Abstract: A simulation scheme of rotational motions for predictions of bump-type gas foil bearings operating at steady-state is proposed. The scheme is based on multi-physics coupling computer aided engineering packages modularized with computational fluid dynamic model and structure elasticity model to numerically solve the dynamic equation of motions of a hydrodynamic loaded shaft supported by an elastic bump foil. The bump foil is assumed to be modelled as infinite number of Hookean springs mounted on stiff wall. Hence, the top foil stiffness is constant on the periphery of the bearing housing. The hydrodynamic pressure generated by the air film lubrication transfers to the top foil and induces elastic deformation needed to be solved by a finite element method program, whereas the pressure profile applied on the top foil must be solved by a finite element method program based on Reynolds Equation in lubrication theory. As a result, the equation of motions for the bearing shaft are iteratively solved via coupling of the two finite element method programs simultaneously. In conclusion, the two-dimensional center trajectory of the shaft plus the deformation map on top foil at constant rotational speed are calculated for comparisons with the experimental results.
Abstract: In this study, a spectral element method (SEM) is employed to predict the free vibration of a Euler-Bernoulli beam resting on a Winkler foundation with elastically restrained ends. The formulation of the dynamic stiffness matrix has been established by solving the differential equation of motion which was transformed to frequency domain. Non-dimensional natural frequencies and shape modes are obtained by solving the partial differential equations, numerically. Numerical comparisons and examples are performed to show the effectiveness of the SEM and to investigate the effects of various parameters, such as the springs at the boundaries and the elastic foundation parameter on the vibration frequencies. The obtained results demonstrate that the present method can also be applied to solve the more general problem of the dynamic analysis of structures with higher order precision.
Abstract: This paper deals with the study of devices for displacement of the mould base of blow-molding machines. The displacement of the mould in the studied case is carried out by a linear actuator, which ensures the descent of the mould base and by extension springs, which return the letter in the initial position. The aim of this paper is to study the inverse dynamics of the device for displacement of the mould base of blow-molding machines and to determine its optimum parameters for higher rate of production. In the other words, it is necessary to solve the inverse dynamic problem to find the equation of motion linking applied forces with displacements. This makes it possible to determine the stiffness coefficient of the spring to turn the mold base back to the initial position for a given time. The obtained results are illustrated by a numerical example. It is shown that applying a spring with stiffness returns the mould base of the blow molding machine into the initial position in 0.1 sec.
Abstract: The reinforcement and repair of concrete structures by bonding composite materials have become relatively common operations. Different types of composite materials can be used: carbon fiber reinforced polymer (CFRP), glass fiber reinforced polymer (GFRP) as well as functionally graded material (FGM). The development of analytical and numerical models describing the mechanical behavior of structures in civil engineering reinforced by composite materials is necessary. These models will enable engineers to select, design, and size adequate reinforcements for the various types of damaged structures. This study focuses on the free vibration behavior of orthotropic laminated composite plates using a refined shear deformation theory. In these models, the distribution of transverse shear stresses is considered as parabolic satisfying the zero-shear stress condition on the top and bottom surfaces of the plates without using shear correction factors. In this analysis, the equation of motion for simply supported thick laminated rectangular plates is obtained by using the Hamilton’s principle. The accuracy of the developed model is demonstrated by comparing our results with solutions derived from other higher order models and with data found in the literature. Besides, a finite-element analysis is used to calculate the natural frequencies of laminated composite plates and is compared with those obtained by the analytical approach.
Abstract: The impact of the storm leads to accidents even in the case of vessels that meet the computed safety criteria for stability. That is why, in order to clarify the causes of the accident and shipwreck, it is necessary to study the dynamics of the ship under the complex sudden impact of external forces. The task is to determine the movement and landing of the ship in the complex and sudden impact of external forces, i.e. when the ship's load changes over a relatively short period of time. For the solution, a technique was used to study the ship's dynamics, which is based on the compilation of a system of differential equations of motion. A coordinate system was adopted for the equation of motion of the hull and the determination of external forces. As a numerical method of integration, the 4th order Runge-Kutta method was chosen. The results of the calculation show that dynamic deviations were lower for high-altitude vessels. The study of the movement of the hull under a difficult situation is performed: receiving of cargo, impact of a flurry of wind and subsequent displacement of the cargo. The risk of overturning and flooding was assessed.
