Multibody System Modeling of Flexible Twist Beam Axles in Car Suspension Systems

2011 ◽  
Author(s):  
Tariq Z. Sinokrot ◽  
William C. Prescott ◽  
Maurizio Nembrini ◽  
Alessandro Toso
Keyword(s):  
Multibody System ◽  
System Modeling ◽  
Equations Of Motion ◽  
Mode Shapes ◽  
Flexible Beam ◽  
Element Analysis ◽  
Suspension Systems ◽  
Car Suspension ◽  
Flexible Component ◽  
System Approaches

Dynamic simulation techniques that are based on Multibody system approaches have become an important topic in studying the performance of various mechanical components that comprise an automotive system. One of the challenging issues in such studies is the ability to properly account for the flexibility of certain parts in the system. One example where this is important is the design of twist beam axles in car suspension systems where twisting deformations are present. These deformations are geometrically nonlinear and require a special handling. In this paper two multibody system approaches that are commonly used in overcoming such problem are examined. The first method is a sub-structuring technique that is based on the popular method of component mode synthesis. This method is based on dividing the flexible component into sub-structures, in which, the linear elastic structural theory is sufficient to describe the deformation of each sub-structure. Using this method the deformation of the beam is described using the mode shapes of vibration of each sub-structure. The equations of motion, in this case, are written in terms of the system’s generalized coordinates and modal participation factors. In the second method a Multibody System (MBS) solver and an external nonlinear Finite Element Analysis (FEA) solver are coupled together in a co-simulation manner. The nonlinear FEA solver, in this case, is used in modeling the deformation of the twist beam. The forces due to the nonlinear deformations of the flexible beam are communicated to the MBS solver at certain attachment points where the flexible body is attached to the rest of the multibody system. The displacements and velocities of these attachment points are calculated by the MBS solver and are communicated back to the nonlinear FEA solver to advance the simulation. The two methods described above will be reviewed in this paper and an example of a twist beam axle in a car suspension system model will be examined twice, once using the sub-structuring method, and once using the co-simulation method. The numerical results obtained using both methods will be analyzed and compared.

A Comparison of Different Multibody System Approaches in the Modeling of Flexible Twist Beam Axles

2011 ◽  
Author(s):  
Tariq Z. Sinokrot ◽  
William C. Prescott ◽  
Maurizio Nembrini ◽  
Alessandro Toso
Keyword(s):  
Multibody System ◽  
Synthesis Method ◽  
Component Mode Synthesis ◽  
Elastic Theory ◽  
Mode Shapes ◽  
Flexible Body ◽  
Element Analysis ◽  
Linear Elastic ◽  
Linear Elastic Theory ◽  
Flexible Component

One of the challenging issues in the area of flexible multibody systems is the ability to properly account for the geometric nonlinear effects that are present in many applications. One common application where these effects play an important role is the dynamic modeling of twist beam axles in car suspensions. The purpose of this paper is to examine the accuracy of the results obtained using four common modeling methods used in such applications. The first method is based on a spline beam approach in which a long beam is represented using piecewise rigid bodies interconnected by beam force elements along a spline curve. The beam force elements use a simple linear beam theory in approximating the forces and torques along the beam central axis. The second approach uses the well known method of component mode synthesis that is based on the linear elastic theory. Using this method the deformation of the beam, which is modeled as one flexible body, is defined using its own vibration and static correction mode shapes. The equations of motion are, in this case, written in terms of the system’s generalized coordinates and modal participation factors. The linear elastic theory is used again in the third approach using a slightly different technique called the sub-structuring synthesis method. This method is based on dividing the flexible component into sub-structures, in which, the method of component mode synthesis is used to describe the deformation of each substructure. The fourth approach is based on a co-simulation technique that uses a Multibody System (MBS) solver and an external nonlinear Finite Element Analysis (FEA) solver. The flexibility of any body in the multibody system is, in this case, modeled in the external nonlinear FEA solver. The latter calculates the forces due to the nonlinear deformations of the flexible body in question and communicates that to the MBS solver at certain attachment points where the flexible body is attached to the rest of the multibody system. The displacements and velocities of these attachment points are calculated by the MBS solver and are communicated back to the nonlinear FEA solver to advance the simulation. The four approaches described are reviewed in this paper and a multibody system model of a car suspension system that includes a twist beam axle is presented. The model is examined four times, once using each approach. The numerical results obtained using the different methods are analyzed and compared.

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.

