Natural Frequency Formula: Why It Matters

Key Takeaways

●  Learn how to utilize the natural frequency formula. 

●  Gain a greater understanding of the importance of a device or system’s natural frequency.

●  Learn about how to change a device or system’s natural frequency.

Using the natural frequency formula, one can calculate a tuning fork’s natural frequency.

The terms “nature” and “natural” share a common definition: “Existing in or initiated by nature; not produced or affected by humankind.” As simple as that sounds, nature is one of our greatest sources of inspiration, constantly driving innovation within electronics and science.

One such naturally occurring phenomenon is frequency, which encompasses various fields including the field of electronics. When discussing electronic devices and systems, we often use natural frequency terms. We can calculate these terms using a formula. 

What Is Natural Frequency and How Does It Work?

An object's natural frequency is the frequency or rate that it vibrates naturally when disturbed. Objects can possess more than one natural frequency and we typically use harmonic oscillators as a tool for modeling the natural frequency of a particular object.

We can apply an unnatural or forced frequency to an object that equals the natural frequency of an object. In cases such as this, we are in effect creating resonance, i.e., oscillations at the object’s natural frequency. If this occurs in certain structures, the oscillations will continue to increase in magnitude, thus resulting in structural failure.

When a system’s oscillations are equivalent to its natural frequency, it forms motion patterns. We call these certain characteristic frequencies an object’s normal mode. Moreover, natural frequency consists of various primary factors and they are as follows:

• We call the frequency in which an object naturally vibrates, its natural frequency.

• We can utilize harmonic oscillators as tools to model an object’s natural frequency.

• Natural frequencies are those that occur naturally when we disturb an object in a physical manner, whereas objects that vibrate in accordance with the application of a particular rate are called forced frequencies.

• If we apply a forced frequency that is equivalent to an object’s natural frequency, the object will encounter resonance.

Calculating The Formula

Visualize a spring with a ball, which represents mass, attached to its end. While the ball and spring are stationary, the spring only partially stretches out, and our simple harmonic oscillator is in a position of equilibrium. Therefore, the tension from the springs is equal to the gravitational force that pulls the ball (mass) downward.

Once we move the ball away from its position of equilibrium, there are two possible outcomes:

1. It adds more tension to the spring, i.e., it is stretched downwards.

2. It provides gravity the opportunity to pull the ball downward devoid of the tension from the counteracting spring, i.e., you push the ball upward.

Regardless of which action you take, the ball will begin to oscillate about the equilibrium position.

This oscillating frequency is the natural frequency, and we measure it in Hz (hertz). In summary, this will provide the oscillations per second depending on the spring's properties and the ball's mass.

Now, we will use the above example to calculate the natural frequency of a simple harmonic oscillator. When calculating the natural frequency, we use the following formula:

f = ω ÷ 2π

Here, the ω is the angular frequency of the oscillation that we measure in radians or seconds. We define the angular frequency using the following formula:

ω = √(k ÷ m)

This, in turn, adjusts our formula to the following:

f = √(k ÷ m) ÷ 2π

f is the natural frequency

k is the spring constant for the spring

m is the mass of the ball

We measure the spring constant in Newtons per meter. A spring with a higher constant is stiffer and requires additional force to extend.

As we calculate the natural frequency utilizing the above formula, we must initially determine the spring constant for the system. Typically, we obtain this value by conducting tests. However, for this example, we will use 150 N/m to represent k and 2 kg to describe our mass m.

Now, we will utilize these values by performing the steps of the calculation:

f = √(150 N/m ÷ 2 kg) ÷ 2π

f = √(75 Hz) ÷ 2π

f = 8.66 Hz ÷ 2π

f = 8.66 Hz ÷ 2(3.14)

f = 8.66 Hz ÷ 6.28

f = 1.3789 Hz

per

f = 1.38 Hz

The natural frequency is 1.38 Hz, which translates into the system oscillating nearly one and a half times per second.

The Importance of Calculating Natural Frequencies

We typically consider the natural frequencies and mode shapes to be the single most critical property of virtually any system. As you might imagine, excessive vibrations in any system lead to structural and functional issues.

The reason for this is the natural frequencies can match with a system's resonant frequencies. For example, if you employ a time-varying force to a system and select a frequency equivalent to one of the natural frequencies, this will result in immense amplitude vibrations that risk putting your system in jeopardy.

This is why when designing a mechanical system it’s important to calculate and ensure the natural frequencies of vibration are far greater than any possible excitation frequency that your system is likely to encounter.

Natural Frequency in PCB and Electronics Design 

While natural frequency is often associated with mechanical systems, it is equally important in printed circuit board (PCB) design. PCBs are not perfectly rigid, they behave like flexible mechanical structures that can vibrate under real-world conditions.

These vibrations may come from environmental sources such as shock, acoustic noise, or internal factors like cooling fans and nearby mechanical components. When these forces match a PCB's natural frequency resonance can occur, amplifying motion and increasing the risk of failure.

Uncontrolled vibration can negatively impact electronic systems in several ways. Repeated stress cycles cause solder joint fatigue, while sensitive or heavy components face an increased risk of physical damage. Connectors can loosen or develop degraded contacts over time, and slight physical deformation of traces can led to signal integrity issues that are difficult to diagnose and costly to fix.

Methods for Shifting Natural Frequencies

In general, the following are rules that allow natural frequency shifting and minimizing the vibrational response of a system:

• To increase the natural frequency, add stiffness.

• To decrease the natural frequency, add mass.

• An increase in the damping diminishes the peak response, however, it broadens the response range.

• A decrease in the damping raises the peak response, however, it narrows the response range.

• Lessening forcing amplitudes mitigates response at the resonance frequency.

The natural frequency value is one of the single most critical parameters or properties of any system. The need for designers to be cognitive of its exact frequency point is crucial to a system's functionality, performance, and lifecycle.

To control natural frequency in PCB systems, engineers can:

• Increase stiffness with thicker materials or added supports.

• Optimize component placement to distribute mass evenly

• Introduce damping materials to reduce vibration amplitude

• Use simulation tools to predict and mitigate resonance early

By considering both mechanical and electrical behavior, designers can significantly improve system reliability.

The natural frequency formula affords the ability to calculate the natural frequency of this simple harmonic oscillator.

Designing for natural frequency early in the process is critical to avoiding resonance-related failures and ensuring long-term system reliability. For more complex designs, a full-featured PCB Design and Analysis Platform provides deeper insight into both mechanical and electrical behavior with a full suite of simulation tools.  Allegro X PCB Layout integrates with robust analysis tools to give you a complete design and simulation platform, allowing you to evaluate the electrical behavior of linear and nonlinear circuits as you prepare to create your PCB layout.