Subharmonic Resonance Of Order Research Articles

Nonlinear vibration of the axle box front cover of high-speed train and its effect on connecting bolts

In order to explore the causes for failure of connecting bolts between the axle box front cover and axle box body on high-speed train, field test was carried out firstly in this work. The results showed that the wheel had 20th-order polygonization and the front cover vibrated drastically with the dominant frequencies of 290 Hz and 1160 Hz at the speed of 300 km/h, which were 1/2 and 2 times the rotational frequency 580 Hz of 20th-order polygonal wheel, respectively. Secondly, bench test was carried out to reproduce the unique and drastic vibration phenomenon of the field test, it was proved that the wheel polygonization took responsibility for the drastic vibration of the front cover. OMA (Operational Modal Analysis) and theoretical analysis were made next to certify existence of nonlinear vibration (subharmonic resonance of order 1/2 and superharmonic resonance of order 2) of the front cover under 20th-order polygonization. Finally, the cause of bolt failure and dynamic characteristic of the front cover resulted from different wheel polygon order was studied, results showed that relative vibration existed between axle box body and front cover. The relative vibration enlarged when the front cover was in nonlinear vibration under 20th-order wheel polygonization, which definitely caused a larger dynamic transverse load for the bolts and accelerated its vibration fatigue.

Size-dependent secondary resonance of a piezoelectrically laminated bistable MEMS arch resonator

This paper deals with the size-dependent nonlinear dynamics of a bistable fully clamped resonant shallow microarch under subharmonic resonance of order 1/2. The microarch is laminated between thin piezoelectric layers and subjected to a combination of piezoelectric and electrostatic actuations. Nonlinearities associated with clamped ends, curvature and electrostatic actuation are taken into account in the formulation of the problem. Size effect is considered using the so-called strain gradient theory, and the result is compared with the classical theory. The frequency response is numerically obtained using the shooting technique and in some cases analytically using multiple time scales method. The effect of piezoelectric actuation and the initial rise on the resonant behavior of the system are investigated in the pre and post-buckling configurations as well as the bistability band of the microarch. Applying a DC voltage to the piezoelectric layers imposes a longitudinal compression or tension force along the microbeam depending on the polarity of the voltage. As a result, the initial rise and bending stiffness of the microarch changes. This property is used as a frequency tuning tool in this paper. It is shown that the resonance frequency of the microarch resonator can be tuned in both forward and backward directions thanks to appropriate piezoelectric actuation either in the pre-buckling or the post-buckling configuration. It is also revealed that taking advantage of piezoelectric actuation, the bandwidth of the static bistable band can be adjusted effectively. The results of this paper can be used for designing MEMS resonators.

Experimental investigation into the nonlinear dynamics of a bistable laminate

An experimental study of the single-well and twin-well, also referred to as intra-well and inter-well, respectively, nonlinear dynamics of a bistable composite laminate is presented. An asymmetric four-ply [0/90/0/90] carbon fiber laminate with two cylindrical stable equilibria supported at its center and free at all boundaries is used for the experimental testing. The mechanical bistability makes the laminate able to snap from one stable equilibrium to the other when displacement reaches a critical value. This snapthrough motion is highly nonlinear and associated with large-amplitude vibrations. This property opens chances for bistable laminates in morphing and energy-harvesting applications. An electromechanical shaker is used to excite the laminate at its center by a controlled amplitude and frequency excitation. The dynamic response of the laminate is measured using a Polytec laser vibrometer. In addition, four macro-fiber composite actuators are attached to the laminate to measure the output voltage due to vibration and hence assess its energy-harvesting capability. The laminate’s natural frequencies and damping under small-amplitude excitations that would match the natural frequencies of an underlying linear system are experimentally identified. A primary resonance excitation of the first bending mode is carried out using amplitude sweep and frequency sweep. It is shown that in both cases, the response starts as a small-amplitude single-well vibration around the static equilibrium position. Varying the excitation amplitude or frequency, the response exhibits a period-doubling bifurcation associated with higher levels of vibration. As the excitation conditions are varied further, the laminate snaps from one equilibrium position to the other. This snapthrough motion is characterized to be of a frequency that is half the natural frequency of the excited mode. It is shown that the period-doubling cascade is responsible for the escape from the single-well to the twin-well response. A secondary Hopf bifurcation that creates a period-three motion and a chaotic snapthrough were also identified. Two low-frequency interesting modes that are attributed to the elasticity of the midpoint support, referred to as rocking modes, were identified using a hammer. These modes got repeatedly excited via subharmonic resonance of order two. In terms of the energy harvesting, the snapthrough motion was found to greatly enhance the energy extraction capability. The frequency band of significant power generation was found to be about 25% of the excitation frequency, which is a good ratio for a single excitation frequency.

