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.

Dynamic Response of Multi-Cylinder Engines As Spatial Elastic Linkage Systems

1996 ◽  
Author(s):  
Cemil Bagci ◽  
Siva K. Rajavenkateswaran
Keyword(s):  
Nonlinear Differential Equations ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Lumped Mass ◽  
Time Histories ◽  
Solving Equations ◽  
Finite Line ◽  
Exponential Method ◽  
Motion Effect ◽  
Industrial Use

Abstract Dynamics of multi-cylinder engines have been performed by the conventional methods using lumped pure torsional systems. This article offers a finite element method of performing elastodynamic analysis of multi-cylinder engines considering them as spatial linkage systems with true spatial geometries of the crankshaft and the linkage loops. An engine can have any number of cylinders with linear offsets and angular orientations relative to each other. A three-dimensional finite-line element with isoparametric joint irregularity freedoms is developed and used. Consistent or lumped mass systems can be used. Elastodynamics of engines is considered in two forms: (a) kinetoelastostatics (KES) where all forces and torques acting on the system are considered except the vibratory motion effect; (b) kinetoelastodynamics (KED) where the forced and damped equations of motion of the system are solved. Matrix exponential method of solving equations of KED motions are presented and used. It is proven to be a very efficient and stable technique for the solutions of large systems of linear and nonlinear differential equations of any order. After solving for the generalized coordinates, time histories of the neutral coordinate displacements, forces, moments, stresses, bearing forces, and generated torque are determined for as many work cycles as desired. A generalized computer program performing KED and KES studies of any multi-cylinder engine is made available for industrial use. KED and KES analyses of a four-cylinder automobile engine are performed.

Experimental and Numerical Modal Analysis of Composite Laminated Shells

2019 ◽  
Vol 161 (A1) ◽  
Keyword(s):  
Heat Dissipation ◽  
Mass Matrix ◽  
Experimental Studies ◽  
Free Vibration Analysis ◽  
Mode Shapes ◽  
Radius Of Curvature ◽  
Lumped Mass ◽  
Auxiliary Equipment ◽  
Mass Model ◽  
Laminated Shells

The presence of cut outs at different positions of laminated shell component in marine and aeronautical structures facilitate heat dissipation, undertaking maintenance, fitting auxiliary equipment, access ports for mechanical and electrical systems, damage inspection and also influences the dynamic behaviour of the structures. The aim of the present study is to establish a comprehensive perspective of dynamic behavior of laminated deep shells (length to radius of curvature ratio less than one) with cut-out by experiments and numerical simulation. The glass epoxy laminated composite shell has been prepared in the laboratory by resin infusion. The experimental free vibration analysis is carried out on laminated shells with and without cut-out. The mass matrix is developed by considering rotary inertia in a lumped mass model in the numerical modeling. The results obtained from numerical and experimental studies are compared for verification and the consistency between mode shapes is established by applying modal assurance criteria.

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.

A Unified Frequency Equation and the Motion Analysis of a Moving Flexible Beam Carrying a Concentrated Mass

1999 ◽  
Author(s):  
S. Park ◽  
J. W. Lee ◽  
Y. Youm ◽  
W. K. Chung
Keyword(s):  
Natural Frequencies ◽  
Equations Of Motion ◽  
Frequency Equation ◽  
Open Loop ◽  
Mode Shapes ◽  
Flexible Beam ◽  
Lumped Mass ◽  
Concentrated Mass ◽  
Forcing Function ◽  
The Mathematical Model

Abstract In this paper, the mathematical model of a Bernoulli-Euler cantilever beam fixed on a moving cart and carrying an intermediate lumped mass is derived. The equations of motion of the beam-mass-cart system is analyzed utilizing unconstrained modal analysis, and a unified frequency equation which can be generally applied to this kind of system is obtained. The change of natural frequencies and mode shapes with respect to the change of the mass ratios of the beam, the lumped mass and the cart and to the position of the lumped mass is investigated. The open-loop responses of the system by arbitrary forcing function are also obtained through numerical simulations.

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.

scholarly journals A concise nodal-based derivation of the floating frame of reference formulation for displacement-based solid finite elements

2019 ◽  
Vol 49 (3) ◽  
pp. 291-313 ◽  
Author(s):  
Andreas Zwölfer ◽  
Johannes Gerstmayr
Keyword(s):  
Finite Element ◽  
Mass Matrix ◽  
Equations Of Motion ◽  
Frame Of Reference ◽  
Body Motion ◽  
Reference Formulation ◽  
Lumped Mass ◽  
Floating Frame Of Reference ◽  
Mass Approximation ◽  
Floating Frame

