scholarly journals Nonlinear Dynamic Modeling and Analysis of an L-Shaped Multi-Beam Jointed Structure with Tip Mass

Materials ◽  
2021 ◽  
Vol 14 (23) ◽  
pp. 7279
Author(s):  
Jin Wei ◽  
Tao Yu ◽  
Dongping Jin ◽  
Mei Liu ◽  
Dengqing Cao ◽  
...  
Keyword(s):  
Differential Equations ◽  
Nonlinear Dynamic ◽  
Natural Frequencies ◽  
Great Influence ◽  
Dynamic Responses ◽  
Mode Shapes ◽  
Nonlinear Odes ◽  
Global Mode ◽  
Orthogonality Relations ◽  
Tip Mass

A dynamic model of an L-shaped multi-beam joint structure is presented to investigate the nonlinear dynamic behavior of the system. Firstly, the nonlinear partial differential equations (PDEs) of motion for the beams, the governing equations of the tip mass, and their matching conditions and boundary conditions are obtained. The natural frequencies and the global mode shapes of the linearized model of the system are determined, and the orthogonality relations of the global mode shapes are established. Then, the global mode shapes and their orthogonality relations are used to derive a set of nonlinear ordinary differential equations (ODEs) that govern the motion of the L-shaped multi-beam jointed structure. The accuracy of the model is verified by the comparison of the natural frequencies solved by the frequency equation and the ANSYS. Based on the nonlinear ODEs obtained in this model, the dynamic responses are worked out to investigate the effect of the tip mass and the joint on the nonlinear dynamic characteristic of the system. The results show that the inertia of the tip mass and the nonlinear stiffness of the joints have a great influence on the nonlinear response of the system.

A Discretization Approach to Modeling Capacitive MEMS Filters

2007 ◽  
Author(s):  
Bashar K. Hammad ◽  
Ali H. Nayfeh ◽  
Eihab Abdel-Rahman
Keyword(s):  
Boundary Value Problem ◽  
Closed Form ◽  
Natural Frequencies ◽  
Multiple Scales ◽  
Boundary Value ◽  
Mode Shapes ◽  
Algebraic Equations ◽  
Nonlinear Odes ◽  
Global Mode ◽  
Wide Range

We present a reduced-order model and closed-form expressions describing the response of a micromechanical filter made up of two clamped-clamped microbeam capacitive resonators coupled by a weak microbeam. The model accounts for geometrical and electrical nonlinearities as well as the coupling between them. It is obtained by discretizing the distributed-parameter system using the Galerkin procedure. The basis functions are the linear undamped global mode shapes of the unactuated filter. Closed-form expressions for these mode shapes and the coressponding natural frequencies are obtained by formulating a boundary-value problem (BVP) that is composed of five equations and twenty boundary conditions. This problem is transformed into solving a system of twenty linear homogeneous algebraic equations for twenty constants and the natural frequencies. We predict the deflection and the voltage at which the static pull-in occurs by solving another boundary-value problem (BVP). We also solve an eigenvalue problem (EVP) to determine the two natural frequencies delineating the bandwidth of the actuated filter. Using the method of multiple scales, we determine four first-order nonlinear ODEs describing the amplitudes and phases of the modes. We found a good agreement between the results obtained using our model and the published experimental results. We found that the filter can be tuned to operate linearly for a wide range of input signal strengths by choosing a DC voltage that makes the effective nonlinearities vanish.

A Hybrid Finite Element-Analytical Approach for Dynamic Analysis of a Micro-Resonator Driven by Electrostatic Combs

2012 ◽  
Vol 226-228 ◽  
pp. 708-713
Author(s):  
Mi Tao Song ◽  
Deng Qing Cao
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Differential Equations ◽  
Nonlinear Dynamic ◽  
Analytical Approach ◽  
Dynamic Responses ◽  
The Finite Element Method ◽  
Global Mode ◽  
Hybrid Finite Element ◽  
Micro Resonator

