Computational Methods in Engineering

---

The importance of the equipment and secondary systems in seismic design and performance evaluation is well recognized and has been the subject of many studies. In all of these studies, earthquake is considered as a single component, and in most of them the primary system is considered as shear building. Most attention has been concentrated on the response of secondary system and its response spectrum. In this paper, the transfer function for absolute acceleration of the secondary system is obtained. The squared modulus of transfer function relates the power spectral density function of the input (excitation) to the output (response), which is useful in the study of the various dynamic parameters of the system. In addition to transfer function, the autocorrelation and power spectral density function of absolute acceleration of the secondary system are obtained. Earthquake is considered as a multi-component system and the necessary formulation is developed for the calculation of these functions as well as the critical angle with and without interaction between the two systems. The damping of the system is considered as proportional in the decoupled analysis, and nonproportional in the coupled analysis. The formulation developed has been illustrated by considering a ten-story torsional builing. Various parameters such as eccentricity, correlation between components, tuning interaction and nonproportional damping are studied. Results show that eliminating the effect of multicomponentness of earthquake can cause large errors especially at large eccentricities. Keywords: transfer function, Random vibration, secondary systems, critical angle, interaction, nonproportional damping

---

There is a worldwide interest in the proper design of embankment dams to resist earthquake loadings. For the first time in Iran, a complete ambient vibration survey due to low-level loads such as wind, machinery activities, low level tectonic activities, and water exit from bottom outlet was performed on Marun embankment dam. These kinds of ambient vibration tests are suitable for manifesting the lower vibration modes of the dam body. Using different signal processing methods such as Power Spectra Density, the results of in-situ tests have been used to evaluate the natural frequencies, mode shapes and modal damping of the dam body. Besides ambient vibration tests, the 3-D modal analysis of the dam body was performed using ANSYS software. The foundation and abutment flexibility effects on dynamic characteristics of the dam body was investigated and the dynamic soil properties were used from Engineer’s report and some empirical relations. Also initial shear modulus of the dam body and foundation materials were evaluated by refraction survey. In this paper, the test procedures, related signal processing results, numerical analysis results and its comparison with the dynamic characteristics of the dam body obtained from the full-scale dynamic tests will be presented. Finally, calibrating procedures of the numerical model (based on increasing the accuracy of dam body geometry, soil and rock material parameters and foundation and abutment flexibility) will be discussed. Keywords: Embankment Dam, Dynamic Characteristics, Ambient Vibration Test, Modal Analysis

---

Based on classical plate theory, standard and spectral finite element methods are extended for vibration and dynamic stability of axially moving thin plates subjected to in-plane forces. The formulation of the standard method earned through Hamilton’s principle is independent of element type. But for solving numerical examples, an isoparametric quadrilateral element is developed using Lagrange interpolation functions. The spectral method is, in fact, the solution of motion equation for an axially moving plate. Although this method has some limitations concerning boundary condition of plate and in-plane forces, it leads to an exact solution of free vibration and stability of plates travelling on parallel rollers. The method can be used as a benchmark of accuracy of other numerical methods.

---

An approximate numerical mthod is presented for analysis and determination of modal characteristics in straight, pretwisted non-unifom helicopter blades. The analysis considers the coupled flapwise bending (out of plane), chordwise bending (in plane), and torsion vibration of both rotating and non-rotating blades. The proposed method is based on the integral expansion of Green functions (structural influence functions) to develop the equations of motion for a clamped-free blade. Several examples are presented in various states such as flapwise bending, coupled bending-bending, coupled bending-torsion, and coupled bending-bending-torsion vibration analysis. The results obtained were compared with available numerical results in the literature. A modal testing and modal analysis were also carried out on a typical helicopter blade in static condition and the results were compared with the numerical ones. The results indicate that the proposed method is fast and robust and can be used for modeling of turbomachine blades, aircraft propellers and helicopter rotor blades.

