Dynamics of Inertia-Variant Flexible Systems Using Experimentally Identified Parameters

1986 ◽  
Vol 108 (3) ◽  
pp. 358-366 ◽  
Author(s):  
A. Shabana
Keyword(s):  
Differential Equations ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Analytical Techniques ◽  
Mode Shapes ◽  
Constraint Equations ◽  
Direct Numerical Integration ◽  
Parameter Estimation Techniques ◽  
Modal Coordinates ◽  
Algebraic Constraint

In this investigation modal parameters (frequency, damping, and mode shapes) which are determined experimentally using parameter estimation techniques are employed to simulate and predict the dynamic behavior of flexible multibody systems which consist of interconnected rigid and flexible components. The system differential equations of motion and algebraic constraint equations describing mechanical joints in the system are first identified using analytical techniques. Dynamic parameters such as mass, damping, and stiffness coefficients that appear in the system differential equations are then identified using a set of experimentally measured data. Mode shapes which are the result of the experimental identification are used to write the physical elastic coordinates of selected nodal points on the flexible body in terms of a reduced set of modal coordinates. The nonlinear differential and algebraic constraint equations are then written in terms of mixed sets of coupled reference and modal coordinates. These equations are integrated numerically using a direct numerical integration technique coupled with Newton–Raphson type iterations in order to check on constraint violations. The formulation developed is numerically exemplified using a three-dimensional dune buggy vehicle model.

Application of Lifting-Surface Theory to the Prediction of Hydroelastic Response of Hydrofoil Boats

1968 ◽  
Vol 12 (04) ◽  
pp. 286-301
Author(s):  
C. J. Henry
Keyword(s):  
Equations Of Motion ◽  
Three Dimensional ◽  
Numerical Procedure ◽  
Mode Shapes ◽  
Operator Technique ◽  
Elastic Mode ◽  
Surface Theory ◽  
Lifting Surface ◽  
Modes Of Vibration ◽  
Elastic Modes

In this report a theoretical procedure is developed for the prediction of the dynamic response elastic or rigid body, of a hydrofoil-supported vehicle in the flying condition— to any prescribed transient or periodic disturbance. The procedure also yields the stability indices of the response, so that dynamic instabilities such as flutter can also be predicted. The unsteady hydrodynamic forces are introduced in the equations of motion for the elastic vehicle in terms of the indicia I pressure-response functions, which are de rived herein from lifting-surface theory. Thus, the predicted vehicle-response includes the effects of three-dimensional unsteady flow conditions at specified forward speed. The natural frequencies and elastic modes of vibration of the vehicle and foil system in the absence of hydrodynamic effects are presumed known. A numerical procedure is presented for the solution of the downwash integral equations relating the unknown indicial pressure distributions to the specified elastic-mode shapes. The procedure is based on use of the generalized-lift-operator technique together with the collocation method.

scholarly journals Re-Evaluating the Classical Falling Body Problem

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 553 ◽  
Author(s):  
Essam R. El-Zahar ◽  
Abdelhalim Ebaid ◽  
Abdulrahman F. Aljohani ◽  
José Tenreiro Machado ◽  
Dumitru Baleanu
Keyword(s):  
Differential Equations ◽  
Body Problem ◽  
Inertial Frame ◽  
Analytic Solution ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Rotating Frame ◽  
Dimensional Model ◽  
Three Dimensional Model

This paper re-analyzes the falling body problem in three dimensions, taking into account the effect of the Earth’s rotation (ER). Accordingly, the analytic solution of the three-dimensional model is obtained. Since the ER is quite slow, the three coupled differential equations of motion are usually approximated by neglecting all high order terms. Furthermore, the theoretical aspects describing the nature of the falling point in the rotating frame and the original inertial frame are proved. The theoretical and numerical results are illustrated and discussed.

