Embed Size (px)
Citation preview
www.gatep
athsh
ala.co
m
Sample
copy
BEST CLASSES &STUDY MATERIAL FOR AEROSPACE ENGINEERING– GATE 2017 GATE PATHSHALA
FREE DAMPED VIBRATIONS 3.1 INTRODUCTION Damping is the resistance offered by a body to the motion of a vibratory system. The may be applied by a liquid or solid internally or externally. For example, the motion of the wheels in water is resisted by the water itself, the wheels of car are put to some resistance /friction because of road surface while moving on it. Because of this resistance vibrations die out over a cycles of motion. Mass, stiffness and damping are the main characteristics of a vibratory system. Out of three the first two are the inherent properties of the system. If the value of damping is mechanical systems, it will be having negligible influence on the natural frequency of the vibratory system has some energy which is dissipated during the motion. At the start of vibratory motion the amplitude of vibration is maximum which goes on decreasing and lost completely with the passage of time. The rate of decreasing the amplitude depends upon amount of damping. The main advantage of providing damping in mechanical systems is just to control amplitude of vibration so that the failure occurring because of resonance may be avoided. 3.2 TYPES OF DAMPING There are mainly four types of damping used in mechanical systems :
Viscous damping Coulomb damping Structural damping Non-linear, Slip or interfacial damping.
3.2.1 Viscous Damping When the system is allowed to vibrate in a viscous medium, the damping is called viscous. Viscosity is the property of a fluid by virtue of which it offers resistance to the motion of one layer over the adjacent one. It can be explained by Fig. 3.1 where two plates are separated by fluid film of thickness t. upper plate is allowed to move parallel to the fixed plate with a velocity x. The net force F required for maintaining the velocity x of the plate is expressed as
F = x ---------- (3.2.1) Fig 3.1 Viscosity
www.gatep
athsh
ala.co
m
Sample
copy
BEST CLASSES &STUDY MATERIAL FOR AEROSPACE ENGINEERING– GATE 2017 GATE PATHSHALA
where A = area of the plate t = thickness of the fluid film μ = coefficient of absolute viscosity of the film The force F can also be written as F = cx c = μA/t ................ (3.2.2) where c is viscous damping coefficient. The main components o£-a viscous damper are cylinder, piston and viscous fluid. There is clearance between the cylinder walls and the piston. More the clearance, more will be the velocity of the piston in the viscous fluid and it will offer less value of viscous damping coefficient. The basic system is shown in Fig. 3.2. The damping force is opposite to the direction of velocity.
The damping resistance depends on the pressure difference on the both sides of the piston in the viscous medium. Figure 3.3 shows the example of free vibrations with viscous damping. The equation of motion for the system can be written as mx + cx + kx = 0 ..............(3.2.3) The differential equations used in the analysis are simple and can be solved easily. This type of damping is widely used in engineering. Energy Dissipation in Viscous Damping For a vibratory body some amount of energy is dissipated because of damping. This energy
Fig 3.2
www.gatep
athsh
ala.co
m
Sample
copy
BEST CLASSES &STUDY MATERIAL FOR AEROSPACE ENGINEERING– GATE 2017 GATE PATHSHALA
dissipation can be per cycle. For a viscously damped system the force F is expressed as
F = cx = c
where x =
Work done dW ≡ F. dx = c . dx
The rate of change of work per cycle i.e.