A rigid multibody method for free vibration analysis of beams with variable axial parameters

2016 ◽  
Vol 23 (1) ◽  
pp. 131-146 ◽  
Author(s):  
Aleksandar Nikolić ◽  
Slaviša Šalinić
Keyword(s):  
Multibody System ◽  
Free Vibration Analysis ◽  
Mode Shapes ◽  
Constant Thickness ◽  
Flexible Beam ◽  
Cantilever Beams ◽  
New Approach ◽  
Transverse Vibrations ◽  
Tip Mass ◽  
Rigid Multibody

This paper presents a new approach to the problem of determining the frequencies and mode shapes of Euler–Bernoulli tapered cantilever beams with a tip mass and a spring at the free end. The approach is based on the replacement of the flexible beam by a rigid multibody system. Beams with constant thickness and exponentially and linearly tapered width, as well as double-tapered cantilever beams are considered. The influence of the tip mass, stiffness of the spring, and taper on the frequencies of the free transverse vibrations of tapered cantilever beams are examined. Numerical examples with results confirming the convergence and accuracy of the approach are given.

Vibrations of a Rotating Beam

1966 ◽  
Vol 11 (4) ◽  
pp. 17-24 ◽  
Author(s):  
Jay L. Lipeles
Keyword(s):  
Boundary Conditions ◽  
Coordinate System ◽  
Equations Of Motion ◽  
Cone Angle ◽  
Elastic Stiffness ◽  
Degree Of Freedom ◽  
Mode Shapes ◽  
Flexible Beam ◽  
Rotor Speed ◽  
Classical Formula

The object of this paper is to analyze the coupled flatwise (out of plane) edgewise (in plane) vibrations of a beam rotating about one if its ends. The hub is assumed motionless and the vibration is considered to occur about a large deflected position. That is, coning and lagging angles are allowed to be large. The equations of motion are derived by a combination of techniques. The kinetic energy of the system is expressed in terms of coordinates lying in the beam. This is done by making use of four coordinate transformations that relate the beam coordinate system to a fixed coordinate system. The inertial load distribution is obtained by application of the first two terms of Lagrange's equation. These loads are used to compute the bending moment distribution which is substituted into Euler's beam bending equation. The equations of motion are solved by assuming a solution in the form of a linear combination of orthogonal modes. These equations are multiplied by the jth mode shape and integrated over the beam length. There are four terms resulting; the mass and elastic stiffness terms form a diagonal array and the Coriolis' and centrifugal spring terms form a full array. These equations may be solved by easily available matrix techniques. The modes chosen for the solution are the normal modes of the nonrotating beam. The advantage of this choice is that each of the modes already satisfies the problem boundary conditions. Since the non‐rotating modes are a good approximation to the rotating modes the series converges rapidly and can be cut off after a few terms. Several sample problems are worked out. First, the beam is assumed rigid and free to flap. The classical formula for flapping frequency is verified with the addition that the terms due to large cone and lag angles are included. Second, the same problem is done except that instead of the flapping degree of freedom the lagging degree of freedom is analyzed. The classical formula for lagging is also verified for the zero cone angle. When the cone angle is large this degree of freedom becomes statically unstable. Third, the above problem is redone for the coupled lagging — flapping degrees of freedom. Fourth, a flexible beam is assumed with zero cone and lag angles. Mode shapes and frequencies are computed as a function of rotor speed. It is shown that as rotor speed increases the beam mode shapes and frequencies approach those of a chain. That is, the elastic stiffness becomes negligible relative to the centrifugal stiffness. The advantages of the formulation developed in this paper (in addition to allowing consideration of large coning and lagging angles) are: 1) that the terms that involve rotor speed (the centrifugal spring and the Coriolis coupling) have that parameter as a factor multiplying the whole matrix so that if frequencies and modes are required over a range of rotor speeds the centrifugal and Coriolis' terms need only be calculated once; 2) at large rotor speeds the Myklestad analysis has difficulty converging but in this procedure, because the non‐rotating modes already satisfy the boundary conditions, there is no difficulty in convergence.

