Dynamic Analysis
Description
Any existing model used for static linear elastic analysis of a framework or shell finite element structure may be converted to a dynamic analysis model by adding structure material densities and/or lumped masses (if necessary). This dynamic analysis control is then used to analyse to obtain natural frequencies and mode shapes ('Normal Modes Analysis').
For further details of the theory refer to the topics Dynamic Analysis Introduction and Dynamic Analysis Basic Theory.
Procedure
The full general procedure for dynamic analysis is:
Construct the structure as normal, ensuring support in all degrees of freedom.
Input structure property data, including structure material densities.
Densities may be defined through:
- Structure property forms: Define via the referenced material property density field.
- Design Beams and sections: Densities are defined along with the beam or section. The structure mass applied for a design beam or section during dynamic analysis is that calculated for the beam or section cross-section (including concrete, steel section, reinforcement and tendon masses) as it appears at mid-length on each structure member.
- Advanced Properties - Modified Density: If necessary the densities specified through structure property forms, or for design sections or design beams, may be over-ridden by applying a modified density to sets of members.
Add any superimposed dead loading applicable to a dynamic analysis using the 'Define Lumped Mass' form (accessed through the Basic Loads Navigation Pane).
Add any implicitly defined loads (for example a nodal loading or a vehicle or lane loading).
Implicit loads will be divided by acceleration due to gravity ('g') to become equivalent lumped masses which will have the same value in all three global directions. For example, a lane loading of 5kN/m2 applied vertically downwards will be converted into a mass of 0.51T/m2 applied in all global directions.
This enables the user to represent, for example, the mass of vehicles parked on a bridge deck. (Note: The mass is applied at the level of the deck rather than at the vehicle centroid. This will generally not much affect the vertical modes normally of interest.)
Enter the Compilations Navigation Pane through the Compilations Group Button, select the Add drop-down menu and select the item 'Dynamic'. This creates a Dynamic Compilation and opens the corresponding Compile Loading Patterns-Dynamic form.
Factors can be applied in the Compile Loading Patterns-Dynamic form to reduce or remove the effect of explicit or implicit nodal loadings This may be used, for example, to reduce the effect of the mass of a vehicle in the longitudinal direction where it is parked with its brakes off.
Access this dynamic analysis control through Calculate | Dynamic Analysis from the Main Menu.
Select a dynamic combination to be analysed.
Undertake a Sturm sequence check at a realistic frequency to find how many modes of vibration (i.e. natural frequencies) exist below that frequency.
Select a realistic number of modes and undertake modal analysis.
Quit this form using OK.
Access the Tabular Results form (accessed using File | Results from the Main Menu). Normally, following dynamic analysis the dynamic results are shown by default. However, if not, they are accessed by setting 'Result Type' to Dynamic.
Select a dynamic compilation in the 'Name' field. Tick the 'Show participation factors' checkbox. The results table now displays mode frequencies, eigenvalues and participation factors (for the selected dynamic compilation). Check both that the number of modes in the results matches that predicted by the 'Sturm sequence check' and that the percentage mass participating is sufficient.
Uncheck the 'Show participation factors' checkbox. This enables the 'Result For' drop down menu so that one mode may be selected. It also enables the 'Effect' drop down menu so that either 'Deflected shape' (with animation or video if required) or 'Displacements and Rotations' (in a selected direction) may be viewed.
Using this form
This form is divided into two panels. The upper panel defines the dynamic compilation that is to be analysed and it should be used first.
The 'Compilation for dynamic analysis' drop-down menu lists all dynamic compilations (created using the 'Compile Loading Patterns-Dynamic' form). The menu additionally always contains the compilation 'C0: Structural mass only' that comprises only the masses defined by material densities specified via structural properties. A green bullet placed alongside each item in the menu indicates that the item has been, and remains, modally analysed. Only the dynamic compilation that has been selected and so is shown in the 'Compilation for dynamic analysis' field may be considered or analysed at any given time.
The 'Combination Factors' box contains two fields. Loading from the lumped masses included in the compilation currently shown in the 'Compilation for dynamic analysis' field is multiplied by the entry in the 'Non-structural mass' field. Loading from structural mass (dynamic compilation 'C0: Structural mass only') defined by material densities specified in section properties is multiplied by the entry in the 'Structural mass' field. Then the two sets of mass loadings are summed to create the total mass loading used for dynamic analysis. By default both 'Combination Factors' fields are set to 1 (in which case loadings are applied as entered).
The 'Natural Frequency' (lower) panel should be used in two distinct stages (and only after the upper panel is set). The first stage is to undertake a Sturm sequence check at a specified frequency (by selecting the 'Sturm sequence check at' radio button, setting the required frequency in the corresponding field and then clicking the Analyse button). Following the Sturm sequence check the 'Num. of mode shapes' field displays the number of mode shapes (of this structure and using the mass-loading defined in the upper panel) whose natural frequencies are lower than the Sturm sequence check frequency. In a few instances, if the specified frequency is almost a natural frequency, the check will fail and a slightly different frequency should then be specified.
