In this research, the nonlinear rolling motion of ships is studied. After obtaining the governing equation of roll motion, the method of Multiple scales perturbation technique is applied to solve the nonlinear differential equation. The ship response is studied with and without harmonic excitation. In order to validate the responses obtained by the method of Multiple scales, the response was compared with the numerical solution. Finally, the effects of damping coefficient and restoring arm on the frequency response function and resonance frequency have been studied.

Fluid contact structures as a branch of the Fluid–Structure Interaction (FSI) are among the physical models that have been able to present themselves as a new example of dynamic systems due to their very rich dynamics and Pay close attention to scientists. The dynamic behavior of two rigid straight articulated pipes conveying fluid is studied. The flow rate in the pipe is harmonic. The numerical results are compared with the present method and Runge–kutta 4th order for validation and an acceptable match between them is obtained. The method of Multiple time scales is used to drive the time response and phase plane curves. The influence of the initial velocity , ratio fluid mass per total fluid mass and mass of pipes and flow frequency on the time response and phase plane curves are examined. The results show, by increasing "u" _"0" and and decreasing the system is closer to loss of stability and increasing dynamic chaos.

Unbalance in rotating machines causes malfunction of the system operation and may lead to its failure.Therefore, the sources for imbalance should be investigated, identified, and measured to solve the mentioned challenges. Rotating unbalance appears when the geometric and the inertia axes of the rotor do not coincide, and as a result this causes self- excited vibrations. One of the methods to control and reduce the unbalances is utilizing automatic ball balancer (ABB). In previous studies, the stability and the dynamic behavior of ABB have been mostly investigated using numerical methods, and the perturbation methods are applied only for stability analysis. Because of the advantages of the analytical methods in studying the dynamics of the systems, in the present study, for the first time the dynamic behavior as well as the stability of a rotor equipped with an ABB is analyzed by the Multiple scales method. To this end, nonlinear equations of the systems are derived using the Lagrange’s equations and, firstly, the Multiple scales method is applied to investigate the stability of system and then the response of the system is achieved considering one and two terms of approximation. The results demonstrate that the stability analysis using the Multiple scales method and the first method of Lyapunov lead to the same results. Moreover, the responses obtained by the Multiple scales method and the mostly used numerical method, Runge-Kutta technique, are in a good agreement.

In this paper, a new technique for multi-scale smoothing of a free-form3-D surface is presented. This technique is a non-trivial generalization of the Curvature Scale Space (CSS) representation for 2-D contours. The (SS shape descriptor has been selected to be a part of the MPEG-7 package of standards. Complete triangulated models of 3-D objects are constructed (through fusion of range images) and. then, described at Multiple scales. This is achieved by convolving local parameterizations of the surface with 2-D Gaussian filters iteratively. The method, presented here for local parameterization makes use of semigeodesic or geodesic polar coordinates as a natural and efficient way of sampling the local surface shape. It is demonstrated that smoothing techniques using semi geodesic coordinates and geodesic polar coordinates produce similar results. The smoothing eliminates surface noise and details gradually. During the smoothing process, some surfaces can become very thin locally. Application of decimation followed by refinement removes very small or thin triangles and segments the modified surfaces into parts which are then smoothed separately. The technique presented here for 3-D multi-scale surface smoothing is independent of the underlying triangulation. It is also argued that the proposed technique is preferable to volumetric smoothing or level set methods. Since it is applicable to incomplete surface data which occurs during occlusion. Also surfaces that are not simply connected or have holes do not pose any problem. Furthermore due to employing 2-D convolutions rather than 3-D. this method is more efficient than other techniques.

In this study, the method of Multiple scales is used to perform a nonlinear vibration analysis of a mechanical system in two cases; with dry and lubricated clearance joints. In the dry contact case, the Lankarani-Nikravesh model is used to represent the contact force between the joined bodies. The surface elasticity is modeled as a nonlinear spring-damper element. Primary resonance is discussed and the effect of the clearance size and coefficient of restitution on the frequency response is presented. Then, a frequency analysis is done using the Fast Fourier Transform. A comparison between the Lankarani-Nikravesh and Hunt-Crossly contact force models is made. The results obtained numerically and analytically had an acceptable agreement. It is observed that decreasing the clearance size changes the frequency response in the primary resonance analysis. Furthermore, Hunt-Crossly contact force model showed a slightly more dissipative effect on the response. In the lubricated joint case, a linear spring and a nonlinear damper based on the Reynolds equation developed for Sommerfeld’ s boundary conditions are used to model the lubricant behavior. It is shown that only the fluid stiffness has influence on the amplitude of the steady state response and the fluid does not make any effect on the response frequencies after the transient response vanishes. The steady state response frequency for both dry and lubricated cases depends on the linear natural frequency corresponding to the pendulum oscillation. In the primary resonance analysis, increasing the dynamic lubricant viscosity decreases the amplitude in the vicinity of the linear natural frequency as expected.

