Wednesday, June 22, 2005

Finite Element (2) Dynamic Analysis

Time Dependant Dynamic Analyses: Modelling Impulse Problems

    If the excitation applied to a structure is impulsive rather than harmonic, many modes contribute to the response and it becomes more appropriate to use direct integration methods rather than modal analysis. There are a large number of applications where transient analyses are necessary. Many structures are subject to time varying loads such as impluse, blast, impact & seismic loadings. Transient dynamic analysis determines the time-response history of a structure subjected to a forced displacement function. The structure may behave linearly, or in some cases, friction, plasticity, large deflections or gaps may produce nonlinear behavior. Once the time response history is known, complete deflection and stress information can be obtained for specific times.

    The first step in any dynamic analysis should be the determination of the frequencies and shapes of the natural vibration modes. In a 3-D structure there are three dynamic degrees of freedom (DDOF) for every unrestrained node with non-zero mass and there is potentially a natural vibration mode for each DDOF. Thus, there are usually many potential vibration modes in a typical structure, but usually only a small number of vibration modes with the lowest frequencies that are of interest. In a multi-storey building, for example, it might be only a few in each of two horizontal directions, plus one or two torsional modes that have to be considered.

Frequency & Transient Analysis Differences
    While frequency analyses take place in the frequency domain, transient analyses are studies in the time domain. It is always possible to go from the time domain to the frequency domain via a fourier transform. Correspondingly, a change from the frequency domain to the time domain may be achieved by implementing an inverse fourier transform.

    Due to mathematical difficulties, solutions in the frequency domain can only be linear in nature. Therefore if the application requires a solution that is a non-linear function of time, then a time domain analysis must be carried out. The solution can subsequently be projected to the frequency domain if required. Frequency analyses can be solved using no boundary conditions, while transient analyses must be fully constrained.

Transient Solutions: Modal & Direct
    There are usually two approaches one can take when carrying out a transient analysis, modal solutions or purely direct solutions.

    The modal approach involves evaluating the relevant natural frequencies of a structure first. Once this is carried out, the response is converted to the time domain and is included in the evaluation of transient response of the structure. Modal analyses are usually used where there many natural frequencies within the operation range. It is important to evaluate the natural frequencies above and below that of the analysis range. This is due to the fact that, in practice, there is never just one distinct mode of vibration due to an excitation, but one dominant mode with a range of additional harmonics from the adjacent upper and lower modes.

    Direct solutions are used to evaluate the response of a structure within a very narrow frequency range of interest, and are usually used for models that subject to high frequency impulses. The solution is purely transient, no frequency extraction is carried out first.

The Solution Approach
    As a transient load is applied, the solution must follow the response of the structure. To achieve this, the overall time period being studied is divided into a number of linear time pieces, each one being referred to as a time step. The successful implementation of any time domain analysis is dependant on a suitable number of time steps being selected. If the time step is too large, portions of the response (such as spikes) could be missed or truncated. On the other hand, if the time step is too small, the analysis will become excessively long or even prohibitive.

    The progression of the solution from one time step to the next is achieved by implementing time integration techniques. Despite many packages providing automatic time stepping estimates, the full response of the structure may not be captured, and manual intervention will be required. If there is a discontinuity in your automatically time stepped results, chances are there is a spike in the response that is not being fully captured.

Stepping Schemes: Time Integration
    Many time stepping algorithms have been developed, each having their advantage over others under certain circumstances. However, three main types of solution dominate, backward difference (Implicit), central difference (Crank-Nicolson) & forward difference (Explicit). The explicit & implicit techniques are often referred to as Euler's rule & the backwards Euler's rule respectively.

    Explicit schemes, which are conditionally stable (stability of solution not guaranteed), find the response at the end of the time step in terms of the conditions at the start of the time step. In other words, the calculation of the solution at time (t+Dt) is obtained by considering the situation at time t. The advantage of this approach is that the underlying system of equations that comprise the model (stiffness matrix, capacitance matrix, flexibility matrix) does not have to be solved at each time step. Furthermore, the material & time matrices can be diagonalised to become uncoupled, and so the solution can be calculated explicitly. Very fast calculations of individual time steps can be achieved as no matrix factorisation is required. However, the technique is much less stable than the implicit method, so very small time steps must be used to ensure an appropriate solution.

