Vibrations Two Marks Questions With Answers

TAGORE ENGINEERING COLLEGEDEPARTMENT OF AERONAUTICAL ENGIEERING AE6602 VIBRATIONS AND ELEMENTS OF AEROELASTICITY TWO MARKS QUESTIONS WITH ANSWERS Unit I 1. Find the length of a simple pendulum for which the period of oscillation is equal to 1 second. Time period of oscillations t = 1 s 𝑔 Natural frequency ωn=√( ) 𝑙 2𝜋 𝑔 =√( ) 𝑡 𝑙 2𝜋 9.81 = √( ) l =0.248 m. 1 𝑙 2. State Alembert’s principle A body which is not in static equilibrium by virtue of some acceleration which it possesses can be brought to static equilibrium by introducing on it the inertia force which can be considered to be an extra external force. This inertia force is equal to mass times the acceleration of the body and acts through the centre of gravity of the body in the direction opposite to that of the acceleration. Ʃ(𝐹) − 𝑚𝑥̈ = 0 3. State about viscous damper. This is the most important type of damping and occurs for small velocities in lubricated sliding surfaces, dashpots with small clearances etc. Eddy current damping is also of viscous nature. The amount of damping resistance will depend upon the relative velocity and upon the parameters of the damping system. For a particular system the damping resistance is always proportional to the relative velocity. 4. Define simple harmonic motion.(SHM) A motion which repeats itself after equal interval of time is called as a periodic motion. “A periodic motion of a particle whose acceleration is always directed towards the mean position and is proportional to its distance from the mean position is called as a simple harmonic motion.” (OR) SHM can also be defined as “The motion of the projection of a particle moving round a circle with uniform angular velocity on a diameter.” 5. Recall resonance. The vibration of the system when the frequency of the external force (ω) is equal to the natural frequency of the system (ωn). The amplitude of vibration at resonance becomes excessive. 6. Derive the equivalent spring constants for two springs in parallel and series combinations. Springs in series: Consider a system having two springs k 1 and k2 in series. If theses springs are replaced by an equivalent spring of stiffness keq, then the total static deflection of the body in two cases under the same  load must be the same. The total deflection in the actual case is the sum of deflections in the individual springs. Each spring has the same force mg acting on it. 𝑚𝑔 𝑚𝑔 𝑚𝑔 Δst = = + 𝑘𝑒𝑞 1 1 1 = + keq In the case of series springs, the reciprocal of the equivalent spring stiffness is equal to the sum of the reciprocals of individual spring stiffnesses. Springs in parallel: The springs are said to be parallel when the absolute deflection in each of the individual springs is equal to the deflection of the system. Consider a system having springs in parallel combinations. The total load supported is equal to the sum of the loads carried by individual springs Let the spring k1 be carrying a load m1g and the spring k2 a load m2g. mg= m1g+ m2g if keq is the stiffness of equivalent spring,then the total deflection of the body in both cases must be the same. 𝑚𝑔 𝑚𝑔 Δst = = = = = 𝑘 keq = k1+k2 7. Differentiate damped and undamped vibrations. If the vibratory system has a damper, the motion of the system will be opposed by it and the energy of the system will be dissipated in friction. This type of vibration is called damped vibration. On contrary, the system having no damper is known as undamped vibration. 8. Define free, damped and forced vibrations.  Free vibrations: The vibrations of a system because of its own elastic properties. No external exciting force acts in this case. Damped vibrations: If the vibratory system has a damper, the motion of the system will be opposed by it and the energy of the system will be dissipated in friction. This type of vibration is called damped vibration Forced vibrations: The vibrations which the system executes under an external periodic force. The frequency of vibration in this case is the same as that of excitation. 