scholarly journals TO THE QUESTION OF ANALYSIS OF EQUATIONS OF MOTION OF A RIGID BODY DURING THE MECHANICAL OSCILATIONS

2021 ◽  
Vol 5 (3) ◽  
Author(s):  
Iryna Bernyk
Keyword(s):  
Differential Equation ◽  
Rigid Body ◽  
Natural Frequency ◽  
Equations Of Motion ◽  
Oscillation Amplitude ◽  
Current Position ◽  
Homogeneous Differential Equation ◽  
Inhomogeneous Differential Equation ◽  
Oscillation Process ◽  
Undamped Oscillations

Depending on the current position of the mass in different areas of the spring deformation during the oscillation process the values that determines the natural frequency of free continuous oscillations have opposite signs. It is defined by the change in the direction of acceleration of the mass in these areas, which makes it possible to determine a single inhomogeneous differential equation of the oscillation process in different areas of the movement of the mass. When the oscillation amplitude is much less than the static position of the mass, this inhomogeneous differential equation represents a homogeneous differential equation of free undamped oscillations.

scholarly journals TO THE QUESTION OF ANALYSIS OF EQUATIONS OF MOTION OF A RIGID BODY DURING THE MECHANICAL OSCILATIONS

2021 ◽  
Author(s):  
Igor Dobrov ◽  
Andriy Semichev
Keyword(s):  
Differential Equation ◽  
Rigid Body ◽  
Natural Frequency ◽  
Equations Of Motion ◽  
Oscillation Amplitude ◽  
Current Position ◽  
Homogeneous Differential Equation ◽  
Inhomogeneous Differential Equation ◽  
Oscillation Process ◽  
Undamped Oscillations

 Depending on the current position of the mass in different areas of the spring deformation during the oscillation process the values that determines the natural frequency of free continuous oscillations have opposite signs. It is defined by the change in the direction of acceleration of the mass in these areas, which makes it possible to determine a single inhomogeneous differential equation of the oscillation process in different areas of the movement of the mass. When the oscillation amplitude is much less than the static position of the mass, this inhomogeneous differential equation represents a homogeneous differential equation of free undamped oscillations.

scholarly journals New Treatment of the Rotary Motion of a Rigid Body with Estimated Natural Frequency

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
A. I. Ismail
Keyword(s):  
Rigid Body ◽  
Natural Frequency ◽  
Numerical Solutions ◽  
Equations Of Motion ◽  
Geometric Interpretation ◽  
Parameter Method ◽  
Large Parameter ◽  
Regular Precession ◽  
New Treatment ◽  
The Stability

In this paper, the problem of the motion of a rigid body about a fixed point under the action of a Newtonian force field is studied when the natural frequency ω = 0.5 . This case of singularity appears in the previous works and deals with different bodies which are classified according to the moments of inertia. Using the large parameter method, the periodic solutions for the equations of motion of this problem are obtained in terms of a large parameter, which will be defined later. The geometric interpretation of the considered motion will be given in terms of Euler’s angles. The numerical solutions for the system of equations of motion are obtained by one of the well-known numerical methods. The comparison between the obtained numerical solutions and analytical ones is carried out to show the errors between them and to prove the accuracy of both used techniques. In the end, we obtain the case of the regular precession type as a special case. The stability of the motion is considered by the phase diagram procedures.

Closed-Form Forced Response of a Damped, Rotating, Multiple Disks/Spindle System

1997 ◽  
Vol 64 (2) ◽  
pp. 343-352 ◽  
Author(s):  
I. Y. Shen
Keyword(s):  
Differential Equation ◽  
Rigid Body ◽  
Closed Form ◽  
Equations Of Motion ◽  
Rotating Disk ◽  
Forced Response ◽  
Vibration Response ◽  
Matrix Differential Equation ◽  
Spindle System ◽  
Matrix Differential

This paper is to study forced vibration response of a rotating disk/spindle system consisting of multiple flexible circular disks clamped to a rigid spindle supported by two flexible bearings. In particular, the disk/spindle system is subjected to prescribed translational base excitations and externally applied loads. Because of the bearing flexibility, the rigid spindle undergoes infinitesimal rigid-body rocking and translation simultaneously. To model real vibration response that has finite resonance amplitudes, the disks and the bearings are assumed to be viscously damped. Equations of motion are then derived through use of Rayleigh dissipation function and Lagrange’s equation. The equations of motion include three sets of matrix differential equations: one for the rigid-body rocking of the spindle and one-nodal-diameter disk modes, one for the axial translation of the spindle and axisymmetric disk modes, and one for disk modes with two or more nodal diameters. Each matrix differential equation contains either a gyroscopic matrix or a damping matrix or both. The causal Green’s function of each matrix differential equation is determined explicitly in closed form through use of matrix inversion and inverse Laplace transforms. Closed-form forced response of the damped rotating disk/spindle system is then obtained from the causal Green’s function and the generalized forces through convolution integrals. Finally, responses of a disk/spindle system subjected to a concentrated sinusoidal load or an impulsive load are demonstrated numerically as an example.

