Vibration of Continuous Media - Report Example

Vibration of Continuous Media First principles From the geometrical interpretation of differentiation, the definition of derivative of f(x) is a derivative. If the derivative exists for each point of the function, then it is defined as the derivative of the function f(x). Assuming f(x) is a real valued function, the function defined by lim{h right 0}{{f(x+h)~-~f(x)}/{h}} For this function, if the derivative exists at each point along the curve, then f’(x) = lim{h right 0}{{f(x+h)~-~f(x)}/{h}} This definition of derivative of f(x) is the First Principle of Derivatives. The gradient function is the function f’(x) or {dy}/{dx} is called Differentiation is the process of finding the gradient value of a function at whichever point on the curve, and the derivative of f (x). There are diverse ways of demonstrating the derivative of a function: {dy}/{dx}, {df(x)}/{dx}, f’(x), y’, {delta y}/{delta x}, and {y over} Example f(x + h)  =  5(x + h)  =   = = 5 f’(x)  =     =  5 Therefore f(x) = 5x; and f’(x) = 5. Deriving an equation from first principles implies that the equation is systematically proven based on the original principles of physics and mathematics which were postulated as a result of the basic researches and inventions that were postulated by scholars. There are the basic laws and theories of science which are recognized the world over and they form the basis upon which the later day scientific applications are based. The contemporary applications came as a result of manipulations of the initial principles of science. They are currently based on the hybrid equations theories and principles which hail from the maiden principle fronted by the scientific scholars. Putting an equation to conform to the initial principles of science is writing it in first principles. In regard to the equation in question, which is the cable equation; its first principles are the Newton equation and the Hookes equation. Newton is a renowned scholar who came up with numerous scientific principles including the principle of gravitational pull. It is from his principle of force that the cable equation veered. The Newton equation on force is as written below:- The equation forms the stepping stone of the cable equation. The Hookes equation on the other hand is as written below:- In regard to Newton’s equation, it follows that the equation for any kind of motion in respect to the weight of an object which is at the arbitrary location cited as x+h is effectively given or computed by a concerted manipulation which entails getting to equate these two desired forces as fronted form the first principle equations: In this case it is evident that entire time-dependence framework of the desired u(x) has with no doubt been made to be effectively explicit. Based on the above proven outcome, it follows that in the event that the array or list of weights weights in question in this computation which are arbitrary and hence indeed any other arbitrary eights do consist of inevitable N weights which as such are spaced evenly and stretched over the entire length L that is equal to Nh of the entire mass M that is equal to Nm, as well as to the sum total of the constant of the spring which is indeed of the desired array K that is equal to k/N. based on this conceptualizations, it follows that we can be able to comfortably write the entire above equation in the form that follows below: In regard to the stipulated equations, it is possible to compound the equations by making use of basic limits for the equation. Taking the limit N → ∞, h → 0 and assuming smoothness one gets: This is manifested by the vibration mechanism of a catapult as well. As the toy trigger is being pulled, tension is applied on the trigger bar, and this relies on newton’s law. The ball is given at first a high pushing force by the gun which makes it vibrate; this also relies on newton’s law. The system uses a number of scientific principles of operation, all of which revolve around the three Newton laws of motion. As the ball is fired from the catapult, it goes against the law of gravity to fly up in air; this is an itinerary of the Newton’s laws. As the ball lands on the ground it is aided by the force of gravity to land down. The motion of the ball on the air forms a trajectory movement which is also a function off the Newton’s Laws. The ball is given a traction power by the spring of the toy which accelerates it towards the destination. The weight of the ball and the force with which the ball is pushed are the functions of the acceleration, which is the second law of Newton. As the child continuously shoots the