scholarly journals Nonlinear interactions in deformable container cranes

2015 ◽  
Vol 230 (1) ◽  
pp. 5-20 ◽  
Author(s):  
Andrea Arena ◽  
Walter Lacarbonara ◽  
Matthew P Cartmell
Keyword(s):  
Rigid Body ◽  
Internal Resonance ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Higher Order ◽  
Mode Shapes ◽  
Lagrange Equations ◽  
Internal Resonances ◽  
Container Cranes ◽  
Bending Mode

Nonlinear dynamic interactions in harbour quayside cranes due to a two-to-one internal resonance between the lowest bending mode of the deformable boom and the in-plane pendular mode of the container are investigated. To this end, a three-dimensional model of container cranes accounting for the elastic interaction between the crane boom and the container dynamics is proposed. The container is modelled as a three-dimensional rigid body elastically suspended through hoisting cables from the trolley moving along the crane boom modelled as an Euler-Bernoulli beam. The reduced governing equations of motion are obtained through the Euler-Lagrange equations employing the boom kinetic and stored energies, derived via a Galerkin discretisation based on the mode shapes of the two-span crane boom used as trial functions, and the kinetic and stored energies of the rigid body container and the elastic hoisting cables. First, conditions for the onset of internal resonances between the boom and the container are found. A higher order perturbation treatment of the Taylor expanded equations of motion in the neighbourhood of a two-to-one internal resonance between the lowest boom bending mode and the lowest pendular mode of the container is carried out. Continuation of the fixed points of the modulation equations together with stability analysis yields a rich bifurcation behaviour, which features Hopf bifurcations. It is shown that consideration of higher order terms (cubic nonlinearities) beyond the quadratic geometric and inertia nonlinearities breaks the symmetry of the bifurcation equations, shifts the bifurcation points and the stability ranges, and leads to bifurcations not predicted by the low order analysis.

Mechanical Behavior of an Electrostatically Actuated Microplate

2003 ◽  
Author(s):  
Xiaopeng Zhao ◽  
Eihab M. Abdel-Rahman ◽  
Ali H. Nayfeh
Keyword(s):  
Differential Equations ◽  
Natural Frequencies ◽  
Finite Dimension ◽  
Equations Of Motion ◽  
Galerkin Approximation ◽  
Mode Shapes ◽  
Design Parameters ◽  
Electrostatically Actuated ◽  
Linear Eigenvalue Problem ◽  
Electrically Actuated

We present a nonlinear model of electrically actuated microplates. The model accounts for the nonlinearity in the electric forcing as well as mid-plane stretching of the plate. We use a Galerkin approximation to reduce the partial-differential equations of motion to a finite-dimension system of nonlinearly coupled second-order ordinary-differential equations. We find the deflection of the microplate under DC voltage and study the pull-in phenomenon. The natural frequencies and mode shapes are then obtained around the deflected position of the microplate by solving the linear eigenvalue problem. The effect of various design parameters on both the static response and the dynamic characteristics are studied.

scholarly journals Analytical model and energy harvesting analysis of a vibrating slender rod with added tip mass in three-dimensional space

2021 ◽  
Author(s):  
Marek Borowiec ◽  
Marcin Bochenski ◽  
Grzegorz Litak ◽  
Andrzej Teter
Keyword(s):  
Energy Harvesting ◽  
Dimensional Space ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Electrical Circuit ◽  
Mode Shapes ◽  
3D Space ◽  
Harvesting Systems ◽  
Load Resistor ◽  
Tip Mass

AbstractIn the paper, a new 3D energy harvesting system is provided. This work discussed the Lagrange approach to derive the differential equations of motion in the case of energy harvesting systems. An electromechanical system consists of a mechanical resonator, a piezoelectric transducer and electrical circuit with the load resistor. A flexible slender rod clamped at the bottom and loaded by the tip mass is proposed as the resonator. Moving in the 3D space, it enables the system to avoid the gravitational potential barrier of the straight vertical shape in case of buckling. This paper investigates the response of the rod deflection and the root mean square power output of selected vibration mode shapes with an attached tip mass.