    Implicit schemes, which are unconditionally stable, find the response at the end of the time step in terms of the conditions at the end of the time step. In other words, the calculation of the solution at time (t+Dt) is found by considering the response at time (t+Dt). An important point to note is that the solution at each time step involves matrix factorisation (evaluating the system of equations that comprise the model), which is a computationally intensive process. Despite this disadvantage, implicit schemes are often used, as the solution is inherently reliable & robust. Implicit analyses allow much larger time steps than the others, and so the solution can be obtained with fewer calculation increments. As implicit schemes are always stable, the time step length is governed by considerations of accuracy alone.

    The Crank-Nicolson approach evaluates the next step of the solution by using the prediction at the centre of the time step. As with the backward difference scheme, this is an implicit solution which is conditionally stable (results in an oscillatory solution if the critical time step for stability is exceeded). The central difference method is more accurate than both the purely implicit or explicit techniques since neither favours the response at the start or end of the time step.

Response Spectrum Analysis
    Response spectrum analysis (RSA) is a procedure for computing the statistical maximum response of a structure to a ground bourne excitation. Each vibration mode considered may be assumed to respond independently as a single-degree-of-freedom system. Design guideline codes specify response spectra that determine the base acceleration applied to each mode according to its period (the number of seconds required for a cycle of vibration). The design response spectrum is then usually obtained by multiplying the basic acceleration coefficient by a factor based on required structural performance, risk & location.

    Having determined the response of each vibration mode to the excitation, it is necessary to obtain the response of the structure by combining the effects of each vibration mode. Because the maximum response of each mode will not necessarily occur at the same instant, the statistical maximum response, where damping is zero, is taken as the square root of the sum of the squares of the individual responses.

    Response spectrum analysis produces a set of results for each excitation load case which is in the form of an envelope. All results are absolute values, each value represents the maximum absolute value of displacement, moment, shear, etc. that is likely to occur during the event which corresponds to the input response spectrum.

Concepts associated with Dynamic Strucural Analyses
    SHAKEDOWN ANALYSIS: If load intensities on a structure remain sufficiently low, the response of the body is purely elastic (with the exception of stress singularities). If the load intensities become sufficiently high, the instantaneous load-carrying capacity of the structure becomes exhausted (unconstrained plastic flow and damage evolution occurs) & collapses.
    If the plastic strain increments in each load cycle are of the same sign then, after a sufficient number of cycles, the total strains (and therefore displacements) become so large that the structure departs from its original form and becomes unserviceable. This phenomenon is called incremental collapse or ratchetting.
    If the strain increments change sign in every cycle, they tend to cancel each other and total deformation remains small leading to alternating plasticity. In this case, however, the material at the most stressed points begins to fails due to low-cycle fatigue.
    If, after some time plastic flow and damage evolution cease to develop further and the accumulated dissipated energy in the whole structure remains bounded such that the structure responds purely elastically to the applied variable loads, one says that the structure shakes down

    FLUTTER is a dynamic instability that involves coupling of aerodynamic forces and elastic and inertial forces of the structure. In a flow, an oscillating structure generates unsteady aerodynamic forces. These unsteady aerodynamic forces introduce coupling into the structure and cause phase shifts between the motions of the structure (degrees of freedom). The speed of the flow affects the amplitude ratios and phase shifts between the various degrees of freedom in such a way that energy is extracted from the airstream. At the critical airspeed, the energy dissipated is exactly equal to the available structural damping. At speeds greater than the critical speed, the extracted energy dissipated is less than available structural damping and the motion is divergent.


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12:41 am  
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Anonymous Vibration Solutions said...

When calculating dynamic response, my experience is that only a few modes (up to six) really contribute to the response. As the mode number increases, the accuracy ot the predicted natural frequencies and mode shapes decreases.

7:04 am  
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