Dynamic Analysis and Fatigue Life Prediction of a Very Flexible Component in Multibody System

2006 ◽  
Vol 321-323 ◽  
pp. 1597-1600
Author(s):  
Ji Won Yoon ◽  
Kab Jin Jun ◽  
Tae Won Park
Keyword(s):  
Finite Element ◽  
Fatigue Life ◽  
Multibody System ◽  
Beam Theory ◽  
Computational Method ◽  
Deformation Analysis ◽  
Flexible Beam ◽  
Combined System ◽  
Absolute Nodal Coordinates ◽  
Flexible Component

Recently, the finite element absolute nodal coordinate formulation(ANCF) was developed for large deformation analysis of flexible bodies in multi-body dynamics. This formulation is based on finite element procedures and the general continuum mechanics theory to represent elastic forces. In this paper, a computational method, which predicts the dynamic and structural properties of a very flexible beam in a multibody system, is presented based on Euler-Bernoulli beam theory and ANCF. In order to consider the dynamic interaction between a continuous large deformable beam and a rigid multibody system, a combined system equations of motion was derived by adopting absolute nodal coordinates and rigid body coordinates. The efficiency and reliability of the computational results are verified by comparison with a commercial program. These methods can be applied for predicting the dynamic stress and fatigue life of the wire harness used in a robot system. The process of predicting the fatigue life using the proposed method in this paper may be applied to continuous mechanical parts of various dynamic systems.

Parametric Resonances of a Spinning Cyclic Symmetric Rotor Assembled to a Flexible Stationary Housing via Multiple Bearings

2012 ◽  
Author(s):  
W. C. Tai ◽  
I. Y. Shen
Keyword(s):  
Rigid Body ◽  
Equations Of Motion ◽  
Housing System ◽  
Mode Shapes ◽  
Element Analysis ◽  
Linear Elastic ◽  
Rotor Bearing ◽  
Symmetric Rotor ◽  
Cyclic Symmetric ◽  
Inertia Forces

This paper is to present findings from a theoretical study on free vibration and stability of a rotor-bearing-housing system. The rotor is cyclic symmetric and spinning at constant speed, while the housing is stationary and flexible. Moreover, the rotor and housing are assembled via multiple, linear, elastic bearings. For the rotor and the housing, their mode shapes are first obtained in rotor-based and ground-based coordinate systems, respectively. By discretizing the kinetic and potential energies of the rotor-bearing-housing system through use of the mode shapes, a set of equations of motion appears in the form of ordinary differential equations with periodic coefficients. Analyses of the equations of motion indicate that instabilities could appear at certain spin speed in the form of combination resonances of the sum type. To demonstrate the validity of the formulation, two numerical examples are studied. For the first example, the spinning rotor is an axisymmetric disk and the housing is a square plate with a central shaft. Moreover, the rotor and the housing are connected via two linear elastic bearings. Instability appears in the form of coupled vibration between the stationary housing and spinning rotor through three different formats: rigid-body rotor translation, rigid-body rotor rocking, and elastic rotor modes that present unbalanced inertia forces or moments. For the second example, the rotor is cyclic symmetric in the form of a disk with four evenly spaced slots. The housing and bearings remain the same. When the rotor is stationary, natural frequencies and mode shapes predicted from the formulation agree well with those predicted from a finite element analysis, which further ensures the validity of the formulation. When the cyclic symmetric rotor spins, instability appears in the same three formats as in the case of axisymmetric rotor. Number of instability zones, however, increases because the cyclic symmetric rotor has more elastic rotor modes that present unbalanced inertia forces or moments.

scholarly journals Dynamic Finite Element Analysis of Flexible Double Wishbone Suspension Systems with Different Damping Mechanisms

2020 ◽  
Author(s):  
Alaa Adel Rahman ◽  
Ayman E Nabawy ◽  
Ayman M Abdelhaleem ◽  
Soliman S Alieldin
Keyword(s):  
Finite Element ◽  
Vibration Isolation ◽  
Mechanical Design ◽  
Constitutive Relations ◽  
Equations Of Motion ◽  
Element Model ◽  
Vehicle Speed ◽  
Element Analysis ◽  
Suspension Systems ◽  
Suspension Model

Suspension systems in running vehicles keep the occupants comfortable and isolated from road noise, disturbances, and vibrations and consequently prevent the vehicle from damage and wearing. To attain comfortable and vibration isolation conditions, both material flexibility and damping should be considered in the considered suspension model. This paper presents an incremental finite element model to study and analyze the dynamic behavior of double wishbone suspension systems considering both material flexibility and damping effects. The flexibility of the suspension links are modeled with plane frame element based on Timoshenko beam hypothesis (TBH). On the other hand, the flexibility of joints connecting the suspension links together and with the vehicle chassis is modeled with the revolute joint element. To incorporate the damping effect, viscoelastic, viscous and proportional damping are considered. An incremental viscoelastic constitutive relations, suitable for finite element implementation, are developed. The developed finite element equations of motion are solved using the Newmark technique. The developed procedure is verified by comparing the obtained results with that obtained by the developed analytical solution and an excellent agreement is found. The applicability and effectiveness of the developed procedure are demonstrated by conducting parametric studies to show the effects of the road irregularities profiles, the vehicle speed, and the material damping on the transverse deflection and the resultant stresses of suspension system. Results obtained are supportive in the mechanical design, manufacturing processes of such type of structural systems.

