Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 52, 1, pp. 81-92, Warsaw 2014 INFLUENCE OF SUPPORT COMPLIANCES ON STABILITY AND LIMITING SLENDERNESS RATIO OF COMPRESSION HELICAL SPRINGS Jacek Krużelecki, Damian Szubartowski Cracow University of Technology, Institute of Applied Mechanics, Kraków, Poland e-mail: jacek.kruzelecki@pk.edu.pl; tszubartowski@o2.pl In the paper, the influence of support compliances on the stability and the limiting slen- derness ratio of compression helical springs was investigated. To analyze the stability of helical springs, the concept of an equivalent column was applied and a general system of supports described by three independent compliances was utilized. In the general case, the solution obtained, which describes the critical force, depends on two rotational compliances and a lateral compliance of a transverse shift of the support. The influence of the individual compliances on the critical load and on the limiting slenderness ratio was studied. Simple approximation formulas were proposed to describe these relationships. Key words: helical spring, stability, support compliance, slenderness ratio 1. Introductory remarks Helical springs, specially compression ones, fulfil different functions in the technique: do the work, store energy, alleviate the impact, raise or take vibration and many others. Moreover, they may be subjected to loss of stability. A helical spring should be treated as a spatially curved bar. Such an approach, based on Kirchhoff-Clebsh equations and geometrically non-linear theory, was developed by Nikolai (1955). It was applied to the problems of stability of helical springs, among others, by Cherny- shev (1946), Haringx (1948), Olenov (1977), Polishchuk (1977), Kernchen (1979). The equations of this type were also used by Ponomariev (1948), Taborrak and Xiong (1989) and others to derive equations of stability for the general behaviour of loadings. They are also used for de- termination of displacements in springs with high deformability. The current configuration of a spring under loading determined in this way can be treated as a precritical state (Chernyshev, 1958; Sutyrin, 1980). There are also attempts to build a finite element and, in this way, to solve the problem of stability taking into account geometrical nonlinearities. Let usmention here, for example, the paper byMottershead (1982) on the dynamic stability. However, inmost commer- cial finite element programs such an element does not exists. All the above papers deal with helical coil springs with a constant pitch. Geometrically linear theory was developed by Trostel (1957) in connection with springs of arbitrary shape. This approach was used in the papers byCzerwiński (1973) – parabolic spring, Czerwiński (1975) – arbitrary geometry of a spring, Szafran-Gądek (1981) – elliptical spring, to determine displacements. Lin and Pisano (1988) studied the geometry of helical springs of arbitrary shapes. An exact stability analysis of springs based on the theory of spatially curved bars is compli- cated and difficult in practical applications. Hence, in most engineering applications, a concept of an equivalent column is introduced. Such a column must account for compressibility of the axis and shear effect. First papers using the concept of an equivalent column to stability analysis of helical springs are due to Hurlbrink (1910) and Grammel (1924). They took into account only the effect of 82 J. Krużelecki, D. Szubartowski change in the column length. They showed that there exists a limiting slenderness ratio below which no bifurcation is possible at all. Biezeno and Koch (1925) pointed out that the effect of shear should also be taken into account in the stability analysis of helical springs.However, they overestimated the impact of shear effects, and in consequence, concluded that any spring, even short,maybe subjected to buckling.The correct solution, based on the concept of the equivalent columnandassumingzeropitchangle (helix slope),was obtainedbyHaringx (1942), Ponomariev (1958) and others. These solutions have confirmed the existence of the limiting slenderness ratio, which was also verified experimentally. Suitable solutions can be found in several monographs, for example byTimoshenko andGere (1961),Wahl (1963) and in application studies such as by Żukowski (1954) or Branowski (1989. Differences betweenBiezeno’s andHaringx-Ponomariev’s solutions result fromdifferent esti- mations of the shear effect. This problemwas investigated in detail by Ziegler (1982). He showed that for helical springs the so-called modified approach is the proper way of taking the shear effect into account, whereas Biezeno andKoch (1925) applied Engesser’s approach. Equivalent columns were also introduced for buckling analysis under combined compression and torsion. We can mention here papers by Ziegler and Huber (1950) and Satoh, Kunoh and Mizuno (1988). In all the above papers, the study of stability was based on linear theory. Nonlinear theory of stability and equivalent column was used byMizuno (1960) to study postcritical behavior of cylindrical springs with a constant pitch. All the above mention papers are based on the simplifying assumption of a very small pitch angle: this anglewas assumed tobezero.Amoregeneral equivalent columnallowing for arbitrary helix slope isduetoBerdychevskyandSutyrin (1983), but that conceptproposed for general non- linear problems is rather complicated and has not been applied to stability analysis. Krużelecki and Życzkowski (1990) proposed a new,more refined and exact concept of an equivalent column for buckling of helical springs of arbitrary shape. It accounts for a helix angle and possible buckling in two planes and also takes into consideration axial compressibility and shear effects. It ensures high accuracy of results also. Such an equivalent column, described by compression, bending and shearing rigidities, gives the possibility