# Earthquake Engineering Questions and Answers – Free Vibration analysis of MDOF System

This set of Earthquake Engineering Multiple Choice Questions & Answers (MCQs) focuses on “Free Vibration analysis of MDOF System”.

1. The expansion of the determinant of the equation |K – ω2M| = 0, gives which of the following equation?
a) General equation
b) Specific equation
c) Characteristic equation
d) Orthogonal equation

Explanation: The expansion of the determinant of the equation |K – ω2M| = 0, yields an algebraic equation of Nth order in ω2, which is known as the characteristic equation. The roots of characteristic equation are known as the eigenvalues and the positive square root of these eigenvalues are known as the natural frequencies (ωi) of the MDOF system.

2. What are the eigenvalues for stable structural systems with symmetric and positive definite stiffness?
a) Complex and positive
b) Real and negative
c) Complex and negative
d) Real and positive

Explanation: For stable structural systems with symmetric and positive definite stiffness and mass matrices the eigenvalues are always real and positive. Eigenvalues are the roots of the characteristic equation. For each eigenvalue the resultant synchronous motion has a distinct shape and is known as natural mode shape.

3. One of the most commonly used normalization procedure is to constrain a length measure of the eigenvector to be unity.
a) True
b) False

Explanation: There are two most commonly used normalization procedures. The first one is to constrain a length measure of the eigenvector to be unity. In the second one we assume the amplitude of synchronous motion at the first degree of freedom as unity.

4. What do we call the roots of characteristic equation?
a) Time period
b) Frequencies
c) Eigenvalues
d) Acceleration

Explanation: The roots of characteristic equation are known as the eigenvalues and the positive square root of these eigenvalues are known as the natural frequencies (ωi) of the MDOF system. The expansion of the determinant of the equation |K – ω2M| = 0, yields an algebraic equation of Nth order in ω2, which is known as the characteristic equation.

5. What are the ways in which a component would deform when they vibrate at the natural frequency?
a) Drift shapes
b) Mode shapes
c) Inter-story drifts
d) Deformation shapes

Explanation: Mode shapes are the ways or shapes in which a component would deform when it vibrates at the natural frequency. The normal modes shapes are as much a characteristic of the system as the eigenvalues are. They depend on the stiffness and inertia.

6. Which equation is used to determine the elements for any Eigen-vectors {φ(i)}?
a) {φ(i)}TM{φ(i)} = 1
b) {φ(i)}TM{φ(i)} = 0
c) {φ(i)}TV{φ(i)} = 1
d) {φ(i)}TK{φ(i)} = 0

Explanation: For any eigen-vector {φ(i)} it is possible to determine the elements of {φ(i)} such that {φ(i)}TM{φ(i)} = 1. This type of normalisation which uses mass/inertia matrix (M) is known as mass renormalization and the mode shape resulting from it is known as mass orthonormal mode shape.

7. The orthogonality property of mode shapes leads to which theorem?
a) Modal contraction theorem
b) Eigenvalue theorem
c) Eigenvector theorem
d) Modal expansion theorem

Explanation: The modal expansion theorem is derived from the orthogonality property of the mode shapes. An important property of mode shapes is that they are mutually orthogonal with respect to the stiffness and mass matrices. The theorem states that any vector x in N-dimensional vector space can be represented as a linear combination of mode-shape vectors.

8. What do we call the phenomenon, if the modulus of all eigenvalues is smaller than 1, but the algorithm simulates a damped motion?
a) Logarithmic damping
b) Algorithmic damping
c) Characteristic damping
d) General damping

Explanation: If the modulus of all eigenvalues is smaller than 1, stability is ensured but the algorithm simulates a damped motion. This phenomenon is referred to as algorithmic damping. To achieve a satisfactory performance, we must demand the presence of complex eigenvalues with modulus of 1.

Sanfoundry Global Education & Learning Series – Earthquake Engineering.