Structural Dynamics: Introduction

Question 1

What are the Euler-Bernoulli assumptions?

Answer 1

1. The cross-section is infinitely rigid in its own plane (does not change shape) -> u(x1,x2,x3) = u(x1)
2. The cross-section of a beam remains plain after deformation. The axial displacement could be composed by one translation and two rotations.
3. The cross-section remains normal to the deformed axis of the beam. The angles are equal to the slope of the deformed axis.

Question 2

What is the bending problem in the most general case?

Answer 2

There is a coupling between the axial force, the moments and the displacements (u’, w’’, v’’). The decoupling between the axial and bending problems is obtained if the reference line of the beam is at the E-weighted centroid of the section. The two bending problems are decoupled so that the E -weighted moment of inertial between the two bending problems is zero.

Question 3

What are the limits of the Euler Bernoulli Beam model?

Answer 3

Real beams are free to warp to comply with the boundary conditions. The shear stress at the corner must go to zero. No local twist at the corner is possible. Warping depends on the geometry of the section.

If a twist moment varies along the length of the beam, warping displacement varies along the beam and torsion is accompanied by tension or compression of longitudinal fibers.

Question 4

What is the Timoshenko model?

Answer 4

The simplest model to recover an average effect of warping is to consider that sections do not remain perpendicular to the beam axis. An independed rotation angle for the section is defined θ.

Question 5

What are the properties of modal forms?

Answer 5

1. Orthogonality with respect to the mass distribution
2. Orthogonality with respect to the stiffness distribution

Question 6

What is the Rayleigh quotient?

Answer 6

It is the division of elastic and inertia forces.

Question 7

What are proper orthogonal modes?

Answer 7

Proper orthogonal modes are form functions that are mutually orthogonal with respect to mass and stiffness distributions. Consequently, proper orthogonal models can be used to decouple the equations of motion for systems where only elastic and inertia forces are considered.

Question 8

How can the Ritz-Galerkin approximation be used for dynamic problems?

Answer 8

For any generic uknown displacement field u, it can be approximated as a linear combination of shape functions Ni(x) and generalized coordinates qi(t).
1. Linear independence, that means that exist a linear combination with coefficient ai so that ΣaiNi(x) different from zero
2. Completeness. It must satisfy all kinematic BC and the error should go to zero for shape functions going to inf.

Question 9

What is the difference between Ritz-Galerkin and FEM.

Answer 9

1) Ritz-Galerkin
*Shape functions with global support
*Matrices M,K are full
* When geometry is complex the definition of appropriate function may be difficult
*The prediction of local effects may require the inclusion of local shape functions which may be difficult to define
*With well defined shape functions the convergence is fast

2) FEM
*Shape functions with local support
*Matrices M,K are sparts
*Easier to apply to complex geometries
*The prediction of local effects may be obtained with local refinement of the number of elements
*Availability of lots of tools