ball with the toy from different destinations and obtains different results, he will effectively learn that the magnitude of the distance which the ball will travel is dependent upon the amount of force applied in triggering the gun. This will slowly sink into the head of the child as he or she prepares to understand that the distance travelled is a function of the acceleration given to the ball, which in turn is a function of both the weight of the ball and the amount of force released from the toy springs by pressing it down. 2. Apply common boundary conditions for the transverse vibration of beams to obtain natural frequencies. The present examination concerns a disc of variable thickness of whose flexural stiffness $D$ varies with the radius $r$ according to the law $D=D_0 r^m$, where $D_0$ and $m$ are constants. The problem of finding boundary conditions for clasping this disc, which are inaccessible to straight observation, from the natural frequencies of its axisymmetric flexural oscillations, is measured. The problem in demand belongs to the class of opposite problems and is a very natural difficulty of identification of boundary conditions. The search for the unknown conditions for clasping the disc is equivalent to finding the span of the vectors of unknown conditions constants. It is shown that this inverse problem is well posed. Two theorems on the stability and a theorem on uniqueness of the solution of this problem are verified, and a process for instituting the unknown circumstances for fastening the disc to the walls is indicated. An estimated formula for defining the unknown conditions is attained using first three natural frequencies While the behavior of an oscillating bubble in an open volume has been thoroughly studied during the past several decades interactions between micro bubbles and a surrounding tube or vessel are not fully established. If the bubble radius is adequately small in comparison to the wavelength of an acoustic field, and the acoustic amplitude is low, then the resulting response of a bubble in an essentially open volume can be shortened to that of a spherically symmetric, linear oscillator (one-degree-of-freedom). Neglecting the pressure on the surface, the natural frequency of a bubble in a liquid is inversely proportionate to the radius of the bubble. This spherically symmetric model can be used to approximate the behavior of a bubble in a vessel only if the bubble radius is much smaller than the vessel radius. For the opposed extreme, if the bubble radius is comparable to the vessel radius, a one-dimensional linear model of a cylindrical bubble can be used to approximate the behavior of the bubble (Sassaroli and Hynynen, P. 3235). In an inelastic tube for current simulation results using a boundary essential method for a bubble in an inelastic tube that indicates that the big bubble estimation accurately calculates the bubble’s natural frequency when the ratio of the radius of the bubble to that of the tube is greater than 0.2. In a more recent work, a hypothetical model for a bubble oscillation in a rigid elastic tube was applied, where two frequencies are established with the lower one converging to the direction of the rigid tube value for enhancing stiffness of the tube (Martynov et al., P. 2970). Types of vibrations Vibrations are oscillations of objects or part of objects round a given point of equilibrium periodically or at random. They are mechanical phenomenon which are used to the advantage of designers and in many other instances are a disadvantage. Forced vibration takes place when an external force is intentionally applied periodically on a mechanical system. Free vibration takes place when only a start up force is applied on a mechanical system and it sets off to oscillate on its own continuously. Vibration absorbers are used in order to eliminate or minimize undesired vibrations. They work by the principle of absorption and isolation. There are different kinds of vibration absorbers used. One kind is the passive vibration absorber. These are made up of a spring and a damper. The vibration is softened by the spring which absorbs the energy. The oscillation is finally ended by the damper. Another kind of vibration absorbers is the dynamic absorber. This comprises of springs and an absorber mass. There is yet another kind of vibration absorber which is known as the impact vibration absorber. This works more like a pendulum. The pendulum mechanism absorbs the vibrations. It is majorly used in buildings and structures. A typical vibration absorber is the Negative-Stiffness-Mechanism passive vibration absorber. It is abbreviated as