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ST7-1.10 Linear

ST7-1.10.10 Linear / Statics

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ST7-1.10.10.1 Assumptions of Linear Analysis
Linear analysis is subject to three fundamental assumptions:
1. Linear Material Assumption
The material behaves according to Hookes Law, wherein stress is considered to be linearly proportional to strain.
2. Linear Geometry Assumption
The response of the system assumes that the displacements are small, even if that turns out not to be the case. The main point about this assumption is that deformation of the system has no effect on its stiffness.
3. Known Boundary Conditions Assumption
The boundary conditions are known and fixed at the start of the analysis and do not change with loading or deformation.

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ST7-1.10.10.2 Rigid Body Modes and Singularity Warning in Static Solvers
This Webnote examines the warning messages that can indicate rigid body motion and/or stiffness matrix singularity in a static solution.

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ST7-1.10.10.3 General Model Troubleshooting
Before analysis, all the data in a finite element model should be carefully checked to ensure that:
1. it is representative of the structure being modelled;
2. it is applicable to the required analyses; and
3. the assumptions and approximations made are suitable.
At the completion of the analysis, and before in-depth interpretation of the analysis results, the solver messages should be studied. Finally, the results should be checked for reasonableness. This Webnote introduces the Strand7 features and tools that can be used for checking and troubleshooting finite element models, so that the above procedures can be more easily performed.

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ST7-1.10.10.4 Inertia Relief
Inertia Relief Analysis is used whenever the applied loads are to be reacted by the inertia of the structure rather than the more common situation where the loads are reacted by the restrained nodes. In a structure such as a floating ship, it is not always obvious where one should apply restraints. For such situations, we can use the Inertia Relief Freedom Cases in Strand7. The following summarises the equilibrating inertial restraint, and the basic steps required for performing such...

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ST7-1.10.10.6 Linear Buckling Analysis
In order to gain an appreciation for the process of performing a linear buckling analysis and to be able to interpret the results, it is important to keep in mind the fundamental question that the linear buckling solver answers, namely: The linear buckling problem can be represented by the following relationship: which can be rearranged as the following eigenvalue problem: where is the global stiffness matrix of the structure; is the geometric stiffness matrix, which depends...
ST7-1.10.20 Linear / Dynamics

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ST7-1.10.20.1 Modelling a Rotating Body
A linear static analysis due to a rotating load is to be performed with Strand7. The analysis is required to determine whether a turning chuck satisfies stress and deformation criteria under rotating operational conditions. The geometry of a turning chuck has been provided in the accompanying IGES file. This Webnote outlines the procedure for importing and manipulating the CAD geometry, automeshing, applying non-structural mass to the model, applying a rotational velocity of 3000 RPM about the...

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ST7-1.10.20.2 Harmonic Response Analysis
The aim of this Webnote is to introduce the Harmonic and Natural Frequency Solvers, and some basic concepts in modal analysis. This is done with reference to a very simple cantilever beam model, which is harmonically excited at its outboard end. Ways of introducing damping to the system in Strand7 are then demonstrated, and a comparison is made with a full system transient solution, highlighting the computational and conceptual benefits of the modal approach for this problem type. The harmonic...

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ST7-1.10.20.3 PSD Analysis of a Wind Load
This Webnote examines the procedure of defining wind PSD on structures. A preliminary step in determining the response of a structure to the fluctuating conditions is to define a spectrum of wind speeds. where In this particular example, the following ambient conditions are assumed. Roughness parameter k = 0.05; Mean Wind Speed = 5 m/s. Using these conditions Davenports equation generates the following wind speed PSD: Prior to running the Strand7 solvers, care must be taken to...

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ST7-1.10.20.4 Harmonic Response of a Rotating Unbalanced Mass
In this Webnote the Harmonic and Natural Frequency solvers are used to assess the effect of a rotating inertial force such as normally arises from a rotating out-of-balance mass. This is introduced by a model of a large fan cowling, housing two out-of-balance counter-rotating fans. A Rayleigh damping model is used. This Webnote makes reference to the model file, ST7-1.10.20.4 cowling.st7.  A common source of vibration in mechanical components arises from rotating imbalances. (1)  ...