Abstract: Pull-in instability is a nonlinear and crucial effect that is important for the design of microelectromechanical system devices. In this paper, the appropriate electrostatic voltage range is determined by measuring fluid flow pressure via micro pressure sensor based microbeam. The microbeam deflection contains two parts, the static and perturbation deflection of static. The second order equation regarding the equivalent stiffness, mass and damping matrices based on Galerkin method is introduced to predict pull-in instability due to the external voltage. Also the reduced order method is used for solving the second order nonlinear equation of motion. Furthermore, in the present study, the micro capacitive pressure sensor is designed for measuring special fluid flow pressure range. The results show that the measurable pressure range can be optimized, regarding damping field and external voltage.
Abstract: Dynamic analysis of composite doubly curved panels with variable thickness subjected to different pulse types using Generalized Differential Quadrature method (GDQ) is presented in this study. Panels with variable thickness are used in the construction of aerospace and marine industry. Giving variable thickness to panels can allow the designer to get optimum structural efficiency. For this reason, estimating the response of variable thickness panels is very important to design more reliable structures under dynamic loads. Dynamic equations for composite panels with variable thickness are obtained using virtual work principle. Partial derivatives in the equation of motion are expressed with GDQ and Newmark average acceleration scheme is used for temporal discretization. Several examples are used to highlight the effectiveness of the proposed method. Results are compared with finite element method. Effects of taper ratios, boundary conditions and loading type on the response of composite panel are investigated.
Abstract: Performance of a Hamiltonian based particle method in simulation of nonlinear structural dynamics is subjected to investigation in terms of stability and accuracy. The governing equation of motion is derived based on Hamilton's principle of least action, while the deformation gradient is obtained according to Weighted Least Square method. The hyper-elasticity models of Saint Venant-Kirchhoff and a compressible version similar to Mooney- Rivlin are engaged for the calculation of second Piola-Kirchhoff stress tensor, respectively. Stability along with accuracy of numerical model is verified by reproducing critical stress fields in static and dynamic responses. As the results, although performance of Hamiltonian based model is evaluated as being acceptable in dealing with intense extensional stress fields, however kinds of instabilities reveal in the case of violent collision which can be most likely attributed to zero energy singular modes.
Abstract: This paper revisits the free vibration problem of delaminated composite beams. It is shown that during the vibration of composite beams the delaminated parts are subjected to the parametric excitation. This can lead to the dynamic buckling during the motion of the structure. The equation of motion includes time-dependent stiffness and so it leads to a system of Mathieu-Hill differential equations. The free vibration analysis of beams is carried out in the usual way by using beam finite elements. The dynamic buckling problem is investigated locally, and the critical buckling forces are determined by the modified harmonic balance method by using an imposed time function of the motion. The stability diagrams are created, and the numerical predictions are compared to experimental results. The most important findings are the critical amplitudes at which delamination buckling takes place, the stability diagrams representing the instability of the system, and the realistic mode shape prediction in contrast with the unrealistic results of models available in the literature.
Abstract: In this paper, we present an analytical method for
analysis of nano-scale spherical shell subjected to thermo-mechanical
shocks based on nonlocal elasticity theory. Thermo-mechanical
properties of nano shpere is assumed to be temperature dependent.
Governing partial differential equation of motion is solved
analytically by using Laplace transform for time domain and power
series for spacial domain. The results in Laplace domain is
transferred to time domain by employing the fast inverse Laplace
transform (FLIT) method. Accuracy of present approach is assessed
by comparing the the numerical results with the results of published
work in literature. Furtheremore, the effects of non-local parameter
and wall thickness on the dynamic characteristics of the nano-sphere
are studied.
