are studied when the internal flow velocity and the axially moving speed of the pipe are small. For linear responses, the Kelvin–Voigt viscoelastic damping has great influences on the second and third modes of the system. For the non-linear dynamic, the results are rich and changeful, including the first and second principal parametric resonances, the secondary resonance, the combination resonance, period-1 motion, quasi-periodic motion, and chaotic motion. Finally, the influence of several key system parameters on the non-linear responses is analyzed.

Traditional and non-traditional active nonlinear vibration absorber with time delay combination feedback for hard excitation

In this work, vibration suppression of a base excited nonlinear spring–mass primary system subjected to simultaneous external hard harmonic and parametric excitations is carried out by using a modified traditional and non-traditional active nonlinear vibration absorber (ANVA). The ANVA consists of a mass, time delayed damper and a PZT stack actuator in series with a linear and nonlinear spring. The ANVA utilizes various combinations of time delayed displacement, velocity and acceleration feedback gains of the primary system for vibration reduction. The governing coupled nonlinear equations of motion for the system with weighted modal matrix approach have been derived and solved by the method of multiple scales (MMS) for simultaneous superharmonic, principal parametric and primary resonance with 1:1 internal resonance conditions. The reduced equations obtained from MMS are solved using Newton’s method, which is then compared with the numerical methods, showing good agreement. The trivial state instability regions are also analyzed for various feedback gains. The parametric analyses are carried out for various system parameters, viz., variation in the nonlinear stiffness, time delay in the damper, changes in the amplitude of excitations, time delay in the feedbacks, stiffness of the absorber and various combinations of feedback gains through different feedbacks. From these parametric analyses, stable and unstable regions of operating frequencies are obtained at which the system response amplitude is minimum for a wide range of operating frequencies. Further, it is shown that 100% vibration reduction of the system can be achieved for a certain range of operating frequencies.

Nonlinear dynamic responses of functionally graded graphene platelet reinforced composite cantilever rotating warping plate

In this study, the nonlinear dynamic behaviors are investigated for the functionally graded graphene platelet (FGGP) reinforced composite rotating warping blade under combined the aerodynamic force and centrifugal force. The blade is simplified to a FGGP reinforced composite rotating warping cantilever plate with a rectangular cross-section. Four distribution patterns of the graphene platelets (GPLs) along the thickness of the plate are considered when establishing the dynamic model. The effective material parameters, Poisson's ratio, mass density and Young's modulus of the FGGP reinforced composite rotating warping cantilever plate are calculated based on the modified Halpin-Tsai model and mixture rule. Considering the warping effect, the dynamic governing equations of motion for the rotating FGGP reinforced composite cantilever plate subjected to the aerodynamic force and centrifugal force are established by using the first-order shear deformation theory and Hamilton's principle. Galerkin approach is exploited to transform the partial differential equations of motion to the ordinary differential equations of motion the FGGP reinforced composite rotating warping cantilever plate. Four-dimensional autonomous averaging equations of the system are obtained by using the asymptotic perturbation method. The parameter analyzes on the amplitude-frequency and force-amplitude responses of the FGGP reinforced composite rotating warping cantilever plate are carried out by considering the 1:2 internal resonance and principal parametric resonance. The stability of the steady solution of the autonomous system is discussed. The influences of the GPL distribution patterns, rotating speed, aerodynamic force and damping on the chaotic dynamics are investigated for the FGGP reinforced composite rotating warping cantilever plate.

Internal resonance in a MEMS levitation force resonator

In this work, we explore internal resonances in a levitation force microelectromechanical system-based actuator assuming flexible cantilever and clamped–clamped microbeam configurations. The levitation force is generated through a special arrangement of two-side stationary charged substrates and a central grounded stationary strip. The design assumes as well an upper flexible microbeam strip. The DC part of the excitation voltage pushes up the moving microbeam away from its lower stationary electrode. A superimposed harmonic AC voltage lets the flexible strip vibrate around its equilibrium position. Energy exchange among the computed lower vibration frequencies and their respective frequency and force response curves are explored for possibilities of principal parametric and internal resonances interactions. The generated responses are computed using the bifurcation toolbox MatCont. The effect of Von Karman nonlinearity to power the energy exchange within vibration modes is explored and dominant factors in the MEMS design for resonances are found from a two-parameter bifurcation analysis. Our results allow optimizing the micro-actuator device performance for future applications.

