9.3 Torsional vibration of shafts

Torsional vibration of shafts is a crucial concept in mechanical systems. It involves the twisting motion of shafts around their axis, which can lead to stress, fatigue, and potential failure if not properly managed.

Understanding torsional vibration is essential for designing and analyzing rotating machinery. This topic covers equations of motion, natural frequencies, mode shapes, and response analysis techniques, providing a foundation for addressing vibration issues in real-world applications.

Equations of motion for torsional vibration

Fundamentals of torsional vibration

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• Torsional vibration involves twisting of a shaft or rod about its longitudinal axis resulting in angular displacement, velocity, and acceleration
• Newton's Second Law for rotational systems forms the basis for deriving the equation of motion, considering moment of inertia and torsional stiffness
• General form of torsional vibration equation expressed as $J\frac{d^2\theta}{dt^2} + k\theta = T(t)$
  • J represents moment of inertia
  • k denotes torsional stiffness
  • θ signifies angular displacement
  • T(t) indicates applied torque
• Torsional stiffness (k) calculated using shaft's material properties and geometry $k = \frac{GJ}{L}$
  • G represents shear modulus
  • J denotes polar moment of inertia
  • L signifies shaft length
• Multi-degree-of-freedom systems require matrix methods for equation formulation, accounting for coupling between shaft sections and attached masses

Incorporating damping and advanced considerations

• Damping effects added to equation of motion with term proportional to angular velocity $J\frac{d^2\theta}{dt^2} + c\frac{d\theta}{dt} + k\theta = T(t)$
  • c represents damping coefficient
• Rotary inertia effect becomes significant for shorter, thicker shafts (crankshafts, turbine rotors)
• Non-linear effects may arise in systems with large angular displacements or material non-linearities (composite shafts)
• Gyroscopic effects considered in high-speed rotating shafts (turbomachinery, propeller shafts)

Natural frequencies and mode shapes

Calculating natural frequencies

• Natural frequencies determined by solving characteristic equation derived from homogeneous form of equation of motion
• Single-degree-of-freedom system natural frequency calculated as $\omega_n = \sqrt{\frac{k}{J}}$
• Multi-degree-of-freedom systems require solving eigenvalue problem $[K - \omega^2M]\Phi = 0$
  • K represents stiffness matrix
  • M denotes mass matrix
  • Φ signifies eigenvectors (mode shapes)
• Rayleigh's quotient estimates natural frequencies using assumed mode shapes, providing upper bound for fundamental frequency
• Effect of rotary inertia on natural frequencies significant for shorter, thicker shafts (propeller shafts, turbine blades)

Understanding mode shapes

• Mode shapes describe relative angular displacements of different shaft parts at each natural frequency, representing vibration patterns
• Orthogonality property of mode shapes utilized to decouple equations of motion in modal analysis techniques
• Nodal points in mode shapes indicate locations of zero displacement (fixed ends, points of symmetry)
• Higher mode shapes exhibit more complex patterns with multiple nodes along shaft length (guitar strings, transmission shafts)

Torsional vibration response analysis

Boundary conditions and their effects

• Common boundary conditions include fixed-free, fixed-fixed, and free-free configurations, each affecting natural frequencies and mode shapes
• Fixed-free boundary condition represents shaft with one end clamped and other end free to rotate (drill strings, car driveshafts)
• Fixed-fixed boundary condition constrains both shaft ends against rotation, resulting in higher natural frequencies (short connecting shafts)
• Free-free boundary condition allows rotation at both ends, often used in experimental modal analysis (suspended shafts for testing)
• Transfer matrix methods employed to analyze complex shaft systems with multiple sections and varying boundary conditions

Response analysis techniques

• Frequency response function (FRF) characterizes torsional vibration response under harmonic excitation for different boundary conditions
• Torsional critical speeds crucial in rotating machinery, where excitation frequency matches natural frequency, potentially leading to resonance (turbine shafts, propeller shafts)
• Time domain analysis techniques used for transient response evaluation (sudden torque applications, impact loads)
• Forced vibration analysis considers external torques and their frequency content (engine crankshafts, wind turbine drivetrains)

Shaft geometry and material properties influence

Geometric factors affecting torsional vibration

• Shaft length directly affects torsional stiffness, with longer shafts having lower stiffness and lower natural frequencies
• Cross-sectional geometry, particularly polar moment of inertia, significantly influences vibration characteristics (solid vs. hollow shafts)
• Non-uniform shafts (stepped shafts, varying cross-sections) require complex analysis methods (finite element analysis, transfer matrix techniques)
• Stress concentrations (keyways, fillets) locally modify torsional stiffness and affect overall vibration characteristics

Material properties and environmental effects

• Shear modulus (G) plays crucial role in determining torsional stiffness and natural frequencies
• Density of shaft material affects mass distribution and moment of inertia, influencing dynamic response
• Temperature effects on material properties, particularly shear modulus, lead to changes in torsional vibration behavior (high-temperature applications, transient thermal conditions)
• Material damping properties influence vibration decay and system response (composite shafts, polymer-based materials)