Modal and harmonic analysis of three-dimensional wind turbine models

2016 ◽  
Vol 40 (6) ◽  
pp. 518-527 ◽  
Author(s):  
Takwa Sellami ◽  
Hanen Berriri ◽  
A Moumen Darcherif ◽  
Sana Jelassi ◽  
M Faouizi Mimouni
Keyword(s):  
Finite Element ◽  
Wind Turbine ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Operating Conditions ◽  
Dynamic Responses ◽  
Mode Shapes ◽  
Element Analysis ◽  
Micro Turbine

In this article, the dynamic responses of wind turbine systems are analytically and numerically investigated. For this purpose, analytic differential equations of motion of wind turbine components subjected to vibration (the blades, the nacelle, and the tower) are solved. This allows determining their dynamic characteristics, mode shapes, and natural frequencies. Two models of two three-dimensional (3D) micro-turbine that are created by the finite element method are set up using the new version of the academic finite element analysis software ANSYS. The first wind turbine is a standard micro three-bladed turbine and the second one is a micro six-bladed Rutland 504. Their natural frequencies and mode shapes are identified based on the modal analysis principle to check the validity of designed models. Dynamic behaviors at several operating conditions of wind turbines are established. Then, spectrum graphs of the structures along x-, y- and z-axis are analyzed.

scholarly journals Effect of Thrust on the Structural Vibrations of a Nonuniform Slender Rocket

2020 ◽  
Vol 25 (2) ◽  
pp. 29
Author(s):  
Desmond Adair ◽  
Aigul Nagimova ◽  
Martin Jaeger
Keyword(s):  
Natural Frequencies ◽  
Equations Of Motion ◽  
Approximate Solutions ◽  
Mode Shapes ◽  
Algebraic Equations ◽  
Structural Vibrations ◽  
Unknown Parameters ◽  
One Dimensional ◽  
Vibration Problems ◽  
Nonuniform Beam

The vibration characteristics of a nonuniform, flexible and free-flying slender rocket experiencing constant thrust is investigated. The rocket is idealized as a classic nonuniform beam with a constant one-dimensional follower force and with free-free boundary conditions. The equations of motion are derived by applying the extended Hamilton’s principle for non-conservative systems. Natural frequencies and associated mode shapes of the rocket are determined using the relatively efficient and accurate Adomian modified decomposition method (AMDM) with the solutions obtained by solving a set of algebraic equations with only three unknown parameters. The method can easily be extended to obtain approximate solutions to vibration problems for any type of nonuniform beam.

Rotor-Dynamics of Different Shaft Configurations for a 6 kW Micro Gas Turbine for Concentrated Solar Power

2016 ◽  
Author(s):  
A. Arroyo ◽  
M. McLorn ◽  
M. Fabian ◽  
M. White ◽  
A. I. Sayma
Keyword(s):  
Gas Turbines ◽  
High Speed ◽  
Dynamic Models ◽  
Solar Power ◽  
Critical Issue ◽  
Rotor Dynamics ◽  
Mode Shapes ◽  
Concentrated Solar Power ◽  
Element Analysis ◽  
Rotor Dynamic

Rotor-dynamics of Micro Gas Turbines (MGTs) under 30 kW have been a critical issue for the successful development of reliable engines during the last decades. Especially, no consensus has been reached on a reliable MGT arrangement under 10 kW with rotational speeds above 100,000 rpm, making the understanding of the rotor-dynamics of these high speed systems an important research area. This paper presents a linear rotor-dynamic analysis and comparison of three mechanical arrangements of a 6 kW MGT intended for utilising Concentrated Solar Power (CSP) using a parabolic dish concentrator. This application differs from the usual fuel burning MGT in that it is required to operate at a wider operating speed range. The objective is to find an arrangement that allows reliable mechanical operation through better understanding of the rotor dynamics for a number of alternative shaft-bearings arrangements. Finite Element Analysis (FEA) was used to produce Campbell diagrams and to determine the critical speeds and mode shapes. Experimental hammer tests using a new approach based on optical sensing technology were used to validate the rotor-dynamic models. The FEA simulation results for the natural frequencies of a shaft arrangement were within 5% of the measurements, while the deviation for the shaft-bearings arrangement increased up to 16%.

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