---

Investigation of vertical vibrations of a railway turnout is important in designing track components under moving loads of trains. In this paper, the turnout is simulated by a linear finite element model with modal damping. A section of the turnout has a length of 36 sleeper spans surrounding the crossing. Rails and sleepers are modeled with uniform Rayleigh- Timoshenko beam elements. The rails are connected via railpads (linear springs) to the sleepers, which rest on an elastic foundation. The rolling stocks are discrete systems of masses, springs, and dampers. By passing the trains at a constant speed, only vertical dynamics (including roll and pitch motions) is studied. The wheel-rail contact is modeled using a non-linear Hertzian spring. The train-track interaction problem is solved numerically by using an extended state space vector approach in conjunction with modal superposition for the turnout. The results show that the rail discontinuity at the frog leads to an increase in the wheel-rail contact force. Both smooth and irregular transitions of the wheels from the wing rail to the crossing nose have been examined for varying speeds of the vehicle. Under perfect conditions, the wheels will change quite smoothly from rolling on the wing rail to rolling on the nose. The impact at the crossing will then be small, giving a maximum wheel-rail contact force which is only 30--50 per cent larger than the static contact force. For uneven transitions, the severity of the impact loading at the crossing depends strongly on the train speed. The increase in the contact force, as compared with the static force, is in the order of 100 per cent at 70 km/h and 200 per cent at 150 km/h.

---

: In this paper, a nonlinear model of clamped-clamped microbeam actuated by electrostatic load with stretching and thermoelastic effects is presented. Free vibration frequency is calculated by discretization based on DQ method. Frequency is a complex value due to the thermoelastic effect that dissipates the energy. By separating the real and imaginary parts of frequency, quality factor of thermoelastic damping is calculated. Both stretching and thermoelastic effects are validated against the results of the reference papers. The variations of thermoelastic damping versus elasticity modulus, coefficient of thermal expansion and geometrical parameters such as thickness, gap distance, and length are investigated and these results are compared in the linear and nonlinear models for high values of voltage. Also, this paper shows that since for high values of electrostatic voltage the linear model reveals a large error for calculating the thermoelastic damping, the nonlinear model should be used for this purpose.

---

The main objective of this paper is to provide an applied algorithm for analyzing propeller-shaft vibrations in marine vessels. Firstly an underwater marine vehicle has been analyzed at different speed in unsteady condition using the finite volume method. Based on the results of this analysis, flow field of marine vehicle (wake of stern) and velocity inlet to the marine propeller  is extracted at different times. Propeller inlet flow field is applied in the boundary element code and using this code, marine propeller has been analyzed in unsteady state. In continue, main / lateral forces and moments over the propeller are extracted. Then the data obtained from the boundary element code alongwith exact geometry of the propeller and shaft have been studied, using finite element code. Natural and forced frequency of the propeller have been determined in various modes of vibration. According to obtained data from Finite Element Method (FEM) numerical analysis, maximum displacement of propeller is for displacement of the propeller tip in forced vibration state

---

: The gear systems are widely used in industry to transmit the power or change the direction of the torque. Due to the extensive usage of the gears, the detailed designing and the subsequent maintenance of these systems are more and more evident. System recognition can be achieved through modeling the system, investigating the system behavior, and comparing the results obtained through the model with the actual system behavior. Up to now, the effect of dry friction has not been taken into account in nonlinear vibration analysis and modeling of a cracked one-stage gear power transmission system. In this paper, the nonlinear vibration of a pair of cracked spur-gear system in presence of dry friction, static transmission error, clearance and time-variant mesh stiffness is investigated. To this end, the time-variant mesh stiffness of an intact tooth is calculated analytically. Then, the tooth root crack is modeled as a cracked cantilever beam. The governing nonlinear equation of motion is extracted accordingly, and in order to consider the effect of dry friction, the governing equation solved by Rung- Kutta method in three separate time spans. Finally, the frequency response and bifurcation diagrams are used to study the effect of the friction and tooth root crack on the nonlinear vibration behavior of the system.

---

Having two stable configurations and no need to any permanent energy sources for remaining in each of these stable states, bi-stable composite plates have gained many applications. This paper has concentrated on control and dynamic response of cross ply bi-stable composite plates (0.90). To do this, using Hamilton principle , Rayleigh-Ritz method, and a MATLAB programme specifically designed for this study, have been applied in order to extract  the governing equation of motions in plates. Then, in order to control the large vibration of the cross ply bi-stable plate, a fuzzy controller was proposed using a fuzzy logic and its prformance was simulated by Simulink in Matlab environment. In order to simulate the real conditions on the controller performance, the effect of disturbances and time delay on the responses of controller were also investigated.

