# Mechanical Vibrations Questions and Answers – Free Vibration of Single Degree of Freedom Systems

This set of Mechanical Vibrations Multiple Choice Questions & Answers (MCQs) focuses on “Free Vibration of Single Degree of Freedom Systems”.

1. Free Vibrations occur in the absence of an external excitation
a) Yes
b) No

Explanation: Free Vibrations are oscillations from a system equilibrium position that occurs in the absence of an external excitation.

2. When the block is drawn from its equilibrium position, a potential energy developed in the—-

a) Spring
b) Block
c) Spring is kx2
d) Block is kx2/2

Explanation: When the block is displaced a distance of x from its equilibrium position , a potential energy kx2/2 is developed in the spring.

3.The differential equation governing free vibrations of a linear system are derived in the form——
a) meq x.. + ceq x. + keq x = 0
b) ceq x.. + meq x. + keq x = 0
c) meq x.. – ceq x. + keq x = 0
d) meq x.. + keq x. + ceq x = 0

Explanation: Free Vibrations of SDOF system are described by a homogenous second-order ordinary differential equation. The differential equation governing free vibrations of a linear system are derived and shown to have the form meq x.. + ceq x. + keq x = 0

4. Free Vibrations of a Single Degree of Freedom system are described by a ——————-
a) Homogenous First order differential equation
b) Homogenous Second order ordinary differential equation
c) Cartesian equation
d) First order differential equation

Explanation: Free Vibrations of SDOF system are described by a homogenous second-order ordinary differential equation . The independent variable is time and while the dependent variable is the chosen generalized coordinate.

5. Free Vibrations are the result of a kinetic energy imparted to the system
a) True
b) False

Explanation: Free Vibrations are a result of displacement from the equilibrium position that leads to a difference in potential energy from the system’s equilibrium position or it is a result of kinetic energy imparted to the systems.

Common data questions from 6 to 8

The differential equation governing free vibrations of a linear system are derived in the form
meq x.. + ceq x. + keq x = 0

6. The Second derivative term is due to—–
a) External forces
b) Velocity
c) Inertia Forces
d) Displacement

Explanation: The Second derivative term of meq x.. + ceq x. + keq x = 0 is due to the inertia forces or effective forces of the system.

7. The first derivative term is due to ——-
a) Velocity
b) Viscous damping
c) Elastic forces
d) Displacement

Explanation: The First derivative term of meq x.. + ceq x. + keq x = 0 is due to presence of viscous damping in the system.

8. The Zeroth derivative term is due to
a) Velocity
b) Viscous damping
c) Elastic forces
d) Displacement

Explanation: The Zeroth derivative term of meq x.. + ceq x. + keq x = 0 is due to presence of elastic forces in the system.

9. The differential equation governing free vibrations of a linear system are derived in the parameters damping ratio, natural frequency is———–
a) x.. – 2 ζ ωn x. – ωn2 x = 0
b) x.. + 2 ζ ωn x. – ωn2 x = 0
c) x.. – 2 ζ ωn x. + ωn2 x = 0
d) x.. + 2 ζ ωn x. + ωn2 x = 0

Explanation: The differential equation in terms of parameters damping ration, natural frequency is
x.. + 2 ζ ωn x. + ωn2 x = 0 . It is the standard form of differential equation for single degree of freedom systems.

Common data questions from 10 to 12
The differential equation governing free vibrations of a linear system are derived in the form
meqx.. + ceq x. + keq x = 0

10. If the energy method is used to derive the differential equation, the Second derivative term is due to—–
a) External forces
b) Kinetic Energy
c) Inertia Forces
d) Displacement

Explanation: If the energy method is used to derive the differential equation, the second derivative term is a result of the system’s kinetic energy.

11. The first derivative term is due to ——-
a) Viscous friction forces
b) Viscous damping
c) Elastic forces
d) Displacement

Explanation: If the energy method is used to derive the differential equation, the First derivative term is a result of the work done by the viscous friction forces.

12. The Zeroth derivative term is due to
a) Potential Energy
b) Viscous damping
c) Elastic forces
d) Displacement

Explanation: If the energy method is used to derive the differential equation, the zeroth derivative term is a result of the system’s potential energy.

Common data for question 13 & 14

The differential equation governing the single degree of freedom system is 5x.. + 4 x. + 7 x = 0

13. Find the natural frequency of motion
a) 3.183
b) 2.183
c) 1.183
d) 0.183

Explanation:

```The natural frequency of motion is given by ωn = √ ( keq/meq)
Substituting the values, we have  ωn  = √(7/5)

14. Find the damping ratio
a) 0.008
b) 0.338
c) 0.118
d) 0.228

Explanation:

```The damping ratio is given by ζ  = ceq / ( 2 √(k<sub>eq</sub>m<sub>eq</sub>) )
Substituting the values, we have , ζ  = 4 / ( 2 √ (7×5))
= 0.338```

15. The differential equation governing the angular oscillation of the compound pendulum is

a) m (L2/3) ӫ – mg (L/2) sin ө
b) m (L2/3) ӫ + mg (L2/2) sin ө
c) m (L/3) ӫ + mg (L/2) sin ө
d) m (L2/3) ӫ + mg (L/2) sin ө

Explanation: let ө(t) be the counter clockwise angular displacement of the rod measure from its equilibrium position. summing moments about O ( at rigid ) using the free body diagram, we obtain differential equation is m (L2/3) ӫ + mg (L/2) sin ө.

Sanfoundry Global Education & Learning Series – Mechanical Vibrations.

To practice all areas of Mechanical Vibrations, here is complete set of Multiple Choice Questions and Answers.

If you find a mistake in question / option / answer, kindly take a screenshot and email to [email protected]