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University of Tripoli

Faculty of Engineering

Department of Mechanical

and Industrial Engineering

Applied Mechanics Division

project supervisor:

Dr. Mohammed Ali Hjaji

preperation:

Muna jamal ben saed

CONTANTS

TORSIONAL VIBRATION OF SHAFTS UNDER HARMONIC LOADINGS

INTRODUCTION

Introduction

  • A shaft is defined as a rotating or non-rotating mechanical part

used for the transmission of power. it can carry gears, pulley

and sprockets to transmit motion.

  • shafts in such applications are subjected to various dynamic

torsional loadings and eventually torsional vibrations.

  • The most common type of vibration is the torsional vibration

due to resonance phenomenon.

  • Accurate prediction of natural frequencies is critical for a

successful design of rotary and non-rotary systems.

  • Under harmonic twisting moments, the steady state dynamic

component of the total response is sustained for a long time

and is thus of great importance in fatigue design.

  • In contrast, the transient component of total response tends

to dampen out quickly and is thus of no importance in

assessing the fatigue life of the shaft.

Applications of Rotating and Non-rotating Shafts

APPLICATIONS

1. Shaft mining, as shown in Figure which is a form of underground mining where shafts are pushed vertically from top to bottom to excavate the ores and minerals.

Shaft mining

agriculture

2. Shafts are used in agriculture such as PTO drive shafts "Power Take-Off shaft" which transfers the power from the tractor to the required equipment.

3. Shafts are used as axle in automotive transmission. they rotates the wheels and supports the weight of your vehicle. Axles are essential components of any vehicle

transmission

4. transmission shafts can be found in transmission gearbox. All gearboxes have an output shaft-side that carries the energy from the an input shaft-side. The input shafts are attached directly to the motor and controls the motor speed as shown in Figure.

gearbox

5. in turbomachines, the turbine and compressor components are mated by a shaft. Figure illustrates a single-shaft gas turbine with one shaft connecting the compressor and turbine components

turbomachines

6. Shafts used as rotating shaft element can be found in helicopters as illustrated in Figure.

helicopters

7. Shafts in a motor shown in Figure are the cylindrical component that exits the motor and its housing.

motors

8. Shafts used in the hydraulic transmission motion as in Figure.

hydraulic transmission

Research Objective

2. the exact closed-form solutions for dynamic response of torsional shaft with different boundary conditions.

3. the quasi-static response of shaft under various torsional harmonic forces.

OBJECTIVES

1. the governing torsional equation of motion of shafts and the corresponding boundary conditions using both, Newton’s approach and Hamilton’s variational principle.

Under torsional harmonic forces, The aim of the present study is to conduct the following:

The present theoretical formulation is based on the following main assumptions:

4. Displacements, strains and rotations are

assumed small,

1. The present formulation is valid to shafts having circular solid/hollow cross-sections,

ASSUMPTIONS

5. The Damping effect is not included in

the present solution,

2. The shaft cross-section is assumed to

remain perfectly rigid in its own plane

throughout deformation,

3. The shaft material is assumed to

remain linearly elastic during deformation,

6. The present solution captureds only

the steady state dynamic response.

Kinematics functions:

KINIMATICS FUNCTIONS

HAMILTON'S PRINCIPLE

General Closed form Solution:

SOLUTION

Cantilever shaft under Harmonic Torsions:

Cantilever shaft

The general exact solution was also conducted for a simply-supported, and free-free shaft:

simply-supported,

free-free shaft

NUMERICAL EXAMPLES

NUMERICAL EXAMPLES

Example (1): Cantilever Solid Steel shaft under end harmonic torsion

EXAMPLE 1

Static analysis for torsional displacement of cantilever shaft :

Plot 1

Steady state dynamic analysis for torsional displacement of cantilever shaft under end harmonic torsional loading:

Plot 2

Example (2) - Simply-supported shaft under distributed harmonic Torsional loading

EXAMPLE 2

Static response of simply supported shaft under distributed harmonic twisting moment

Plot 1

Steady state torsional response modes for simply supported shaft under distributed harmonic twisting moment

Plot 2

Example (3) - Cantilever Hollow Shaft under distributed harmonic twisting moment

EXAMPLE 3

Torsional natural frequency of cantilever hollow shaft

Plot 1

first four steady state mode shapes for the torsional vibration response of the cantilever hollow shaft under distributed harmonic torsional moment

Plot 2

Summary, Conclusion and recomendations

conclusion

&

discussion

Summary and Conclusions

1. The exact closed-form formulated in

this study is competent to efficiently capture the static and steady state dynamic responses of the shafts.

2. Moreover, it is skillfully able to extort both natural frequencies and related mode shapes of the shaft.

2. The solution developed in the present

study consistently predicts the torsional

response of shafts when compared to

exact solutions .

Summary and Conclusions

include the effect of adding static compressive forces beside the harmonic twisting moments.

1

Proposed Future Developments

capture the static and dynamic analyses of shafts made of composite materials

formulate a family of exact shape functions which based on the exact homogeneous solution of the governing torsional equation to formulate a finite beam element for the shafts.

5

2

The methodology developed can be extended to

4

3

capture the static and steady state responses of a rotating shafts.

to capture the dynamic response of shafts under effect of combined flexural-torsional harmonic forces.

Proposed Future Developments

thank you all

thanks

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