Subharmonic Resonance Due to Gap Between Geometric and Magnetic Centers of Rotor Supported by Superconducting Magnetic Bearing

Superconducting magnetic bearings (SMBs) have various merits because of stable levitation without physical contact. However, the system's low damping causes vibration with a large amplitude near the critical rotational speed, and the nonlinearity of the electromagnetic force causes complicated vibrations. Moreover, if magnetization distribution of a rotor is not uniform, the rotor's dynamical behavior can be more complicated because a superconducting bulk in a superconductive state traps the nonuniform magnetic field, which has significant influence on restoring force acting on the rotor. Therefore, it is necessary to consider influence of magnetic unbalance of the rotor on the whirling of a rotor supported by a SMB. This paper deals with nonlinear dynamics of a rotor, which has a gap between the geometric and magnetic centers of the rotor and which is supported by a SMB. We derived the equations of motion of the rotor and discussed whether subharmonic resonances of order 1/2 and order 1/3 occur by considering nonlinear terms of the electromagnetic force. By numerical calculation based on the Rung-Kutta method, we clarified that subharmonic resonances of order 1/2 can occur without disturbance, whereas subharmonic resonances of order 1/3 can occur only with disturbance. Moreover, we performed nonlinear analysis by means of the harmonic balance method. We also carried out experiments, which verified the analytical and numerical predictions of occurrence of subharmonic resonances.

Vibration Reduction of Rotor Supported by Superconducting Magnetic Bearing Utilizing Electromagnetic Shunt Damper

A superconducting magnetic bearing can levitate a rotor without control and is expected to be applied to flywheel energy storage systems. However, because the levitation force has nonlinearity, the rotor can show nonlinear vibrations such as subharmonic resonance and superharmonic resonance. Therefore, it is necessary to suppress the amplitude of nonlinear vibration of the rotor. Recently, an “electromagnetic shunt damper,” which can replace a dynamic vibration absorber, has been researched. The purpose of this study is to investigate whether an electromagnetic shunt damper can suppress the nonlinear vibration of a rotor supported by a superconducting magnetic bearing. We obtained the system's governing equations via an analytical model and nondimensionalized the equations. Linear analysis was performed, and we confirmed that an electromagnetic shunt damper is effective against linear systems. Furthermore, numerical calculations via the Rung-Kutta method were carried out. The obtained results show that an electromagnetic shunt damper can suppress the vibration amplitude of primary resonance and subharmonic resonance of order 1/2.

Perturbation analysis of an electrostatic Micro-Electro-Mechanical System (MEMS) subjected to external and non-linear parametric excitations

Electrostatic micro-electro-mechanical system (MEMS) is a special branch with a wide range of applications in sensing and actuating devices in MEMS. In this paper the perturbation analysis of the electrostatically actuated MEMS resonant sensors which represented by a modifled Du‐ng - Van der Pol equation subjected to weakly non-linear parametric and external excitations is studied by using a perturbation technique (multiple time scales). Harmonic resonance and subharmonic resonances of order ( 1 and 1 ) are investigated. For each resonances we obtain the modulation equations in the amplitude and phase, steady state solutions, frequency-response equations and stability conditions are determined. Efiects of difierent parameters on the system behavior are investigated numerically. Results are presented graphically and discussion is provided.

Nonlinear oscillation of a cantilever FGM rectangular plate based on third-order plate theory and asymptotic perturbation method

Nonlinear dynamic analysis of a cantilever functionally graded materials (FGM) rectangular plate subjected to the transversal excitation in thermal environment is presented for the first time in this paper. Material properties are assumed to be temperature dependent. The nonlinear governing equations of motion for the FGM plate are derived based on Reddy’s third-order plate theory and Hamilton’s principle. The first two vibration mode functions satisfying the boundary conditions of the cantilever FGM rectangular plates are chosen to be the admissible displacement functions. Galerkin’s method is utilized to convert the governing partial differential equations to a two-degree-of-freedom nonlinear system including quadratic and cubic nonlinear terms under combined external excitations. The present study focuses on resonance case with 1:1 internal resonance and subharmonic resonance of order 1/2. The asymptotic perturbation method is employed to obtain four nonlinear averaged equations which are then solved by using Runge–Kutta method to find the nonlinear dynamic responses of the plate. It is found that chaotic, periodic and quasi-periodic motions of the plate exist under certain conditions and the forcing excitations can change the form of motions for the FGM rectangular plate.

