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Residual unbalance in hand-held power tool rotors transmits undesirable vibrations to the hand of its operator. These vibrations can be effectively suppressed using a one plane automatic dynamic balancer (ADB). This balancing device consists of several balls constrained to move inside a sealed cylindrical ball-race unit partially filled with oil. One of the hand-held power tools is an angle grinder. This study introduces the design of an ADB for eliminating vibration of a grinder based on the achieved design parameters. A physical model of the system is derived for a Jeffcott rotor with an ADB. Utilizing Lagrange's method, the nonlinear equations of motion for an autonomous system in polar coordinate system is derived. Further, the equilibrium position and the linear variational equations are obtained by the perturbation method. Moreover, the dynamic stability of the system in the neighborhood of the equilibrium positions is investigated by the Routh-Hurwitz criteria. The results of the stability analysis provide the design requirements for the ADB to achieve balancing of the system. In addition, time responses are presented by the generalized-alpha method. Employing the modal analysis method the equivalent damping and stiffness coefficients are achieved. Finally, the ADB is designed and manufactured by solving the equations of motion governed on identified unbalanced grinder. To evaluate and identify the performance of the ADB, vibration levels are measured in cases of with balancer, without balancer, and are compared with a typical commercial ADB.

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Abstract- In this study, dynamic stability of a system consisting of three torsionally elastic shafts with different rotation axises is analyzed. The system stability have been investigated by means of a three degree-of-freedom model in a spatial coordinate (three dimensional). Each shaft carrying a rigid disk at one end and have been linked through two Hooke's joints. Equations of motion for the system were derived. These equations are linearised. After linearization of the differential equations are shown to consist of a set of Mathieu–Hill equations. Their stability are analyzed by means of a monodromy matrix method. Finally dynamic stability regions have been shown on different system parameters such as rotational velocity, misalignment angle’s of shaft axis, stiffness and rigidity of shafts. The stability charts constructed on various parameters. It was observed that with increasing inertia disk ratio and decreasing Hooke's joint angle, the stable region increases. Keywords: Dynamic Stability, Shaft System, Torsional Vibration, Hooke’s Joint

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Analytical solution for the dynamic stability analysis of functionally graded piezoelectric materials (FGPM) circular plates has been presented based on Love-Kirchhoff hypothesis and the Sander’s non-linear strain-displacement relation. The FGPM plate assumed to be gradded across the thickness. The material properties of the FGPM plate assumed to vary continuously through the thickness of the plate according to a power law distribution of the volume fraction of the constituent materials. The plates are subjected to a radial loading and electric field in the normal direction. Bolotin’s method has been employed to obtain the dynamic instability regions. The effect of plate parameters such as thickness–radius ratios, power index, as well as electric field and state loads on instability behavior of the plate is comprehensively investigated.The functionally graded composite material plays a significant role in changing the unstable regions and the buckling loads.

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The major concern in Shallow arches behavior under lateral loading is their instability at a critical load, which can make the structure to collapse or displace to another stable configuration, a phenomenon called snap through. By introduction of functionally graded materials in recent years, and incorporating them into this problem, interesting results can be obtained which can give structures with favorable stability properties. In this work, dynamic stability of the hinged-hinged functionally graded shallow arch under implusive loading is investigated. Material properties vary through the thickness by power law. Nonlinear governing equations are derived using Euler-Bernoulli beam assumption and equations of motion are expressed by a nonlinear differential-integral equation. The solution utilizes a Fourier form of response. The procedure of analysis of dynamic stability that is followed in this work uses the total energy of the system and the Lyapunov function in the phase space that consists of essentially three steps: First, one finds all the possible equilibrium configurations of the shallow arch. Next, the local dynamic stability of each of the equilibrium configurations is studied.. Last, when the preferred configuration from which a snap through may occur is locally stable and when there is at least one other locally stable equilibrium configuration, then we proceed to find a sufficient, condition for stability against snap through. The effect of gradation on stability and critical load of the arch is investigated in detail.

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Oil journal bearings are one of the most common parts of high load carrying rotating machine. Stability of these bearings can be affected by various stimulus such as changes in loading and lubrication conditions. Therefore, identification of the dynamic response of journal bearings can improve the control and fault detection process of rotor-bearings systems and prevent them from placing in critical operation condition. Since past, the mass unbalance of rotor is proposed as an effective factor on the dynamic behavior and long life of bearings. For this reason, in this research the effects of this parameter on the stability of hydrodynamic two lobe noncircular journal bearing with micropolar lubricant is investigated based on the nonlinear dynamic model. To achieve this goal, the governing Reynolds equation is modified with respect to micropolar fluid theory and the equations of rotor motion are derived considering the mass unbalance parameter. The static and dynamic pressure distributions of the lubricant film and the components of displacement, velocity and acceleration of the rotor are obtained by simultaneous solution of the Reynolds equation and the equations of rotor motion. Investigation of results in terms of dynamic trajectory, power spectrum, bifurcation diagram and Poincare map show that the dynamic behavior of two lobe bearings appears in different manner with variation of mass unbalance of rotor. The response of analyzed dynamic system include converge oscillations to the equilibrium point, periodic, KT periodic and quasi periodic behavior and also divergent disturbances which leads to collision between the rotor and bearing.

