# Parametric Stability Analysis of a Parabolic-Tapered Rotating Beam Under Variable Temperature Grade

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## Abstract

### Purpose

The static and dynamic stability of a parabolic-tapered beam of circular cross-section subjected to an axial alive load and rotating in the X-Y plane about the Z-axis is analyzed cosidering a variable temperature grade along the centroidal axis of the beam as the beam is in steady state condition.

### Methods

The stability is analyzed for clamped-clamped and pinned-clamped boundary conditions. The parametric instability regions are acquired by means of Saito-Otomi conditions. The consequences of variation parameter, revolution speed, temperature grade and boundary conditions on the instability regions are examined for dynamic load and static buckling loads for 1st, 2nd and 3rd modes and are represented by a number of graphs.

### Results

The results divulge that the stability is increased by increasing revolution speed; however, increase in thermal grade and the variation parameter leads to destabilize the converging system for all boundary conditions.

### Conclusions

This research can be useful for vibration isolation of rotating non-uniform beams with high surrounding temperature and moderate rotational speeds and the design of rotor blades with high strength to weight ratio by choosing the suitable parameters obtained from this computational analysis.

## Keywords

Static and dynamic stability Rotating beam Parabolic taper Constant and variable temperature grades## List of Symbols

- \( A\left( x \right) \)
Cross-sectional area of any standard segment

- \( A (\eta ) \)
Non-dimensional cross-sectional area of any standard segment

*A*Cross-sectional area of the base

- \( \left( {a,0} \right) \)
Location of the focus of the parabola

*B*_{0}Radius of the hub

- \( b_{0} \)
Dimensionless hub radius (\( {{B_{0} } \mathord{\left/ {\vphantom {{B_{0} } l}} \right. \kern-0pt} l} \))

- \( d\left( x \right) \)
Diameter of any standard segment

- \( d (\eta ) \)
Non-dimensional diameter of any standard segment

- \( d_{l} \)
Diameter of the far end section

- \( E\left( x \right) \)
Young’s modulus at any standard segment

- \( E \)
Young’s modulus at the base

*f*Frequency of external excitation

- \( \overline{f} \)
Non-dimensional excitation frequency

- \( I\left( x \right) \)
Moment of inertia of any standard segment

- \( I (\eta ) \)
Non-dimensional moment of inertia

*I*Area moment of inertia of the base

*l*Span of the beam

- \( m (\eta ) \)
Mass variation function

*N*Angular speed about

*Z*-axis*N*_{0}Revolution speed parameter

- \( S (\eta ) \)
Elastic modulus variation function

- \( T (\eta ) \)
Moment of inertia variation function

*t*Time

- \( W_{0} \)
Static load along axial direction

- \( W_{1} \)
Alive load along axial direction

- \( w\left( \tau \right) \)
Non-dimensional external load

- \( w_{0} \)
Non-dimensional static load along axial direction

- \( w_{1} \)
Dimension less axial alive load

- \( \Delta \left( {x,t} \right) \)
Deflection along the transverse direction

- \( \psi \)
Parameter for temperature grade

- \( \psi (\eta ) \)
Temperature grade or thermal gradient variation function

- \( \alpha \)
Coefficient of thermal expansion of the beam material

- \( \beta^{*} \)
Variation or taper parameter (\( = {a \mathord{\left/ {\vphantom {a D}} \right. \kern-0pt} D} \))

- \( \gamma \)
\( = {l \mathord{\left/ {\vphantom {l D}} \right. \kern-0pt} D} \)

- \( \xi \)
Dimensionless transverse displacement

- \( \eta \)
Non-dimensional length \( ({x \mathord{\left/ {\vphantom {x l}} \right. \kern-0pt} l} ) \)

- \( \tau \)
Dimensionless time

- \( \rho \)
Mass density of the beam material

- \( \delta_{0} \)
Reference temperature

- \( \delta \)
Temperature at the base

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