Ae2403 - Vibrations and Elements of AeroelasticitySlip Test 11. Define Simple Harmonic motion.ANS: In mechanics and physics, simple harmonic motion is a type of periodic motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement. Simple harmonic motion can serve as a mathematical model of a variety of motions, such as the oscillation of a spring. In addition, other phenomena can be approximated by simple harmonic motion, including the motion of a simple pendulum as well as molecular vibration. Simple harmonic motion is typified by the motion of a mass on a spring when it is subject to the linear elastic restoring force given by Hooke's Law. The motion is sinusoidal in time and demonstrates a single resonant frequency. In order for simple harmonic motion to take place, the net force of the object at the end of the pendulum must be proportional to the displacement.

2. State the DAlemberts Principle.Ans: The principle states that the sum of the differences between the forces acting on a system of mass particles and the time derivatives of the momenta of the system itself along any virtual displacement consistent with the constraints of the system, is zero. Thus, in symbols d'Alembert's principle is written as following,

whereis an integer used to indicate (via subscript) a variable corresponding to a particular particle in the system,

is the total applied force (excluding constraint forces) on the -th particle,

is the mass of the -th particle,

is the acceleration of the -th particle,

together as product represents the time derivative of the momentum of the -th particle, and

is the virtual displacement of the -th particle, consistent with the constraints

3. Explain the term Aeroelasticity.ANS: Aeroelasticity is the branch of physics and engineering that studies the interactions between the inertial, elastic, and aerodynamic forces that occur when an elastic body is exposed to a fluid flow. Although historical studies have been focused on aeronautical applications, recent research has found applications in fields such as energy harvesting [1] and understanding snoring.[2] The study of aeroelasticity may be broadly classified into two fields: static aeroelasticity, which deals with the static or steady response of an elastic body to a fluid flow; and dynamic aeroelasticity, which deals with the bodys dynamic (typically vibrational) response. Aeroelasticity draws on the study of fluid mechanics, solid mechanics, structural dynamics and dynamical systems. The synthesis of aeroelasticity with thermodynamics is known as aerothermoelasticity, and its synthesis with control theory is known as aeroservoelasticity.

4. What must be the length of the simple pendulum for the period of oscillation to be equal to 1 second?ANS: T= ( 2* PI ) * Sqrt( l/g) => 5. Define Free, Damped and Forced vibrations.ANS: Refer class notes6. Define Period, rms amplitude and frequency of Simple Harmonic Motion.ANS: Refer class notes7. What are the causes for Vibrations.ANS: Refer class notes8. What are the effects of Vibrations.ANS: Refer class notes9. Mention different ways to prevent Vibration.ANS: Refer class notes10. Differentiate discrete system and continuous system.ANS: Refer class notes11. What is Resonance?ANS: Refer class notes12. Mention three major types of vibrations with illustrations.ANS: Refer class notes13. Draw the free body diagram of a simple pendulum and derive its natural frequency.ANS: Refer class notes14. Derive the equivalent spring constants for two springs in Series and Parallel.ANS: Refer class notes15. Using energy method, find the frequency of simple spring mass system.ANS: K.E = m (dx/dt)2 toatal P.E, U= -mgx + k ( x/2 + st )x ; U= k x2Total Energy = Total P.E + K.E =const => m (dx/dt)2 + k x2 = constDiff w.r.to t => m d2y/dx2 + k x = 0 => n = sqrt( k/m)16. Two SHMs represented by 3 sin (5t) and 4 cos(5t) are superposed one over the other, what are the frequency , amplitude and phase difference if any of the resultant motion.ANS: Freq= 5, Amp= sqrt(3*3 +4*4) = 5, Phase diff= tan-1(4/3)17. Two SHMs represented by 5 sin (15t) and 8 cos(15t) are superposed one over the other, what are the frequency , amplitude and phase difference if any of the resultant motion.Ans : Similar to previous one18. Derive the equivalent spring constants for a cantilever with a spring mass attached at the free end.ANS: Refer class notes19. Derive the equivalent spring constants for stepped bar with three different Cross sections.ANS: Refer class notes20. Determine the natural frequency of rotating shaft for which the rotating mass assumed to be lumped at the mid point.Ans : Clue : Simply supported beam with load at the mid point.