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In this paper, dynamic stability of a laminated composite beam subjected to a tip follower force is investigated. Using elementary theory of bending, Euler-Bernoulli beam theory and Classical Lamination Theory (CLT), bending moment of laminated composite beam is calculated with respect to it’s extensional, bending and bending-extensional coupling stiffness matrices, A, B and D, and dynamic stability equation of laminated beam is established. Due to similarity between this equation and isotropic stability equation, as an assumption, isotropic beam boundary conditions are used for composite beam. Cantilever- free boundary conditions are used and a closed form solution is established. Flutter instability problems for symmetric and un– symmetric laminated beams are solved by this method and results are compared with finite element results in literatures. Considering the simplicity of the present method, results show good agreement with the finite element method. Finally dynamic stability behavior of laminates with different stacking sequences are investigated by present method and effect of different parameters such as fiber orientation, number of layers, and stacking sequence, on the flutter load and corresponding frequency of symmetric and un -symmetric laminates are investigated.

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One of the remarkable concerns in Shallow arches’ behavior under lateral loading is snap through, a phenomenon which can make the structure collapse or displace to another stable configuration. Introducing functionally graded materials in recent years led to some interesting results, for instance, using functionally graded materials in shallow arches can give structures with favorable stability properties. In this work, we investigate dynamic stability of the pined-pined functionally graded sinusoidal shallow arch under impulsive loading. Material properties vary through the thickness by power law function. Nonlinear governing equations are derived using Euler-Bernoulli beam assumption and equations of motion are expressed by a nonlinear differential-integral equation. The solution utilizes a Fourier form of response. The procedure to analyze dynamic stability followed here uses total energy of the system and Lyapunov function in the phase space. We find the stable region against dynamical snap through under material properties’ variation through the thickness of shallow arch. We also proceed to find the sufficient critical load in order to make the dynamical snap through occur. The results are analyzed in detail and illustrated in some diagrams.

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The dynamic stability of single-walled carbon nanotubes (SWCNT) and double-walled carbon nanotubes (DWCNT) embedded in an elastic medium subjected to combined static and periodic axial loads are investigated using Floquet–Lyapunov theory and bounded solution theory. An elastic Euler- Bernoulli beam model is utilized in which the nested slender nanotubes are coupled with each other through the van der Waals (vdW) interlayer interaction. The Galerkin’s approximate method on the basis of trigonometric mode shape functions is applied to reduce the coupled governing partial differential equations to a system of the extended Mathieu-Hill equations. Applying Floquet–Lyapunov theory and Rung-Kutta numerical integration method with Gill coefficients, the influences of number of layer, elastic medium, exciting frequency and combination of exciting frequency on the instability conditions of SWCNTs and DWCNTs are investigated. A satisfactory agreement can be observed by comparison between the predicted results of Floquet–Lyapunov theory with bounded solutions theory ones. Based on results, increasing the number of layers, and elastic medium, dynamic stability of SWCNTs and DWCNTs surrounding elastic medium increase. Moreover, the instability of CNTs increases by increasing the exciting frequency.

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In this paper, instability due to occurrence of parametric resonance in transverse vibration of a rectangular plate on an elastic foundation under passage of continuous series of moving masses is examined as a model of bridge-moving loads interaction. The extended Hamilton’s principle is employed to derive the partial differential equation of motion. Subsequently, the governing partial differential equation is transformed into a set of ordinary differential equations by the Galerkin procedure. Considering local, Coriolis and centripetal acceleration components of the moving masses in the analysis leads to appearance of time-varying mass, damping and stiffness matrices in the coefficients of the governing equation. The passage of continuous series of moving masses along the rectilinear path results in a parametrically excited system with periodic coefficients. Applying incremental harmonic balance method as a semi-analytical method to the governing equations, stability of the system is investigated for a wide range of masses and velocities of the passing loads and different boundary conditions of the plate. Moreover, effect of the foundation stiffness on stability of the plate is examined. Results indicate that using clamped supports for the edges of entrance and departure of masses over the plate’s surface leads to formation of an instability tongue in the parameters plane which does not appear for the case of using simply supports. Also, it is observed that critical velocities of the moving masses will be increased by escalation the foundation stiffness. Numerical simulations confirm the accuracy of the semi-analytical results.

