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**“.

**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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**Author Bio:**

**Mr. Vineesh Vijayan**, Application Engineer

**CADFEM India**

**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**.