Application of nonlinear resonances for the diagnostics of closing cracks in rodlike elements

The vibrations of elastic bodies with closing cracks are essentially nonlinear. As a specific feature of these vibrations, one can mention the manifestation of so-called nonlinear effects, e.g., nonlinear (i.e., sub- and superharmonic) resonances and the nonlinearity of vibrations for these resonances. The proposed method for the evaluation of the parameters of cracks (their sizes and location) is based on the analysis of the nonlinearity of vibrations in the neighborhood of a superharmonic resonance of order 2/1 and/or a subharmonic resonance of order 1/2 in the case of variation of the site of application of the driving force because, as follows from the results of numerical and experimental investigations, the level of nonlinearity of the vibrations of rods with closing cracks for nonlinear resonances depends not only on the parameters of the crack but also on the site of application of the driving forces.

Crack Detection in Beams by Means of the Driving Force Parameters Variation at Non-Linear Resonance Vibrations

A closing crack causes the dynamic behavior of a vibrating system to be significantly non-linear and, consequently, the appearance of non-linear resonances (that is super- and subharmonic one). The main idea of the proposed procedure of crack location and size estimation is based on the determination of the vibration response non-linearity around the superharmonic resonance of order 2/1 and subharmonic resonance of order 1/2 at different parameters of driving force. First parameter of driving force under investigation is its point of application. It is shown both numerically, with the use of the finite element model of a cracked beam, and experimentally that the level of non-linearity of vibration response at non-linear resonances is strongly dependent not only on the crack size but also on the driving force application point along the cracked beam. Moreover, the abrupt change of the non-linearity of vibration response when driving force is applied close to the crack clearly indicates its location. In such a way the procedure of damage detection is proposed to estimate both the crack size and location in beams. Second parameter is the level of driving force asymmetry. Addition of the static component to the harmonic driving force varies the state of crack making crack more or fully open or closed. As a result the level of non-linearity of vibration response at any non-linear resonance varies from maximal value (in absence of static component) to practically zero value (when the static component of driving force is so large that crack becomes permanently open or closed at vibration). In such a way the presence of crack can be detected without preliminary information on the vibration response of a structure in the intact state.

505 不連続ばね特性を用いた非線形回転軸系の制振

In practical rotating machinery, the nonlinear spring characteristics appear due to various causes. As the results, the resonances occur in various roational speeds in addition to the vicinity of the major critical speed. In the passive vibration control method, it is difficult to adjust system parameters which satisfy the optimal vibration suppression condition in the whole rotational speed. In the active vibration method, using control devices are usually complicated and costly. This paper shows the method utilizing discontinuous spring characteristics, which was proposed to suppress the harmonic resonance at the major critical speed in a linear rotor system, is also effective to suppress nonlinear resoances. The vibration suppression of the harmonic resonance and the subharmonic resonance of order 1/2 are studied. However, some problems exist when ths method is applied in a nonlinear rotor system. The validity of the proposed method is also verified by experiments.

Chaotic Vibrations and Internal Resonance Phenomena in Rotor Systems. Case that the Critical Speeds of a Combination Resonance and Subharmonic Resonances of Order 1/2 are close.

In rotor systems, if the gyroscopic moment is small, a forward natural frequency and a backward natural frequency almost satisfy a condition of 1 : (-1) internal resonance. In such systems, the critical speeds of the forward and the backward subharmonic resonances of order 1/2 and the combination resonance are close in the vicinity of twice the major critical speed and internal resonance phenomena may appear. This study clarifies dynamical characteristics of nonlinear phenomena due to internal resonance, such as steady-state oscillations, almost periodic motions, bifurcation phenomena, and two kinds of chaotic vibrations, at the rotational speed of twice the major critical speed.

