Understanding Stiffness

1. Definition: What is Stiffness?

Stiffness is a fundamental physical property that describes the extent to which an object or structure resists deformation or deflection in response to an applied force. In the context of vibration analysis, stiffness (often denoted by the letter ‘k’) is one of the three key properties, along with mass (m) and damping (c), that govern the vibrational behavior of any mechanical system.

A component with high stiffness will deflect very little under a given load, while a component with low stiffness will deflect significantly. For example, a thick, short steel bar has high stiffness, while a long, thin rubber band has very low stiffness.

2. The Critical Role of Stiffness in Vibration

The stiffness of a system is a primary factor in determining its natural frequencies. The natural frequency is the frequency at which a system will oscillate if it is disturbed and then allowed to vibrate freely. The relationship is defined by the basic formula:

Natural Frequency (ωn) ≈ √(k / m)

Where ‘k’ is the stiffness and ‘m’ is the mass. This relationship shows that:

• Increasing stiffness will increase the natural frequency.
• Decreasing stiffness will decrease the natural frequency.
• Increasing mass will decrease the natural frequency.

3. Stiffness and Resonance

This relationship is critically important because of the phenomenon of resonance. Resonance occurs when a forcing frequency (like the running speed of a machine) matches one of the system’s natural frequencies. When this happens, the vibration amplitude is dramatically amplified, often leading to premature wear and catastrophic failure.

Understanding stiffness is therefore essential for diagnosing and solving resonance problems:

• Problem Diagnosis: If a machine is in resonance, the analyst knows that the forcing frequency is too close to a natural frequency.
• Solution Design: To fix the problem, the analyst must change the system’s natural frequency. Since it is often difficult to change the mass of a machine or the forcing frequency (its running speed), the most common solution is to change the stiffness. By adding braces, gussets, or improving the machine’s foundation, the stiffness of the system is increased. This raises the natural frequency, moving it away from the forcing frequency and eliminating the resonance condition. A Frequency Response Function (FRF) measurement is used to confirm the change in natural frequency.

4. Stiffness in Machinery Diagnostics

Changes in stiffness can also be a direct indicator of a developing fault:

• Looseness: A loose mounting bolt or a crack developing in a machine’s frame or foundation represents a significant loss of local stiffness. This will cause the machine’s vibration amplitude to increase. In the FFT spectrum, mechanical looseness often generates a series of harmonics (1X, 2X, 3X, etc.) of the running speed.
• Soft Foot: This condition, where a machine foot does not sit flat on its base, creates a distorted and non-linear stiffness profile, which can lead to high vibration and make alignment difficult.
• Bearing Wear: As a bearing wears, the clearance between the rolling elements and the races increases. This can be viewed as a decrease in the overall stiffness of the rotor support system, which can lower the rotor’s critical speeds.

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