Short Answer:

Mass has an inverse effect on the natural frequency of a vibrating system. When the mass of a system increases, its natural frequency decreases because a heavier body takes more time to complete one vibration. On the other hand, when the mass decreases, the natural frequency increases, and the system vibrates faster.

In simple words, mass represents the inertia of a body, which resists motion. A larger mass causes slower vibrations due to higher inertia, while a smaller mass vibrates more quickly. Thus, natural frequency is inversely proportional to the square root of the system’s mass.

Detailed Explanation :

Effect of Mass on Natural Frequency

Mass is one of the key parameters that determine how a mechanical system vibrates. It represents the amount of matter and directly relates to the inertia of the body. In vibration systems, inertia opposes the motion of the system when displaced from its equilibrium position.

Natural frequency is the frequency at which a system tends to vibrate freely when disturbed from its mean position. It depends mainly on two factors — stiffness (k) and mass (m) — of the system. When the stiffness is constant, increasing the mass causes a reduction in the system’s natural frequency, and decreasing the mass raises it.

This relationship is essential in mechanical and structural design, as incorrect mass distribution can lead to unwanted vibration, resonance, or instability in machines and structures.

Mathematical Relation Between Mass and Natural Frequency

Let us consider a simple spring–mass system, where a mass  is attached to a spring of stiffness . When the mass is displaced slightly and released, it vibrates freely due to the spring’s restoring force.

According to Hooke’s Law,

where,

•  = restoring force (N),
•  = stiffness of the spring (N/m),
•  = displacement from mean position (m).

According to Newton’s second law,

Equating both forces,

This is the differential equation of free vibration.

The general solution of this equation is:

where,

•  = amplitude of vibration,
•  = natural angular frequency (rad/s),
•  = phase angle.

Now, the natural angular frequency is given by:

The natural frequency (f_n) in hertz is:

This equation clearly shows that natural frequency is inversely proportional to the square root of mass (m).

Interpretation of the Relationship

1. When the mass increases, the term  decreases, which reduces the natural frequency.
2. When the mass decreases, the term  increases, which raises the natural frequency.

Hence, heavier systems vibrate more slowly (low frequency), and lighter systems vibrate faster (high frequency).

This happens because increasing the mass increases the system’s inertia — the resistance to motion — so it requires more time to complete each cycle of vibration.

Physical Explanation

To understand the effect physically:

• In a vibrating system, two forces interact — the inertial force due to mass and the restoring force due to stiffness.
• The inertia tries to resist motion, while the restoring force tries to bring the system back to equilibrium.
• A higher mass means a stronger inertial effect, which slows down the response of the system.
• Therefore, it takes longer for the system to return to its mean position, resulting in a lower frequency of vibration.
• Conversely, a smaller mass reduces inertia, allowing the system to oscillate faster and increasing the natural frequency.

This inverse relationship between mass and natural frequency applies to all vibrating systems — mechanical, structural, and even electrical analogies.

Example Calculations

Example 1:
Let the stiffness  and the mass .

Now, if the mass increases to :

This shows that increasing the mass from 10 kg to 40 kg reduces the natural frequency from 3.18 Hz to 1.59 Hz (half the value).

Examples in Real-Life Mechanical Systems

1. Vehicle Suspension Systems:
   When more passengers or cargo are added (increasing total mass), the suspension’s natural frequency decreases, making the ride feel softer. When the vehicle is unloaded (less mass), the suspension vibrates faster, resulting in a stiffer ride.
2. Machine Foundations:
   Heavy machinery has low natural frequencies because of their large mass. Engineers adjust the foundation’s mass to tune the system’s frequency away from excitation frequencies.
3. Bridges and Towers:
   Structures with greater mass vibrate at lower frequencies. Designers consider mass distribution to avoid resonance with wind or earthquake vibrations.
4. Rotating Equipment:
   In turbines and engines, changing the mass of rotating parts changes their natural frequencies, which affects stability and performance.
5. Musical Instruments:
   In string instruments, heavier strings produce lower natural frequencies (lower pitch), while lighter strings produce higher frequencies (higher pitch).

Influence of Mass in Multi-Degree-of-Freedom Systems

In complex systems like beams, turbines, and engines, multiple masses and stiffnesses exist. Each mode of vibration has its own natural frequency.

• Increasing overall mass generally decreases all the natural frequencies.
• Uneven mass distribution can cause different parts of the system to vibrate at different frequencies, leading to imbalance or resonance problems.

Therefore, careful control of mass and stiffness distribution is essential in multi-degree-of-freedom vibration analysis.

Practical Importance of Mass in Design

1. Avoiding Resonance:
   By controlling mass, engineers ensure that the system’s natural frequency does not match the external excitation frequency.
2. Vibration Reduction:
   Increasing mass is one of the simplest methods to lower the natural frequency and minimize vibration in structures.
3. Dynamic Balancing:
   Proper mass distribution helps in maintaining stability and reducing vibration in rotating systems.
4. Testing and Analysis:
   Measuring natural frequency helps determine whether changes in mass affect system behavior or integrity.
5. Safety:
   Understanding mass influence prevents excessive vibrations that may lead to fatigue failure or discomfort.

Conclusion

Mass has an inverse relationship with natural frequency. As the mass of a vibrating system increases, the natural frequency decreases because the system becomes heavier and slower in response. Conversely, reducing mass increases the natural frequency, making the system vibrate faster. This relationship, expressed as , is vital in mechanical design to ensure safety, stability, and comfort. By properly balancing mass and stiffness, engineers can control vibrations and prevent resonance in machines and structures.