Energy and Variational Methods with Applications


Prerequisite/Recommended prerequisite for participation in the module

The module adds to the knowledge obtained in 1st Semester.

Content, progress and pedagogy of the module

Learning objectives


  • Have gained an in-depth understanding of energy and variational methods and classical principles of stationarity to derive governing equations of statics and dynamics, and their application for solution of general problems in structural mechanics, including:
    • Energy methods and energy minimization principles as the foundation of the finite element method
    • Variational methods (methods of Ritz, Galerkin, Rayleigh and Rayleigh-Ritz)
    • Different applications of those such as analysis of statics, buckling and vibration of beams, plates and similar
    • Dynamics of discrete multi-dof and multi-body mechanical systems
    • Exact and approximate solutions to the natural frequencies and modal analysis problems for multi-dof/continuous vibration systems
  • Approximate methods for nonlinear mechanical vibrations


  • Be able to apply energy and variational methods for the solution of problems in statics and dynamics involving discrete and continuous, multi-rigid-body and multi-dof vibrational mechanical systems
  • Be able to adequately simulate and analyze dynamics of linear and non-linear mechanical systems


  • Be able to apply energy minimization / maximization principles to derive the relationships between stresses, strains, displacements, material properties, and external effects (e.g., tractions and volume forces) in the form of balance of the kinetic and potential energies and the work done by internal and external forces.
  • Be able to use the variational calculus as a convenient and robust tool for formulating the governing equations of statics and dynamics of rigid and solid bodies in applied mechanics.
  • Be able to apply the energy and variational methods to find approximate analytical and numerical (e.g. finite element) solutions of complex problems in statics, stability and dynamics of mechanical systems
  • Be able to formulate equations of motion for multi-body mechanical systems and for discrete multi-dof and continuous vibration systems using Lagrange and Newton-Euler equations.
  • Be able to understand and analyze the dynamic behavior (mode shapes and eigenfrequencies) of linear vibration systems
  • Be able to use appropriate (e.g., harmonic balance) methods to analyze behavior of nonlinear vibration systems.

Type of instruction

The teaching is organized in accordance with the general form of teaching. Please see the programme cirruculum §17.

Extent and expected workload

Since it is a 5 ECTS course module the expected workload is 150 hours for the student.



Name of examEnergy and Variational Methods with Applications
Type of exam
Written or oral exam
Assessment7-point grading scale
Type of gradingInternal examination
Criteria of assessmentThe criteria of assessment are stated in the Examination Policies and Procedures

Facts about the module

Danish titleEnergi- og variationsmetoder med anvendelser
Module codeM-DMS-K2-4
Module typeCourse
Duration1 semester
Language of instructionEnglish
Empty-place SchemeYes
Location of the lectureCampus Aalborg
Responsible for the module


Study BoardStudy Board of Mechanical Engineering and Physics
DepartmentDepartment of Materials and Production
FacultyFaculty of Engineering and Science