scholarly journals Vibration analysis of a sandwich cylindrical shell in hygrothermal environment

2021 ◽  
Vol 10 (1) ◽  
pp. 414-430
Author(s):  
Chunwei Zhang ◽  
Qiao Jin ◽  
Yansheng Song ◽  
Jingli Wang ◽  
Li Sun ◽  
...  
Keyword(s):  
Cylindrical Shell ◽  
Natural Frequency ◽  
Equations Of Motion ◽  
Single Layer ◽  
Functionally Graded ◽  
Sandwich Shell ◽  
Continuous Change ◽  
Multilayered Structures ◽  
Cylindrical Sandwich Shell ◽  
Vibrational Behavior

Abstract The sandwich structures are three- or multilayered structures such that their mechanical properties are better than each single layer. In the current research, a three-layered cylindrical shell including a functionally graded porous core and two reinforced nanocomposite face sheets resting on the Pasternak foundation is used as model to provide a comprehensive understanding of vibrational behavior of such structures. The core is made of limestone, while the epoxy is utilized as the top and bottom layers’ matrix phase and also it is reinforced by the graphene nanoplatelets (GNPs). The pattern of the GNPs dispersion and the pores distribution play a crucial role at the continuous change of the layers’ properties. The sinusoidal shear deformation shells theory and the Hamilton’s principle are employed to derive the equations of motion for the mentioned cylindrical sandwich shell. Ultimately, the impacts of the model’s geometry, foundation moduli, mode number, and deviatory radius on the vibrational behavior are investigated and discussed. It is revealed that the natural frequency and rotation angle of the sandwich shell are directly related. Moreover, mid-radius to thickness ratio enhancement results in the natural frequency reduction. The results of this study can be helpful for the future investigations in such a broad context. Furthermore, for the pipe factories current study can be effective at their designing procedure.

scholarly journals Closed-form time derivatives of the equations of motion of rigid body systems

2021 ◽  
Author(s):  
Andreas Müller ◽  
Shivesh Kumar
Keyword(s):  
Rigid Body ◽  
Closed Form ◽  
Multibody Systems ◽  
Equations Of Motion ◽  
Alternative Formulation ◽  
Dynamics Of Rigid Body ◽  
Control Forces ◽  
And Control ◽  
Derivatives Of ◽  
Insight Into

AbstractDerivatives of equations of motion (EOM) describing the dynamics of rigid body systems are becoming increasingly relevant for the robotics community and find many applications in design and control of robotic systems. Controlling robots, and multibody systems comprising elastic components in particular, not only requires smooth trajectories but also the time derivatives of the control forces/torques, hence of the EOM. This paper presents the time derivatives of the EOM in closed form up to second-order as an alternative formulation to the existing recursive algorithms for this purpose, which provides a direct insight into the structure of the derivatives. The Lie group formulation for rigid body systems is used giving rise to very compact and easily parameterized equations.

A Hybrid Highly Parallelizable Algorithm for the Dynamics of Rigid Body Chain Systems

1999 ◽  
Author(s):  
Shanzhong Duan ◽  
Kurt S. Anderson
Keyword(s):  
Rigid Body ◽  
Degrees Of Freedom ◽  
High Efficiency ◽  
Equations Of Motion ◽  
Constraint Forces ◽  
Dynamics Of Rigid Body ◽  
A Chain ◽  
Explicit Determination ◽  
Order Algorithm

Abstract The paper presents a new hybrid parallelizable low order algorithm for modeling the dynamic behavior of multi-rigid-body chain systems. The method is based on cutting certain system interbody joints so that largely independent multibody subchain systems are formed. These subchains interact with one another through associated unknown constraint forces f¯c at the cut joints. The increased parallelism is obtainable through cutting the joints and the explicit determination of associated constraint loads combined with a sequential O(n) procedure. In other words, sequential O(n) procedures are performed to form and solve equations of motion within subchains and parallel strategies are used to form and solve constraint equations between subchains in parallel. The algorithm can easily accommodate the available number of processors while maintaining high efficiency. An O[(n+m)Np+m(1+γ)Np+mγlog2Np](0<γ<1) performance will be achieved with Np processors for a chain system with n degrees of freedom and m constraints due to cutting of interbody joints.

