In random vibration analysis, accurately modeling the dynamic behavior of a system is crucial to predicting its response under stochastic excitations. A discrete damper (like a shock absorber) is an essential component in this context, as it helps simulate the energy dissipation mechanisms present in real-world systems. By incorporating a discrete damper into the model, engineers can effectively control resonant peaks, reduce vibration amplitudes, and improve system stability. This enhances the accuracy of the analysis and provides insights into optimizing the design while ensuring a more realistic representation of damping effects.
In this blog entry, we demonstrate how to create damper elements in Random Vibration Analyses using material damping and beam elements. This approach provides a workaround for solver limitations that prevent the direct inclusion of element damping in Random Vibration Analysis.
To better understand this process we can consider a single mass damped oscillator, this is the same simple model we study in any vibration course. Because we know the solution for this oscillator we can compare the presented approach results with this analytical response.
Under a harmonic force excitation, is possible to create the highly known Amplitude ratio vs Frequency ratio diagram.
The Amplification amplitude can be calculated as:
The maximum value for amplitude is:
By assigning some numerical values for this example, is possible to define one base curve for further comparison.
This approach involves conducting two Full Harmonic Analyses. Due to the impossibility to include the damper elements in Random Vibration Analysis these Full Harmonic will characterize the dynamic behavior of the system. The reference model includes the damper, defined as desired for inclusion in the final Random Vibration Analysis with the actual shock absorber constant. The second Harmonic model uses the same geometry, but the damper is replaced with a solid beam. The material properties, including damping, are tuned to match the dynamic response of the reference model.
In this example, the reference model is expected to closely match the theoretical displacement amplitude, which it does effectively. While this may seem unnecessary since the solution is already known, the method becomes essential when solving complex models without analytical data. In such cases, the reference model's response serves as a benchmark for tuning the beam model.
Finite element model - Reference:
We can create a finite element model to perform the first Full Harmonic Analysis, representing the theoretical behavior of the system using a mass block, a spring, and a damper. Pay close attention to the numerical values for damping and stiffness. The block is modeled as rigid, with boundary conditions allowing only vertical displacement. The block's density is adjusted to achieve the previously defined mass.
Now, everything is set to perform the Random Vibration Analysis. Is possible to include the material properties from the analyzed beam in a real application. Note that the Damping factor found is exclusive for the geometric parameters of the created beam, that is why using real application dimensions would be useful. Imagine an hypothetical application of an arm allowed to rotate around a pinhole and attached to a spring and a damper. The load will be transferred via the spring end attached to ground.
The damper is included using the beam element and attached to the body arm and ground using two rotational joints.
Here, a vertical acceleration PSD (Power spectral density) curve is applied and two models are compared. The Damped model with the beam (left) and a second model with no damping after suppressing the beam body (right). Is clear how the damped model has less displacement response under the same spectral load. That shows the effect of the material damping methodology applied to this application model.
In this graph is possible to appreciate the peak attenuated by the damper element included into the model.
Incorporating a discrete damper into random vibration analysis enables a more accurate representation of energy dissipation and system dynamics under stochastic excitations. By leveraging material damping and beam elements, engineers can overcome solver limitations and achieve a practical workaround for including damping effects. The iterative tuning of the beam's material properties ensures the modeled dynamic response aligns closely with theoretical predictions, providing a reliable method for complex systems without analytical solutions. This approach highlights the importance of meticulous modeling and parameter optimization in enhancing the predictive accuracy and real-world applicability of finite element analyses in random vibration scenarios.
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