# Earthquake Engineering Questions and Answers – Fundamental Frequency of MDOF System

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

1. For any arbitrary vector {u}, representing a displacement configuration of a multi degree of freedom system, what is the Rayleigh quotient (ρ) where Mu and Ku are mass and stiffness matrix respectively?
a) $$ρ = \frac{u^T K_u}{u^T M_u}$$
b) $$ρ = \frac{u^T M_u}{u^T K_u}$$
c) $$ρ = \frac{u K_u}{u M_u}$$
d) $$ρ = \frac{u M_u}{u K_u}$$

Explanation: For a given case when the vector u represents the amplitudes of the harmonic oscillations of the multi-DOF system or the Rayleigh quotient (ρ) corresponds to square of the frequency of harmonic oscillations.

2. Why is the Stodola Method used?
a) To find generalized velocity
b) To find generalized acceleration
c) To find generalized displacement
d) To find relative velocity

Explanation: The Stodola method can be viewed as an iterative solution of a system of simultaneous equations to arrive at that configuration of generalized displacements for which the inertia forces are exactly balanced by the elastic forces in the structural members.

3. To Which mode does the Stodola Iteration method first converge?
a) Lowest
b) Highest
c) Lowest unless the chosen trial vector exactly resembles a higher mode.
d) Highest unless the chosen trial vector exactly resembles the lowest mode

Explanation: The iteration method described earlier will always converge to the lowest mode, unless the chosen trial vector exactly resembles a higher natural mode. Therefore, to determine the higher modes using iteration procedure, it is necessary to sweep out all the lower modes.

4. What is the approach of geometrical interpretation of sweeping to determine trial vector which is orthogonal to all previously determined eigenvector called?
a) Vector Division
b) Vector Purification
c) Vector Inflation
d) Vector Partition

Explanation: A geometrical interpretation of the process of sweeping is to determine a trial vector which is orthogonal to all the previously determined eigenvectors and this approach is known as vector purification/deflation. This process is part of the Stodola Method.

5. What is the property of Rayleigh quotient of being stationary in the neighborhood of the natural modes of system?
a) Maximax Property
b) Minimax Property
c) Median Property
d) Modal Property

Explanation: Rayleigh quotient is a global minimum for the fundamental mode and global maximum for the highest mode of vibration—also known as the minimax property of Rayleigh quotient.

6. Which equation is used to determine the Sweeping Matrix for determining the first mode?
a) $$S = I – \frac{1}{\{φ^{(i)}\}^T M\{φ^{(i)}\}} \{φ^{(i)}\} \{φ^{(i)}\}^T M$$
b) $$S = I – \frac{1}{\{φ^{(i)}\}^T K\{φ^{(i)}\}} \{φ^{(i)}\} \{φ^{(i)}\}^T K$$
c) $$S = \frac{1}{\{φ^{(i)}\}^T M\{φ^{(i)}\}} \{φ^{(i)}\} \{φ^{(i)}\}^T M – I$$
d) $$S = \frac{1}{\{φ^{(i)}\}^T K\{φ^{(i)}\}} \{φ^{(i)}\} \{φ^{(i)}\}^T K – I$$

Explanation: The Sweeping matrix is determined to get the modes from a trial vector. In practice, the coefficient matrix of the eigenvalue problem is post-multiplied by the sweeping matrix and the resulting updated coefficient matrix is used in the iteration procedure to converge to the n + 1th mode.

7. Stodola method starts with a choice of a trial vector.
a) True
b) False

Explanation: Stodola method starts with the choice of a trial vector, say, {φ(0)}. Pre-multiplying {φ(0)} by the dynamical matrix, D (D = K-1 M) yields another vector {φ(1)}, which is an improved estimate of the eigenvector. An estimate of the eigenvalue is obtained by taking the ratio of any element of new vector {φ(1)} to the corresponding element of the trial vector. If {φ(1)} were a true eigenvector, this ratio would be constant for any choice of the element of these vectors.

Sanfoundry Global Education & Learning Series – Earthquake Engineering.

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