# Earthquake Engineering Questions and Answers – Forced Vibration Analysis of MDOF System

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

1. What is equation for Forced Vibration Analysis of a multi degree of freedom system that gives the set of differential equations in terms of displacement u?
a) Mü + Cu̇ + Ku = f
b) Mü + Cü + Kü = f
c) Mü + Cu̇ + Ku = 0
d) Mü + Cu̇ + Ku = fu

Explanation: The set of N-coupled and non-homogenous equations is given by equation Mü + Cu̇ + Ku = f where u is displacement, u̇ is the velocity, ü is the acceleration, M is mass, C is damping and K is stiffness. is the external vibration that is applied. These equations are extremely difficult to solve manually and thus a suitable transformation is required to solve them.

2. Which method is used to transform a set of N coupled non homogenous differential equations to set of N uncoupled non homogenous differential equations?
a) Stodola Method
b) Mode-superposition method
c) Absolute sum method
d) Square root of sum of squares method

Explanation: The Mode-superposition method is used to transform N coupled differential equations to N uncoupled differential equation. This is done to solve the set of equations for a forced vibration system.

3. Which equation represents the set of equations due to excitation by support motion where, is displacement relative to ground displacement, üg is ground acceleration and r is vector of influence coefficients?
a) Mü + Cu̇ + Ku = Mrüg
b) Mü + Cu̇ + Ku = – Mrüg
c) Müg + Cu̇ + Ku = – Mrüg
d) Mü + Cu̇ + Ku = – Mrü

Explanation: This equation is similar to the standard forced vibration equation i.e. Mü + Cu̇ + Ku = f where the force (f) is equal to (- Mrüg). The ith element of vector r represents the displacement of ith degree of freedom due to a unit displacement of the base.

4. How is the mode participation calculated if M is mass matrix, r is a vector of influence coefficients and {ϕ(i)} is the eigen value of ith mode?
a) $$Γ_i=\frac{Mr}{M\{ϕ^{(i)}\}}$$
b) $$Γ_i=\frac{\{ϕ^{(i)}\}^T}{\{ϕ^{(i)}\}^T \{ϕ^{(i)}\}}$$
c) $$Γ_i=\frac{\{ϕ^{(i)}\}^T Mr}{\{ϕ^{(i)}\}^T M\{ϕ^{(i)}\}}$$
d) $$Γ_i=\frac{M}{M\{ϕ^{(i)}\}}$$

Explanation: $$Γ_i=\frac{\{ϕ^{(i)}\}^T Mr}{\{ϕ^{(i)}\}^T M\{ϕ^{(i)}\}}$$ is obtained using the mode superposition method for system of excitation by support motion for a MDOF. This equation differs from the equation of motion of a SDOF system excited by support acceleration üg by a scaling factor for the excitation.

5. Which assumption is used in the Absolute Sum Method for calculating maximum response in MDOF system?
a) The maximum of each modal coordinate occurs at the same instant of time
b) The minimum of each modal coordinate occurs at the same instant of time
c) The maximum of each modal coordinate occurs at the different instant of time
d) The minimum of each modal coordinate occurs at the different instant of time

Explanation: The absolute sum method of modal combination provides a very conservative estimate of the maximum response in physical coordinates because the time of occurrence of maxima in each mode is generally different.

6. What is the formula of Square Root of Sum of Squares (SRSS) method used for estimating maximum response in physical coordinate system??
a) $$v_{i,max} = [∑_{r=1}^{N} (q_{r,max} ϕ_i^{(r)})^2]^{1/2}$$
b) $$v_{i,max} = ∑_{r=1}^{N} (q_{r,max} ϕ_i^{(r)})$$
c) $$v_{i,max} = [∑_{r=1}^{N} (q_{r,max} ϕ_i^{(r)})^{1/2}]^2$$
d) $$v_{i,max} = ∑_{r=1}^{N} (q_{r,min} ϕ_i^{(r)})$$

Explanation: The Square root of sum of squares (SRSS) method does not assume that maxima of each modal coordinate occurs at the same instant which is assumed in Absolute Sum method. It also assumes that the natural frequencies are not closely spaced.

7. How to decide the number of modes to be in response calculation?
a) All modes having natural frequency more than or equal to highest frequency in excitation are included and At least 90% of the total mass of the structural system should be included in the dynamic response computation
b) All modes having natural frequency less than or equal to highest frequency in excitation are included and At least 99% of the total mass of the structural system should be included in the dynamic response computation
c) All modes having natural frequency less than or equal to highest frequency in excitation are included and At least 99% of the total mass of the structural system should be included in the dynamic response computation
d) All modes having natural frequency less than or equal to highest frequency in excitation are included and At least 90% of the total mass of the structural system should be included in the dynamic response computation

Explanation: On a structure, dynamic loads (like earthquake motions, ocean waves, wind forces etc.) have their maximum energy concentrated in low frequencies (generally less than 35 Hz), thus the higher modes (with higher natural frequencies) are not excited by the low frequency forces.

8. The correction due to higher mode response truncation is called as the missing mass correction.
a) True
b) False

Explanation: Sometimes in spatial structures like piping, it is necessary to consider the higher mode participation as the truncation error may be significant during the design. The missing mode correction is the static correction term to account for higher mode response.

Sanfoundry Global Education & Learning Series – Earthquake Engineering.