7th International Scientific Conference TRANSBALTICA 2011 PROJECTING ELASTOMERIC SHOCK ABSORBERS WITH ADJUSTABLE STIFFNES. V. Gonca, Y. Shvab Riga Technical University, Institute of Mechanics
ELASTOMERIC SHOCK ABSORBER MODEL z P k 2 Fig. 1. Principle of operation of elastomeric shock absorbers with adjustable stiffness Problem: � For the elastomeric elements of complex configuration or consisting of well contacting parts made of various materials which does not permit the application of variational methods of calculation, who effectively used in the calculation of elastomeric elements simple geometry.
DECOMPOSITION OF THE VOLUME OF THE PRODUCTS ON A SUBAREAS � Deformations are assumed to be small. It is proposed to use the variational principle proposed V.Prager using discontinuous displacement function and modified for weakly compressible and incompressible materials. � Let's consider the elastomeric element with the complex geometrical form with volume V and the area of a surface F: N N ∑ ∑ = = V V F F (1) n n = = = = n 1 n n n 1 1 1 where: V n - one-coherent regular subareas; N - number of subareas, received as a result of crushing elastomeric element; F - total surface area of all subareas elastomeric element. � The surface limiting n-th subarea, looks like = n + n + Γ F F F (2) σ n u n where: n – loading surface; F σ n – surface of the fixing; F u Γ n – surface area of contact subarea partitioning elastomeric element.
MATHEMATICAL MODEL 3 G ∇ u + s + f = 2 0 (3) � Equation of equilibrium: i i i + µ 2 ( 1 ) − µ (4) 3 ( 1 2 ) = u s � Volumetric deformation: j j + µ , 2 ( 1 ) � Deformations: ε ij = 0,5 (u i,j + u j,i ) (5) � Stress: σ ij = G [u i,j + u j,I + 3µ /(1+µ) � s �δ ij ] (6) � Forces boundary conditions: G[u i,j + u j,I + 3µ /(1+µ) s δ ij ] n j = P i (7) = u i u � Displacements boundary conditions: (8) 0 i
THE CONDITIONS OF CONTINUITY ON CRUSHING SURFACE � On the crushing surface Γ n must be satisfied the conditions of continuity: + n = n u u 1 - of displacements: (9) i i σ σ n n = = − − σ σ n + n + m m m m 1 1 - and of stresses: - and of stresses: (10) (10) ij ij j j ij ij j j � Contact surface Γ n may be: a) artificial in the geometric decomposition of the elastomeric element b) natural, that is, a volume V composed of different materials. c) = a) + b)
In determining the integral characteristics, of type “force – settlement”, of the elastomeric element boundary problem (3) - (8) (without crushing the elastomeric element in the subareas) easier to solve a variational method using the Ritz procedure for the functional: ( ) ( ) µ − µ = ∫ 1 3 9 1 2 n n n n + n n 2 J u s G u u u u + n n − n n s u s dV ( , ) [ (11) ] ( ) i i j j i i j j i i i + µ , , , , , + µ 2 2 1 4 1 V n Using the variational principle of V.Prager and applying the method of undetermined Lagrange multipliers with a functional (11) be the boundary value problem (3) - (8) with the conditions of multipliers with a functional (11) be the boundary value problem (3) - (8) with the conditions of the joining (9) and (10) replaced by the variational problem with discontinuous function of demand on the surfaces of the partition Γ n for the functional: µ { } ″ N N − 1 3 ∑ ∑ ∫ = n n − n + n J u s J u s G u u + n δ n Γ n s m u d * ( , ) ( , ) [( ) (12) + i i n i j j i i j j i n , + µ , 1 = = n n Γ 1 1 n n where in each subare: n – displacement; u i s n - function of hydrostatic pressure; G n - modulus of elasticity in shear; µ n - Poisson's ratio.
Calculate model(Fig. 2). Z P � k 1 V 2 II h r k 2 I V 1 Ø2a Ø 2b Fig.2. Elastomeric shock absorbers with adjustable stiffness calculate model.