Abstract: In this article, the radial displacement error correction
capability of a high precision spindle grinding caused by unbalance
force was investigated. The spindle shaft is considered as a flexible
rotor mounted on two sets of angular contact ball bearing. Finite
element methods (FEM) have been adopted for obtaining the
equation of motion of the spindle. In this paper, firstly, natural
frequencies, critical frequencies, and amplitude of the unbalance
response caused by residual unbalance are determined in order to
investigate the spindle behaviors. Furthermore, an optimization
design algorithm is employed to minimize radial displacement of the
spindle which considers dimension of the spindle shaft, the dynamic
characteristics of the bearings, critical frequencies and amplitude of
the unbalance response, and computes optimum spindle diameters
and stiffness and damping of the bearings. Numerical simulation
results show that by optimizing the spindle diameters, and stiffness
and damping in the bearings, radial displacement of the spindle can
be reduced. A spindle about 4 μm radial displacement error can be
compensated with 2 μm accuracy. This certainly can improve the
accuracy of the product of machining.
Abstract: This paper studies a mathematical model based on the
integral equations for dynamic analyzes numerical investigations of a
non-uniform or multi-material composite beam. The beam is
subjected to a sub-tangential follower force and elastic foundation.
The boundary conditions are represented by generalized
parameterized fixations by the linear and rotary springs. A
mathematical formula based on Euler-Bernoulli beam theory is
presented for beams with variable cross-sections. The non-uniform
section introduces non-uniformity in the rigidity and inertia of beams
and consequently, more complicated equilibrium who governs the
equation. Using the boundary element method and radial basis
functions, the equation of motion is reduced to an algebro-differential
system related to internal and boundary unknowns. A generalized
formula for the deflection, the slope, the moment and the shear force
are presented. The free vibration of non-uniform loaded beams is
formulated in a compact matrix form and all needed matrices are
explicitly given. The dynamic stability analysis of slender beam is
illustrated numerically based on the coalescence criterion. A realistic
case related to an industrial chimney is investigated.
Abstract: Structural failure is caused mainly by damage that
often occurs on structures. Many researchers focus on to obtain very
efficient tools to detect the damage in structures in the early state. In
the past decades, a subject that has received considerable attention in
literature is the damage detection as determined by variations in the
dynamic characteristics or response of structures. The study presents
a new damage identification technique. The technique detects the
damage location for the incomplete structure system using output
data only. The method indicates the damage based on the free
vibration test data by using ‘Two Points Condensation (TPC)
technique’. This method creates a set of matrices by reducing the
structural system to two degrees of freedom systems. The current
stiffness matrices obtain from optimization the equation of motion
using the measured test data. The current stiffness matrices compare
with original (undamaged) stiffness matrices. The large percentage
changes in matrices’ coefficients lead to the location of the damage. TPC technique is applied to the experimental data of a simply
supported steel beam model structure after inducing thickness change
in one element, where two cases consider. The method detects the
damage and determines its location accurately in both cases. In
addition, the results illustrate these changes in stiffness matrix can be
a useful tool for continuous monitoring of structural safety using
ambient vibration data. Furthermore, its efficiency proves that this
technique can be used also for big structures.
Abstract: New design of three dimensional (3D) flywheel system
based on gimbal and gyro mechanics is proposed. The 3D flywheel
device utilizes the rotational motion of three spherical shells and the
conservation of angular momentum to achieve planar locomotion.
Actuators mounted to the ring-shape frames are installed within the
system to drive the spherical shells to rotate, for the purpose of steering
and stabilization. Similar to the design of 2D flywheel system, it is
expected that the spherical shells may function like a “flyball” to store
and supply mechanical energy; additionally, in comparison with
typical single-wheel and spherical robots, the 3D flywheel can be used
for developing omnidirectional robotic systems with better mobility.
The Lagrangian method is applied to derive the equation of motion of
the 3D flywheel system, and simulation studies are presented to verify
the proposed design.
Abstract: New design of three dimensional (3D) flywheel system
based on gimbal and gyro mechanics is proposed. The 3D flywheel
device utilizes the rotational motion of three spherical shells and the
conservation of angular momentum to achieve planar locomotion.
Actuators mounted to the ring-shape frames are installed within the
system to drive the spherical shells to rotate, for the purpose of steering
and stabilization. Similar to the design of 2D flywheel system, it is
expected that the spherical shells may function like a “flyball” to store
and supply mechanical energy; additionally, in comparison with
typical single-wheel and spherical robots, the 3D flywheel can be used
for developing omnidirectional robotic systems with better mobility.