9. State about logarithmic decrement. The logarithmic decrement represents the rate at which the amplitude of a free damped vibration decreases. It is defined as the natural logarithm of the ratio of any two successive amplitudes. It is found from the time response of underdamped vibration 10. Distinguish between vibrometer and accelerometer. Vibrometer Accelerometer Used to measure the displacement of a Used to measure the acceleration of a vibrating vibrating body body Designed with low natural frequency Designed with high natural frequency Known as low frequency transducer Known as high frequency transducer Heavy in construction Very light in construction 11. Distinguish between longitudinal, lateral and torsional vibrations. Consider a weightless bar of length l whose one end is fixed and other end carries a disc as shown in fig. The system may have one of the below mentioned three types of free vibrations. fig(a) fig(b) fig(c) Longitudinal vibrations: When the particles of a bar or disc move parallel to the axis of the shaft, then the vibrations are known as longitudinal vibrations as shown in fig.(a). The bar is elongated and shortened alternately and thus the tensile and compressive stresses are inducted in the bar. The motion of spring mass system is longitudinal vibrations.  Transverse Vibrations: When the particles of the bar or disc move approximately perpendicular to the axis of the bar, then the vibrations are known as transverse vibrations as shown in fig.(b) In this case, bar is straight and bent alternately. Bending stresses are induced in the bar. Torsional Vibrations : When the particles of the bar or disc get alternately twisted and untwisted on account of vibratory motion of suspended body, it is said to be undergoing torsional vibrations as shown in fig. (c). In this case, torsional shear stresses are induced in the bar 12. Find the natural frequency of a simple spring mass system using energy method. 1 Kinetic energy T = 2 1 Potential energy U = 𝑘x2 2 Using energy method, T + U = constant 1 1 ∴ + 𝑘x2 = constant 2 2 Differentiating with respect to time, m𝑥̇ 𝑥̈ + kx𝑥̇ = 0 𝑘 𝑘 𝑥̈ + 𝑥=0 ωn2 = 𝑚 𝑚 ωn = √𝑘/𝑚 13. List out the causes of vibrations. (i) Unbalanced forces in the machine: these forces are produced from within the machine itself. (ii) Dry friction between the two mating surfaces: This produces what are known as self-excited vibration. (iii) External excitations: Excitations may be periodic, random or the nature of an impact produced external to the vibrating system (iv) Earth quakes: These are responsible for the failure of many buildings, dams, etc. (v) Winds: cause the vibration of transmission and telephone lines under certain conditions. 14. State the effects of vibrations. Excessive stresses, undesirable noise, looseness of part, partial or complete failure of parts. 15. List out the methods to reduce vibrations.  Removing the causes of vibrations  Putting the screens if noise is objectionable  Placing the machinery on proper isolators  Using shock absorbers  Using dynamic vibrations absorbers 16. Define amplitude and phase angle of simple harmonic motion  The maximum displacement of a vibrating body from the mean position is called the amplitude of vibrations. Phase angle is the angle between two rotating vectors representing simple harmonic motions of the same frequency. 17. Consider a simple single degree of freedom system. The undamped frequency of the system will be (larger/ smaller/ the same as) the damped natural frequency. Ans : larger Unit II 1. Explain about principal modes of vibrations. The motion where every point in the system executes harmonic motion with one of the natural frequencies of the system is called the principal modes of vibration. A system having two degrees of freedom can vibrate in two principal modes of vibrations corresponding to its two natural frequencies. 2. State about vibration absorbers. If a particular system is having large vibrations under its excitation, this vibration can be eliminated by coupling a properly designed auxiliary spring mass system to the main system. This forms the principle of undamped dynamic vibration absorber where the excitation is finally transmitted to the auxiliary system, bringing the main system to rest. 3. Explain Hamilton’s principle. It is a generalization of the principle of virtual displacements to dynamics of systems of particles, rigid bodies or deformable solids. The application of this principle leads directly to the equation of motion for both discrete and continuous systems. General form of Hamilton’s principle, 𝑡 ∫𝑡 2(𝛿𝐿 + 𝛿𝑊𝑛𝑐 ) dt = 0 (A) 1 Where, L=Lagrangian function = T - U T = Kinetic energy U = potential energy Wnc= Work done by non-conservative forces t1, t2 are the two instants of time For a conservative system, 𝛿𝑊𝑛𝑐 = 0 ∴ eqn (A) reduces to , 𝑡 ∫𝑡 2 𝛿𝐿 𝑑𝑡 = 0 (B) 1 Hamilton’s principle for the conservative system or motion states that “the motion of a particle acted on by conservative forces between two arbitrary instants of time t1 and t2 is such that the line integral over the Lagrangian function is an extremum for the path motion.” 