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ST7-1.10.20.5 Linear Transient Analysis
The linear transient dynamic solver is used to calculate the time history of the dynamic response of a structure subjected to any arbitrary forcing function and initial conditions, be it machinery vibration, wave loading, or random earthquake excitation. The dynamic equilibrium can be obtained by solving: t + t + t =(t)  where = global mass matrix (comes from the material density, volume and applied mass attributes) = global damping matrix (comes from spring-damper...

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The natural frequency of a structure may change due to the applied load. This behaviour is exemplified by the classic example of a guitar string, which vibrates at higher frequencies as it is tightened. The tension in the string stiffens it in the lateral direction. This Webnote looks at how to perform simple natural frequency analysis with and without the loading effects.  Some important parameters of the natural frequency solver are listed below. The most important ones are bold. ...

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ST7-1.10.20.7 Using Modal Damping in Harmonic Response Analysis
Modal damping is useful for situations where different regions of the model have different damping characteristics. Based on the natural frequency results, modal damping combines these different regions based on excitation in each mode. The spectral and harmonic solvers can then use these combined damping ratios, which are calculated for each natural frequency. This can be used to include the overall effect of localised damping in a modal analysis, which is not otherwise possible. This Webnote...

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ST7-1.10.20.9 Spectral Response Analysis
Spectral response analysis is typically used for calculating the linear elastic response of a structure subjected to random dynamic loading. In Strand7, random dynamic loading can be expressed in terms of either response spectrum or power spectral density (PSD) curves. Furthermore, the applied excitation can be either a base excitation (displacement, velocity or acceleration) or a dynamic load.  Suppose that an oscillator with a single degree of freedom (SDOF) is excited by an external action...

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ST7-1.10.20.10 Modal Mass, Stiffness and the Mass Participation Factor
This Webnote discusses the relationships between the following dynamic parameters and how they can be obtained from Strand7. Modal mass and Engineering modal mass Modal stiffness and Engineering modal stiffness Eigenvalues and Eigenvectors Mass participation factor Readers of this Webnote might also benefit from reading ST7-1.10.20.11 Mode Shape Scaling.  An introductory description of the use and application of each of these quantities in structural dynamics is provided below....

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ST7-1.10.20.12 Vibrational Analysis of a Wine Glass
The harmonic oscillation of a wine glass is a familiar phenomenon to many people. In this Webnote we recreate the harmonic behaviour of a Riedel Vinum Shiraz / Syrah glass. The natural frequency solver is used to recreate the tone generated by rubbing a moistened finger around the rim of the glass, and the linear transient solver is used to recreate the frequency spectrum seen in response to an impulse load. In both cases Strand7 results are compared with experimental results.

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ST7-1.10.20.13 Response Spectrum and Power Spectral Density
This Webnote compares two common techniques of dynamic analysis for stochastic inputs, Spectral analysis with a known Response Spectrum, and Spectral analysis with an input Power Spectral Density, against full transient solutions. The Spectral Response solver in Strand7 takes two different types of input to determine the response of the structure to random excitations: a Response Spectrum, or a Power Spectral Density function. In this Webnote, we demonstrate these different approaches...

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ST7-1.10.20.20 Static and Dynamic Stiffness
A common transfer function measurement is the Frequency Response Function (FRF). The aim of this Webnote is to give a brief overview of some of these functions, and to take a more detailed look at dynamic stiffness, which is a particular FRF.

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ST7-1.10.20.22 Compression-Only Footfall Harmonic Response Analysis
Footfall analysis on pedestrian-loaded structures is often performed using the Harmonic Response solver, which is used to determine the steady-state response of a structure to a given sinusoidal forcing load. Whilst it is tempting to simply apply the peak footfall load in such an analysis, this approach assumes that the footfall load acts in tension as well as compression, when in reality the step load is more like a half-sine wave.
The assumption that the load acts in both tension and compression is not necessarily a conservative one. There is more to the difference between a half-sine wave and a fully-reversing sine wave than the load magnitude. Whenever a discontinuous acceleration is applied, it can excite higher frequency modes than the frequency of the applied footfall load. Thus the footfall, while nominally acting around 2.0 Hz, can also excite natural frequencies with harmonics around 4.0 Hz, 6.0 Hz, 8.0 Hz, 10 Hz etc. (n x fstep).
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