Analytical and Finite Element Study for Friction Pendulum with Parameterized Sliding Surfaces

Transcription

1 Proceeings of the Worl Congress on Engineering Vol III WCE, July - 6,, Lonon, U.K. Analytical an Finite Element Stuy for Friction Penulum with Parameterize Sliing Surfaces Anrea Amalia Mina, Member, IAENG, Gilbert-Rainer Gillich Member, IAENG, Clauiu Mirel Iavornic, Member, IAENG an Petru Florin Mina, Member, IAENG Abstract The new approach propose by the authors in this paper consists of implementing friction penulums having the sliing surface profile base on a polynomial function of superior orer. This seismic isolation system permits, for each surface, a greater flexibility in controlling the oscillations of the upper structure, in terms of isplacements or issipate energy, by using three parameters to control the forceisplacement relation. Inex Terms isolation system, issipate energy, friction penulum, finite element analysis F I. INTRODUCTION or safety an integrity reasons against earthquakes, effective an reliable techniques are esire. The optimum solution against earthquake effects is to control an limit the transfer of energy from the groun to a superstructure that means to issipate this energy []. For these reasons, the actual tren is represente by using the friction penulum. This is a stury passive evice, capable to sustain significant loas an to assure the control of oscillating structures []. Friction penulums can be classifie, accoring to the number of sliing surfaces, in mono-armature or multiarmature [3]. While classical friction penulum bearings contain one or more cylinrical or spherical sliing surfaces, each having a raius of curvature R an friction coefficient µ, they provie just pairs of two parameters controlling the ynamic behaviour of a structure isolate by this kin of evice. The friction penulum system with one concave spherical shape sliing surface is presente in [] while the Manuscript receive March 8, ; revise April 6,. This work was supporte in part by the Managing Authority for Sectorial Operational Program for Human Resources Development (MASOPHRD), within the Romanian Ministry of Labor, Family an Equal Opportunities by co-financing the project Doctoral Scholarships investing in research-evelopment-innovation for the future (DocInvest) POSDRU/7/.5/S/7683. A. A. Mina is with the Economical Engineering Department, Eftimie Murgu University of Resita, 385 Romania. ( a.mina@uem.ro). G. R. Gillich is with the Mechanical Engineering Department, Eftimie Murgu University of Resita, 385 Romania. ( gr.gillich@uem.ro). C. M. Iavornic is with the Mechanical Engineering Department, Eftimie Murgu University of Resita, 385 Romania. ( c.iavornic@gmail.ro). P. F. Mina is with the Mechanical Engineering Department, Eftimie Murgu University of Resita, 385 Romania. ( f.mina@uem.ro). ouble concave bearing was presente in [5]. A new base isolator, calle a variable curvature friction penulum system, was introuce by Tsai, Chiang an Chen in [6]. The ifference between the variable curvature friction penulum system an the friction penulum system is that the isolator s raii can be lengthene with the increase of the isolator isplacement. Pranesh an Sinha introuce in [7] an [8] a variable curvature friction penulum system with the sliing surface base on an ellipse. The major axis of the ellipse can be lengthene with increase of sliing isplacement. They have mae analysis in terms of behaviour of the isolate structure, without being analyze the behaviour of the friction penulum itself. The new approach propose by the authors in this paper consists of implementing friction penulums having the sliing surface profile base on a polynomial function of superior orer. This seismic isolation system permits, for each surface, a greater flexibility in controlling the oscillations of the upper structure, in terms of isplacements or issipate energy, by using three parameters to control the force-isplacement relation. Namely, these parameters are the friction coefficient µ, an two characteristic values of the curve on which the sliing surface is constructe. Opposite to classical systems, where the contact between the slier an the sliing surface assure almost uniformly istribute contact stress even for metallic components, the new evices make use of an elastic element between the slier an the sliing surface. A quasi-uniform pressure istribution on the sliing surface can be obtaine in this way. Static an ynamic analyses were performe using the finite element metho confirming that the use of an elastomeric element is suitable for this purpose. II. MATHEMATICAL MODEL OF SIMPLE FRICTION PENDULUM WITH SLIDING SURFACES GENERATED BY POLYNOMIAL FUNCTIONS In the case of the simple friction penulum, with cylinrical or spherical sliing surface, the control of the movement can be mae with two parameters: the raius of the penulum R an the friction coefficient. The height variation h expresse in terms of an then in terms of isplacement u, is: ISBN: ISSN: (Print); ISSN: (Online) WCE