NSM. This is significantly unique in comparison to the classical linear absorbers. This particular absorber is capable of impacting all excited modes by suppressing them. It has a friction characteristic with a special shape. This influences in a small way the effect of suppressing excited modes in their totality. The influence is majorly done by a static friction adjusted threshold. The threshold is adjusted appropriately. It is specifically vital in absorbing and damping sub-Hertz vibrations. They can create quite easily compact and freedom of six-degree-of- systems giving them low natural frequencies. The load on the absorber is a forces P which acts by compression. Weight W also surpresses the spring by compression. This is at the position of operation of the absorber’s isolator. This is schematically represented in the first diagram. The isolator will have its stiffness given as K=KS-KN. Here KS represents the stiffness of the spring and KN represents the magnitude or amount of a negative stiffness. This is a function by mathematical approach of length of the damper’s bars together with the load P. It is possible to make the stiffness of the isolator to tend towards zero and at the same time the spring is still supporting the weight W. The above call for proper design of vibration elements which is a key element of engineering and is carried out to any piece of engineering equipment or piece of machinery to ensure that it ultimately conforms to all the desired parameters. In the event that an engineering component is made haphazardly with no consideration of design essentials, it would lead to ultimate losses in terms of the resources spent on making it, being wasted as it shall break down, or it shall lead to losses in terms of the component, causing harm to human beings or even to their property. Poorly designed components also fall in the same boat as the above mentioned components, hence it is vital that a proper design is done on all components and at the right time and manner. Design is done so as to produce engineering components which are of the right height, width, length; the right mass, weight, color, texture and made from the right material. The Negative-Stiffness-Mechanism passive vibration absorber uses typically three isolators which are stacked in series to one another: It has a tilt-motion isolator which is placed on top of yet another isolator, the horizontal-motion one. This is on top of another isolator known as the isolator of vertical-motion. In place of bars, this absorber uses flexures. A tilt flexure is used to serve as the isolator or damper of tilt-motion. Vertical-stiffness is adjusted using an adjustment screw. This is typically used for adjusting the forces of compression which act on the flexures of negative-stiffness. They in turn change the vertical stiffness. A vertical and steady load screw for adjustments is used for the purpose of adjusting and varying weight loads. This is achieved by raising and also by lowering the height of the base of the spring that supports. This keeps the flexures intently straight, and not bent at any instance during the operating positions. These dampers have a wide range of applications. They typically enable instruments which are vibration-sensitive to damp the undesired vibrations. These instruments include scanning probe microscopes which cannot tolerate vibrations at all. The other sensitive instrument in which this damper is used is the micro-hardness tester. This tester would give very wrong readings if no damper were used in it. The scanning electron microscopes are yet another instrument which is sensitive to vibrations. It as well uses this damper to alleviate vibrations. This damper enables these sensitive instruments to operate even in extremely unfavorable conditions of vibrations. This extreme conditions are common in areas like the top floors of tall buildings. Measurements from experiments of the frequencies that resonate, obtained from two bubbles in a stiff/rigid tube exhibited the same trends as models. However, they were approximately 14% higher for the natural frequency of the in-phase, and 20% higher for the single bubble and out-of-phase frequencies. The experimental outcomes that were previously obtained with a balloon model as opposed to agarose cylinders were only 5% greater than the numerical expectations of one bubble in a rigid tube. The reason for the increased difference is most likely due to the enhanced effect of the agarose rigidity on the response of the bubble. Plot of the 4 natural frequencies f versus elastic modulus E for 2 bubbles (R10=R20=1 cm) intermediate from the tube center (z1,z2=±1.5 cm) in a compliant tube with rtube=1.27 ... 