Simple Modeling and Analysis for Crankshaft Three-Dimensional Vibrations, Part 2: Application to an Operating Engine Crankshaft

1995 ◽  
Vol 117 (1) ◽  
pp. 80-86 ◽  
Author(s):  
T. Morita ◽  
H. Okamura
Keyword(s):  
Stiffness Matrix ◽  
Dynamic Stiffness ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Inertia Force ◽  
Connecting Rod ◽  
Automobile Engine ◽  
Dynamic Stiffness Matrix ◽  
Modeling And Analysis ◽  
Engine Crankshaft

The modeling and analysis procedures with the dynamic stiffness matrix method described in Part 1 were applied to a crankshaft system, consisting of crankshaft, front pulley, flywheel, piston, and connecting rod, under firing conditions. For firing conditions, (7) one half of the reciprocating masses consisting of the piston, piston pin, and connecting rod small end, and (2) rotating masses of the connecting rod big end mass, were attached to the two ends of the crankpin, taking account of the rigidity of the connecting rod. The excitation forces were calculated from the gas force and the inertia force due to the reciprocating masses. By solving the equations of motion derived in the form of the dynamic stiffness matrix, we calculated the three-dimensional steady-state vibrations of the crankshaft system under firing conditions. A crankshaft system for a four-cylinder in-line automobile engine was used for the analysis. We calculated the influence of the mass and moments of inertia of the front pulley on the behavior of the crankshaft vibrations and the excitation induced at the crankjournal bearings. Calculated values were compared with experimental results.

Simple Modeling and Analysis for Crankshaft Three-Dimensional Vibrations, Part 1: Background and Application to Free Vibrations

1995 ◽  
Vol 117 (1) ◽  
pp. 70-79 ◽  
Author(s):  
H. Okamura ◽  
A. Shinno ◽  
T. Yamanaka ◽  
A. Suzuki ◽  
K. Sogabe
Keyword(s):  
Stiffness Matrix ◽  
Rectangular Cross Section ◽  
Dynamic Stiffness ◽  
Three Dimensional ◽  
Free Vibrations ◽  
Mode Shapes ◽  
Dynamic Stiffness Matrix ◽  
Modeling And Analysis ◽  
Coupling Behavior ◽  
Film Stiffness

To simplify the analysis of the three-dimensional vibrations of automobile engine crankshafts under firing conditions, the crankshaft was idealized by a set of jointed structures consisting of simple round rods and simple beam blocks of rectangular cross-section. The front pulley, timing gear, and the fly-wheel were idealized by a set of masses and moments of inertia. The main journal bearings were idealized by a set of linear springs and dash-pots. For each constituent member, the dynamic stiffness matrix was derived (in closed form) from the transfer matrix. Then the dynamic stiffness matrix for the total crankshaft system was constructed, and the natural frequencies and mode shapes were calculated. The modeling and analysis procedures were applied to the analysis of free vibrations of four kinds of crankshafts: single cylinder, three-cylinder in-line, four-cylinder in-line, and V-six engines. The different coupling behavior of the three-dimensional vibrations in the planar-structure and the solid-structure crankshaft is discussed, and the influence of the bearing oil film stiffness on the crankshaft natural frequency is also analyzed.

A New Finding on the Dynamic Stiffness Matrices of Asymmetric and Axisymmetric Shafts

2001 ◽  
Author(s):  
Francesco A. Raffa ◽  
Furio Vatta
Keyword(s):  
Stiffness Matrix ◽  
Dynamic Stiffness ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Rotational Inertia ◽  
Dynamic Stiffness Matrix ◽  
Rayleigh Beam ◽  
Principal Moments Of Inertia ◽  
Stiffness Matrices ◽  
New Finding

Abstract In this paper the dynamic stiffness method is developed to analyze a rotating asymmetric shaft, i.e. a shaft whose transverse section is characterized by dissimilar principal moments of inertia. The shaft is modeled according to the Rayleigh beam theory including the effects of both translational and rotational inertia, and gyroscopic moments. The mathematical description is carried out in a reference system rotating at the shaft speed and is based on the exact solution of the governing differential equations of motion. The exact expressions of the shaft displacements are utilized for deriving the 8 × 8 complex dynamic stiffness matrix of the shaft. A new relationship is obtained which links the dynamic stiffness matrix of the asymmetric shaft to the 4 × 4 real dynamic stiffness matrix of the axisymmetric shaft.

