PROJECTING ELASTOMERIC SHOCK ABSORBERS WITH MOVING SIDE STOP. Gonca, V.; Shvab, J.; & Noskov, S.

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1 8th Internatonal DAAAM Baltc Conference "INDUSTRIAL ENGINEERING" Aprl 01, Tallnn, Estona PROJECTING ELASTOMERIC SHOCK ABSORBERS WITH MOVING SIDE STOP Gonca, V.; Shvab, J.; & Noskov, S. Abstract: In the desgn of Vbraton solator t s often necessary to ensure a reducton (or ncrease) of the value of frequency η r, that requres the ablty to calculate the stffness characterstcs of elastomerc Vbraton solator n the low fnte and medum fnte deformatons. A smlar problem arses n the desgn and calculaton of equfrequent rubber-metal compensatng devces, whch fnd applcaton n varous felds of engneerng and constructon ndustres, effectvely replacng the hydro- pneumatc - sprng compensatng devces, workng under axal stress-stran. In ths case, the stffness characterstc of force settlement P = P (Δ), even for small fnte deformatons, wll be non-lnear (or pecewse lnear). In ths ttle proposed a method for determnaton of rgdty dependence "Force - Settlement" for shock-absorbng elements wth absolutely rgd movng (parallel to the vertcal axs z) vertcal sde stops beng under pressure, and t s taken nto account low compressblty of materal of rubber layers. Receved solutons can be used to fnd the dependence force - settlement of cylndrcal shock absorbers, as well as n projectng such shock absorbers. Keywords: rubber, shock-absorber, weak compressblty, sde stops, stffnes. 1. INTRODUCTION To reduce the harmfulness of vbratons n engneerng structures are wdely used rubber-metal vbraton solators of dfferent geometry, wth unquestonable advantages [] compared to vbraton solators made from other materals. In the theory of solaton of vbratons attenuaton coeffcent transfers power from the source of vbraton through the vbraton solators on the base s ntroduced for the frequency η > 0,5 η r, where η r mass s resonance frequency of vbraton source [1] : ( c / M ) 0.5 ηr = (1) π where: c - stffness of vbraton solator; M - mass of vbratng mechansm. The man task n the desgn of elements of vbraton solator s to provde basc reducton of the resonance frequency below η r frequency range of vbroactve dsturbng forces. At the same tme statc and dynamc dsplacements of vbraton source do not shall to exceed the permssble values. For below resonance modes: ( g / ) 0.5 η = 0 r () π where: g - acceleraton of gravty; Δ 0 statc settlement of the elastc elements of vbraton solator, f we assume that the stffness of vbraton solator does not depends on the frequency and the load value. From () yelds, that only for small deformatons ncrease of permssble settlement /Δ 0 / decreases the fundamental

2 frequency η r (snce there s a lnear relatonshp for the rgdty characterstcs of the "force - settlement") of elastomerc solator [1,3]. For large deformatons stffenng dependence "force - settlement" of elastomerc vbraton solator has sgnfcantly non-lnear behavor of the hard type [4] and from () yelds, that frequency η r can grow agan. For elastomerc vbraton solators mnmum attanable frequency η r, dependng on the physcal and mechancal propertes of elastomerc materal, s [] : 8 Hz - for unrenforced elastomers, 5 Hz - for rubbermetal elements workng under compresson, 3 Hz - for rubber-metal elements workng under a shft loadng. In the desgn of Vbraton solators s often necessary to ensure a reducton (or ncrease) of the value of frequency η r, that requres the ablty to calculate the stffness characterstcs of elastomerc Vbraton solator n the low fnte and medum fnte deformatons. A smlar problem arses n the desgn and calculaton of equfrequent rubber-metal compensatng devces, whch fnd applcaton n varous felds of engneerng and constructon ndustres. In ths case, the stffness characterstc of force settlement P = P (Δ), even for small fnte deformatons, wll be nonlnear (or pecewse lnear). To solve ths problem for the functon P = P (Δ), n the general case, t s necessary to desgn shock absorber that ensures the condton: dp ( ) 1 = f ( P( )) d P( ) (3) For the specal case f(p(δ)) = const = C, then from equaton (3) we obtan the soluton for P(Δ): P ( ) = B exp( C ) (4) The constant of ntegraton B s determned from addtonal condtons. For example, f shock absorber s mult-layered, an addtonal condton can be mnmzaton of weght of elastomerc materal, provdng the requred ntal settlement Δ 0 of shock absorber. In ths case, from (4): B = P( 0 ) exp( C 0 ) (5) and to wrte down the shock absorber stffness characterstcs P(Δ): [ C( )] P ( ) = P( 0 ) exp 0 (6) wll provde the desred balance between rgdty and the level of shock absober loadng. If constant С = ρ /g (where: ρ - the natural frequency of equpment that s equpped wth the projected shock absorber; g - acceleraton of gravty), equaton (6) descrbes the stffness characterstcs P (Δ), whch provdes a descrpton of equfrequent characterstc of compensatve shock absorber. It can be several desgn [1, 4] for solutons. In one of the features (cylndrcal shock absorbers wth fxed absolutely rgd sde stops) the technque of calculaton of such shocks absorbers s descrbed.. PROBLEM DESCRIPTION In ths paper we propose a desgn of shock absorber wth rgd sde boards, whch can move parallel to the drecton of the appled compressng force. Snce due to lateral movement of stops changes free surface of elastomerc layer n a shock absorber, the stffness of shock absorber changes. In contrast to the shock absorbers wth fxed sde stops (fxed stops help to mplement only the ncrease of stffness n the process of loadng), the movng sde stops make t possble to ncrease or decrease stffness (from ntal stffness) n the process of loadng. Stffness of shock absorber can vary from the hard shock (free lateral surface of the elastomerc layer of shock absorber s zero, the precptaton s only due to the weak compressblty of the elastomer) to the stffness of the shock absorber wth no sde stops (heght of the free elastomerc layer of a shock absorber s the heght of the elastomerc layer). Then t can be three cases of movng of sde supports. In the frst case, the sde stops are movng at the tme when the shock absorber s not loaded (fg.1).