2. The basic acoustic model comprises one or two bubbles filled with air within a cylindrical tube filled with liquid, submerged in a tank containing liquid. The dimensions of the tank are set to match conditions of experiment(radius of 0.1275 m and a height of 0.355 m) or selected to be adequate in order to minimize effects of boundary conditions (height of 1.4 m and radius of 0.5 m). unless stated differently, the tube has a radius rtube=1.27 cm, and the radius is 1 cm, a thickness ttube=0.3175 cm, and a length of L=20 cm. Assuming time harmonic motion with angular frequency ω, the subsequent eigenvalue difficulty is solved for the free vibrations frequencies and mode shapes. The overriding equation solved through each subdomain of fluids is ∇2p+ω2c2sp=0, Where c is the speed of sound, p is the pressure. The equation above is solved using specific boundary conditions a domain, which is axisymmetric. On the boundaries of the tanks, a boundary that is sound soft, where p= 0 is applied. A boundary that is sound hard, ∂p/∂n=0, is used to the surfaces of the in elastic tube. On the symmetry axis, ∂p/∂r=0 at r=0. More so, perforation of the tympanic membrane frequently occurs because of infection, high impulsive sound pressure, and external trauma, such as that connected to an explosion. Many diverse surgical techniques are vital in the repair of the ossicles and tympanic membrane. Clinical operations such as tympanoplasty are a solution in the repair of damaged ossicles and tympanic membrane, therefore improving hearing and decreasing the chance of infection. The membrane is replaced or repaired with the use of graft constituents, from either artificial sources or the patient’s body. Free transverse vibration of a thin elastic beam the equation considered as continuum is a derivation of Bernoulli-Euler assumptions. They include an axis x does not undergo any extension it is located along the longitudinal axis, which Is neutral perpendicular cross sections remain plane during the process of deformation material is elastic, linear and homogeneus The equation is usually in the form; EI  + A  = p (x, t) (5.319) where x is a coordinate (longitudinal), v is a transversal dislocation of the beam in y direction, perpendicular to x, t is time, I is the planar moment of inertia of the cross section, A is the cross sectional area, E is the Youngs modulus, and is the density. On the right hand side of the equation there is the loading p (x, t) - generally a function of time and space- acting in the xy level. For free transverse vibrations, we have zero on the right-hand side of Eq. If bending stiffness EI is independent of space and time coordinates the equation can look like +   = 0. (5.320) Assuming the stable formal vibration is in a harmonic form v(x, t) = V(x)cos(t - ) (5.321) Thin cantilever beam natural frequency determination Assuming the beam is clamped on the left, which is the origin of the coordinates, so x= 0 at that point. Then the length of the beam is L, which is on the right hand side coordinate x=L. Boundary conditions at the ends that are clamped require that the first derivation with the displacement be equal to zero, that is, V(x) = 0, (5.325)   = 0. (5.326) At the free end of the beam the shear force T(x) and the bending moment M(x) are equal to zero. The shear force is proportional to the first derivative with regard to space, and the moment is proportional to the second derivative of curvature (displacement). Therefore, the boundary conditions are M(L) = EI   = 0, (5.327) T(L) = EI   = 0. (5.328) The program VCR char_eq_beam.m computes the derivatives of the assumed solution Eq. (5.328) with respect the to spatial coordinate, substitutes the boundary conditions values into Eq. (5.329) to (5.332), forms the same system of four equations with unknowns C1 to C4 and regulates the organization matrix in the form [0 1 0 1 ] [lam 0 lam 0 ] [sinh(lam*L)*lam^2 cosh(lam*L)*lam^2 -sin(lam*L)*lam^2 -cos(lam*L)*lam^2] [cosh(lam*L)*lam^3 sinh(lam*L)*lam^3 -cos(lam*L)*lam^3 sin(lam*L)*lam^3] It is clear that lam stands for . It is obvious that a homogeneous system of equations has a non-trivial explanation only if its determinant is equal to zero. Therefore, the program computes the determinant of the above matrix that, after simplifications done by the program itself, appears in the form 1+cosh(lam*L)*cos(lam*L). Therefore, the frequency equation for a thin cantilever beam is coshL cosL + 1 = 0. (5.329) The frequency equation realized by the program VCRfreq_eq_beam_1.m. Look through it carefully. clear syms lam x C1 C2 C3 C4 L FD; V = ... C1*sinh(lam*x)+C2*cosh(lam*x)+C3*sin(lam*x)+C4*cos(lam*x); V1 = diff(V,x); V2 = diff(V1,x); V3 = diff(V2,x); V0 = subs(V,x,0) V10 = subs(V1,x,0) V2L = subs(V2,x,L) V3L = subs(V3,x,L) FD(1,1)=subs(V0,C2,C4,0,0); FD(1,2)=subs(V0,C2,C4,1,0); FD(1,3)=subs(V0,C2,C4,0,0); FD(1,4)=subs(V0,C2,C4,0,1); FD(2,1)=subs(V10,C1,C2,C3,C4,1,0,0,0); FD(2,2)=subs(V10,C1,C2,C3,C4,0,1,0,0); FD(2,3)=subs(V10,C1,C2,C3,C4,0,0,1,0); FD(2,4)=subs(V10,C1,C2,C3,C4,0,0,0,1); FD(3,1)=subs(V2L,C1,C2,C3,C4,1,0,0,0); FD(3,2)=subs(V2L,C1,C2,C3,C4,0,1,0,0); FD(3,3)=subs(V2L,C1,C2,C3,C4,0,0,1,0); FD(3,4)=subs(V2L,C1,C2,C3,C4,0,0,0,1); FD(4,1)=subs(V3L,C1,C2,C3,C4,1,0,0,0); FD(4,2)=subs(V3L,C1,C2,C3,C4,0,1,0,0); FD(4,3)=subs(V3L,C1,C2,C3,C4,0,0,1,0); FD(4,4)=subs(V3L,C1,C2,C3,C4,0,0,0,1); FD freq_det = det(FD) freq_det = collect(freq_det,lam^6); freq_det = freq_det/lam^6; freq_det = simplify(freq_det); freq_det = freq_det/2; disp(frequency determinant is ) disp([freq_det]) To determine natural frequencies of a cantilever beam we have to compute the roots of the frequency equation (5.333). To recalculate the - values into natural frequencies, the relations (5.327) and (5.334) should be used leading to =          i = 1, 2, 3,... (5.331) The equations of motion need the specifications for boundary conditions on each edge. This is reflected in Hamilton’s principles below. The in-plane deflection u1 or a modified Kirchoff share resultant T21 The transverse deflection u3 or a modified Kirchoff shear resultant V23 The second in-plane deflection, u2 or the resultant force N22 And the slope B2 or the bending moment M22 Similar to the beams, free edges make a shell less stiff than clamped edges, and other boundary motions, which provide interim values of how stiff the edges are (Soedel, P. 88). The formula below shows possible modes of vibration of a circular drumhead that are radially symmetric. Then, the function does not depend on the angle and the wave equation simplifies to We can seek for solutions in separated variables, substituting this in the equation above and dividing both sides by results to Common boundary conditions in fluid mechanics 1. No-slip at the wall. This boundary condition suggests that the velocity of a fluid in contact with the wall is the same as that of the wall. Walls are always immobile; therefore, the fluid velocity is zero. In drag flows, the velocity of the wall is limited and the wall velocity is equal to the fluid velocity. vp |     at the boundary = V   wall     (1) 2. Symmetry. In other flows, there is always a plane of symmetry. Given the fact that the field of velocity is the same on both sides of symmetric plane, t velocity must pass through a maximum or minimum at a plane of symmetry. Therefore, the boundary condition to apply is that the first velocity derivative is equal to zero at the plane of symmetry.    vp  xm       at the boundary =0     (2) 3. Stress continuity. When a fluid forms one of the boundaries of the flow, the stress pattern is continuous from one fluid to another. Consequently, for a viscous fluid in contact with an inviscid very low or zero viscosity fluid), this means, at the boundary, the pressure in the viscous fluid is the same as the pressure in the inviscid fluid. Subsequently, the inviscid fluid cannot support any shear stress (zero viscosity), it means that the pressure is zero at this boundary. The boundary condition between air and a fluid such as a polymer, for example, would mean that the shear stress in the polymer at the boundary would be zero. jk |     at the boundary =0     (3) Instead, if 2 viscous fluids meet and form a flow interface, this same boundary condition would necessitate that the pressure in one fluid equal the pressure in the other at the interface. jk(fluid 1) |     at the boundary = jk (fluid 2) |     at the boundary     (4) 4. Velocity continuity. When a fluid forms one of the boundaries of the flow as described above, the velocity is also continuous from one fluid to another.   vp(fluid 1) |     at the boundary = vp (fluid 2) |     at the boundary     (5 Dirichlet boundary conditions Dirichlet boundary conditions are the most frequently used type of boundary conditions. This category imposes a distinct value of a function on the boundary. It is easy to consider a condition like this, [Math Processing Error] of a domain [Math Processing Error] for a boundary [Math Processing Error]. There are two elementary ideas on the implementation of such conditions: 1. Overwrite degrees of autonomy with the anticipated value (strong implementation) 2. Present punishment terms as boundary integrals (weak implementation) FEAT2 provisions the "strong" application method (method 1) out-of-the-box. Dirichlet boundary conditions are continuously "discretized" into a boundary condition configuration t_discreteBC, which realizes nothing apart from collecting all points of freedom on the boundary composed of their corresponding values. In a second step, the t_discreteBC configuration is applied to vectors and matrices to transform their content (Ervedoza, Sylvain & Zuazua). 