We often come across the dynamics or vibration problems in most of the product development across all industries. We have gained enough knowledge through the degree of Bachelors or Masters, with an example of the famous engineer disastrous design of “Tacoma Narrows Bridge-1940”, that considering the dynamics effects in design is quite critical. Generally, a dynamics problem is categorized into linear or nonlinear dynamics. This blog is related to the linear dynamics and nonlinear dynamics will be discussed in length in another blog. The most commonly heard terminology which we stumble upon in the design of such vibrating products is “natural frequencies“, “resonance“, “eigenmodes” etc. We have many linear dynamics simulation techniques such as “Harmonic Analysis“, “Random Vibration“, “Response Spectrum“, “Transient Analysis“, “Acoustic Analysis“, etc based on the loading it undergoes and also on the output we are interested in. But all these dynamics simulations start with the first step called “Modal Analysis“.
What makes something unique is an identity of its own, right? What if I say natural frequency is the identity of a body? Let’s see why. Assume you are imparting energy (striking) to a body. What will happen is obvious, the body will start to vibrate. But the vibrational frequency will be in its natural frequencies. It doesn’t matter how much or where you give the energy it will always vibrate in its natural frequencies, period. This is why the base of any dynamic analysis is said to be modal analysis as it shows its identity or behavior.
The modal analysis calculates natural frequencies and mode shapes of the designed model. It’s the only analysis that doesn’t require any input excitation or loads, which also makes sense, as mentioned before natural frequencies are independent of the excitation loads. Natural frequency depends only on two things mass and stiffness.
Why do we need to do the modal analysis?
Yes, we can find natural frequencies, and mode shapes through modal analysis but why to do it? Let me explain that through an example, when you ride a motorcycle, sometimes you might have experienced vibration in your handle. Some bikes have too much vibration that, you can’t even use your rear mirror. This is because the handle’s natural frequency is matching with engines RPM.
Structure borne noise: A structure makes peak noise when it’s vibrating in its natural frequency.
Whirling of shafts: To find the critical RPM of a shaft so that, crossing the critical speed will be done with precautions to avoid resonance.
Avoiding resonance is always preferred, thus shifting or modifying natural frequency can help to control, above mentioned issue. Natural frequency depends on only two parameters, modifying them will help to design a dynamic body part.
As an engineer, I always go in search of the physical meaning rather than proving by an equation. But at times equations are unavoidable.
Now let us see how we can get the natural frequency of a system.
Excitation force Equation of any dynamic system can be represented as
Every natural frequency is associated with each mode shape and mode shape represents the displacement behavior. There is enough material on the internet which explains mode shape. Right now, let’s concentrate on the “Mass Participation Factor” other important information from modal analysis, which would assist us in the simulation of other linear dynamic simulations. What is mass participation in general? Consider a 3 massed system as shown below.
The figure shown above is a mode of the system. Here masses participating in the X direction are M1 and M2 and thus total mass participation is M1 +M2. The only difference in FEA is that instead of lumped masses, the system will be discretized by elements.
Now let’s see ANSYS modal Analysis. Here I’m taking a cantilever beam, with 0.8478 kg. Having a cross-section of 3*60*600.
As you can see from the participation factor calculation. I have solved the first 12 natural frequencies. In this table, ANSYS solver calculates only frequency and Participation factor ie 1st and 4th column. Every other column is calculated from these two values. Let’s look at the most important column one by one.
As you can see this ratio is in direct comparison with total mass, which gives a better clarity on how much of the total mass participation is extracted. In other words, the sum of the effective masses in each direction should equal the total mass of the structure. But clearly, this depends on the number of extracted modes.
The same logic is for rotational mass participation. But here you might get ratios bigger than 1. Then, there can be confusion as mass participating is more than the total mass of the system? The answer is, participation factors for rotational DOFs are calculated about the global origin (0,0,0). The calculated effective mass essentially contains a moment arm (effective mass multiplied by the distance from centroid). Thus, the effective mass for the rotational DOFs can be greater than the actual mass and the participation factors can be greater than 1.0.
Ideally, the ratio of effective mass to total mass can be useful for determining whether or not a sufficient number of modes have been extracted for the “Mode Superposition” for further analysis. Modal superposition is a solution technique for all linear dynamic analyses. Any dynamic response can be calculated as the sum of the mode shapes, that are weighted with some scaling factor (modal coefficient). Also, the time step of the transient simulation is calculated based on the most influential mode in the modal analysis.
Modal analysis is usually done without considering any force or working environment. This assumption is taken by considering mass and stiffness to be constant. But at times this doesn’t work, like a pressurized vessel or pipe. As compressive or tensile force changes the stiffness. In such cases, we have to do a prestressed Modal Analysis.
In some cases, like rotating shafts with motors, turbines, the natural frequency will not match with the working condition. This is because, for different RPM, stiffness also changes,(Centrifugal force changes). In these cases, we need to consider the rotational speed of the shaft. In ANSYS Modal Analysis (Rotodynamic Analysis) rotational speed with Coriolis effect can be directly given to find the critical speed of the shaft.
Modal analysis is a prerequisite for all linear dynamics analyses like Harmonic, Random vibration, and Response spectrum analysis. Hence Modal Analysis plays a huge role wherever you are trying to find out vibration characteristics of a design. Thereby it is relevant for most industries like Rotating machines, construction, automobile, Machine tool, agricultural equipment, and Mining.
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Mr. Vineesh Vijayan, Application Engineer
Mr. Vineesh Vijayan has done his Master of Technology in Machine Design. With good expertise in Structural Simulation currently, he is contributing his efforts to CADFEM India as an Application Engineer.