3 In ths case, a pecewse nonlnear stffness characterstcs force-settlement (fg.4) jumps along the axs of settlement correspondng to the change of heght of the sde stops n tme. At ths pont force s constant, but by decreasng the free surface of the elastomerc layer settlement ncreases. Fg. 1. Shock absorber wth movng sde stops In ths case we obtan a lnear stffness characterstc force-settlement (fg. ). Fg.. Dependence force settlement for shock absorber wth movng sde stops. Frst case. In the second case sde stops are movng step by step dependng on the loadng of the shock absorber (fg. 3). Fg.4. Dependence force settlement for shock absorber wth movng sde stops. Second case, (a) soft stffness characterstc, (b) hard stffness characterstc In thrd case, the heght of the sde stops does not change contnuously, dependng on the load (fg. 5) Fg. 3. Shock absorber wth movng sde stops. Sde stops movng step by step. Fg.5 Dependence force settlement for shock absorber wth movng sde stops. Thrd case.

4 3. SOLUTION To obtan the analytc dependence of P (Δ) force-settlement for small fnte strans (up to 10% - 15%) shock absorber dvded nto two parts: Part I ((h - k (P)) z h), we have an axsymmetrc compresson. In Part II (0 z (h - k (P)) volume compresson. The soluton we obtan takng nto account the weak compressblty of the elastomer. For the frst part of shock absorber the dependence of the "force - settlement" P(Δ) n the lght of weak compressblty of rubber, we fnd usng the prncple of mnmum total potental energy of deformaton U(u 1,w 1 ) [1,] : U ( u1, w1 ) = J ( u1, w1 ) P 1 (7) where: J = π G h b h 0 u + r u r w + + r 1 u w 3 µ u u + + s + + z r 1+ µ r r w 9 µ (1 µ ) s r dr dz z 4 (1+ µ ) (8) Where: G - s modulus of rgdty, μ - s Posson's rato. Choosng the dsplacement functons u 1 (r, z), w 1 (r, z), respectvely, along the axes r and z: z z * u1 = A1 r (1 )(1 ), h = h k( P) * h h 3 1 z ( h + h *) w1 ( z ) + h hh * h * 1 + h * 3 z 4h * + 3hh * 3h = (9) that satsfy the geometrc boundary condtons: u 1 ( r, h ) = 0, u 1( r, h k( P)) = 0 w 1 ( r, h ) = 1, w 1 ( r, h k( P) ) = 0 (10) From the prncple of mnmzng the functonal U(u 1,w 1 ) the dependence of the "force - settlement" P(Δ 1 ): P ( h k( P)) 1 = [ π b G 1, + 1,5α + 1 µ 1+ 3 α µ, α = b/ ( h - k(p) ) (11) For the second part of the shock absorber, assumng that the moblty of the sde constrant s absolutely rgd, the dependence of the force - of sedment P (Δ II ), wthout takng nto account the frctonal forces on the contact surface elastomer metal, defned as n the case volumetrc compresson [1] : 3Ph ( 1 µ ) II = πb G ( 1+ µ ) (1) For all the shock absorber dependence force settlement P (Δ), wth (9) and (10) takes the form: = 1 + II (13) When usng the dependences (9) - (13), note that they are obtaned for small fnte deformatons, when: 0 1/ (h k(p)) 0,1 0,15 (14) The thckness of a free (wthout sde boards) elastomerc layer h 0 (P = 0) n the process of loadng can vary: 0 h0 ( P = 0) = h k( P = 0) h (15) Based on the requrements of explotaton the shock absorber can vary the thckness of the elastomerc layer s free dependng on the sze of loadng, or can be gven any requrement to change the stffness characterstcs of the shock absorber. For the two algorthms condton of smallness of fnal deformaton must be satsfed (11). The most smple to constructng dependence of force - settlement, If you specfy the ntervals for changng the load P and the thckness of the free layer of elastomerc shock absorber. In ths case,

