Vibration Localization In Linear Cracked Rectangular Thin Plate And Nonlinear Dual-oscillator System

Vibration in mechanical systems is inevitable due to the dynamic load during running,and may be transmitted from the excitation source to entire system.Also,the periodicity or symmetry of real structures can be broken by the manufacturing errors,flaws of the material or fatigue fractures.This can result in the vibration energy of the system,which should be transmitted to the entrie system,being confined in only part of the system.This phenomenon is called vibration localization.Vibration localization not only changes the original mode shapes and results in mode localization,but also affects the energy transmission between different parts of the system.In this work,vibration localization in linear and nonlinear systems is investigated.A linear,rectangular thin plate is chosen as the representative linear system,and the mode localization caused by through cracks in such plates is investigated.It is found that the localized area might contact its surroundings due to amplitude amplified by localization.Then,a simplified impact model,a nonlinear vibro-impact dual-oscillator system,which is feasible in study and realizable in experiment,is proposed to investigate the vibration localization phenomenon in a nonlinear system.The energy localization during the energy transmission in the dual-oscillator system caused by slight asymmetry of the supporting springs is discussed.The main research contents are organized as follows:1.A solution method is presented for vibration analysis of through-cracked rectangular plates.A domain decomposition technique is employed in combination with a set of admissible functions which were previously proposed for triangular and rectangular plates with elastic boundary conditions.The continuity conditions at interconnecting boundaries of the subdomains are enforced by translational and rotational springs with zero or infinite stiffness.The Rayleigh-Ritz method is employed to determine the generalized coordinates,and corresponding modal frequencies and shapes.Numerical examples are presented for plates with various crack configurations and boundary conditions.The current results are compared with those available in the literature to verify their convergence and correctness.The main advantage of the proposed method lies in its applicability to various combinations of boundary conditions and crack configurations.This method is then employed to determine the modal characteristics of simply supported,square plates by focusing on the effects of various crack parameters on frequency veering,mode splitting and mode localization.The degree of localization is evaluated quantitatively through a scalar localization measure.Numerical results show that when the crack is longer or closer to the nodal line,or there are stronger boundary constraints,the mode shapes are more affected,and the localization occurs more easily.2.Then,vibration mode localization caused by more complicated cracks,V-shaped through cracks,in rectangular plates is discussed.The solution method above is modified for the free vibration analysis of rectangular plates containing V-shaped through cracks.A domain decomposition technique is employed in combination with two sets of modified Fourier series.All the rectangular and triangular subdomains are transformed into unit square domains and right-angled isosceles triangular domains,respectively The RayleighRitz method is used to determine the natural frequencies,and the mode shapes are then constructed by superposition of the admissible functions.Finally,this method is applied to determine the modal characteristics of plates with V-shaped cracks by focusing on the effects of various crack parameters on frequencies and mode localization.It is found that the frequencies and mode shapes are more affected by longer crack lengths and smaller included angles,and that mode localization is more likely to occur under these conditions.It is found that a V-shaped crack can introduce a cantilever-triangular-like substructure in the plate and the vibration amplitude of this substructure is greatly amplified,which might produce contact with the surroundings.3.Then,a simplified impact model,a nonlinear vibro-impact dual-oscillator system is proposed,which is is feasible to study and realizable in experiment.The energy localization during energy transmission in the dual-oscillator system,caused by slight asymmetry of the supporting springs,is discussed.This system is composed of two otherwise linear,viscously damped,single-degree-of-freedom oscillators,one of which is driven by harmonic base motion.A semi-analytical solution of the oscillator equations of motion,in which the times of impact are determined by bisection,is used to simulate the responses while the parameters of the system are varied.The existence of a stable 1:1 resonance is investigated using Floquet theory,and predictions are made about the ranges of excitation frequency and amplitude leading to such a response.It is found that the assumption of a coefficient of restitution smaller than 1(i.e.,modeling energy loss during impact)increases the range of stable solutions.A transmission coefficient is defined to quantify the efficiency of energy transfer between the oscillators,and it is found that the energy transmission efficiency from the higher-frequency oscillator to the lowerfrequency one is much greater than the transmission efficiency for energy flow from the lower-frequency to the higher-frequency oscillator.There is an optimal excitation amplitude for maximum energy transfer in higher-to-lower-frequency transmission,while such an optimum doesn’t exist for transmission in the opposite direction.Localization always happens in the lower-frequency oscillator no matter the excitation amplitude or excitation frequency.4.Finally,vibration localization in through-cracked rectangular thin plates and in a dual-oscillator system were validated in experiments.First,three rectangular thin plates,whose length and width are 820 mm by 520 mm,were used in experiments.Each plate contained a through crack with different locations.The natural frequencies and mode shapes were tested with all clamped boundary condtion.The results were in good agreement with numerical predictions and the conclusions about localization were experimentally confirmed.Then,a dual-oscillator experimental platform was designed and built up.The energy localization during the energy transmission within this system was experimentally examined,and good agreement was again observed between experimental and numerical results.The conclusions drawn in numerical case studies were validated in experiments.