Modal Analysis to Interpret Localization Phenomena in Two Nonlinear Tuned Mass Dampers

2021 ◽  
Author(s):  
Yuji Harata ◽  
Takashi Ikeda
Keyword(s):  
Steady State ◽  
Modal Analysis ◽  
Equations Of Motion ◽  
Harmonic Excitation ◽  
Mode Shapes ◽  
Response Curves ◽  
Tuned Mass Dampers ◽  
Second Mode ◽  
Steady State Solutions ◽  
Modal Coordinates

Abstract This study investigates localization phenomena in two identical nonlinear tuned mass dampers (TMDs) installed on an elastic structure, which is subjected to external, harmonic excitation. In the theoretical analysis, the mode shapes of the system are determined, and the modal equations of motion are derived using modal analysis. These equations are demonstrated as forming an autoparametric system in which external excitation directly acts on the first and third vibration modes, whereas the second vibration mode is indirectly excited due to the nonlinear coupling with the other modes. Van der Pol’s method is employed to obtain the frequency response curves for both physical and modal coordinates. The two TMDs vibrate in phase for the first and third modes, but vibrate out of phase for the second mode. Consequently, when all modes appear, the two TMDs may vibrate at different amplitudes, i.e., localization phenomena may occur because the TMD motions are expressed by the summation of motions for all modes. The numerical calculations clarify that the localization phenomena may occur in the two TMDs when all three modes appear simultaneously. Moreover, there are two steady-state solutions of the harmonic oscillations for the second mode with identical amplitudes; however, their phases differ by π. Hence, which TMD vibrates at higher amplitudes depends on which of these two steady-state solutions for the phase.

Modal Analysis of Ballooning Strings With Small Sag

1999 ◽  
Author(s):  
Rongjun Fan ◽  
Sushil K. Singh ◽  
Christopher D. Rahn
Keyword(s):  
Steady State ◽  
High Speed ◽  
Natural Frequencies ◽  
Rotation Speed ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Mode Shapes ◽  
Theoretical Predictions ◽  
Nonlinear Equations Of Motion

Abstract During the manufacture and transport of textile products, yarns are rotated at high speed and form balloons. The dynamic response of the balloon to varying rotation speed, boundary excitation, and disturbance forces governs the quality of the associated process. Resonance, in particular, can cause large tension variations that reduce product quality and may cause yarn breakage. In this paper, the natural frequencies and mode shapes of a single loop balloon are calculated to predict resonance. The three dimensional nonlinear equations of motion are simplified via small steady state displacement (sag) and vibration assumptions. Axial vibration is assumed to propagate instantaneously or in a quasistatic manner. Galerkin’s method is used to calculate the mode shapes and natural frequencies of the linearized equations. Experimental measurements of the steady state balloon shape and the first two natural frequencies and mode shapes are compared with theoretical predictions.

Critical Speed Analysis of Multi-Cylider Engines As Spatial Elastic Linkage Systems

1996 ◽  
Author(s):  
Sirihari Kurnool ◽  
Cemil Bagei
Keyword(s):  
Critical Speed ◽  
Mass Matrix ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Mode Shapes ◽  
Lumped Mass ◽  
Constant Frequency ◽  
Elastic Mechanism ◽  
Finite Line ◽  
Crankshaft Rotation

Abstract A multi-cylinder engine is a cluster of slider-crank linkages. Presently used conventional pure torsional shaft models predict results far from the results predicted considering actual three-dimensional linkage and crankshaft geometries. Pure torsional model doesn’t sense the variation in frequency with the variation in engine geometry. It predicts one constant frequency value for each mode; it does not permit the use of flexible bearings. Article offers a finite element method for performing frequency and critical speed analysis of multi-cylinder engines considering three-dimensional geometries of the linkage loops, crankshaft, and the crankshaft throws, as a spatial elastic mechanism system. Any number of cylinders in any angular orientations with respect to each other may be considered. A three-dimensional flexural finite-line element with isoparametric joint freedom irregularity is developed and used to formulate the eigenvalue equations of motion for the system. Consistent mass matrix as well as lumped mass matrix methods can be used. The element can be restrained to perform coupled torsional and flexural or pure torsional frequency analysis of geared rotor model of engines and shafts on many rigid or flexible bearings. Geared connections can also be considered flexible. A generalized computer program is made available for industrial use. It determines frequencies, mode shapes and critical speed bands of an engine for complete crankshaft rotation for as many modes as desired. The frequency and critical speed analysis of a four-cylinder MGB automobile engine with in-plane crank throws, with and without bearing flexibilities, is performed and the results are compared with those obtained using the conventional pure torsional shaft model. Geared tandem ship drive system is studied to test the reliability of the developments.

