Mündliche Prüfungn List Of Questions (21SS) PDF

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Preliminaries: Analytical Dynamics for Point Masses 1. Discuss why the Newton equations are not practical and why the virtual work principle is more appropriate. Introduce the concept of degrees of freedom and the projected Newton form for a system of point masses. 2. Explain the principle of virtual power. Derive the Lagrange equations and Hamilton’s principle. Illustrate on a simple example how they are applied. 3. Discuss how Lagrange equations and Hamilton’s principle can be written in the presence of additional constraints.

Two-Dimensional Dynamics of a Rigid Body 4. Introduce the virtual displacement, velocity and acceleration in a 2D rigid body and show how the virtual work yields the Newton-Euler equations in 2D. Show that Newton-Euler equations in 2D can also be obtained with Lagrange equations.

3D Kinematics and Finite Rotations 5. Describe the spherical motion of a point and discuss the properties of a rotation matrix. Explain the Theorem of Euler. 6. Explain how a rotation matrix can be defined in terms of rotation axis and angle. Introduce the exponential map. 7. Derive the absolute velocity in a spherical motion, introducing the definitions of angular velocities in the material and spatial frame. Discuss the relation between angular velocities and the rotation axis and angle. Derive the expression of absolute accelerations. 8. Discuss the necessity for parametrizing finite rotations. Discuss the parametrization in terms of cartesian rotation vector and the one with Euler. 9. Introduce the notion of infinitesimal rotations and pseudo-parameters. Explain the relation to angular velocities.

Dynamics of a Rigid Body in Space 10. Introduce the virtual displacement, velocity and acceleration in a 3D rigid body and show how the virtual work yields the Newton-Euler equations in 3D when considering infinitesimal angular displacements. Discuss the form of Newton-Euler equations found for a 3D rigid body when considering variations of rotation parameters. 11. Show that the Newton-Euler equations can also be found from Hamilton’s principle. 12. Explain how the Newton-Euler equations are used in practice. (The theorems of linear and angular momentum have not been covered and are not exam material)

Systems of Rigid Bodies 13. Introduce the block notation for the equations of motion of unconstrained rigid bodies. Discuss the constraints imposed by lower-pair links.

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14. Explain how the first and second order equations of motions are written using Lagrange multipliers for a constrained system of rigid bodies. Illustrate the use of Lagrange multipliers by writing the constraints and the Jacobian of the constraints for a hinge linkage. 15. Explain the Augmented Lagrange multiplier formulation. Discuss the penalized Lagrange approached and its relation to flexible links. Explain how the equations of motion can be found for a minimal set of coordinates in first and second order.

Flexible Bodies: Floating Frame Formulation 16. Describe the kinematics of a flexible body in a floating frame. Derive the kinetic and deformation energy. 17. Explain how the dynamic equations can be found if a node is taken as reference point. Discuss the mass matrix in that case. 18. Explain the idea of choosing the center of mass as reference point (Tisserand axis) and discuss important practical issues of the approach.

Time-integration 19. Describe shortly the idea of time integration (use for instance the Newmark formulas). What is the difference between implicit and explicit? 20. Explain how the Newmark scheme (explicit and implicit) can be applied to unconstrained non-linear systems. 21. Discuss the techniques to apply Newmark scheme (explicit and implicit) to constrained non-linear systems.

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