2 Proceeings of the Worl Congress on Engineering Vol III WCE, July - 6,, Lonon, U.K. h R cos R cos arcsin u () R The friction force value is Nsignu () The signum function is equal to - or + epening on whether u is negative or positive, respectively. If we consier the motion in the positive irection of Ox axis, we can take signu, so F f expresse in terms of is Gcos, where G is the vertical loaing resulting from the structure. To simplify calculations, we consier further only the positive Ox axis, i.e. positive values of u, because the curve is symmetrical to the vertical Oz axis. Fig.. A simple friction penulum scheme with the forces acting on circular sliing surface On the interval,u the issipate energy ue to friction is: or u uu u u Gcosu G cosu (3). By changing the variable u R sin, one obtain u R cos, an consequently the integral calculate on,u becomes an integral calculate on,, hence the issipate energy becomes: E GR cos () where arcsin u R (5) Solving the integral from (), we obtain: GR sin cos (6),u, The issipate energy on the interval expresse in terms of imum isplacement u is: u u GRarcsin R u R R (7) Fig.. A friction penulum scheme with polynomial sliing surface Norms limit the value of the angle to 36 ; for the polynomial functions having b, b,5 an b we calculate the imum isplacements u an the imum heights h calculate for ifferent values of a, presente in the table below. b,5 TABLE I u a h,,5,5 5,3,57,5 66,3,8,9 8,75,3,3 7,3,97, 9 6,5,,3 8,8,, 5 83,,96, 8 75,5,6, 89 For a circular sliing surface with the raius R 5, for 36 we have etermine the values of u =,93 an h,98. For the friction penulum with the sliing surface generate by polynomial functions we consier the equation of the curve that generates the sliing surface as: b y a u (8) ISBN: ISSN: (Print); ISSN: (Online) WCE

3 Proceeings of the Worl Congress on Engineering Vol III WCE, July - 6,, Lonon, U.K. Fig. 3. Graphical representation of ifferent curves that generates the sliing surface In Fig. 3 we have shown the graphical representation of a circular arc with the raius R 5 (with otte line) an the arcs of equations,5 y, 75 u y,u (with ashe line), (with soli line) an (with ashot line). The issipate energy is: where u uu y,8u (9) Nsign u is the friction force. If we consier the motion in the positive irection of Ox sign u, thus F G cos. axis, we can take Hence, the issipate energy is: u u Gcosu G cosu f () At the point A we have b tg h a b u instantaneous isplacement in terms of is: b u tg a b an consequently b b u tg ab b cos. The () The integral on [,u ] becomes an integral on, ], where is: [ so: b arctg a b u () b b b G tg ab b cos (3) b If we enote b an c we have: b ab b for the case that b : c G F,,,cos cos where F,,,cos t t b t cos for b we have: () cos sin ln cos sin E G a t (5) In Fig. we have represente the issipate energies for the ifferent sliing surfaces generate by the curves represente in Fig. 3. One can observe that for higher values of u there are significant changes in the issipate energy for the analyse cases. Fig.. The issipate energies for the ifferent sliing surfaces The kinetic energy of the structure of mass m is given by mv Ek where v is the velocity. We enote by E the imum potential energy, given by: p Ep mgh So, if friction is not consiere E E E (6) k p p mv mgh mgh an thus the imum velocity, attene for u is v g h h (7) ISBN: ISSN: (Print); ISSN: (Online) WCE

