Natural Frequency Analysis of a Shaft Rotor System

Abstract:

This comprehensive lab report delves into the intricate process of analyzing the natural frequencies of a simple shaft rotor system. Utilizing both the Holzer Method and Finite Element Analysis (FEA), this study aims to determine the system's natural frequencies and compare them to results obtained from ANSYS simulations. The report provides in-depth insights into the methodology, results, and discussions surrounding this important engineering analysis.

1. Introduction

The mechanical behavior of rotating shaft rotor systems is of utmost importance in various engineering applications.

Knowledge of the natural frequencies of these systems is crucial to avoid resonance and ensure their safe and efficient operation. In this report, we explore the methods used to determine the natural frequencies of a simple shaft rotor system comprising a shaft and two rotors.

The primary objectives of this study are to:

• Employ the Holzer Method to find natural frequencies.
• Compare the results obtained through the Holzer Method with those obtained from ANSYS simulations.
• Apply Finite Element Analysis (FEA) to calculate mass and stiffness matrices and determine natural frequencies.
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Understanding the natural frequencies and mode shapes of a mechanical system like this is crucial for various applications, including machinery design, vibration analysis, and structural engineering.

2. Methodology

2.1 Holzer Method

The Holzer Method is an iterative trial-and-error approach employed to identify the natural frequencies of mechanical systems. It seeks frequencies that result in zero external torque or compatible boundary conditions, which correspond to the natural frequencies of the system.

This versatile method can be applied to a range of systems, including spring-mass systems, beams, and rotor systems.

In our case, we focus on a shaft rotor system with three angular displacements: θ₁, θ₂, and θ₃, and the system's frequency represented by ω.

The formula for inertia torque of the first disc is given as:

^J₁θ₁¨¨ = ^J₁ω²θ₁

During harmonic motion, the torque acting on shaft 1 results in a twist defined by:

^J₁ω²/K₁ = θ₁ - θ₂

With the value of θ₂ known, we can calculate the inertia torque of the second disc as ^J₂ω²θ₂. The sum of the first two inertia torques acts on shaft 2, causing it to twist by:

(J₁ω²θ₁ + ^J₂ω²θ₂)/K₁ = θ₂ - θ₃

This iterative process allows us to calculate the amplitudes and torques of each disc. By repeating these calculations for various ω values, we can pinpoint the natural frequencies where the sum of torques is zero. The corresponding angular displacements, θ_i, represent the mode shapes of the system.

2.2 Finite Element Analysis (FEA)

Finite Element Analysis (FEA) is a powerful numerical technique used to calculate mass and stiffness matrices for complex mechanical systems. In our case, FEA is employed to determine these matrices for the shaft rotor system under investigation.

The process involves:

1. Calculating gear dimensions based on known data, such as the number of teeth, torques, and RPM of the gears.
2. Using suitable gear module assumptions to derive the necessary parameters, including pitch circle diameter, pitch circle radius, and module.
3. Considering material properties, such as steel, to calculate moment of inertia for the gears.
4. Determining shaft dimensions, including length and diameter, with a Factor of Safety (FOS) of 2.5.

Once the gear and shaft dimensions are known, the mass and stiffness matrices can be established. These matrices are essential for calculating the natural frequencies of the system using mathematical equations and numerical methods.

3. Results and Discussion

3.1 Holzer Method Results

The natural frequencies obtained using the Holzer Method are presented in Table 1:

3.2 ANSYS Simulation Results

The ANSYS simulations provided the following natural frequencies:

3.3 Finite Element Analysis (FEA) Results

The application of Finite Element Analysis (FEA) yielded the following natural frequencies in rad/s, along with their corresponding frequencies in Hz:

The results demonstrate the agreement between the Holzer Method, ANSYS simulations, and FEA, providing a comprehensive understanding of the natural frequencies of the shaft rotor system. The discrepancies observed can be attributed to simplifications and assumptions made in the analyses.

4. Conclusion

This study has provided a detailed exploration of the methods employed in the analysis of natural frequencies for a simple shaft rotor system. The Holzer Method, ANSYS simulations, and Finite Element Analysis (FEA) were compared, highlighting the strengths and weaknesses of each approach.

ANSYS simulations demonstrated superior accuracy, particularly at higher critical frequencies, due to its advanced modeling capabilities. However, it is essential to recognize the iterative nature and limitations of analytical methods like the Holzer Method when applied to complex systems.

FEA serves as a valuable alternative for systems with multiple degrees of freedom (MDOF), offering a more comprehensive understanding of the mechanical behavior. In applications like the one analyzed here, FEA proves to be an indispensable tool for achieving accurate results.