Combining the finite element method and the analytical method, a hybrid finite element-analytical approach is established to calculate the nonlinear dynamic responses of a micro-resonator driven by electrostatic combs accurately for the purpose of programmed dynamical simulations and great shortening of workloads. The spatially discretized equations obtained by using the analytical undamped global mode functions to the nonlinear integro-partial differential equations and the ordinary differential equations of the micro-resonator, in which the coefficients are estimated by the discrete global mode shapes from the finite element method, are used to calculate the nonlinear dynamic responses of the micro-resonator. The results are compared with those merely based on the analytical mode functions of the micro-resonator, which shows that they can reach high accuracy when the elements in the micro-resonator are sufficiently small.

scholarly journals Free Vibrations and Nonlinear Responses for a Cantilever Honeycomb Sandwich Plate

2018 ◽  
Vol 2018 ◽  
pp. 1-12 ◽  
Author(s):  
Junhua Zhang ◽  
Xiaodong Yang ◽  
Wei Zhang
Keyword(s):  
Differential Equations ◽  
Nonlinear Dynamic ◽  
Sandwich Plate ◽  
Natural Frequencies ◽  
Ritz Method ◽  
Shear Deformation Theory ◽  
Dynamic Responses ◽  
Chaotic Motions ◽  
Honeycomb Sandwich ◽  
Honeycomb Sandwich Plate

Dynamics of a cantilever honeycomb sandwich plate are studied in this paper. The governing equations of the composite plate subjected to both in-plane and transverse excitations are derived by using Hamilton’s principle and Reddy’s third-order shear deformation theory. Based on the Rayleigh–Ritz method, some modes of natural frequencies for the cantilever honeycomb sandwich plate are obtained. The relations between the natural frequencies and the parameters of the plate are investigated. Further, the Galerkin method is used to transform the nonlinear partial differential equations into a set of nonlinear ordinary differential equations. Nonlinear dynamic responses of the cantilever honeycomb sandwich plate to such external and parametric excitations are discussed by using the numerical method. The results show that in-plane and transverse excitations have an important influence on nonlinear dynamic characteristics. Rich dynamics, such as periodic, multiperiodic, quasiperiodic, and chaotic motions, are located and studied by the bifurcation diagram for some specific parameters.

Dynamic Analysis of a Micro-Resonator Driven by Electrostatic Combs

2011 ◽  
Author(s):  
M. T. Song ◽  
D. Q. Cao ◽  
W. D. Zhu
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Dynamic Response ◽  
Natural Frequencies ◽  
Nonlinear Partial Differential Equations ◽  
Rigid Bodies ◽  
Mode Shapes ◽  
Flexible Beams ◽  
Global Mode ◽  
Micro Resonator

The dynamic response of a micro-resonator driven by electrostatic combs is investigated in this work. The micro-resonator is assumed to consist of eight flexible beams and three rigid bodies. The nonlinear partial differential equations that govern the motions of the flexible beams are obtained, as well as their boundary and matching conditions. The natural matching conditions for the flexible beams are the governing equations for the rigid bodies. The undamped natural frequencies and mode shapes of the linearized model of the micro-resonator are determined, and the orthogonality relation of the undamped global mode shapes is established. The modified Newton iterative method is used to simultaneously solve for the frequency equation and identify repeated natural frequencies that can occur in the micro-resonator and their multiplicities. The Gram-Schmidt orthogonalization method is extended to orthogonalize the mode shapes of the continuous system corresponding to the repeated natural frequencies. The undamped global mode shapes are used to spatially discretize the nonlinear partial differential equations of the microresonator. The simulation results show that the geometric nonlinearities of the flexible beams can have a significant effect on the dynamic response of the micro-resonator.

Wave-Induced Vibrations of an Axial-Loaded Immersed Timoshenko Beam Carrying an Eccentric Tip Mass with Rotary Inertia

2010 ◽  
Vol 54 (01) ◽  
pp. 15-33
Author(s):  
Jong-Shyong Wu ◽  
Chin-Tzu Chen
Keyword(s):  
Closed Form ◽  
Timoshenko Beam ◽  
Forced Vibration ◽  
Natural Frequencies ◽  
Closed Form Solution ◽  
Form Solution ◽  
Rotary Inertia ◽  
Mode Shapes ◽  
Mode Superposition Method ◽  
Tip Mass

