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The dynamic response of a structure under time dependent loadings can be assessed by considering a linear combination of the mode shapes. Hence, the Natural Frequency calculations are the basis of the dynamic module and need to be carried out prior to any of the analyses in the dynamic analysis module.
The Dynamic module is accessed through Masterframe, by going to Analysis>Dynamic Analysis. The basic layout of the screen is shown below.
Prior to running the mode shape analysis, mass needs to be assigned to the model. The software can assign the mass of the structure using the self-weight of the frame. This is done by ensuring that the load groups which represent the self-weights of the structure are included in the analysis. This is done by by adding a load factor in the appropriate Load Factor Groups.
To access the load combinations, click on the .png)
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The self-weight of the structure is included by adding a factor of 1.0 to the appropriate Load Groups. Amending load factors in this table does not affect the load combinations set up within Masterframe.
For floor vibration design, it is also recommended to include 10% of the live load in the mass of the structure. While it is possible to amend the live load of the model in MasterFrame to reduce the live loads to just 10% of the values used in the static analysis, this can be achieved by adding a factor of 0.1 to the live load groups in the Mass for Dynamic analysis load factor table.
The Density, for use with the self-weight of line elements, and the self-weight load groups can be amended under the Member Self Weight Mass tab. However, unlike the Load Factors, any changes made to the density will also take effect in Masterframe, so potentially affecting the MasterFrame static analysis.
One of the main considerations when running the Natural Frequency analysis is the range over which the natural frequencies are calculated. The number of possible mode shapes (including possible repeated eigenvalues) of a structure is equal to the number of degrees of freedom of the structure. This is because the number of degrees of freedom matches the size of the stiffness matrix and an n x n matrix has n (possibly repeated) Eigenvalues and associated Eigenvectors. The more mode shapes to be found, the longer the analysis will take. For large model, particularly model will FE surfaces, this number of degrees of freedom can be large and so the subsequent mode shape analysis can take significant time to analyse if it were to find all mode shapes. For this reason, it is not advised to select the "Natural Frequencies (0-1000Hz)" option.
The "Frequencies in a Range" option allows a minimum and maximum natural frequency limit to be selected, such that the software will calculate the mode shapes within this range. This can, therefore, reduce the time needed for the analysis. However, care needs to be taken when selecting the range, to ensure that modal frequencies which would contribute to the solution of the dynamic analysis are being excluded. In general, for a footfall analysis, the suggested upper range would be 3-4 times the 4th harmonic of the walking frequency, which would give an upper frequency of 30-40Hz.
A mode shape analysis can be run for 2D plane frames and grillages as well as for 3D frames. The advantage of using 2D or a grillage analysis is the reduced analysis time, which results in the reduction in the stiffness matrix due to the elimination of zero stiffness terms.
Once the required analysis has been set-up, click on the .png){kind=small}
icon.
When the analysis completes, the central table will be populated with the mode shapes information. A typical layout is shown below:
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The information given in the tabulated values is:
Mode - the order of the mode shape, arranged from lowest to highest frequency
Frequency (Hz) - the frequency of the mode shape
MPX (%) - mass participation percentage in the x-direction
MPY (%) - mass participation percentage in the y-direction
MPZ (%) - mass participation percentage in the z-direction
Max DOF - the largest displacement or rotation in the mode shape occurs at the highlighted node number
The table of mode shapes can be scrolled through using the scroll bar or mouse wheel. The table of values can also be reorders, by using the mouse to click on the title of the column. Selecting the column title will arrange the values in that column in ascending order, while clicking the title again will re-order in descending order.
A row in the tabulated mode shapes can be highlighted by selecting the row using the mouse pointer and left clicking. Once a row is highlighted, the results panel on the right hand side of the screen will populate. These values are normalised and represent the shape of the selected mode shape. The largest DOF given in the MAX DOF column will have value 1.0 in the Results table.
In addition to the above, the mode shape can be animated by clicking on the .png){kind=small}
icon. The scale (magnitude of the deflection ) and speed controls can be used to modify the animation.
Mass Participation
The Mass Participation is a measure of the mass which is mobilised in a mode shape. This can be used to determine how significant a particular mode shape is likely to be when considering periodic motion in a particular direction, since the modes with large mass participation in a particular direction are likely to feature in the linear combination of mode shapes that make up the solution to a forcing function acting in that direction. However, where the mode shape is highly symmetrical and sinusoidal, the mass can be equally distributed about the neutral position, so the sum of the mass is zero. Therefore, in certain circumstance, modes which are significant in the solution to a certain forcing function can have low or zero mass participation. Therefore, care needs to be taken if using the mass participation to identify particular modes shapes for use in the Vibration Design.
Total Degrees of Freedom
The is the total number of degrees of freedom in the entire model. The number of degrees of freedom indicate the size of the n x n square stiffness matrix. The number of degrees of freedom will give some indication of how long the analysis of the mode shapes will take.
Total Model Mass (kg)
This is the mass of the model used in the last analysis. The mass will be determined by the factors input in the Mass Load Combination table. The mass used in the dynamic module can be checked by creating a loadcase in Masterframe with the safe load factors applied to the same load groups and then reviewing the total support reaction in the static analysis. The total reaction in MasterFrame will be in kilonewtons (kN). To convert the mass in kg, the software uses an acceleration due to gravity of 9.81 m/sē
Total Mass Participation
In addition to the Total Model Mass, the software notes the total mass participation in each global axis direction. If any of the mass participation values are low, this may be indicative of not enough mode shapes being captured in the frequency range. However, edge or nodal restraints may prevent periodic motion in a particular axis, in which case this would then also return a low mass participation in that direction.