The Lagrangian method is applied to derive the equation of motion of
the 3D flywheel system, and simulation studies are presented to verify
the proposed design.
Abstract: In this paper we consider the equation of motion for
the F (R, T) gravity on their property of conformal invariance. It
is shown that in the general case, such a theory is not conformal
invariant. Studied special cases for the functions v and u in which
can appear properties of the theory. Also we consider cosmological
aspects F (R, T) theory of gravity, having considered particular case
F (R, T) = μR+νT^2. Showed that in this case there is a nonlinear
dependence of the parameter equation of state from time to time,
which affects its evolution.
Abstract: In this paper, a numerical algorithm using a coupled Galerkin-Differential Quadrature (DQ) method is proposed for the solution of dam-reservoir interaction problem. The governing differential equation of motion of the dam structure is discretized by the Galerkin method and the DQM is used to discretize the fluid domain. The resulting systems of ordinary differential equations are then solved by the Newmark time integration scheme. The mixed scheme combines the simplicity of the Galerkin method and high accuracy and efficiency of the DQ method. Its accuracy and efficiency are demonstrated by comparing the calculated results with those of the existing literature. It is shown that highly accurate results can be obtained using a small number of Galerkin terms and DQM sampling points. The technique presented in this investigation is general and can be used to solve various fluid-structure interaction problems.
Abstract: In this paper, the non-linear free axisymmetric vibration of a thin annular plate made of functionally graded material (FGM) has been studied by using the energy method and a multimode approach. FGM properties vary continuously as well as non-homogeneity through the thickness direction of the plate. The theoretical model is based on the classical plate theory and the Von Kármán geometrical non-linearity assumptions. An approximation has been adopted in the present work consisting of neglecting the in-plane deformation in the formulation. Hamilton’s principle is used to derive the governing equation of motion. The problem is solved by a numerical iterative procedure in order to obtain more accurate results for vibration amplitudes up to 1.5 times the plate thickness. The numerical results are given for the first axisymmetric non-linear mode shape for a wide range of vibration amplitudes and they are presented either in tabular form or in graphical form to show the effect that the vibration amplitude and the variation in material properties have significant effects on the frequencies and the bending stresses in large amplitude vibration of the functionally graded annular plate.
Abstract: It has been shown that a load discontinuity at the end of
an impulse will result in an extra impulse and hence an extra amplitude
distortion if a step-by-step integration method is employed to yield the
shock response. In order to overcome this difficulty, three remedies
are proposed to reduce the extra amplitude distortion. The first remedy
is to solve the momentum equation of motion instead of the force
equation of motion in the step-by-step solution of the shock response,
where an external momentum is used in the solution of the momentum
equation of motion. Since the external momentum is a resultant of the
time integration of external force, the problem of load discontinuity
will automatically disappear. The second remedy is to perform a single
small time step immediately upon termination of the applied impulse
while the other time steps can still be conducted by using the time step
determined from general considerations. This is because that the extra
impulse caused by a load discontinuity at the end of an impulse is
almost linearly proportional to the step size. Finally, the third remedy
is to use the average value of the two different values at the integration
point of the load discontinuity to replace the use of one of them for
loading input. The basic motivation of this remedy originates from the
concept of no loading input error associated with the integration point
of load discontinuity. The feasibility of the three remedies are
analytically explained and numerically illustrated.
Abstract: Nonlinear response behaviour of a cracked RC beam under harmonic excitation is analysed to investigate various instability phenomena like, bifurcation, jump phenomena etc. The nonlinearity of the system arises due to opening and closing of the cracks in the RC beam and is modelled as a cubic polynomial. In order to trace different branches at the bifurcation point on the response curve (amplitude versus frequency of excitation plot), an arc length continuation technique along with the incremental harmonic balance (IHBC) method is employed. The stability of the solution is investigated by the Floquet theory using Hsu-s scheme. The periodic solutions obtained by the IHBC method are compared with these obtained by the numerical integration of the equation of motion. Characteristics of solutions fold bifurcation, jump phenomena and from stable to unstable zones are identified.