Under the specified assumptions for the equation of motion, the closed-form solution for the natural frequencies and associated mode shapes of an immersed "Euler-Bernoulli" beam carrying an eccentric tip mass possessing rotary inertia has been reported in the existing literature. However, this is not true for the immersed "Timoshenko" beam, particularly for the case with effect of axial load considered. Furthermore, the information concerning the forced vibration analysis of the foregoing Timoshenko beam caused by wave excitations is also rare. Therefore, the first purpose of this paper is to present a technique to obtain the closed-form solution for the natural frequencies and associated mode shapes of an axial-loaded immersed "Timoshenko" beam carrying eccentric tip mass with rotary inertia by using the continuous-mass model. The second purpose is to determine the forced vibration responses of the latter resulting from excitations of regular waves by using the mode superposition method incorporated with the last closed-form solution for the natural frequencies and associated mode shapes of the beam. Because the determination of normal mode shapes of the axial-loaded immersed "Timoshenko" beam is one of the main tasks for achieving the second purpose and the existing literature concerned is scarce, the details about the derivation of orthogonality conditions are also presented. Good agreements between the results obtained from the presented technique and those obtained from the existing literature or conventional finite element method (FEM) confirm the reliability of the presented theories and the developed computer programs for this paper.

scholarly journals Influence of Piezoelectric Performance on Nonlinear Dynamic Characteristics of MFC Shells

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-15 ◽  
Author(s):  
Xiangying Guo ◽  
Pan Jiang ◽  
Dongxing Cao
Keyword(s):  
Electric Field ◽  
Nonlinear Dynamic ◽  
Piezoelectric Properties ◽  
Great Influence ◽  
Dynamic Responses ◽  
Aerodynamic Forces ◽  
External Excitation ◽  
Structural Dynamic ◽  
Laminated Shells ◽  
Laminated Shell

Based on the structures of unmanned aerial vehicle (UAV) wings, nonlinear dynamic analysis of macrofiber composite (MFC) laminated shells is presented in this paper. The effects of piezoelectric properties and aerodynamic forces on the dynamic stability of the MFC laminated shell are studied. Firstly, under the flow condition of ideal incompressible fluid, the thin airfoil theory is employed to calculate the effects of the mean camber line to obtain the circulation distribution of the wings in subsonic air flow. The steady aerodynamic lift on UAV wings is derived by using the Kutta–Joukowski lift theory. Then, considering the geometric nonlinearity and piezoelectric properties of the MFC material, the nonlinear dynamic model of the MFC laminated shell is established with Hamilton’s principles and the Galerkin method. Next, the effects of electric field, external excitation force, and nonlinear parameters on the stability of the system are studied under 1 : 1 internal resonance and the effects of material parameters on the natural frequency of the structure are also analyzed. Furthermore, the influence of the aerodynamic forces and electric field on the nonlinear dynamic responses of MFC laminated shells is discussed by numerical simulation. The results indicate that the electric field and external excitation have great influence on the structural dynamic responses.

Development of an Analytical Model for Beams With Two Dimples in Opposing Direction

2017 ◽  
Author(s):  
Mofareh Ghazwani ◽  
Kyle Myers ◽  
Koorosh Naghshineh
Keyword(s):  
Differential Equations ◽  
Opposite Direction ◽  
Natural Frequencies ◽  
Mode Shapes ◽  
Modal Strain Energy ◽  
Noise And Vibration ◽  
Various Boundary Conditions ◽  
Strategic Approach ◽  
Value Model ◽  
Good Agreement

Structures such as beams and plates can produce unwanted noise and vibration. An emerging technique can reduce noise and vibration without any additional weight or cost. This method focuses on creating two dimples in the same and opposite direction on a beam’s surface where the effect of dimples on its natural frequencies is the problem of interest. The change in the natural frequency between both cases have a different trend. The strategic approach to calculate natural frequencies is as follows: first, a boundary value model (BVM) is developed for a beam with two dimples and subject to various boundary conditions using Hamilton’s Variational Principle. Differential equations describing the motion of each segment are presented. Beam natural frequencies and mode shapes are obtained using a numerical solution of the differential equations. A finite element method (FEM) is used to model the dimpled beam and verify the natural frequencies of the BVM. Both methods are also validated experimentally. The experimental results show a good agreement with the BVM and FEM results. A fixed-fixed beam with two dimples in the same and opposite direction is considered as an example in order to compute its natural frequencies and mode shapes. The effect of dimple locations and angles on the natural frequencies are investigated. The natural frequencies of each case represent a greater sensitivity to change in dimple angle for dimples placed at high modal strain energy regions of a uniform beam.

