# STRUCTURAL DYNAMICS PDF

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Fundamentals of Structural Dynamics. 1 Course description. Aim of the course is that students develop a “feeling for dynamic problems” and. Faculté des Sciences Appliquées. Twelve Lectures on Structural Dynamics. André PREUMONT. Active Structures Laboratory. Department of Mechanical. (i) Idealisation of material such as homogeneity, isotropy and idealisation of structural behaviour such as linearly elastic, elastic and nonlinear. (ii) Idealisation of.

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1. Structural Dynamics, Dynamic Force and Dynamic System. Structural Dynamics. Conventional structural analysis is based on the concept of statics, which can. STRUCTURAL DYNAMICS BASICS: Real-life structures are subjected to loads which vary with time. Except self weight of the structure, all other loads vary. References: R.W. Clough and J. Penzien 'Dynamics of Structures' A.K.. Chopra 'Dynamics of Structures: Theory and Applications to Earthquake.

Hence this vibration occurs without the external force. Forced Vibration: If the external force is also involved during vibration, then it is forced vibration. Damping: All real life structures, when subjected to vibration resist it.

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Due to this the amplitude of the vibration gradually, reduces with respect to time. In case of free vibration, the motion is damped out eventually.

## VALIDATION OF STRUCTURAL DYNAMICS MODELS AT LOS ALAMOS NATIONAL LABORATORY

Damping forces depend on a number of factors and it is very difficult to quantify them. Degree of Freedom: It is very well known that any mass can have six displacement Components 3 translations and 3 rotations. In most systems, some of these displacements are restrained. The number of possible displacement components is called as Degree of Freedom DoF. Hence DOF also represents minimum number of coordinate systems required to denote the position of the mass at any instant of time.

An overhead tank is considered as an example. This can be modeled as a cantilever column with concentrated mass at top. If we want axial vibration, then only one coordinate y is sufficient. If only the flexural deformation is required then also only one co-ordinate x is required. If both are required, then two coordinates are required. Depending upon the co-ordinates to describe the motion, we have 1. Single degree of freedom system SDoF. Multiple degree of freedom MDoF.

Continuous system. The single bay single storey rigid frame is taken as SDoF based on assumptions. Hence neglected. When this frame vibrates due to lateral load in horizontal direction, the force acting are inertial force, 2 damping force, 3 restoring force, and 4 External force.

If the external force is removed after initial disturbance, the free vibration occurs. Further it will be treated as free damped vibration, if damping is present and if damping is not present it is called free undamped vibration. An SDoF is represented as shown in Figure.

For this system, only one coordinate x is required translation. The solution of this equation gives response of an SDoF system. Free Undamped Vibration: In this case, is zero and C is zero because no damping. Since cosine and sine functions are periodic functions, motion defined by x will also be periodic motion repeats itself after certain interval of time. Reciprocal of T is the frequency which is nothing but number of times motion repeats itself in one second. Moreover, the high contrast of images with less influence on conformation or flatness makes it possible to perform 3D reconstruction from each individual DNA origami Bennett linkage particle.

Thus, any averaging method on the heterogeneous particles was not suitable for analysis of the 3D structure and overall motion. The tilt images of each targeted particles were iteratively aligned to a global center to achieve a final ab initio 3D reconstruction.

Only particles whose overall shape was visible through the entire tilt range were selected. The step-by-step refinement procedure and the intermediate results are shown in Fig.

After low-pass filtering to 8. The overall shape confirmed the success of the designed structure 2. Seven representative tilt images of a targeted particle first column were gradually aligned via an iterative refinement process next five columns.

Scale bars are 50 nm in a and 20 nm in b—d and f—h Full size image Although the resolution of the reconstructed map was insufficient to determine the secondary structure of this Bennett linkage, the overall map can be used as a constraint to flexibly dock an initial model of the Bennett linkage, built using CanDo, 17 , 19 to obtain a new Bennett linkage conformation. By satisfying both the best overall fit to the density map and a minimal overlap among arms within the model, we changed the joint angles between two adjoined arms Fig.

The representative tilt images showed that this Bennett linkage was also visible in each tilt image. However, it exhibited less bending angles between two nearby arms Fig.

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By flexible docking of the Bennett linkage model into this map, a second conformation of a Bennett linkage was obtained Fig. These density maps, with resolutions ranging from 5. Although each DNA helix within the Bennett linkages could not be identified under the current resolution, the spatial location and orientation of each arm of Bennett linkages can be defined based on the 3D density maps. As a result, more conformations were revealed Fig.