� The considered absorber is broken on two parts on border of a thrust block in parallel a shaft or. At use n and s n , it is enough to satisfy to geometrical boundary conditions on external functional (15), choosing u i surface F u . Shock absorber is divided into two subareas (see Figure 1). All functions with an index „1” it is carried to a subarea I, and with an index „2” to a subarea II. We believe that the geometry of the elastomeric layer can not take into account the compressibility of the elastomer, that is, believe that the Poisson coefficient � = 0,5. − = = u r k u r k ( , ) ( , ) 0 1 2 2 1 = = u a z w a z ( , ) ( , ) 0 ≥ ≥ ≥ ≥ − − ≥ ≥ ≥ ≥ − − z z k k z z k k 1 1 1 1 0 0 0 0 2 2 2 2 � The main boundary conditions will be: � The main boundary conditions will be: (13) − k = w r ( , ) 0 1 2 = − ∆ w r k ( , ) 2 1 where: the functions u i and w i - displacement, respectively, on the axis of r and z.
� Dependence “force – settlement” it is defined from the equation of balance of the top base of the absorber: b ∫ π σ = − rdr P 2 = 1 (14) zz z k 2 a � On a surface of splitting of a condition of ideal contact piece look like: = u r u r ( , 0 ) ( , 0 ) 1 2 = w r w r ( , 0 ) ( , 0 ) 1 2 (15) σ = σ r r ( , 0 ) ( , 0 ) rz rz 1 2 σ = σ r r ( , 0 ) ( , 0 ) zz zz 1 2
We choose conveyances u n , w n and function s n whenever possible in the most simple kind with the account only conditions (6) and prospective character of deformation. = − + − − u B rz z k B r a z k ( ) ( )( ) 2 1 1 2 1 ∆ z = − + − − w A r a z k ( )( ) 2 k 1 1 1 s = C 2 2 (16) (16) = − + u B r a z k ( )( ) 1 3 2 = − + w A r a z k ( )( ) 1 2 2 s = C 1 1 where A 1 ,A 2 ,B 1 ,B 2 ,C 1 ,C 2 , � – unknown constants.
Functions (16) have on section height approximately following appearance for conveyances at r = a; b: � for u � for w Z Z u 2 w 2 u 2 w 2 r r u 1 u 1 w 1 w 1 Fig. 2. Expected character of deformation
� After integration it is received that functional J* depends only on unknown constants: ( ) ∗ = ∆ J J A A B B B C C , , , , , , , (17) 1 2 1 2 3 1 2 � From a condition stationarity: ∗ ∂ J = 0 (18) ( ( ) ) ∂ ∂ ∆ ∆ A A A A B B B B B B C C C C , , , , , , , , , , , , , , 1 1 2 2 1 1 2 2 3 3 1 1 2 2 � We receive system of the algebraic equations. For dependence “force – settlement”: Pk D ∆ = 1 1 (19) π − α D Ga 2 2 2 ( 1 ) where: a α = b D, D 1 - determinants of algebraic equations (18), an expression which, due to the complexity of writing, are not given
Then the stiffness of the shock absorber according to k 2 (because k 1 = (h – k 2) ) can be calculated by the formula: π ⋅ ⋅ 2 − α 2 P G a D 2 ( 1 ) = = c ∆ − h k D ( ) 2 1 For absorber: h = 30 mm, a = 20 cm, b = 40 mm, G = 7 · 10 -2 kg/mm 2 , in Figure 3 shows the results (using MatCad): (using MatCad): Fig.3. The stiffness of the shock absorber according to k 2
CONCLUSION � The model of rubber shock absorber offered in-process with adjustable stiffness can have a large range of stiffness regulation. Stiffness of shock absorber can change, from ordinary declivous rubber shock absorber’s stiffness (depends on geometrical sizes and brand of rubber) to absolutely hard support. � The chart of possible stiffness is in-process got for the shock absorber of concrete sizes. From a chart evidently, that stiffness of such shock absorber can change from a 70 к g/mm (k 2 = 0) to absolutely hard support (k 2 = h). � A construction of shock absorber is not difficult and easily realized. � The proposed calculate method allows in case of partition of the investigated area V on � The proposed calculate method allows in case of partition of the investigated area V on subarea V n , using functional (15) and discontinuous functions sought to obtain integrated dependences of type “force – settlement”. Acknowledgement This work has been supported by the European Social Fund within the project «Support for the implementation of doctoral studies at Riga Technical University»
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