Free vibration analysis and post-critical free vibrations of nanocomposite rotating beams reinforced with graphene platelet

2021 ◽  
pp. 107754632110511
Author(s):  
Arameh Eyvazian ◽  
Chunwei Zhang ◽  
Farayi Musharavati ◽  
Afrasyab Khan ◽  
Mohammad Alkhedher
Keyword(s):  
Natural Frequency ◽  
Thermal Environment ◽  
Rotational Velocity ◽  
Equations Of Motion ◽  
Free Vibration Analysis ◽  
Beam Theory ◽  
Weight Fraction ◽  
Free Vibrations ◽  
Graphene Platelet ◽  
The Impact

Treatment of the first natural frequency of a rotating nanocomposite beam reinforced with graphene platelet is discussed here. In regard of the Timoshenko beam theory hypothesis, the motion equations are acquired. The effective elasticity modulus of the rotating nanocomposite beam is specified resorting to the Halpin–Tsai micro mechanical model. The Ritz technique is utilized for the sake of discretization of the nonlinear equations of motion. The first natural frequency of the rotating nanocomposite beam prior to the buckling instability and the associated post-critical natural frequency is computed by means of a powerful iteration scheme in reliance on the Newton–Raphson method alongside the iteration strategy. The impact of adding the graphene platelet to a rotating isotropic beam in thermal ambient is discussed in detail. The impression of support conditions, and the weight fraction and the dispersion type of the graphene platelet on the acquired outcomes are studied. It is elucidated that when a beam has not undergone a temperature increment, by reinforcing the beam with graphene platelet, the natural frequency is enhanced. However, when the beam is in a thermal environment, at low-to-medium range of rotational velocity, adding the graphene platelet diminishes the first natural frequency of a rotating O-GPL nanocomposite beam. Depending on the temperature, the post-critical natural frequency of a rotating X-GPL nanocomposite beam may be enhanced or reduced by the growth of the graphene platelet weight fraction.

The Global Motion of a Rigid Body Subject to Periodic Body-Fixed Torques

1995 ◽  
Author(s):  
X. Tong ◽  
B. Tabarrok
Keyword(s):  
Rigid Body ◽  
Equations Of Motion ◽  
Melnikov Method ◽  
The Body ◽  
Body Motion ◽  
Heteroclinic Orbits ◽  
Coordinate Frame ◽  
Global Motion ◽  
Stable And Unstable Manifolds ◽  
Body Subject

Abstract In this paper the global motion of a rigid body subject to small periodic torques, which has a fixed direction in the body-fixed coordinate frame, is investigated by means of Melnikov’s method. Deprit’s variables are introduced to transform the equations of motion into a form describing a slowly varying oscillator. Then the Melnikov method developed for the slowly varying oscillator is used to predict the transversal intersections of stable and unstable manifolds for the perturbed rigid body motion. It is shown that there exist transversal intersections of heteroclinic orbits for certain ranges of parameter values.

MULTIBODY DYNAMIC MODELING AND FLUTTER ANALYSIS OF A FLEXIBLE SLENDER VEHICLE

2012 ◽  
Vol 12 (06) ◽  
pp. 1250049 ◽  
Author(s):  
A. RASTI ◽  
S. A. FAZELZADEH
Keyword(s):  
Differential Equations ◽  
Rigid Body ◽  
Dynamic Modeling ◽  
Equations Of Motion ◽  
The Body ◽  
Elastic Deformations ◽  
Strip Theory ◽  
Multibody Dynamic ◽  
Flutter Analysis ◽  
Rigid Body Motions

In this paper, multibody dynamic modeling and flutter analysis of a flexible slender vehicle are investigated. The method is a comprehensive procedure based on the hybrid equations of motion in terms of quasi-coordinates. The equations consist of ordinary differential equations for the rigid body motions of the vehicle and partial differential equations for the elastic deformations of the flexible components of the vehicle. These equations are naturally nonlinear, but to avoid high nonlinearity of equations the elastic displacements are assumed to be small so that the equations of motion can be linearized. For the aeroelastic analysis a perturbation approach is used, by which the problem is divided into a nonlinear flight dynamics problem for quasi-rigid flight vehicle and a linear extended aeroelasticity problem for the elastic deformations and perturbations in the rigid body motions. In this manner, the trim values that are obtained from the first problem are used as an input to the second problem. The body of the vehicle is modeled with a uniform free–free beam and the aeroelastic forces are derived from the strip theory. The effect of some crucial geometric and physical parameters and the acting forces on the flutter speed and frequency of the vehicle are investigated.

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