# Finite Element Method Questions and Answers – Dynamic Considerations – Element Mass Matrices

This set of Finite Element Method Multiple Choice Questions & Answers (MCQs) focuses on “Dynamic Considerations – Element Mass Matrices”.

1. What are the different methods of constructing mass matrices of individual elements?
a) Direct, Variational and Template mass lumping
b) Indirect, Translational and Vocational mass lumping
c) Transparent, Translucent and Opaque mass lumping
d) Absolute, Temporary and Persistent mass lumping

Explanation: Mass matrices of individual elements can be constructed by making use of various methods. These methods are broadly classified into three different types. They are Direct mass lumping method, Variational mass lumping method and template mass lumping method.

2. Which of the following is true about direct mass lumping?
a) Cross coupling is considered
b) Cross coupling is ignored
c) Bi-directional torque is ignored
d) Bi-directional torque is considered

Explanation: Direct mass lumping can be defined as a method in which the entire mass of element under consideration is apportioned only to its nodal freedoms.

3. Which of the following is considered as an advantage of the direct mass lumping method?
a) Offers hassle free calculations
b) Offers user friendly inputs
c) Offers computational and storage benefits
d) Ignores point masses

Explanation: In the direct mass lumping method, the goal is to build a diagonally mass matrix. This type of matrix offers a lot of advantages. One of it includes the computational and storage benefits that it has to offer especially in explicit time integration problems.

4. Which of the following is true about variational mass lumping?
a) Kinetic energy is not part of the governing potential
b) Kinetic energy is part of the non governing potential
c) Kinetic energy is not part of the non governing potential
d) Kinetic energy is part of the governing potential

Explanation: Variational mass lumping method is based on the principle of variational formulation. This is achieved by taking kinetic energy as part of the governing potential of the body. This kinetic energy is mathematically expressed as
Te = 1/2 (u.e)TMeu.e

5. Which of the following is not a mass matrix property?
a) Negativity
b) Matrix Symmetry
c) Physical symmetry
d) Conservation

Explanation: Mass matrix properties are considered as certain conditions that are needed for verification and debugging. Negativity is not considered as a mass matrix property. There are 4 different matrix properties. They are Matrix symmetry, physical symmetries, conservation and positivity.
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6. What is meant by physical symmetry?
a) Element symmetry must not be available
b) Element symmetry must be available
c) Matrix symmetry must be available
d) Matrix symmetry must not be available

Explanation: For physical symmetry condition to be satisfied, the element symmetry must be reflected in the mass matrix. If you take into consideration a consistent mass matrix and a diagonally lumped mass matrix of a prismatic bar element; they must be symmetric along the anti-diagonal.

7. Which of the following is a non linear constraint?
a) Conservation
b) Matrix Symmetry
c) Positivity
d) Physical symmetry

Explanation: The constraints under consideration are all mass matrix properties. Of the four given, Positivity is the non linear constraint. This can be checked or verified in two ways – either through the Eigen values of Me, or the sequence of principal errors.

8. Which of the following satisfies the full rank condition? Given degrees of freedom = n, rank = r.
a) n > r
b) n < r
c) n != r
d) n = r

Explanation: When the rank is equivalent to the total degrees of freedom of the element, it is referred to as rank sufficient or full rank matrix. Similarly, a matrix is referred to as rank deficient when the rank is lesser than the degrees of freedom of the element under consideration.

9. A RBD local mass matrix globalizes to the same matrix if all element degrees of freedom are translational and all of them refer to the same global system.
a) False
b) True

Explanation: The given statement is true. The statement refers to the repetition rule that arises due to the repeating block diagonal(RBD). A matrix that can be put in block diagonal form identical blocks is referred to as a repeating block diagonal. The contents and order of the matrix are irrelevant in such a formulation.

10. What is the generalized expression for formulation of a mass matrix?
a) Me = ρ ∫e NTdV
b) Me = ρ ∫e NNdV
c) Me = ρ ∫e NTNdV
d) Me = ρ ∫e NTNdS

Explanation: The generalized expression for formulation of a mass matrix is given by Me = ρ ∫e NTNdV.
where, Me = mass matrix
ρ = material density(considered constant)
N = Nodes
dV = volume element

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