---

Using Vibro-Impact Nonlinear Energy Sinks (VI NESs) is one of the novel strategies to control structural vibrations and mitigate their seismic response. In this system, a mass is tuned on the structure floor, so that it has a specific distance from an inelastic constraint connected to the floor mass. In case of structure stimulation, the displaced VI NES mass collides with the  inelastic constraint and upon impacts, energy is dissipated. In the present work, VI NES is studied when its parameters, including clearance and stiffness ratio, are simultaneously optimized. Harmony search as a recent meta-heuristic algorithm is efficiently specialized and utilized for the aforementioned continuous optimization problem. The optimized attached VI NES is thus shown to be capable of interacting with the primary structure over a wide range of frequencies. The resulting controlled response is then investigated, in a variety of low and medium rise steel moment frames, via nonlinear dynamic time history analyses. Capability of the VI NES to dissipate siesmic input energy of earthquakes and their capabilitiy in reducing response of srtructures effectively, through vibro-impacts between the energy sink’s mass and the floor mass, is discussed by extracting several performance indices and the corresponding Fourier spectra. Results of the numerical simulations done on some structural model examples reveal that the optimized VI NES has caused successive redistribution of energy from low-frequency high-amplitude vibration modes to high-frequency low-amplitude modes, bringing about the desired attenuation of the structural responses.

---

In this paper, free vibration of carbon nanotube-reinforced functionally graded circular plates with hole has been
investigated. Distribution of carbon nanotubes are continuous and the gradual and graded changes of materials through the
plate thickness are considered as volume fraction. Considering the linear and non-linear variation of circular plates through the
radial direction and also considering the proposed function for the thickness, the plate thickness can be convex or concave.
Moreover, the motion equations of plate were obtained based on the third-order shear deformation theory. These equations are
coupled differential equations which can convert Ordinary Differential Equations (ODE) using the Trigonometric series
expansion of displacement fields such that they satisfy the axial symmetry condition. Solving the converted ODE equations is too
difficult. For this reason, the differential quadrature method is employed to solve these equations. The obtained results are
compared with the results reported by other researchers and an excellent agreement is observed between them. Finally, the effects
of different geometric parameters as well as different volume fracture of nanotubes on natural frequency have been studied.

---

In this paper, wave propagation method was applied to detect damage of structures. Spectral Finite Element Method
(SFEM) was used to analyze wave propagation in structures. Two types of structures i.e. rod and Euler-Bernoulli beam were
modelled using spectral elements. The advantage of spectral finite element over conventional Finite Element Method (FEM), in
wave propagation problems, is its accuracy and lower computational time. Two examples of rod and Euler-Bernoulli beam with
embeded concentrated mass were presented to illustrate the superiority of SFEM to FEM. Finally, a cracked beam was modeled
and analyzed using spectral finite elements and the location of the crack was determined using time history response of beam
structure.

---

---

This paper presents a probabilistic assessment on the free vibration analysis of functionally graded material plates, including layers with magneto-electro-elastic properties, using the 3D solution and surrogate models. The plate is located on an elastic foundation and the intra-layer slipping effect is also considered in the analysis by employing the generalized intra-layer spring model. Due to the high computational cost of the 3D solution in calculating the free vibration frequency of the plate, surrogate models are used. The meta models including kriging method, radial fundamental function method and polynomial response surface method are used to construct the surrogate model. For surrogate models training, the results of the three-dimensional solving method are used. The elastic foundation hardness, the intra-layer slipping effect, the material properties index, and the layer densities are considered as the variables with uncertainty. The three-dimensional solution method is validated through a comparison with other available reference. The results obtained through the surrogate models have been compared to those of the 3D solution formulation, showing a good agreement. The effects of some parameters including the elastic foundation hardness, the intra-layer slipping effect, the density of each layer, and the material properties index on the fundamental frequency of functionally graded material plates are investigated. By using three-dimensional solution method and Kriging Surrogate Model, it is shown that the shear and transverse components of elastic foundation hardness and the density of each layer have the greatest effect on the fundamental frequency of the functionally graded material plates.

---

Today many complex models, typically finite element models, have been employed in the analysis of jacket offshore structures. However, these comprehensive models are not readily adopted in engineering practice, especially during the preliminary design stage. As the dynamic analysis of jacket platforms is very complicated, it will be very advantageous to make a simplified computational method to assess dynamic performance of such structures. In this work a refined simplified model has been utilized to calculate dynamic responses of jacket platforms. In this regard, the model simplification based on the vibration modal data and Timoshenko’s beam equation has been employed to overcome the uncertainty problem in modeling. According to the curve fitting method on the set of frequency response functions to derive modal parameters, the concept of power spectrum density has been also used to confirm the proposed computational model.In this regard, first the behavior of the physical model in the frequency domainhas been presented and compared with the spectral results obtained from the simplified model based on Timoshenko beam. Because the modal test of the physical model was performed under the force of white noise, the dynamic responses of the simplified model were also extracted under the force of white noise using MATLAB software. In this paper, an applied mathematical model has been produced, and it has been demonstrated that the refined simplified model can reflect the real structural features.