Energy dissipated ∆E = ∫ (F. dx)dt/ = ∫ c dt/
∆E = ∫ . dt/ ........... (3.2.4) Let us assume the simple harmonic motion of the type x = A sinωt
= ω A cos sinωt Equation (3.2.4) can be written as
∆E = ∫ cω A dt/ = πcωA ---- (3.2.5)
From the above equation, it is clear that the energy dissipation per cycle is proportional to the square of the amplitude of motion: The total energy of a vibrating system can be either maximum of its potential or kinetic energy. The maximum kinetic energy of the system can be written as
E = (K. E) = mx = mω A ........... (3.2.6)
We can find the ratio of ∆E to E. This ratio is known as specific damping capacity of the system. Thus specific damping ratio
β = ∆ =
= 2 = ............. (3.2.7)
which is a constant quantity.
www.gatep
athsh
ala.co
m
Sample
copy
BEST CLASSES &STUDY MATERIAL FOR AEROSPACE ENGINEERING– GATE 2017 GATE PATHSHALA
This relation is very useful in the design of vibratory instruments. The damping materials are rated by their damping capacity β. 3.2.2 Coulomb Damping When one body is allowed to slide over the other, the surface of one body offers some resistance to the movement of the other body on it. This resisting force is called force of friction. Thus force of friction arises only because of relative movement between the two surfaces. Some amount of energy is wasted in overcoming this friction as the surfaces are dry. So it is sometimes known as dry friction. The general expression for coulomb damping is F = μR ... (3.2.8) where μ is the coefficient of friction and RN is the normal reaction. Friction force F is proportional to the normal reaction R N on the mating surface. The system is shown in Fig. 3.5. The friction force acts in a direction opposite to the direction of velocity. The damping resistance is almost constant and does not depend on the rubbing velocity. The three possible conditions of coulomb damping are shown in Fig. 3.5 with mathematical expressions.
Let us consider the leftward movement of the body the equation for which can be written as mx + kx = F ............... ( 3.2.9) The solution of the above equation can be written as
x = B cos t + D sin t + ................. (3.2.10)
where ω = k/m Let us assume the motion characteristics of the system as x = Xo at t = 0 x= O at t = 0
Fig 3.5
Fig 3.5
Adambakkam SRMUniversity,Potheri Shoulinganllure-mail:[email protected]:(+91)9962996817/8939174202 website:www.gatepathshala.comCorporateOffice:#153,II-Floor,Karuneegarstreet,Adambakkam,Chennai-600088
EDUCATIONALSERVICESLLP.CHENNAI
Study Material Details The study material covers the syllabus of respective examination (to the extent of availability) both technical and Non technical completely for Aerospace Engineering. The study material contains
Subject wise and Chapter wise theory,
Supporting example problems,
Previous Questions with Solutions,
Assignments with solutions.
5 Full length Mock test-Papers (for entire syllabus).
The study material is updated time to time depending on changes in the exam pattern, feed back of our beloved students and other subject experts.
The study material will be sent to a student by post in 2 to 3 parcels/installments. Model Papers will be sent in appropriate time so that the student can write those exams with sufficient preparation.
The study material will be sent by post after one week on receiving the application form along with the required fee. The postal coaching commenced from March 1st week onwards. Hence the students are advised to apply as soon as possible to start preparing for GATE-2015 with us.
Fee for Postal Coaching:
GATE Rs 8500/-
Fee for Postal Coaching + Online Test Series:
GATE Rs 11500/-
Registration Details
Interested students can take admission by depositing requisite payment by any of the three given mode ( Cash / Cheque / DD ) of payment in
AXIS Bank
A/C No. 914020012764893
Payable to "GATE PATHSHALA EDUCATIONAL SERVICES LLP."
IFSC CODE : UTIB0001567
Branch : Alandur, Chennai
The DD shall be drawn in favor of “GATE PATHSHALA EDUCATIONAL SERVICES LLP.”, payable at Chennai. A copy of 10th class certificate, two pass port size photos and a copy of college ID or any other ID proof are to be enclosed to the application. The fee can also be paid by cash at our main office at cash counter along with the application form and necessary enclosures.
Adambakkam SRMUniversity,Potheri Shoulinganllure-mail:[email protected]:(+91)9962996817/8939174202 website:www.gatepathshala.comCorporateOffice:#153,II-Floor,Karuneegarstreet,Adambakkam,Chennai-600088
EDUCATIONALSERVICESLLP.CHENNAI