A study on small scale thermal dynamic instability of rotating GPL-reinforced microbeams under principal parametric resonance stimulation of axial and transversal modes regarding the proportional damping

A study on the principal parametric stimulation of rotating microbeams strengthened by means of graphene platelet is presented. The microbeam is exposed to a temperature gradient. On the basis of the assumptions of the hypothesis of the Timoshenko for beams alongside the modified couple stress hypothesis the nonlinear motion equations are achieved. It is supposed that the rotating speed of the microbeam varies harmonically about a mean quantity. The frequency of the harmonic term of the rotating velocity is presumed to be almost two times an axial or a transversal natural frequency. In such situation, the principal parametric agitation is stimulated. Assuming a proportional damping, a least square scheme is employed to define the Rayleigh’s coefficients. The impacts of the rotating speed, the weight fraction as well as the scattering pattern of the graphene platelets, and the damping coefficient on the presented results are examined. The results illuminate that as a consequence of the addition of the damping coefficient, the critical incitement amplitude constant increases more for reinforced microbeams rather than not-reinforced microbeams. Moreover, by means of enlarging the weight fraction of the graphene platelets, the critical incitement amplitude constant develops more for a rotating X microbeam rather than an O microbeam. Additionally, at moderate to high magnitudes of the damping coefficient, the instability area border aligned with the fundamental axial mode is impressed by the graphene platelet scattering pattern although it is invariant for small values of the damping coefficient. Moreover, by the development of the temperature the instability area border aligned with the fundamental transversal mode gets broader, while the critical incitement amplitude coefficient decreases; the both more for on O scattering pattern for the graphene platelet. Furthermore, the instability region boundary is more influenced by the graphene platelet weight fraction, while the critical incitement amplitude coefficient is more impressed by the damping coefficient design value.

Hyperelastic Microcantilever AFM: Efficient Detection Mechanism Based on Principal Parametric Resonance

The impetus of writing this paper is to propose an efficient detection mechanism to scan the surface profile of a micro-sample using cantilever-based atomic force microscopy (AFM), operating in non-contact mode. In order to implement this scheme, the principal parametric resonance characteristics of the resonator are employed, benefiting from the bifurcation-based sensing mechanism. It is assumed that the microcantilever is made from a hyperelastic material, providing large deformation under small excitation amplitude. A nonlinear strain energy function is proposed to capture the elastic energy stored in the flexible component of the device. The tip–sample interaction is modeled based on the van der Waals non-contact force. The nonlinear equation governing the AFM’s dynamics is established using the extended Hamilton’s principle, obeying the Euler–Bernoulli beam theory. As a result, the vibration behavior of the system is introduced by a nonlinear equation having a time-dependent boundary condition. To capture the steady-state numerical response of the system, a developed Galerkin method is utilized to discretize the partial differential equation to a set of nonlinear ordinary differential equations (ODE) that are solved by the combination of shooting and arc-length continuation method. The output reveals that while the resonator is set to be operating near twice the fundamental natural frequency, the response amplitude undergoes a significant drop to the trivial stable branch as the sample’s profile experiences depression in the order of the picometer. According to the performed sensitivity analysis, the proposed working principle based on principal parametric resonance is recommended to design AFMs with ultra-high detection resolution for surface profile scanning.

A New Robust Method to Investigate Dynamic Instability of FTV for the Double Tripod Industrial Driveshafts in the Principal Parametric Resonance Region

The present work aims to design a robust method to detect and certify the deterministic chaos or ergodic process for the forced torsional vibrations (FTV) of a double tripod industrial driveshaft (DTID) in transition through the principal parametric resonance region (PPRR) which is considered by the researchers in the field as one of the most important resonance regions for the systems having parametric excitations. The DTID’s model for FTV considers the following effects: nonuniformities of inertial characteristics of the DTID’s elements, the harmonic torque excitation induced by the asynchronous electrical motor used for a heavy-duty grain mill, and the harmonic reaction torque generated by different granulation of the substance needed to be milled. Based on these aspects, a model of the FTV for the DTID was designed which was a modified, physically consistent model already used by the authors to investigate the FTV of automotive driveshafts (homokinetic transmission). For the DTID elements, the dynamic instability for nonstationary FTV in the PPRR using time–history analysis (THA) was analyzed—THA represents the phase portraits. Time–history analysis is a detection method for possible chaotic dynamic behavior for the nonstationary FTV (NFTV) in transition through PPRR. If this dynamic behavior was seen, a new robust method LEA–PM was created to certify and confirm the deterministic chaos for the NFTV of DTID. The new method, LEA–PM, is composed of the Lyapunov exponent’s approach (LEA) coupled with the Poincaré Map (PM) applied to the global system of differential equations that describe the FTV of DTID in the PPRR. This new robust method, which embeds LEA and PM, LEA–PM, establishes if the mechanical system has a deterministic chaotic dynamic behavior (strange attractor) or an ergodic dynamic process in this resonant region. LEA represents a new method that includes not only the maximal Lyapunov exponent method (MLEM) but also new mathematical criteria that is “the sum of all Lyapunov exponents has to be negative” which, coupled with MLEM, indicates the presence of deterministic chaos (strange attractors). THA–LEA–PM had been used for the NFTV of DTID computing the phase portraits, the Lyapunov exponents, and representing the Poincaré Maps of the NFTV for the DTID’s elements in transition through PPRR, founding deterministic chaos or ergodic dynamic behavior. Based on the obtained results, numerical simulations revealed the pitting manifestations of the DTID’s elements, typical for the geared systems transmission, mentioned recently in experimental data research for the homokinetic transmissions. Using the new robust method, THA–LEA–PM (time–history analysis coupled with LEA–PM) can be used in future research for chaotic dynamic analysis of DTID’s NFTV transition through superharmonic resonances, subharmonic resonances, combination resonances, and internal resonances. Time–history analysis as a detection method for chaos and LEA–PM as a certifying method for deterministic chaos can be integrated as a design tool for DTID’s FTV control of the homokinetic transmission.