4 Proceeings of the Worl Congress on Engineering Vol III WCE, July - 6,, Lonon, U.K. Fig. 5. The evolution of velocities an the heights In Fig. 5 we have presente the evolution of velocities an the heights while the structure moves from the point of imum vertical isplacement to point O ( with u ), for ifferent types of sliing surfaces generate by,5 y, u (with ashe line), y, 75 u (with soli line), y,8u (with ash-ot line) an a circular arc with the raius R 5 (with otte line). If we consier friction, the velocity in point O is given by E v gh h (8) m where E is the issipate energy ue to friction when the structure slies from point A to point O (see Fig. ). In the next pictures (Fig. 6, Fig. 7, Fig. 8 an Fig. 9) we have presente the evolution of velocities, first for the sliing without friction (with ash-ot line), next for the sliing with energy issipation ue to friction (with soli line), for the sliing surfaces generate by a circular arc,5 with the raius R 5 (Fig. 6), y, 75 u (Fig. 7), y, u (Fig. 8) an for y,8u (Fig. 9). Note that ue to friction the structure no longer climbs to the same height it woul climb if it were frictionless. Fig. 7. The evolution of velocities for the sliing surface generate by y, 75 u,5, for the sliing with an without friction Fig. 8. The evolution of velocities for the sliing surface generate by y, u, for the sliing with an without friction Fig. 6. The evolution of velocities for the circular sliing surface, for the sliing with an without friction Fig. 9. The evolution of velocities for the sliing surface generate by y,8u, for the sliing with an without friction ISBN: ISSN: (Print); ISSN: (Online) WCE

5 Proceeings of the Worl Congress on Engineering Vol III WCE, July - 6,, Lonon, U.K. TABLE II III. FINITE ELEMENT STUDY AND RESULTS Earthquake isolation systems are evices locate at the base of overhea structures, which have the aim to reuce the structure oscillating perio uring an earthquake an ecrease the transmitte energy using frictional issipation. Classical isolation systems, with cylinrical or spherical sliing surfaces have the parts mae of stainless steel, which provies urability. During earthquake, the slier an the concave surface of the lower plate maintain contact an consequently a relative uniform stress istribution is assure. For other types of sliing surfaces, limite contact occurs. To prevent this kin of inconvenience, namely to obtain a continuum contact between the slier an the concave plate, it is necessary to insert an elastomeric layer between the slier an the sole, as presente in Fig.. This layer is similar to that use in elastomeric base isolation systems, assuring high portability [9]. So we can avoi incomplete contact (spotte or linear) an obtaining a controlle eformation for fit the concave shape of stationary plate. Fig.. Friction penulum with polynomial generate surface The finite element analysis was mae taking into account the real-scale moel of an earthquake isolation system an transposing the whole loas, contacts an supports to virtual moel using avance nonlinear analysis software. Five curvature cases were analyse, one cylinrical an four base on surfaces generate by polynomials, like that presente in Fig.. Case Pressure. [MPa] Arc.557 Curve.3 Curve.59 Curve 3.55 Curve.33 Contact pressure shape [s] (sole) Three main types of contacts were use for seismic isolation system components: frictionless contact between first concave plate an slier, frictional contact (µ=.5) between secon concave plate an sole, boune contact between elastomeric layer an sole, elastomeric layer an slier respectively. Immeiately after applying the vertical loa, the sole is elastically eforme an in the contact area a contact patch is forme ue to the pressure exerte by the elastomeric layer on the opposite sie. This patch can be quantitatively evaluate using the contact pressure parameter. After secons, uring the time perio in which the horizontal isplacement is impose because the gliing assembly has been isplace, the contact patch becomes irregular with a slight increase of the pressure value in the rear of the sole, as we can see in Table II. The normal stress istribution for entire assembly together with its imum value, at the en of the analyse perio, is presente in the Table III. Case Stress.[MP] TABLE III Normal stress [s] (Assembly) Curve.997 Curve.3 Arc.93 Curve 3.78 Fig.. Different types of sliing surfaces For all cases, uring the first secon a vertical increasing loa was applie, until it reaches the weight of superstructure. Afterwars, until the en of the consiere time interval of secons, a preetermine course (simulating horizontal oscillation uring an earthquake) was impose. Curve.73 Curve.3 From the entire sets of results obtaine using finite element analysis (FEA) we chose to present the variation of the contact pressure between the upper concave plate an the slier for all analyse cases, etermine after an secons respectively. ISBN: ISSN: (Print); ISSN: (Online) WCE

6 Proceeings of the Worl Congress on Engineering Vol III WCE, July - 6,, Lonon, U.K. [] V.A. Zayas, S.S. Low, S.A. Mahin, A simple penulum technique for achieving seismic isolation, Earthquake Spectra 6, , 99. [5] MC. Constantinou, Friction Penulum ouble concave bearing. NEES Report, available at: - DC%Report-DEMO.pf,. [6] CS. Tsai, TC. Chiang, BJ. Chen., Finite element formulations an theoretical stuy for variable curvature friction penulum system, Engineering Structures 5, 3, [7] M. Pranesh, R. Sinha, Aseismic esign of structure-equipament systems using variable frequency penulum isolator, Nuclear Engineering an Design 3,, [8] M. Pranesh, R. Sinha, VFPI: an isolation evice for aseismic esign, Earthquake Engn. Struct. Dyn., 9, [9] C. M. Iavornic, Z.I. Praisach, O. Vasile, G.R. Gillich, V. Iancu, Stuy of Stress an Deformation in Elastomeric Isolation Systems Using the Finite Element Metho, Recent Avances in Signal Processing, Computational Geometry an Systems Theory - ISCGAV', ISTASC', pp. 39-,. Fig.. Contact pressure between the upper concave plate an the slier etermine after an secons respectively This iagram was obtaine evaluating the contact region between upper concave plate an slier after the first time interval of [s] when the loa is vertical an pressing the assembly an then at the en of the secon time interval of [s] when the loa is combine (horizontal an vertical). IV. CONCLUSION For classical earthquake isolation systems the ynamic response control is limite ue to the reuce number of control parameters. In orer to avoi this eficiency, this stuy shows the optimal way to improve the ajustment of elastic characteristics for a large exciter frequency spectrum of the earthquake. The use of plates with concave surfaces generate by polynomials, for the case of friction, ensures higher spee near the equilibrium position, thus better conitions to restore the position prior to the earthquake. Profile curves generate by polynomial functions show the possibilities to achieve the goal of superior, controlle energy issipation. While for the spherical surface the slier has the same shape with the concavity of the plate, for the sliing surfaces generate by other curves we must interpose, between the slier an lower armature, an elastomeric element able to follow the changes of the curvature. Finite element analysis prove that the normal stress values are reuce in case of use this kin of earthquake isolation system; all values for stress magnitue fall in the allowable range. REFERENCES [] T. Hyakua, K. Saito, T. Matsushita, N. Tanaka, S. Yoneki, M. Yasua, M. Miyazaki, A. Suzuki, T. Sawaa, The structural esign an earthquake observation of a seismic isolation builing using Friction Penulum system, Proceeings, 7th International Seminar on Seismic Isolation, Passive Energy Dissipation an Active Control of Vibrations of Structures, Assisi, Italy,. [] Y. B. Yang, T. Y. Lee an I. C. Tsai, Response of multi-egree-offreeom structure with sliing supports, Earthquake Engrg. Struct. Dyn. 9, 739Ð75, 99. [3] P. Roussis, M.C. Constantinou, Uplift-restraining friction penulum seismic isolation system, Earthq. Eng. Struct. Dyn. 35: , 6. ISBN: ISSN: (Print); ISSN: (Online) WCE