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Today, oil journal bearings are widely used as an efficient support for rotary systems in various industries. When these bearings are used by loading in high speed conditions, whirling disturbances in the rotor motion status leading to collisions and abrasion is probable. Designing specific geometric shapes or applying industrial lubricants with different new combinations can affect the journal bearings ability to maintain their dynamic stability in critical situations. From this view, the use of non-circular bearings and non-Newtonian fluids in the field of lubrication has recently been heavily taken into consideration. In the present study by choosing non-Newtonian lubricant simulated by power law fluid model, the effects of design parameters such as eccentricity ratio, aspect ratio and power law index on dynamic stability of noncircular two, three and four lobe bearings are investigated. For this purpose, assuming the limited cycle oscillations of the rotor around the equilibrium point after damping the effects of initial imposed disturbances and using finite element numerical method to solve the governing equations, stability range of the system in form of linear dynamic analysis characteristics is determined based on the whirl frequency ratio and critical mass parameter. The results indicate that by increasing the power law index and decreasing aspect ratio, the dynamic range of bearing support will be developed. Also, by increasing the number of noncircular bearings lobes with power law lubricant and providing the system's positioning conditions in high values of eccentricity ratio, more ability to damping dynamic disturbances can be achieved.

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The lubricant's ability to maintain the dynamic stability of rotor particularly in special conditions such as operating at critical speeds and instantaneous turbulences in loading or lubricant properties is always one of the most prominent characteristics of the journal bearings. Aspect or length to diameter ratio of bearing is an important factor that in different loading conditions will have an obvious effect on the performance of the trapped lubricant film between the rotor surface and bearings shell. So, the effects of aspect ratio on the damping of rotor disturbances with linear and nonlinear dynamic analysis approaches are studied in this research. Initially, the static equilibrium point of the rotor center in noncircular two, three and four lobe bearings space is obtained using the governing Reynolds equation of micropolar lubrication for different values of aspect ratio. Later, assuming the rotor perturbation as the limit cycle oscillations around the equilibrium point, critical mass and whirl frequency ratio are determined as the linear dynamic stability indexes for recognizing the converging disturbances. In nonlinear analysis model, the simultaneous solving of the lubrication and the rotor motion equations in successive time steps with Runge-Kutta method is done to differentiate the converging or diverging rotor perturbations. Results show that decreasing the aspect ratio improves the stability and the chance of controlling disturbances and returning the rotor center to static equilibrium position. Comparison of linear and nonlinear dynamic analysis results also indicates more cautious behavior and limited stability range of linear model in most of investigated cases.

 

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The use of members with non-uniform cross-sections due to the reduction of the number of materials and the weight of the structure is widely used in industrial structures and metal bridges. Buckling is one of the major problems engineers face in the design of axial compression members (columns). For this reason, several researches have been conducted by researchers in the field of column buckling. Most of the previous research is limited to investigating stability and buckling in Non-prismatic elastic columns in the static state. During an earthquake, the structure is subjected to vertical and lateral earthquake loads. To evaluate the dynamic behavior of the structure during an earthquake, the stability and dynamic buckling of the column must be evaluated. The effect of the earthquake's vertical load and the dynamic axial load has an effect on the dynamic stability of the member in the form of the second-order effect of buckling. In this article, the dynamic buckling of a column with a variable section and viscous damper under alternating axial load is investigated in a comprehensive model. The alternating axial load effect is assumed as a cosine function and the viscous damping effect at the end of the member is assumed as a Dirac delta function. The changes in the moment of inertia along the length of the column are considered in three modes: linear, cubic, and fourth-order changes. The constituent differential equation includes column strain energy, second order effect of alternating axial load, inertia per unit length of the column, and damping of a viscous damper. To solve the constitutive equation, first the weak form of the governing differential equation is written. Lagrange interpolation functions are used as the shape function and the Fourier function (proposed by Bolotin) as the dynamic response of the equation. In the next step, the matrices of material hardness, geometric hardness, and mass are extracted. After extracting the above matrices, the eigenvalues (Buckling load factor, natural frequency) of the equation are checked. Muller root finding technique is used by coding in MATLAB software to calculate eigenvalues. For accuracy in calculations, the function of the form of the equation is checked by the Lagrange method with the number of thirty terms. Also, finding the roots of the equation to calculate the eigenvalue is done with a step of 0.05 using Mueller's method. The buckling load coefficient of the column is evaluated for different values of the expansion coefficient and the damping percentage of the viscous damper in different boundary conditions. The results show that the mentioned values have a significant effect on the changes in the buckling load factor in terms of excitation frequency and resonance frequency. Depending on the boundary conditions, increasing the opening factor causes the diagram to move to the right or left side of the dimensionless excitation frequency axis. Also, increasing the damping coefficient of the viscous damper causes the diagram to move to the left side of the dimensionless excitation frequency axis. Dimensionless parameters such as bar coefficient, excitation frequency, and opening coefficient have been used to report the dynamic behavior of the set in all the tables and figures. The results of this research can be generalized for the design of columns under periodic axial load. The results of this article are verified and compared with previous research. There is an acceptable agreement between the results of the present article and previous research.