4. D e f i n e s e m i d e f i n i t e s y st e m . The system having one of their natural frequencies equal to zero are known as semi definite systems. 5. Explain orthogonality principle. For a system with three-degree of freedom the orthogonality principle may be written as, m1 A1 A2 + m2B1B2 +m3C1 C2=0 m1 A2 A3 + m2B2B3 +m3C2 C3=0 m1 A1 A3 + m2B1B3 +m3C1C3=0 Where m1, m2 , m3 are masses. A1, A2 , A3 , B1 , B2 , B3 , C1 , C2 , C3 are the amplitude of vibration of the system. 6. Write down the Lagrange’s equation and state the use of the equation. The general form of this equation in terms of generalized coordinates is, 𝑑 𝜕𝑇 𝜕𝑇 𝜕𝑈 ( ) - 𝜕𝑥 + 𝜕𝑥 = Qj 𝑑𝑡 𝜕𝑥𝑗̇ 𝑗 𝑗 Where, T = total kinetic energy of the system U= total potential energy of the system j= 1, 2, 3 ……… n Qj = generalized external force n = degree of freedom of the system for a conservative system generalized force Q j = 0 , 𝑑 𝜕𝑇 𝜕𝑇 𝜕𝑈 ∴ ( ) 𝑑𝑡 𝜕𝑥𝑗̇ - 𝜕𝑥 + 𝜕𝑥 = 0 𝑗 𝑗 Lagrange’s equation can be used to obtain the equations of motion of the vibratory systems. 7. Explain about static and dynamic couplings. Consider a lathe machine, which can be modeled as a rigid bar with its center of mass not coinciding with its geometric center and supported by two springs k 1 and k2.  The equations of motion can be written as, Case 1: Considering, i.e., point C and G coincides, the equation of motion can be written as, In this case the system is statically coupled and if k1l1=k2l2 , this coupling disappears, and Uncoupled x and θ vibrations are obtained. Case 2: If, k2l2=k1l1 , the equation of motion becomes Hence in this case the system is dynamically coupled but statically uncoupled Case 3: If l1=0, i.e. point C coincide with the left end, the equation of motion will become , Here the system is both statically and dynamically coupled 8. Define normal modes of vibrations. Normal mode vibrations are free vibrations that depend only on the mass and stiffness of the system and how they are distributed. 9. Define Eigen vector and Eigen value Eigenvalues (mathematical entities) directly correspond to square of the frequencies (𝜔𝑛 2) or to natural frequencies (mechanical entities), while eigenvectors are natural modes of vibrations. It can be shown that a discrete system with n degrees of freedom has n pairs of Eigen frequencies and Eigen modes 10. Following an initial disturbance, a two degree of freedom system will always vibrate at one of its natural frequencies. State whether True or False? Ans: true Unit III 1. Differentiate continuous and discrete systems with examples Discrete systems Continuous systems Mass is assumed to act only at certain discrete Mass is assumed to be distributed continuously points throughout the length Has finite number of degrees of freedom Has infinite number of degrees of freedom Has finite number of natural frequencies Has infinite number of natural frequencies Governed by ordinary differential equations Governed by partial differential equations 2. Sketch the first three mode shapes of a simply supported beam. 3. Write the equation of motion for vibration of a string 𝑇 Where , c = √ 𝑚 T = tension in the string m= mass per unit length 4. Draw the first three mode shapes of a stretched string and indicate the corresponding frequencies. T is the tension in the string m is the mass per unit length l is the length of the string  𝜋 𝑇 2𝜋 𝑇 3𝜋 𝑇 ωn1 = √ 𝑙 𝑚 ωn2 = 𝑙 √𝑚 ωn3 = 𝑙 √𝑚 5. Give the boundary conditions for a fixed fixed beam. Deflection and slope at both the fixed ends are zero 6. Write the equation of motion for a shaft subjected to free torsional vibrations. J is the polar moment of inertia I0 is the mass moment of inertia G is the shear modulus θ is the angle of twist