5 the algorthm for constructng the functon P (Δ) wll be the followng. Let the nterval values of the compressve of force P s splt nto N (not necessarly equal) ntervals and set requrements for the change n the thckness of the free elastomerc layer, that s a functon k (P) for each nterval of the force P. For - th nterval: P P P + P 1 (16) Where: P *- step of force P (f these steps are the same, P * = P / N) The thckness of the free (wthout sde boards) elastomerc layer to the -th nterval of loadng takes values: h 1 P P ) h( P = P ) = h( P = P k( P = P ) h ( = ) (17) The total settlement Δ Σ shock absorber on the - th step of loadng s gven by: Σ II = Σ ( k ( P ( ) + ( I ( h 0 k ( P ), P ) ), P )) + (18) where the values of Δ I ((h 0 - k (P ), P *) and Δ II (k (P ), P *) calculated by formulas (11) and (1) and substtutng the values of the requred thckness of the free elastomerc layer s at each stage of loadng. From (8) - (14) that f k (P = P ) s negatve, the thckness of the elastomerc layer s free to grow and shock absorber stffness wll decrease, that s dependence of the force - settlement be precptate a "soft" type. For postve values of k (P = P ) elastomerc layer thckness of free wll decrease and dampng ncreases, that s dependence of the force - settlement be precptate a "hard" type. 4. EXPERIMENT 4.1 Descrpton As an example, consder a step change n stffness of the shock absorber. Parameters of the shock absorber: b = 18 mm, h = 40 mm, μ = 0.493, G = N/m. A cylndrcal rubber shock absorber loaded wth an axal compressve force and the heght of the sde stop has changed step by step, step sze s 6.5 mm. The experment was performed on the test machne Zwck/Roell Z Expermental Results The results obtaned n the experment are shown n Table 1. Force, N k, mm Stffness, N/mm Table 1. The dependence of the stffness of rubber shock absorber on the heght of the sde stops. The expermental results. 4.3 Analytcal Soluton The analytcal soluton consder usng formula (13). The calculaton was made usng "MathCad" software. We have two analytcal solutons. In frst soluton (soluton a ) rubber s consdered as weakly compressble materal. Results are shown n Table. Force, N k, mm Stffness, N/mm Table. The dependence of the stffness of rubber shock absorber on the heght of the sde stops. Analytcal soluton a. In second analytcal soluton (soluton b ), rubber s consdered as non - compressble materal and n formula (13) we use Posson s rato μ = 0.5. Results are shown n Table 3. Force, N k, mm Stffness, N/mm

6 Table 3. The dependence of the stffness of rubber shock absorber on the heght of the sde stops. Analytcal soluton b. 4.4 Comparson of Results Force, N Error ε, % (soluton a ) Error ε, % (soluton b ) Table 4. Comparson of both obtaned results - expermental and analytcal. Then we compare the results obtaned analytcally wth the results obtaned durng the experment and calculate the error. Results are shown n Table CONCLUSION In ths ttle proposed a method for determnaton of rgdty dependence "Force - Settlement" for shock-absorbng elements wth absolutely rgd movng (parallel to the vertcal axs z) vertcal sde stops beng under pressure. It's also taken nto account low compressblty of materal of rubber layers. Receved solutons can be used to fnd the dependence force - settlement for cylndrcal shock absorbers, as well as n projectng such shock absorbers. You can desgn a shock absorber wth a gven nonlnear ( hard or soft ) stffness characterstcs. As an example, consdered the technque of constructng a rubber shock absorber wth varable stffness (the frst example of a soft stffness, the second wth hard stffness). The heght of the sde stops depends on the appled force, and changes step by step. As can be seen from a comparson of the results (Table 4.) neglectng the weak compressblty of rubber leads to a large error when the thckness of rubber decreases. 7. ACKNOWLEDMENTS Ths work has been supported by the European Socal Fund wthn the Project Nr. 009/001/1DP/ /09/APIA/VIAA/ 11 Nanotechnologcal research of the mechancal element surface and nternal structure n mechancal engneerng. Ths work has been supported by the European Socal Fund wthn the project «Support for the mplementaton of doctoral studes at Rga Techncal Unversty». 8. REFERENCES [1] Lavendel E.E. Calculaton of rubber products. Машиностроение, Moscow, [] Lyapunov V.T., Lavendel E.E., Schlyapochnokov S.A. Rubber vbraton solators. L., Судостроение, [3] Lavendel E.E., Dymnkov S.I., Pavlovsks A.A., Snegs M.I. Appled methods of calculatng the product of hghly elastc materals. Znatne, Rga, [4] Gonca V., Švabs J. Projectng Elastomerc Shock Absorbers wth Varable Stffness In Journal of Vbroengneerng. - Vol.1, Issue 3. (010) pp ADDITIONAL DATA ABOUT AUTHORS V. Gonca, J. Shvab, S. Noskov; Rga Techncal Unversty, Ezermalas 6, LV-1006, Rga, Latva e-mals: Vladmrs.Gonca@rtu.lv, Jurjs.Svabs@rtu.lv, Stanslavs.Noskovs@rtu.lv Phone: