Dinamik absorberlerle titreşim kontrolu

Abstract

ÖZET Titreşim kontrolunda kullanılan çeşitli metodlar -arasın da, dinamik titreşim absorberlerinin özel bir yeri vardır. Ba sittir kütle-yay sistemi olarak düşünülen ilk dinamik absorber, araştırmacılar tarafından geliştirilerek, kütle-yay sistemlerin den başka, kiriş, plak ve kabuk titreşimlerinin sönümlenmesinde de başarıyla kullanılmıştır. Ancak, modern teknolojinin günümüz de bilhassa uzay çalışmalarında ulaştığı hız, titreşim kontrolünün önemini arttırmış, araştırmacıları dinamik absorberler üzerinde yeniden düşünmeye sevketmiştir. Bu konuda beliren en dikkate değer görüş, kirişlerin de absorber olarak kullanılabi leceğidir. Bu çalışma, klasik kütle-yay absorberinden sonraki aşama yı, kiriş biçimli dinamik titreşim absorberini araştırmakta; bu yeni tip absorberle, kiriş ve plak titreşimlerinin etkin biçimde kontrol edilebileceğini teorik ve deneysel olarak göstermektedir Çalışmanın birinci bölümü, titreşim kontrolunda kullanılan çeşitli metodlar arasında dinamik absorberin yerini, genel teo risini ve konu üzerindeki araştırmaları özetlemektedir. İkinci bölüm ise, çalışmanın temel konusu olan kiriş bi çimli absorberin, kiriş titreşimlerine uygulanması hakkındadır. Bu bölümde kiriş biçimli absorberin, ana sisteme tabii frekans ve genlik yönünden etkisi araştırılmakta; rezonans tehlikesinin uzaklaştırılması açısından absorber optimizasyonu yapılmaktadır., Üçüncü bölümde, kiriş biçimli dinamik titreşim absorberle rinin plak titreşimlerinin sönümlenmesinde kullanılması amacı il-VI- dikdörtgen plaklar sınır şartları yönünden araştırılmış* uygu lama için seçilen endüstriyel önemi haiz dikdörtgen konsol (üç kenarı serbest, bir kenarı ankastre) plağın tabii frekans ve mod şekilleri üzerinde literatürde mevcut çalışmalar elenmiştir» Dördüncü bölümde, karesel konsol plağın titreşim kontrolü için, mesnetleme şekli değişik, kiriş biçimli iki ayrı absorber geliştirilmiştir. Bunlardan birisi plak ucuna, ankastre; diğeri ise iki taraflı viskoelastik yaylarla bağlanmaktadır. Ti. f- resim probleminin çözümü, yaklaşık varyasyonel metodlardan Ritz Meto du ile yapılmıştır. Çalışmada, titreşim araştırmalarının çoğunda yapılageldi- ği gibi, sönüm elemanı olarak sembolik viskoz damper kullanıl mamış, kompleks elastisite modülü yardımıyla parçaların iç sür tünmelerinden kaynaklanan malzeme sönümü (histeretik sönüm) he-> saba katılmıştır. Bu da Ritz Metodunun verdiği lineer denklem takımının kompleks katsayılı olmasına yol açmıştır. Ortaya çıkan bu matematik problem, nümerik analiz yöntem leriyle bilgisayarda çözülerek, absorber eklenmiş plağın zor lanmış titreşimi incelenmiş; absorber tabii frekansları, kütlesi, « izafi malzeme ve yay sönümleri parametre olarak alınıp, sistem frekans-genlik karekteriş tikleri çıkartılmıştır. Ayrıca, bu parametrelere uygun değerler verilerek optimizasyon sağlanmıştır. Beşinci bölümde, teorik olarak geliştirilen kiriş biçimli dinamik titreşim absorberinin deneysel gerçeklenmesi anlatılmak tadır. Altıncı bölüm, teori ve deney sonuçlarını karşılaştırmakta, sonuçlar bölümü ortaya çıkan bazı önemli noktaları özetlemektedir..VII Metin içine sığdırılmasında güçlük bulunan şekil, tablo ve bilgisayar programları ile, konunun anlaşılmasında yardımcı olacak bazı bilgiler EK'lerde verilmiştir.

SUMMARY With the increase of tendency to high speeds and light weight structures in modern technology, the analysis of vibra tion problems becomes more and more important, especially in mechanical engineering design. Because excessive dynamic stresses may destroy structures during resonance, it is necess ary to eliminate or reduce their severity or, alternatively, to design equipment to withstand their influences. Methods of vibration control may be grouped into three broad categories: 1. Reduction at the source. 2. Isolation. 3. Reduction of the response.IX Among them, reduction of the response is most common and can be realized by three ways:. a) Alteration of Natural Frequency: If the natural fre quency of the structure of an equipment coincides with the fre quency of the applied force or displacement, the vibration condition becomes much worse as a result of resonance. Under such circumstances, if the frequency of the excitation is substantially constant, it is often possible to alleviate the vibration by changing the natural frequency of such structure. b) Energy Dissipation: If the vibration frequency is not constant, the desired reduction of vibration can be achieved by the dissipation of energy to eliminate the severe effects of resonance. c) Auxilliary Masses: Another method of reducing the vibration of the responding system is to attach an auxiliary mass to system; with proper tuning, the mass vibrates and reduces the vibration of the system tö which it is attached. When the auxiliary mass system has a little damping, it is called a `dynamic absorber`. If damping is provided in aux iliary systems, the names `damped absorber` or `auxiliary mass damper` are given to this type of system. In the past three-quarters of the century the dynamic vibration absorber has proven to be a useful device to limit undesirable vibration in hundreds of diverse applications in machine design and structural dynamics. The conventional dynamic absorber comprises a mass that is attached by a single spring and viscous damper to a vibrating mass system. Eversince the concept of the conventional dynamic absorber was described by J. Ormondroyd and J. P. Den Hartog ^X in 1928, much has been written` about the dynamic absorber and, in particular, how it should be tuned and damped; that is, how values for the absorber spring stiffness and the coefficient of viscosity of the damper should be chosen to provide optimum absorber performance. Not only has the attachment of dynamic absorber to resonating items on resilient members been discussed, but the attachment of conventional dynamic absorber to distrib uted mechanical systems such as rods, beams, and plates has been considered; In recent papers, the use of `auxiliary distributed systems` as dynamic vibration absorbers is suggested and employed to reduce excessive vibration of lumped parameter systems and beams. This work considers the use of beams as dynamic vibration absorbers applied to beams and cantilever plates in the presence of harmonic forces. Structural damping is incorporated into the main and auxiliary systems by treating them as having a complex elastic modulus. An analysis is performed and verified experi mentally. In the first chapter, methods of vibration control are groupped and summarize/1. The second chapter considers the use of a double-ended cantilever beam as a dynamic vibration absorber applied to clamped-clamped beams subject to Bernoulli-Euler equation. The absorber is tuned to suppress the fundamental `'mode and the effects of variations of the absorber mass, tuning ratio, and loss factor on the response and natural frequencies are discussed. Optimal tuning and mass ratio are found for a given structural damping factor.XI The third chapter is a preliminary work to apply dynamic absorbers of beam type to internally damped cantilever rectangu lar plates. It can readily be shown that the equation governing small amplitude lateral vibration of a thin plate may be written as DV^W-h phW = q (0.1) where D : flexural rigidity p : mass density h : thickness. q : distributed load It becomes quickly apparent to researchers studying the rectangular plate vibration problem that plates with classical edge conditions (clamped, simply supported, or free) fall into one of two families. In the more easily solved family of problem: are to be found those plates which have at least one pair of opposite edges simply supported. It is shown that exact solution! of the Levy type are easily obtained for these plates. The remaining rectangular plate problems may be considered to fall into a seperate family which may be much more difficult to solve. The difficulty lies in the fact that the Levy type solution cannot be used directly as it is not possible to choose eigenfunctions of one variable which satisfy exactly the boundar conditions at two opposite edges, and permit seperation of vari ables when the solution is substituted into the governing differential equation. The cantilever plate problem which is of strong industrial interest does not immediately lend itselfXII to a Levy type solution. In fact, because of the free edges, it is considered to be one of the more difficult problems to solve. Necessity is emphasized to resort to approximate methods for the cantilever plate and some of them employed by various investigators to find the natural frequencies and nodal patterns are summarized. In the fourth chapter, the procedure developed by Ritz is used to calculate natural frequencies and amplitudes of a square cantilever plate to which distributed parameter dynamic absorbers are attached. In order to develop the equations of motion, Hamilton's principle is used. The elastic and kinetic energies of a plate in terms of plate displacement are as follows: W-Wo(xfy)-6lü)t- (0.2) n a2w 32w 2 a2w 82W V=Ş//{(+_) - 2(1-V){ -..? / 3x2 9y2 dxZ dy2 32W 2 - (- - ) }> dxdy (0.3) 8x3y T = -~ //hp(-^) dxdy (0.4) The Lagrangian is defined as L -= T - V (0.5) and satisfies the Hamilton's principle as follows; in 1 2 8ft dt '- 0 (0.6) tiXIII The direct application of the calculus of variations to this equation leads to the partial differential equation (0.1), Instead of following such a procedure, Ritzfs method consists of assuming the deflection W (x,y) as a linear series of `admissible` functions and determining the coefficients in the series so as to satisfy equation (0.6). For the cantilever plate with the edges parallel to the x-and y-axes, series is taken for W in the form o Wn(x,y) = Z Z C X (x).Y(y) (0.7) o ? _ mn m n m n which functions X and Y must be `admissible`» that is they m n.. > J must satisfy the geometric boundary conditions. When W (x,y) as given by equation (0.7) is substituted in equation (0. 5), the right-hand side becomes a function of coefficients C mn This is extremized by taking partial derivatives with respect to each coefficient and equating them to zero. Thus, a set of linear homogeneous equation is found. The natural frequencies are determined from the condition that the determinant of the system must vanish. *. The accuracy of results and the practicability of computa tions depend to a greater extent upon the set of function that is chosen to represent the plate deflection. In this investiga tion, use is made of the functions which define the normal modes of vibration of uniform beams. To calculate the amplitudes in case of forced vibration, structural damping is introduced by using complex elastic modulus as follows, E*= E(l + /p/27T* (0.8)XIV Analytical investigation arid experiments show that the most troublesome vibration amplitude is experienced during the first symmetric mode of rectangular cantilever plates. To reduce its severity, a cantilever beam is attached to the plate as a vibration absorber. This problem is analyzed by employing the principle of superposition and the solution of the beam; for the absorber force and moment in terms of the vibratory ampli tude of the plate where the absorber is attached. Tuning is determined to suppress the fundamental mode, Then the effects of tuning, mass ratio, and structural damping on the natural frequencies and corresponding amplitudes are investigated. Optimal values are established by varying them until the two peaks in the resultant amplitude curve of the plate lie on the same horizontal. As a second dynamic vibration absorber a flexible beam, joined by way of visco-elastic links at the ends to the plate, is evaluated. The device has the advantage of relying on the interaction between the beam stiffness and the link stiffness for tuning, rather than on the link stiffness alone as for the conventional dynamic absorber. This problem is analyzed by adding the potential energies of the forces transmitted back to the plate in terms of the vibratory amplitudes of the plate where visco-elastic members are attached. With the appropriate selection of the complex spring stiffness, mass and tuning ratio, and structural damping factor, resonances of the plate are suppressed in a uniform and symmetrical manner. In order to verify the analysis, experimental investi gation is performed and the results are presented in the fifthXV chapter. Primary systems are made in the same dimensions as in the theorotical analysis, and experimental absorbers are made to represent as closely as possible the dimensions deter mined in previous chapters. A sinusoidally varying force is supplied by means of an electro-dynamic exciter and measurements are taken by using contactless inductive transducers. The main results of this work can be summarized as fol lows : 1. The procedure developed by Ritz gives accurate upper limits by using a 21-term series based on taking twelve terms for symmetric modes and nine terms for antisymmetric ones. 2. For very low values of structural damping factor the original resonant peak of the primary system is, replaced by two high peaks, one above and one below the original resonant frequency. The new natural frequencies change with varying values of tuning and mass ratio. Damping effect on them is not considerable. With proper tuning and mass ratio dynamic ab sorbers splits the original resonant frequency into two new ones which may be quite effective when constant frequency excitation is of concern. 3. In case the excitation frequency is not constant, the system at new frequencies could also be a subject to significant vibration. If, however, the loss factor is fairly high, the two peaks still occur but their amplitudes are reduced considerably because of energy dissipation. The amplitudes of new resonances vary with tuning, mass ratio, and loss factors, of absorber beams and resilient members supporting them. Optimal values are determined so as to suppress the most troublesome resonance of the plate in a uniform and s ymme t r i c a 1 cianne r.XVI When the absorbers' are very heavily damped their masses can be considered to be attached to the primary system in a rigid manner and no more reduction at the amplitudes may be obtained. The absorbtE beam on visco-elastic links is found more effective when it is tuned to vibrate in bending modes than to vibrate in vertical translational modes, such as in conven tional tuned damper for the same mass and spring stiffness.