209 周辺固定積層扁平殻の1/2分数調波共振(OS2-4 連続体の振動・故障解析)(OS2 振動の解析と制御)

209 周辺固定積層扁平殻の1/2分数調波共振(OS2-4 連続体の振動・故障解析)(OS2 振動の解析と制御)

NON-LINEAR COUPLED TRANSVERSE AND AXIAL VIBRATION OF A COMPLIANT STRUCTURE, PART 2: FORCED VIBRATION

Responses of a beam undergoing both axial and transverse vibration are studied when the beam is subjected to transverse forces. The beam is supported by a torsional spring at the base and has a point mass at the free end. This is a simplified model of a complaint offshore structure. It is assumed that the environmental forces are due to waves and current. The semi-empirical Morison equation is used to model the fluid forces. Waves in this case are assumed to be random and their heights follow the Pierson–Moskowitz spectrum. Borgman's method is used to obtain the wave height from the Pierson-Moskowitz spectrum, and the wave velocities and the accelerations are obtained from the wave height using the Airy linear wave theory. The wave velocities and accelerations are then used in the Morison equation to form the fluid forcing function. As a preliminary study, the harmonic force is used to model the fluid force. When the deterministic harmonic force at various frequencies is applied, subharmonic resonances of order 1/2 are observed. Parametric studies of random forcing are performed by varying current velocity and significant wave height.

Forced Oscillations of a Horizontal Continuous Rotor with Geometric Nonlinearity. Internal Resonance Phenomena at Harmonic Resonances and at Subharmonic Resonances of Order 1/3.

Harmonic resonances and subharmonic resonances of order 1/3 in a continuous rotating shaft with distributed mass are discussed. The restoring force of the shaft has geometric stiffening nonlinearity due to the extension of the shaft centerline. The effect of a distributed lateral force, such as the gravity, is assumed. The possibility of the occurrences of harmonic resonances and subharmonic resonances of order 1/3, the shapes of resonance curves, and internal resonance phenomena are investigated.

Forced Oscillations of a Continuous Rotor with Geometric Nonlinearity. Subharmonic Resonances of Order 1/2 and Internal Resonance.

Subharmonic resonances of order 1/2 of a continuous rotating shaft with distributed mass are discussed. The restoring force of the shaft exhibits geometric stiffening nonlinearity due to the extension of the shaft center line. It is assumed that a distributed lateral force, such as the gravity, acts on the rotor. The possibility of the occurrence of subharmonic resonances, the shapes of resonance curves, and internal resonance phenomena are investigated.

Whirling of a forced cantilevered beam with static deflection. III: Passage through resonance

Nonstationary excitations of slender, elastic, cantilevered beams with equal principal moments of inertia are considered. The excitation frequency is slowly increased or decreased through a resonance of the first mode at a constant rate. Three resonances are investigated: primary resonance, superharmonic resonance of order two and subharmonic resonance of order two. After application of Galerkin's method with three modes, the nonlinear, nonstationary response of the first mode of the beam is determined by two methods: integration of the modulation equations obtained from the method of multiple scales, and direct numerical integration of the temporal equations of motion. Time histories are presented and the effects of excitation amplitude, rate of acceleration or deceleration through resonance, damping and initial conditions of the disturbance on the maximum response are studied. The effect of a persistent random disturbance is also examined. Although the excitation acts in the vertical plane, whirling occurs if the beam is subjected to out-of-plane disturbances.

Subharmonic resonance of order ½ with an asymmetrical restoring force

Excitation of subharmonic resonances is one of the serious vibration problems which often have to be overcome in the development of high speed machinery. Such resonances have been observed, for example, in gas turbines(3), in rotor systems(4) and in the geared drive of a rotating kiln(5).The subharmonic resonance of order 1/2, of a single degree of freedom system with viscous damping, is investigated for the case of a centrifugal disturbing force. The restoring force characteristic is of the non-odd form and its asymmetry is expressed in terms of the static deflection parameter. The Ritz averaging method, with a five term approximation, is used for the theoretical analysis. Numerical results are presented for the undamped case and are compared with analogue computer solutions.The choice of the Ritz averaging method is determined mainly by its adequate accuracy at larger magnitudes of non-linearity. It is also more versatile than other methods and in a previous application(6) to a system with an asymmetrical restoring force it produced satisfactory results.