AbstractThe Floating Frame of Reference Formulation (FFRF) is one of the most widely used methods to analyze flexible multibody systems subjected to large rigid-body motion but small strains and deformations. The FFRF is conventionally derived via a continuum mechanics approach. This tedious and circuitous approach, which still attracts attention among researchers, yields so-called inertia shape integrals. These unhandy volume integrals, arising in the FFRF mass matrix and quadratic velocity vector, depend not only on the degrees of freedom, but also on the finite element shape functions. That is why conventional computer implementations of the FFRF are laborious and error prone; they require access to the algorithmic level of the underlying finite element code or are restricted to a lumped mass approximation. This contribution presents a nodal-based treatment of the FFRF to bypass these integrals. Each flexible body is considered in its spatially discretized state ab initio, wherefore the integrals are replaced by multiplications by a constant finite element mass matrix. Besides that, this approach leads to a simpler and concise but rigorous derivation of the equations of motion. The steps to obtain the inertia-integral-free equations of motion (in 2D and 3D spaces) are presented in a clear and comprehensive way; the final result provides ready-to-implement equations of motion without a lumped mass approximation, in contrast to the conventional formulation.

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.

Elastodynamics of Planar Mechanisms Using Planar Actual Finite Line Elements, Lumped Mass Systems, Matrix-Exponential Method, and the Method of “Critical-Geometry-Kineto-Elasto-Statics” (CGKES)

1979 ◽  
Vol 101 (3) ◽  
pp. 417-427 ◽  
Author(s):  
C. Bagci ◽  
S. Kalaycioglu
Keyword(s):  
Equations Of Motion ◽  
Matrix Exponential ◽  
Time Dependent ◽  
Lumped Mass ◽  
Planar Mechanisms ◽  
Dynamic Stresses ◽  
Coulomb Damping ◽  
Finite Line ◽  
Exponential Method ◽  
Line Elements

The article presents a general method for the elastodynamic analysis of planar mechanisms. It uses planar actual finite line elements (regular and irregular elements given in a companion article) and lumped mass systems to formulate the equations of motion of a mechanism. Damping coefficient matrix can incorporate time dependent viscous or coulomb damping coefficients in addition to the coefficients of velocity dependent internal damping. The forcing vector can incorporate any externally applied time dependent force or torque, inertial forces and inertial torques, any nonlinear viscous or Coulomb damping forces and torques. The matrix exponential method is introduced for the numerical solution of the equations of motion. Matrix displacement method of determining dynamic stresses using the generalized coordinate displacements is given. Elastodynamic analysis of a plane four-bar mechanism is performed for several cycles of kinematic motion, and the dynamic stresses are compared with those obtained by experiments. The method of “Critical-Geometry-Kineto-Elasto-Statics” (CGKES) is proposed for the computation of dynamic stress magnitudes making use of the critical geometry of the mechanism. It requires the analysis of a mechanism at the critical geometry position of the mechanism which is defined by the lowest fundamental frequency of the mechanism. The results predicted by the method of CGKES compares within two percent with the experimental results.

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.

Dynamics of Flexible Manipulator Arms: Alternative Derivation, Verification, and Characteristics for Control

1993 ◽  
Vol 115 (3) ◽  
pp. 394-404 ◽  
Author(s):  
B.-S. Yuan ◽  
W. J. Book ◽  
J. D. Huggins
Keyword(s):  
Mass Matrix ◽  
Equations Of Motion ◽  
Structural Complexity ◽  
Hydraulic Cylinder ◽  
Geometric Constraints ◽  
Flexible Manipulator ◽  
Mode Shapes ◽  
Actuator Dynamics ◽  
Transformation Matrices ◽  
Uniform Cross Section

This work seeks to provide an effective way for developing the dynamics of a multi-link flexible manipulator consisting of rotary joints connecting two links. Kinematics of both the rotary joint motion and the link deformation are described by 4×4 transformation matrices as proposed in previous works (Book, 1984). The link deflection is assumed small so that the link transformation can be composed of summations of assumed link shapes. To determine the appropriate choice of component mode shapes, two essential techniques employed here are experimental and finite element methods. The resulting equations of motion allow the complete nonlinear model to be recursively derived from the Jacobian matrix and the mass properties via symbolic manipulation. Two prototype models of flexible manipulators are used to verify the dynamics with frequency and time responses. This paper contributes several new results: (1) the velocity terms (Coriolis and centrifugal forces) are related to variations in the mass matrix, (2) the skew symmetry of certain useful terms are shown, (3) the system is theoretically demonstrated to be stable with joint P.D. controllers in addition to an experimental approach, (4) practical and effective incorporation of actuator dynamics (hydraulic cylinder) and structural complexity (non-uniform cross section) is achieved through selection of mode shapes, (5) geometric constraints are incorporated through simplified coordinate transformations and (6) the results are verified on two physical cases.

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