scholarly journals Nonlinear pre and post-buckled analysis of curved beams using differential quadrature element method

2019 ◽  
Vol 14 (1) ◽  
Author(s):  
M. Zare ◽  
A. Asnafi
Keyword(s):  
Differential Equations ◽  
Numerical Method ◽  
Equations Of Motion ◽  
Differential Quadrature ◽  
Mode Shapes ◽  
Curved Beams ◽  
Buckling Loads ◽  
Differential Quadrature Element Method ◽  
Quadrature Element Method ◽  
Post Buckling

AbstractThis paper studied the in-plane elastic stability including pre and post-buckling analysis of curved beams considering the effects of shear deformations, rotary inertia, and the geometric nonlinearity due to large deformations. Firstly, the governing nonlinear equations of motion were derived. The problem was solved performing both the static and dynamic analysis using the numerical method of differential quadrature element method (DQEM) which is a new and efficient numerical method for rapidly solving linear and nonlinear differential equations. Firstly, the method was applied to the equilibrium equations, leading to a nonlinear algebraic system of equations that would be solved utilizing an arc length strategy. Secondly, the results of the static part were employed to linearize the dynamic differential equations of motion and their corresponding boundary and continuity conditions. Without any loss of generality, a clamped-clamped curved beam under a concentrated load was considered to obtain the buckling loads, natural frequencies, and mode shapes of the beam throughout the method. To validate the proposed method, the beam was modeled using a finite element simulation. A great agreement between the results was seen that showed the accuracy of the proposed method in predicting the pre and post-buckling behavior of the beam. The investigation also included an examination of the curvature parameter influencing the dynamic behavior of the problem. It was shown that the values of buckling loads were completely influenced by the curvature of the beam; also, due to the sharp change of longitudinal stiffness after bucking, the symmetric mode shapes changed more than it was expected.

Coupling of a State-Space Inflow to Nonlinear Blade Equations and Extraction of Generalized Aerodynamic Force Mode Shapes

1993 ◽  
Vol 46 (11S) ◽  
pp. S295-S304 ◽  
Author(s):  
Donizeti de Andrade ◽  
David A. Peters
Keyword(s):  
Aerodynamic Force ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Flow Distribution ◽  
Mode Shapes ◽  
State Equations ◽  
Force Mode ◽  
Rotor Disk ◽  
Induced Flow ◽  
Wake Model

The aeroelastic stability of helicopter rotors in hovering flight has been investigated by a set of generalized dynamic wake equations and hybrid equations of motion for an elastic blade cantilevered in bending and having a torsional root spring to model pitch-link flexibility. The generalized dynamic wake model employed is based on an induced flow distribution expanded in a set of harmonic and radial shape functions, including undetermined time dependent coefficients as aerodynamic states. The flow is described by a system of first-order, ordinary differential equations in time, for which the pressure distribution at the rotor disk is expressed as a summation of the discrete loadings on each blade, accounting simultaneously for a finite number of blades and overall rotor effects. The present methodology leads to a standard eigenanalysis for the associated dynamics, for which the partitioned coefficient matrices depend on the numerical solution of the blade equilibrium and inflow steady-state equations. Numerical results for a two-bladed, stiff-inplane hingeless rotor with torsionally soft blades show the importance of unsteady, three-dimensional aerodynamics in predicting associated generalized aerodynamic force mode shapes.

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