In this course, we discussed the workflow of mode-superposition (MSUP) analyses, the importance of damping, how many modes to include, and the role of contact and other nonlinearities in linear dynamics. Let us review some key points from each lesson.
Understanding the Mode-superposition Method
- Mode-superposition is a method of using the natural frequencies and mode shapes from a modal analysis to characterize the dynamic response of a structure to transient, harmonic, response spectrum, and random excitations.
- In the mode-superposition method, a linear combination of mode shapes is used to determine the actual displacement vector of the structure.
- For an MSUP workflow, the modal analysis will act as an upstream analysis. Each downstream analysis must use one modal analysis; however, one modal analysis can be used in multiple downstream analyses.
- MSUP workflow can be created in two ways: one is in the Workbench project page and the other is within Ansys Mechanical. To follow the latter method, the “Future Analyses” option under Analysis Settings must be changed to “Mode-superposition” before solving the modal analysis.
- Usually, MSUP analyses are computationally inexpensive compared to modal analysis. Moreover, the results of one modal analysis can be reused in other downstream analyses. These downstream linear dynamic analyses also support pre-stressed modal analysis.
- Damping is an energy-dissipation mechanism that causes vibrations to diminish over time.
- Unlike elastic modulus, damping is not something we can often easily measure since energy dissipation can come from joints and part interactions, fluid interactions, material behavior, and other sources. It is estimated via testing or is specified by industry practice.
- There are three types of damping covered in this lesson: constant damping, alpha damping, and beta damping.
- The constant damping ratio, as its name suggests, is constant for each frequency. It is the ratio of damping to critical damping at a given frequency (mode).
- Alpha damping results in a damping ratio that is inversely proportional to frequency. Because of this, we usually do not use alpha damping since it has adverse effects on very low frequencies.
- Beta damping provides a damping ratio that is proportional to frequency. Consequently, it tends to affect higher-frequency content. Alpha and beta damping together are called Rayleigh damping.
- These damping values may be defined on a system level (“global”), affecting all parts — done via “Analysis Settings”. The damping ratio may be defined on a material basis (“material”), affecting all parts that have that material assigned — done via Engineering Data.
- Effects of damping are cumulative, so if you specify global damping and one part has material damping, then that part sees the effect of the sum of material and global damping.
How Many Modes to Include? (Modal Truncation)
- For any structure, the total number of modes is equal to the number of nodal degrees of freedom (DOFs), but while performing any analysis, we extract far fewer modes than this number due to computational constraints.
- A rule of thumb for modal sufficiency is to compute modes up to 1.5 times the highest frequency in the excitation frequency range.
- After solving any modal analysis, it is always recommended to check the ratio of effective mass to the total mass in the direction of excitation or the direction of the expected response. A ratio above 0.9, or 90%, is generally preferred.
- It is also important to visually review the mode shapes and to analyze the participation factors in the direction of excitation. Understanding which modes have high participation factors is crucial because these are the significant modes that would be used in the dynamic calculations.
- For harmonic and transient analyses, in instances where the localized forces are acting on the structure and causing localized deformation, high-frequency modes may be needed to capture them. The “Residual Vector Method” presents an efficient alternative.
- For response spectrum analysis, in instances where we may extract enough modes to cover a wide frequency range, the effective mass may still be relatively low. Also, for base excitation in a response spectrum analysis, the effective mass of the structure is important in obtaining an accurate response. Hence, the missing mass method is an efficient technique to include the effect of this "missing mass" without having to extract a very large number of modes.
Role of Contact and Non-linearities in Linear Dynamics
- Some important assumptions in a Linear Dynamic Analysis are: It is based on Small Deflection theory which assumesthat the displacements are small enough that the resulting stiffness changes are insignificant, and the area of contact is fixed, which means that contact is either bonded or sliding (no separation) only, as in these contacts the contacting area doesn’t change.
- To account for the nonlinearities in a linear dynamics analysis, linear perturbation can be used. Linear perturbation is a linear analysis with nonlinear prestress effects included.
- Normally in a linear dynamic analysis, if the initial contact status is sliding or sticking, the rough and frictional contact with coefficient of friction greater than zero will get converted to bonded contact and frictionless becomes no separation. If the status is near or far, then all three nonlinear contacts are ignored.
- For the case of linear perturbation about a nonlinear based state, the contact status from the base analysis (not the initial contact status) is used to determine how the contact will behave, following the same rules. In other words, we run a nonlinear static analysis first, then the calculated contacting area from this analysis is used in the modal analysis.
- The behavior of the contact can also be altered in the definition of prestressed environment. The different contact statuses available for the modal analysis are true status, force sticking, and force bonded.
- For material nonlinearities, for metals, only the stiffness near the origin, which is the linear portion of the stress-strain curve is used — the plasticity is ignored.
- For hyperelastic materials, we have two options: If no prestress is considered, one may switch the hyperelastic material with a linear elastic material. Alternatively, to account for the stiffness in a specific deformed configuration, we can first solve a nonlinear static analysis with hyperelasticity, then perform a pre-stressed modal analysis without having to substitute the hyperelastic material for a linear elastic one.
- Turning on "Large Deflection" in the base static analysis, followed by defining the base analysis in the prestress environment, accounts for this geometric nonlinearity effect in the linear perturbation method.