Aircraft Systems: Harmonically Excited Vibration

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Questions and Answers

What is whirling in a rotating shaft?

Whirling is the rotation of the plane made by the bent shaft and the line of centers of the bearings.

What is critical speed in a rotating shaft?

Critical speed is the speed at which the shaft reaches its natural frequency, causing resonance and leading to violent vibration and potential failure.

What is the formula to calculate stiffness (k) in a shaft?

k = 48 EI/Ɩ3, where E is the modulus of elasticity and I is the moment of inertia.

What is the difference between whirling speed and rotational speed?

The whirling speed may not be equal to the rotational speed of the shaft. Signup and view all the answers

What is vibration in mechanical systems?

Vibration is a mechanical phenomenon in which oscillations occur about an equilibrium point. Signup and view all the answers

What are modes of vibration in vibrational analysis?

Modes of vibration are the different types in which the system tries to oscillate naturally without any excitation force. Signup and view all the answers

What is the term used to describe the frequency of oscillation in a system?

Modal frequency or natural frequency Signup and view all the answers

What is the formula for calculating the deflection of a shaft due to its self-weight?

Deflection (𝝙) = 5WƖ4/384 EI Signup and view all the answers

What is the moment of inertia (I) of the shaft calculated as?

I = (𝛑d4)/64 Signup and view all the answers

What is the rate of loading (W) of the shaft calculated as?

W = wself/length Signup and view all the answers

What is the purpose of using ANSYS workbench in this context?

To find the critical frequency for different modes of vibrations under self-weight of shaft and nature of deformation Signup and view all the answers

What is the acceleration of the mass center 'S' in terms of the acceleration of 'G' with respect to 'S'?

as = aG/S Signup and view all the answers

In the equation of motion, what type of damping force is assumed to be acting at 'S'?

Viscous damping force Signup and view all the answers

What is the general classification of whirl in a rotating shaft?

Self-excited motion Signup and view all the answers

How many degrees of freedom (DOF) does the whirl motion have?

2 Signup and view all the answers

What is the relationship between the center of mass 'G' and the geometric center 'S' of the disk?

The center of mass 'G' is eccentric to the geometric center 'S' of the disk Signup and view all the answers

What is the speed at which the disk is rotating?

𝞈 Signup and view all the answers

What is the condition of resonance encountered in a rotating shaft system when the rotation speed coincides with the critical speed?

A condition of resonance is encountered in which the amplitude is restrained only by damping. Signup and view all the answers

What is the relationship between the whirling speed and the rotation speed in a synchronous whirl?

The whirling speed 𝝷 is equal to the rotation speed 𝞈. Signup and view all the answers

What is the direction of the eccentricity line e = SG in relation to the displacement line r = OS?

The eccentricity line e = SG leads the displacement line r = OS by the phase angle 𝝓. Signup and view all the answers

What happens to the center of mass G and the shaft center S at very high speeds 𝞈>>𝞈n?

The center of mass G tends to approach the fixed point O, and the shaft center S rotates about it in a circle of radius e. Signup and view all the answers

What is the expression for the unbalance in terms of the total mass m with eccentricity e?

The unbalance is defined in terms of the total mass m with eccentricity e. Signup and view all the answers

Why must turbines operating above the critical speed be careful during startup and shutdown?

They must run through dangerous speed at resonance each time they are started or stopped. Signup and view all the answers

What is the formula to calculate the deflection (𝝙) of a simply supported shaft?

𝝙 = 5WƖ4/384 EI Signup and view all the answers

What is the critical frequency (fn) of a shaft in terms of the acceleration due to gravity (g) and deflection (𝝙)?

fn = (𝛑/2)((5)(g)/(384 𝝙))1/2 Signup and view all the answers

What is the critical speed (Nn) of a shaft in terms of its critical frequency (fn)?

Nn = 60 fn Signup and view all the answers

What is the formula to calculate the critical frequency (f2) of the second mode of vibration?

f2 = 22fn Signup and view all the answers

What is the purpose of modal analysis in the context of vibration analysis?

Modal analysis is used to determine the natural frequencies, mode shapes, and mode vectors of a structure under vibration excitation. Signup and view all the answers

What is the relationship between the critical speeds (N1, N2, N3) and the corresponding frequencies (f1, f2, f3) of a shaft?

N1 = f1, N2 = f2, N3 = f3 Signup and view all the answers

Study Notes

Whirling Rotating Shaft

Whirling is the rotation of the plane made by the bent shaft and the line of centers of the bearings, resulting from various causes such as mass unbalance, hysteresis damping, gyroscopic forces, and fluid friction in bearings.

Vibration

Vibration is a mechanical phenomenon in which oscillations occur about an equilibrium point, often due to self-weight or externally applied loads.
Resonance occurs when the speed of the rotating shaft reaches its natural frequency, causing violent transverse vibration and potentially leading to shaft failure.

Critical Speed

Critical speed is the rotational speed at which the shaft's natural frequency is reached, causing resonance and potentially leading to failure.
Operating speed should always be lower than the critical speed to avoid resonance.

Shaft Characteristics

Material: structural steel
Diameter: 50 mm
Length: 1000 mm
End conditions: simply supported
Deflection (𝝙) = 5WƖ4/384 EI
Critical frequency (fn) = (𝛑/2)(5g/384 𝝙)1/2
Critical speed (Nn) = 60 fn

Modes of Vibration

Mode 1: f1 = fn = 98.78 Hz
Mode 2: f2 = 22fn = 395.12 Hz
Mode 3: f3 = 32fn = 889.02 Hz
Corresponding critical speeds: N1 = 5926.8, N2 = 23707.2, N3 = 53341.2

Modal analysis is used to study the dynamic characteristics of a structure under vibration excitation.
Natural frequencies, mode shapes, and mode vectors can be determined using the modal analysis workbench.

Synchronous Whirl

Synchronous whirl occurs when the whirling speed (𝝷) is equal to the rotation speed (𝞈).
The equation for the phase angle can be derived from the vector triangle.

Phases of Different Rotation Speed

At very high speeds (𝞈>>𝞈n), the center of mass G tends to approach the fixed point O, and the shaft center S rotates about it in a circle of radius e.

Turbines

Turbines operating above the critical speed must run through dangerous speeds at resonance each time they are started or stopped.

Deflection of Shaft

Deflection (𝝙) = 5WƖ4/384 EI
Equation can be used to calculate deflection, given the shaft's material properties and dimensions.

Theoretical calculations can be compared with computational results using ANSYS Modal Analysis for self-weight deflection.

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Description

Learn about harmonically excited vibration in aircraft systems, specifically whirling rotating shafts. Understand the causes of whirling, including mass unbalance, hysteresis damping, and gyroscopic forces. Test your knowledge on this important topic in aircraft dynamics.