Detection of thermal size-dependent dynamic instability of rotating CNTRC microbeams with damping under parametric resonance excitation

Small-scale dynamic instability analysis of a rotating nanocomposite damped microbeam reinforced with carbon nanotube exposed to a temperature growth is addressed in this research. The modified couple stress theory, along with the Timoshenko beam hypothesis is explored to develop the motion equations. The practical thermo-mechanical characteristics are determined by resorting to the modified rule for mixtures. The Chebyshev–Ritz approach is exerted on the nonlinear motion equations to achieve the discretized form of the motion equations. The influence of structural damping is integrated into the governing equations. Considering a varying rotating speed, a possible instability resulting from principal parametric resonance stimulation is examined using the Bolotin's route and the Floquet hypothesis. The influences of the material length scale constant, distribution type of carbon nanotubes, temperature, and the damping constant on the dynamic instability boundaries are investigated. The outcomes enlighten that a rotating nanocomposite microbeam with O distribution pattern for carbon nanotubes is more sensitive to the changing temperature.

Application of the Chebyshev–Ritz route in determination of the dynamic instability region boundary for rotating nanocomposite beams reinforced with graphene platelet subjected to a temperature increment

The Chebyshev–Ritz route is exploited for the sake of the examination of the principal parametric resonance instability boundary of a rotating nanocomposite beam subjected to a temperature rising. The parametric resonance excitation is because of a time varying rotational speed. It is assumed that the rotating speed oscillates harmonically about a constant value. Resorting to the Halpin–Tsai micromechanical, the practical mechanical and thermal characteristics are estimated. The Newton–Raphson scheme is utilized to determine the prestressed configuration. The structural damping influence is incorporated to the motion equations. By the assumption that the frequency of the harmonically varying term of the rotating speed is approximately twice the one of the natural frequencies, the principal parametric resonance excitation is activated. The Bolotin’s technique alongside the Floquet theory are utilized to define the instability region boundaries. The impacts of various effectual parameters including the graphene platelet weight fraction, and its dispersion pattern, the design value of the damping constant, the boundary condition type, and the rotating speed on the instability region are examined. It is inferred that by the increment of the weight fraction of the graphene platelet as well as the design value of the damping ratio, the critical excitation amplitude increases.

Modeling and Analysis of FBV Movements for Automotive Driveshafts in the PPR Region

This research’s goal is to model and analyze the forced bending vibrating (FBV) movements for the elements of an automotive driveshaft using a perturbation technique, the asymptotic method approach (AMA), in the region of principal parametric resonance (PPR). The PPR region was chosen because the principal parametric resonance region is one of the essential resonance regions. The model of FBV movements for the automotive driveshaft (AD) considers the aspects of the following phenomena: geometric nonuniformity of the AD elements and shock excitation due to the road. To overcome the equations for the FBV movements of the AD elements, all inertia characteristics were reduced to the longitudinal ax of the midshaft using the variation of the geometric moments of inertia with the concurrent axis and Stener’s theorem. The midshaft of the AD was considered a Timoshenko simply supported beam with a concentrated mass at both ends and springs and dampers for linear and rotational movements at both ends. To determine the equations describing the FBV movements of the AD elements, Hamilton’s principle was used. After establishing the equations of motion for each AD element coupled with the specific boundary conditions, the amplitude and the phase angle were computed for stationary and nonstationary motion in the PPR region using the first order of the AMA, and the dynamic instability frontiers were determined based on the same equations. The dynamic behavior of the AD was investigated concerning the variation of the damping ratio and the variation of the parametric excitation coefficient. The AMA coupled with the model of FBV movements for the AD exhibits the future research directions for analyzing FBV movements for the AD in the regions of superharmonic resonances, subharmonic resonances, combination resonances, internal resonances, and simultaneous resonances. Additionally, the AMA can predict the endurance of the AD and design control of car damping systems.