Dynamic Stiffness Analysis of a Beam Based on Trigonometric Shear Deformation Theory

2007 ◽  
Vol 130 (1) ◽  
Author(s):  
Jun Li ◽  
Hongxing Hua ◽  
Rongying Shen
Keyword(s):  
Shear Deformation ◽  
Stiffness Matrix ◽  
Deformation Theory ◽  
Dynamic Stiffness ◽  
Equations Of Motion ◽  
Shear Deformation Theory ◽  
Beam Element ◽  
Mode Shapes ◽  
Dynamic Stiffness Matrix ◽  
Stiffness Analysis

The dynamic stiffness matrix of a uniform isotropic beam element based on trigonometric shear deformation theory is developed in this paper. The theoretical expressions for the dynamic stiffness matrix elements are found directly, in an exact sense, by solving the governing differential equations of motion that describe the deformations of the beam element according to the trigonometric shear deformation theory, which include the sinusoidal variation of the axial displacement over the cross section of the beam. The application of the dynamic stiffness matrix to calculate the natural frequencies and normal mode shapes of two rectangular beams is discussed. The numerical results obtained are compared to the available solutions wherever possible and validate the accuracy and efficiency of the present approach.

Dynamic Stiffness Matrix Method for the Free Vibration Analysis of Rotating Uniform Shear Beams

2009 ◽  
Author(s):  
Dominic R. Jackson ◽  
S. Olutunde Oyadiji
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Stiffness Matrix ◽  
Dynamic Stiffness ◽  
Equations Of Motion ◽  
Free Vibration Analysis ◽  
Power Series Solution ◽  
Dynamic Stiffness Matrix ◽  
Uniform Shear ◽  
Natural Boundary Conditions

The Dynamic Stiffness Method (DSM) is used to analyse the free vibration characteristics of a rotating uniform Shear beam. Starting from the kinetic and strain energy expressions, the Hamilton’s principle is used to obtain the governing differential equations of motion and the natural boundary conditions. The two equations are solved simultaneously and expressed each in terms of displacement and slope only. The Frobenius power series solution is applied to solve the equations and the resulting solutions are also expressed in terms of four independent solutions. Applying the appropriate boundary conditions, the Dynamic Stiffness Matrix is assembled. The natural frequencies of vibration using the DSM are computed by employing the in-built root finding algorithm in Mathematica as well as by implementing the Wittrick-Williams algorithm in a numerical routine in Mathematica. The results obtained using the DSM are presented in tabular and graphical forms and are compared with results obtained using the Timoshenko and the Bernoulli-Euler theories.

COUPLED FREE VIBRATION ANALYSIS OF SHEAR-FLEXIBLE LAMINATED COMPOSITE I-BEAMS

2013 ◽  
Vol 21 (01) ◽  
pp. 1250024
Author(s):  
NAM-IL KIM
Keyword(s):  
Free Vibration ◽  
Shear Deformation ◽  
Stiffness Matrix ◽  
Vibration Analysis ◽  
Dynamic Stiffness ◽  
Equations Of Motion ◽  
Laminated Composite ◽  
Free Vibration Analysis ◽  
Dynamic Stiffness Matrix ◽  
Analytical Technique

The coupled free vibration analysis of the thin-walled laminated composite I-beams with bisymmetric and monosymmetric cross sections considering shear effects is developed. The laminated composite beam takes into account the transverse shear and the restrained warping induced shear deformation based on the first-order shear deformation beam theory. The analytical technique is used to derive the constitutive equations and the equations of motion of the beam in a systematic manner considering all deformations and their mutual couplings. The explicit expressions for displacement parameters are presented by applying the power series expansions of displacement components to simultaneous ordinary differential equations. Finally, the dynamic stiffness matrix is determined using the force–displacement relationships. In addition, for comparison, a finite beam element with two-nodes and fourteen-degrees-of-freedom is presented to solve the equations of motion. The performance of the dynamic stiffness matrix developed by study is tested through the solutions of numerical examples and the obtained results are compared with results available in literature and the detailed three-dimensional analysis results using the shell elements of ABAQUS. The vibrational behavior and the effect of shear deformation are investigated with respect to the modulus ratios and the fiber angle change.

A New Finding on the Dynamic Stiffness Matrices of Asymmetric and Axisymmetric Shafts

2002 ◽  
Vol 124 (4) ◽  
pp. 649-653
Author(s):  
Francesco A. Raffa ◽  
Furio Vatta
Keyword(s):  
Stiffness Matrix ◽  
Dynamic Stiffness ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Rotational Inertia ◽  
Dynamic Stiffness Matrix ◽  
Rayleigh Beam ◽  
Principal Moments Of Inertia ◽  
Stiffness Matrices ◽  
New Finding

In this paper the dynamic stiffness method is developed to analyze a rotating asymmetric shaft, i.e., a shaft whose transverse section is characterized by dissimilar principal moments of inertia. The shaft is modelled according to the Rayleigh beam theory including the effects of both translational and rotational inertia, and gyroscopic moments. The mathematical description is carried out in a reference system rotating at the shaft speed and is based on the exact solution of the governing differential equations of motion. The exact expressions of the shaft displacements are utilized for deriving the 8×8 complex dynamic stiffness matrix of the shaft. A new relationship is obtained which links the dynamic stiffness matrix of the asymmetric shaft to the 4×4 real dynamic stiffness matrix of the axisymmetric shaft.

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