Structural Dynamics

Module aims

This course aims to present available methodologies for solving dynamic problems on aerospace structures, to introduce the underlying theory and basic considerations for the numerical modelling of response of structures to time-dependent loads, including those coming from the aerodynamics in aerial vehicles, and to present computer-based techniques to solve for the linear and non-linear dynamic response of structures

Learning outcomes

On successfully completing this course unit, students will be able to:

Knowledge and understanding

  • understand the underlying concepts of dynamic discretisation and generalised co-ordinates.
  • understand the modal solution of undamped free/forced vibrations and the general dynamic solution for linear undamped forced vibrations.
  • understand the solution procedures for problems in aeroelasticity.

Intellectual skills

  • understand the eigenvalue problem as minimisation of the Rayleigh quotient.
  • identify the mass matrix, the stiffness matrix, the damping matrix and the gyroscopic matrix on the linear form of the structural dynamic equations of motion
  • understand mass and stiffness modelling considerations, and finite element meshing considerations
  • understand explicit and implicit solution methods and their stability.
  • understand the static and dynamic interactions occurring between an elastic wing and the surrounding fluid

Practical skills

  • derive Hamilton’s principle as the fundamental equations in dynamics and to illustrate its application.
  • derive Lagrange's equations from Hamilton’s principle and to illustrate their use. To extend these to include damping.
  • derive the free vibrations of arbitrarily damped and gyroscopic equations of motion, the damped eigenvalue problem and structure of the damped eigenvectors.
  • derive the general forced vibration solution for linear arbitrarily damped systems.
  • derive simplified damping models, model damping, proportional damping, hysteretic damping.
  • derive direct frequency domain solutions.

Transferable skills

  • Computer skills.

Module syllabus

 Degrees of Freedom. Generalised Co-ordinate Systems.
• Hamilton’s principle. Derivation of Hamilton’s Principle. Application to Deriving Differential Equations of Motion and Approximate Solution Methods for both Linear and Non-Linear Systems.
• Lagrange’s equations. Derivation of Lagrange’s Equations and Example of their use. The Matrix Form of the Equations of Motion and Characteristics of the Matrices involved in the Equations.
• Modal solutions. Eigenvalues and Eigenvectors for Free Vibrations. Inclusion of Rigid Body Modes for Free Vibrations. Transient Forced Response of Undamped Systems. Frequency Domain Solutions.
• Modelling. Modelling Considerations for Mass and Stiffness using the Rayleigh Quotient. Number of Modes for a Dynamic Analysis. Simplified damped models, modal and proportional damping. Damping mechanisms, structural (hysteretic) damping. Solution methods for damped systems.
• Time integration. Explicit and Implicit Time Domain Solutions. Assumption of Time Varying of Accelerations. Newmark and central-difference methods. Numerical Stability.
• Static Aeroelasticity. Collar’s triangle. Basic concepts of stability. Typical section problem. Divergence. Control Reversal. 
• Dynamic Aeroelasticity.  Flutter. Unsteady thin aerofoil theory. Circulation Effects. Theodorsen function. General solution methods in linear aeroelasticity


Teaching methods

Lectures , tutorials. Class notes are provided.


Examined Assessment
2-hour written examination in January (100%).
Non-Examined Assessment
Progress Test (peer marked)


Reading list


Module leaders

Professor Silvestre Pinho