In this research, the behavior of nonlinear vibrations of the viscoelastic Euler-Bernoulli beam under the influence of external fluid flow has been studied. The governing equations of motion are obtained by assuming Von-Karman nonlinear strain-displacement relations and considering the interaction between structure and fluid. To consider more realistic hypotheses, contrary to previous researches, the effect of viscoelastic behavior has been evaluated using a more complete and practical model called the Standard linear solid model. After non-dimensionalizing the motion equations, the governing nonlinear differential equations are discretized using the Galerkin method. Then, the system's analytical response is acquired through the method of Multiple time scales. After verifying the results and confirming the semi-analytical method's accuracy with the numerical solution results, different parameters' effect on the system's dynamic behavior has been analyzed. The results indicate that the viscoelastic behavior and the nonlinear model significantly affect the lock-in area and the maximum amplitude of the viscoelastic beam vibrations. In most studies on viscoelastic beams' vibrations, the damping effect in nonlinear terms has been neglected. However, this study demonstrates that the effect of damping on terms related to the nonlinearity of strain fields is substantial.

In this paper, the nonlinear vibrations of a rectangular hyperelastic membrane resting on a nonlinear elastic Winkler-Pasternak foundation subjected to uniformly distributed hydrostatic pressure are investigated. The membrane is composed of an incompressible, homogeneous, and isotropic material. The elastic foundation includes two Winkler and Pasternak linear terms and a Winkler term with cubic nonlinearity. Using the theory of thin hyperelastic membrane, Hamilton’s principle, and assuming the finite deformations, the governing equations are obtained. Also, the kinetic energy, the work of uniform distributed force and pressure, and the effects of damping are determined, according to the strain energy function for neo-Hookean hyperelastic constitutive law. By applying Galerkin’s method, the nonlinear partial differential equation of motion in the transversal direction is transformed to the ordinary differential equations. Then, utilizing the method of Multiple scales, the superharmonic and subharmonic resonances including the 1:3 superharmonic and 3:1 subharmonic, 1:5 superharmonic, and 5:1 subharmonic, 1:7 superharmonic, and 7:1 subharmonic are analyzed. Also, the analytical results are compared with those presented by other researchers. Finally, the effect of the Winkler and Pasternak stiffness, the material properties, and various geometrical characteristics on the superharmonic and subharmonic resonances of the vibration behavior of a rectangular hyperelastic membrane is investigated.

Dynamic behavior of a circular shaft with geometrical nonlinearity and constant spin, subjected to periodic axial load is investigated. The case of parametric combination resonance is studied. Extension of shaft center line is the source of nonlinearity. The shaft has gyroscopic effect and rotary inertia but shear deformation is neglected. The equations of motion are derived by extended Hamilton principle and discretized by Galerkin method. The Multiple scales method is applied to the complex form of equation of motion and the system under parametric combination resonance is analyzed. The attention is paid to analyze the effect of various system parameters on the shape of resonance curves and amplitude of system response. Furthermore, the role of external damping on combination resonance of linear and nonlinear systems is discussed. It will be shown that the external damping has different role in linear and nonlinear shaft models. To validate the perturbation results, numerical simulation is used.

The chatter phenomenon is a kind of unstable self-excited vibration which occurs in most machining processes including grinding. A 3-D nonlinear model of chatter vibrations in grinding process has been presented in this paper. The workpiece has been modeled as a continuous rotating shaft with transverse vibrations and wheel is regarded as a one degree of freedom damped spring-mass system. The equations of motion have been dimensionless by p-Buckingham theory. Then using assumed modes method, partial differential equations of workpiece changed to ordinary differential equations and then the method of Multiple scales is adopted for the approximate solution. Stability regions for workpiece and wheel in various workpiece rotary speeds and various wheel positions drawed and then the effect of various parameters such as rotational speed of the wheel, and the radius of the workpiece have been investigated. Results showed that vibrations in workpiece and in wheel are quite different from each other. When grinding occurs in the middle of the workpiece the possibility of chatter phenomenon is more, while the position of wheel has no effect on the vibration of the wheel.