The second stage is to select the 'Modal Analysis for' radio button and then to enter into the corresponding 'Modes' field the number of modes of vibration for which dynamic analysis is required. Clicking the Analyse button then starts the analysis. When analysis is complete the dot alongside the current 'Compilation for dynamic analysis' becomes green and the Result button is enabled such that clicking on it displays a listing of Eigenvalues, natural frequencies and participation factors for each mode. The number of modes should be checked to ensure that it matches the number predicted by the Sturm sequence check. Deflected shapes for the current 'Compilation for dynamic analysis' may now be viewed in the Tabular Results form.
In some cases inversion by Cholesky decomposition fails due to ill-conditioning or when the structure has insufficient support and a listing describing the error is displayed. The problem might be overcome by checking the 'Stiffness Matrix Conditioning' checkbox and then entering a small value in the range -1E-6 to +1E-3 into the 'Shift by' field.
Form Graphic
Field Help
Compilation for dynamic analysis
This drop-down contains a list of the names of all currently defined dynamic compilations (created using the Compile Loading Patterns-Dynamic form). The list additionally always contains the compilation 'C0: Structural mass only' that comprises only the masses defined by material densities specified in section properties. A green bullet placed alongside each item in the list indicates that the item has been, and remains, modally analysed.
Selecting an item in the list causes that dynamic compilation to become currently active. Only the currently active compilation may be considered or analysed at any given time.
Combination Factors: Non-structural mass
Loading from the lumped masses included in the compilation currently shown in the 'Compilation for dynamic analysis' field is multiplied by the entry in this field before being summed with factored loading from structural mass to create the total mass loading used for dynamic analysis. By default this field is set to 1 (and thus loadings are applied as entered).
Combination Factors: Structural mass
Loading from structural mass (dynamic compilation 'C0: Structural mass only') defined by material densities specified via structural properties is multiplied by the entry in this field before being summed with factored loading from non-structural mass to create the total mass loading used for dynamic analysis. By default this field is set to 1 (and thus loadings are applied as entered).
Sturm sequence check at
The first stage of analysis of a selected dynamic compilation is to undertake a Sturm sequence check at a specified frequency by (i) picking this radio button, (ii) setting the frequency in this field and then (iii) clicking the Analyse button. A scrolling display shows the status of the check and is closed on completion of the check by picking the Done button. Following the Sturm sequence check the Num. of mode shapes field displays the number of mode shapes (of this structure and using the mass-loading defined in the upper panel of this form) whose natural frequencies are lower than the Sturm sequence check frequency.
The principle functions of the Sturm sequence check are:
- To verify that the program has not missed any natural frequencies in the range of results that it has obtained.
- To test multiple eigenvalues.
- To test for 'free degrees of freedom'
Num. of mode shapes
This field is set automatically following a Sturm sequence check. It shows the number of modes of vibration that have a natural frequency lower than that specified in the Sturm sequence check at field.
Modal Analysis for
The final stage of analysis of a selected dynamic compilation is to (i) select this radio button, (ii) enter into the corresponding 'Mode(s)' field the number of modes of vibration for which dynamic analysis is required and then (iii) click on the Analyse button. A scrolling display shows the status of the analysis and is closed on completion by clicking the Done button.
When analysis is complete the bullet alongside the current Compilation for dynamic analysis becomes green and the Result button is enabled such that clicking on it displays a listing of Eigenvalues, natural frequencies and participation factors for each mode. The number of modes should be checked to ensure that it matches the number predicted by the Sturm sequence check and shown in the Num. of mode shapes field. Deflected shapes for the current 'Compilation for dynamic analysis' may now be viewed in the Tabular Results form.
Normalise Eigenvectors
If this box is checked then the scale of the displacements in the results is modified so that the maximum displacement in each mode is changed to 1.0 (net effect of DX, DY and DZ). All other displacements and rotations are factored accordingly so that they are a proportion (smaller than 1.0) of the maximum displacement.
Stiffness Matrix Conditioning
In some cases inversion by Cholesky decomposition fails due to ill-conditioning and a listing describing the error is displayed. The error might be overcome by checking this checkbox and then entering a small value into the Shift by field.
Shift by
In some cases inversion by Cholesky decomposition fails due to ill-conditioning and a listing describing the error is displayed. The error might be overcome by ticking the Stiffness Matrix Conditioning checkbox and then entering an arbitrary small value in the range -1E-6 to +1E-3 into this field.
Entering a 'shift' value affects only the dynamics results. It has no effect on static analyses of the same model.
Large positive shifts affect the accuracy of the results. Negative shifts reduce the stability of the solution.
In applying the shift the program adds (the mass matrix times a multiple of the shift) to the stiffness matrix. This helps to produce a non-singular stiffness matrix. The results presented to the user are adjusted accordingly.
Result
This button is enabled only after 'Modal Analysis' is completed. Clicking on it displays a listing of Eigenvalues, natural frequencies and participation factors.
Analyse
Clicking this button starts either a Sturm sequence check or a Modal Analysis.