A Subharmonic Resonance-Based MEMS Filter

2007 ◽  
Author(s):  
Bashar K. Hammad ◽  
Ali H. Nayfeh ◽  
Eihab Abdel-Rahman
Keyword(s):  
Multiple Scales ◽  
Subharmonic Resonance ◽  
Mode Shapes ◽  
Center Frequency ◽  
Distributed Parameter ◽  
Nonlinear Odes ◽  
Global Mode ◽  
Reduced Order ◽  
Filter Performance ◽  
Mems Filter

We present a novel micromechanical filter exploiting the subharmonic resonance of order one-half to obtain a center frequency twice the fundamental frequency of the primary resonators, an ideal stopband, and a sharp roll-off. The filter is made up of two clamped-clamped microbeam resonators connected by a coupling beam. We discretize the distributed-parameter system using the Galerkin procedure to obtain a reduced-order model composed of two nonlinear coupled ODEs. It accounts for geometrical and electrical nonlinearities as well as the coupling between these two fields. Using the method of multiple scales, we determine four first-order nonlinear ODEs describing the amplitudes and phases of the modes. We use these equations to determine closed-form expressions for the static and dynamic deflections of the filter. The basis functions in the discretization are the linear undamped global mode shapes of the unactuated filter. The filtering mechanism is based on the exploitation of the interval where the trivial response to subharmonic excitations is unstable. We found criteria to tune the effective nonlinearities of the filter to realize a bandpass filter of an ideal stopband rejection and a sharp roll-off. When these criteria are not met, multivalued responses appear and distort the filter performance.

Modal Analysis of Flexible Assembling Fixture for Aircraft Panel Component

2010 ◽  
Vol 97-101 ◽  
pp. 3392-3396
Author(s):  
Li Gang Qu ◽  
Ke Qiang Pan ◽  
Xin Chen
Keyword(s):  
Finite Element ◽  
Modal Analysis ◽  
Dynamic Characteristic ◽  
Natural Frequencies ◽  
Great Influence ◽  
Vibration Characteristics ◽  
Mode Shapes ◽  
Dynamic Vibration ◽  
Structural Weakness ◽  
Modal Vibration

The dynamic characteristic of flexible assembling fixture (FAF) for aircraft panel component is analysed by the method of finite element modal analysis. Consequently, the every order of natural frequencies and mode shapes of given different postures of the FAF are obtained. It structural weakness were pointed out through the analysis results of the modal vibration characteristics. The properties of mass and stiffness of the FAF's components are concurrently calculated, whose optimal matching and harmonizing with each other have great influence on the dynamic vibration characteristics of the FAF. As the results of these analysis, the design improving suggestion for the FAF is put forward.

Dynamic Analysis of an Elevator Traveling Cable Using a Singularity-Free Beam Formulation

2017 ◽  
Vol 84 (4) ◽  
Author(s):  
W. Fan ◽  
W. D. Zhu
Keyword(s):  
Dynamic Analysis ◽  
Multibody Dynamics ◽  
Natural Frequencies ◽  
Vertical Motion ◽  
Dynamic Responses ◽  
Mode Shapes ◽  
Out Of Plane ◽  
Forced Responses ◽  
Free Beam ◽  
Good Agreement

A round elevator traveling cable is modeled using a singularity-free beam formulation. Equilibria of the traveling cable with different elevator car positions are studied. Natural frequencies and the corresponding mode shapes of the traveling cable are calculated and they are in excellent agreement with those calculated by abaqus. In-plane natural frequencies of the traveling cable do not change much with the car position compared with its out-of-plane ones. Dynamic responses of the traveling cable are calculated and they are in good agreement with those from commercial multibody dynamics software recurdyn. Effects of vertical motion of the car on free responses of the traveling cable and those of in-plane and out-of-plane building sways on forced responses are investigated.

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