3. Describe characteristics of vibrations in membranes with examples A membrane is an element that is relatively thin in a given direction in comparison to the other two, and is always flat. It is in a stretching shape a plane, thereby being able to withstand tensile stresses. To the contrary, it is impeccably flexible with regard to bending. The main difference with the transverse vibration is the fact that the transverse vibration is a characteristic of displacement, while this membrane vibration is a function of two variables, which are x and y. Surfaces of membranes constitute speakers, microphones, and other devices components. They may be applicable in the study of two dimensional propagation and wave mechanics. In bioengineering, most tissues in the human body are considered as membranes. This is because of the vibration features of an eardrum in understanding sound. The design of the human year consists of membranes that vibrate to process sound. The vibration characteristics of the tymphanic membrane in the grass frog, measured using a lasser doppler (velocity meter) tested the extent to which the acoustic system of a frog acts as a pressure gradient receiver. This explains how the frog processes sound. The second scenario of characteristics of vibration in membranes is in this experiment process. A conventional settlement and coagulation method was handy in the treatment of fine carbon-loaded wastewater from TV picture tube plants. A large amount of solid waste was created because this method is practically expensive. The presentation of a vibration membrane to compact with this kind of wastewater is manageable and cost effective. The recovered water can be recycled and the concentrated fine considered in the production process. This work presents a parametric study on recovery of water from fine carbon-loaded wastewater by use of a vibration membrane. Some of the yardsticks considered in the experiment were the type of membrane, vibration characteristics, and concentration. The characteristic feautrue of a membrane is without bending resistance so that its restoring forces area s a result of tension only. This is similar to one-dimensional stretched string in general, the formulation of the vibration of a membrane bases on the following assumptions: i. Uniformity and thinness of the membrane ii. The position of equilibrium of the membrane iii. The displacement w at any point on the diagram is towards a direction normal to the plane of reference iv. The displacement w is so small in that the tensions T remains at a constant during vibration. Free vibration of a rectangular membrane z C(0,b0) B(a,b,0) x y b o(0,0,0) A(a,0,0) x Figure 1: The cartesan system of coordinates for an a x b membrane of rectangular shape 4. Definition of fundamental frequency and velocity of wave propagation The fundamental frequency is the lowest frequency produced by any given instrument. It is also called the first harmonic of an instrument. While on the other hand, wave propagation involves the transmission of information from a location to another. The velocity differs depending on the characteristics of the carrier that determines the signal. The diagram below shows wave propagation. References Soedel, Werner. Sound and Vibrations of Positive Displacement Compressors. Boca Raton, FL: CRC Press, 2007. Internet resource Sassaroli E., and Hynynen K. ,“Forced linear oscillations of microbubbles in blood capillaries,” J. Acoust. Soc. Am. 2004. 115(6), 3235–3243.10.1121/1.1738456 [PubMed] [Cross Ref] Martynov S., Stride E., and Saffari N., “The natural frequencies of microbubble oscillation in elastic vessels,” J. Acoust., 2009. Soc. Am. 126, 2963–2972.10.1121/1.3243292 [PubMed] [Cross Ref] Ervedoza, Sylvain, and E Zuazua. Numerical Approximation of Exact Controls for Waves. New York, NY: Springer, 2013. Internet resource. Arnold P.J and Gerald M.N. The vibration absorbers in the modern day application. Australia (2003).Sydney. Henry Kissinger. Passive vibration absorbers and dampers. New Jersey Engineering Society 1992. Patel M.N. Journal of mechanical vibrations and dampers. (2001)I India. New Delhi. Meijaard, J.P; P.J: A family of embedded Runge-Kutta formulae. J Comput Appl Math 6 Read More