Scale bars are 20 nm Full size image Statistical analyses of the conformations Aligning all the conformations of DNA origami Bennett linkages shows a variety of conformations Fig. To quantitatively analyze the conformational dynamics and fluctuations, two sets of bending angles formed within the quadrilateral-shape Bennett linkages were measured, i.

Both sets of angles determined the overall conformation of Bennett linkages. The pink curve was predicted by using kinematic analysis. In contrast, the percentages from kinematic analysis were not like each other or the experimental curves, suggesting the freedom or energies of the conformations were more unevenly distributed than predicted. Discussion Scaffolded DNA origami has proven to be a powerful and efficient technique to fabricate nanostructures. Recently cryo-electron tomography was used to study the structural diversity of supercoiled DNA The limited tilt angle range was responsible for missing wedge of information in Fourier space.

As a result, the final 3D reconstruction often contains certain artifacts, such as elongation, blurring, and distracting caustics To reduce the effects of the missing wedge, the missing wedge data was estimated via computational algorithms reported by numerous groups 23 , 24 , 25 , For example, a computational approach to fill the missing data in 2D electron crystallography was first reported by Agard and Stroud in the s Recently, a simple Fourier angular filter to effectively suppress the ray artifacts in the single-axis tilting projection acquisition scheme was reported by Kovacik et al.

A statistical reconstruction method, sequential maximum a posteriori expectation maximization was used to compensate the missing-wedge effects by Ruotsalainen et al.

An iterative compressed-sensing optimized non-uniform fast Fourier transform reconstruction ICON for missing-wedge restoration was proposed by Sun et al.

Moreover, a generalized Fourier iterative reconstruction algorithm GENFIRE proposed by Miao and colleagues 26 showed a certain capability in reducing the missing wedge artifact to achieve a 3D structure with more isotropic resolution.

All those approaches benefited the 3D reconstruction by reducing the missing wedge effects and providing a relatively isotropic resolution 3D density map.

## 1st Edition

In our IPET, a similar missing wedge filling algorithm was also included for 3D reconstruction correction. Since cosine and sine functions are periodic functions, motion defined by x will also be periodic motion repeats itself after certain interval of time. Reciprocal of T is the frequency which is nothing but number of times motion repeats itself in one second.

This reciprocal is represented as f and it is natural frequency of the system. For example: A in the above equation is the amplitude of motion. Angle is called phase angle.

Equation 10 is harmonic in nature. Determine natural frequency of free vibration of weight.

Calculate the natural angular frequency of the frame shown in figure. Compute also natural period of vibration. In this case, the restoring force in the form of spring force is provided by AB and CD which are columns. The equivalent stiffness is computed on the basis that the spring actions of the two columns are in parallel. Find the natural frequency of the system shown. The mass of the beam is negligible in comparison to the suspended mass. The beam has elastic behavior and hence has spring action.

Further spring attached to the mass also has a restoring force. The deformation of the mass at the centre is equal to deformation of beam at midspan and that of attached spring. This is a d. Hence following cases are possible.

## Table of Contents

The system is said to be over damped. Hence in this case, there is no periodicity or harmonic motion but the damping is so huge, that the system just comes back slowly equilibrium from displaced position.

Case 2: Hence , are real, negative and equal. Even in this case, there is no oscillatory motion. The damping in the system is enough to bring it back to equilibrium. System is said to be Critically Damped. Hence the roots are complex conjugate roots. Using these conditions, c1 and c2 can be computed. Equation 8 describes motion in undamped case. The variation of the displacement under damped free vibration system is as shown.

The amplitude decreases in an exponential manner. This means that damped natural time periods is more that of undamped period. However, this increase in time period is very small and hence for all practical purposes, it is assumed that a small viscous damping will not affect time period of vibration. A platform of weight 18 KN is being supported by four equal columns which are clamped to the foundation. Experimentally, it has been computed that a static force 5KN applied horizontally, to the platform produces a displacement of 2.

Compute the following: Logarithmic decrement: For a free vibration under damped SDoF system, the osciallatory motion is as shown in the figure.For example: Comparison between static and dynamic analysis: The magnitude of the maximum value of magnification factor is very sensitive to the value of.

In this process, an ab initio 3D density map was directly back-projected in Fourier space and served as the initial model.

However, it exhibited less bending angles between two nearby arms Fig. The IPET reconstruction method requires no pre-given initial model, class averaging, or lattice, but can tolerate small tilt errors and large-scale image distortion to achieve an intermediate resolution of 3D reconstruction 1—2 nm for negative staining samples and 3—5 nm for cryo-electron microscopy cryo-EM samples via decreasing the reconstruction image size 5.

Advertisement Hide. Bibliographic information DOI https: The damping in the system is enough to bring it back to equilibrium.

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