to analyze stability of helical springs of arbitrary shape. Most of the papersdealwith simply supportedor clamped springs at both ends. In this paper the influence of modes of supports and their compliances on stability and slenderness limiting ratio is examined. The concept of an equivalent column is used. 2. Equations of stability of compression helical springs The concept of an equivalent column proposed by Hurlbrink (1910), Grammel (1924), Haringx (1942) anddevelopedbyKrużelecki andŻyczkowski (1990) allowsahelical springtobe treated in stability problemsas a columnwith the appropriate rigidities. Because ofmuchhigher sensitivity of shearing on stability of a spring in comparison with a column having a solid cross-section, this effect should be also taken into account. A helical spring – an equivalent column – with an initial length H0 which is loaded by a compressive force P is considered. Because of high compressibility, a helical spring becomes shorter and its actual length is H. In that state described by large displacements with changes of geometry taken into account, a small bending described by a linear theory is imposed, and this way the critical state of the helical spring is investigated. The differential equations of bucking of a spring for any type of support can be written in the following form Influence of support compliances on stability and limiting slenderness ratio... 83 dw dx = θ+χ dθ dx =− M EI dM dx = Q+P dw dx dQ dx =0 (2.1) where xdenotes theactual coordinate along theaxis of the compressed spring (0¬ x ¬ H), w is the lateral displacement, θ is the inclination of a normal to the cross-section against the central axis of the undeformed column (the angle of rotation of a cross-section caused by bending), M is the bendingmoment, Q represents the shear force in the spatial frame. In equation (2.1), the shearing effect is represented by the term χ according to the so-called modified approach discussed by Ziegler (1982) χ = Q+Pθ GA (2.2) The actual bending EI, shearing GA and axial EA rigidities were defined in the paper by Krużelecki and Życzkowski (1990). It turned out that for the problem under consideration we can use a simplifiedmodel of an equivalent column, namely assume that the helix angle is equal to zero. Then, the current rigidities can be written as follows EA =(EA)0 = EIwH0 2π(1+ν)R30n0 EI =(EI)0 H H0 GA =(GA)0 H H0 (2.3) where (EI)0 = H0EIw π(2+ν)R0n0 (GA)0 = H0EIw πR30n0 (2.4) and Iw = πd4/64, d is the diameter of the wire, R0 and n0 denote the initial radius and the original number of coils of the spring, respectively, E stands for the Young modulus and ν is the Poisson ratio. Such formulas describing the rigidities of a spring with the helix angle equal to zero can be also found in the papers by Haringx (1948), Żukowski (1954), Timoshenko and Gere (1961). The set of differential equations (2.1) of the first order can be transformed into one equation of the forth order d4w dx4 +k2 d2w dx2 =0 (2.5) for which the general solution can be written in the form w(x)= Asinkx+Bcoskx+Cx+D (2.6) where k = √ P EI ( 1+ P GA ) (2.7) Helical compression springs used in different technical applications are supported and fa- stened in many different ways. Influence of coil spring mountings on their selected operational properties are presented in the book by Michalczyk and Slawiński (2011). Analysis of modes of supports leads to the conclusion that the ends of a spring should be supported elastically taking one end as a support immovable in the space. Such a general way of support, presented in Fig. 1, can describe possible behavior of the ends of a real spring. A similar attempt to the system of supports for analysis of stability of solid columns was applied by Życzkowski (1988). The coordinate system is associated herewith thebottom supportwhich is treated as immovable one. The quantities C1 and C2 in Fig. 1 characterize the rotational compliances of the lower and upper supports, respectively while C3 describes the lateral compliance of a transverse shift 84 J. Krużelecki, D. Szubartowski Fig. 1. General case of the spring support of the upper end. So, assuming, for example C1 = 0, it denotes that rotation of the lower end of the spring is completely blocked whereas C1 = ∞ means a simply supported end. On the other hand C3 =0 denotes that lateral movement of the upper end is completely blocked while C3 =∞ indicates that later displacement of this end is free. The boundary conditions for the considered general system of supports in the actual confi- guration (compressed spring) can be written as follows w(0)= 0 −C1M(0)= θ(0) C2M(H)= θ(H) C3Q(H)=−w(H) (2.8) Taking into account equations (2.1) and (2.2), boundary conditions (2.8) take the form w(0)= 0 C1EIw ′′(0)−w′(0)−w(H) 1 C3GA =0 C2EIw ′′(H)+w′(H)+w(H) 1 C3GA =0 C3EIw ′′′(H)+C3P ( 1+ P GA ) w′(H)− ( 1+ P GA ) w(H)= 0 (2.9) Boundary conditions (2.9) lead to four homogeneous algebraic equations for which the con- dition of non-trivial solution gives √ H2P EI ( 1+ P GA ){(C3P H −1 )[( 1+ P GA )−1 −EIC1C2P ] +(C1+C2) EI H } sin ( √ H2P EI ( 1+ P GA ) ) − [ 2−PH(C1+C2) (C3P H −1 )] cos ( √ H2P EI ( 1+ P GA ) ) +2=0 (2.10) Equation (2.10) is written in the carrent configuration, which means that it is expressed by the actual rigidities (GA) and (EI) and the actual (after copression) length H of the spring. Since, the critical loading P is “hidden”, in the actual rigidities as well as in the actual length of the spring one has to express (2.10) by the initial quantities. Taking into accout that H = H0 ( 1− P (EA)0 ) (2.11) Influence of support compliances on stability and limiting slenderness ratio... 85 and utilizing (2.3) and (2.4), we obtain λ √ pcr 2+ν 2+2ν ( 1− 1+2ν 2+2ν pcr ) { [1− (1+ψ3)pcr] [ λ2pcrψ1ψ2 2+ν 2+2ν − ( 1− 1+2ν 2+2ν pcr )−1] +ψ1+ψ2 } sin ( λ √ pcr 2+ν 2+2ν ( 1− 1+2ν 2+2ν pcr ) ) − { 2+λ2pcr(ψ1+ψ2)[1− (1+ψ3)pcr] 2+ν 2+2ν } · cos ( λ √ pcr 2+ν 2+2ν ( 1− 1+2ν 2+2ν pcr ) ) +2=0 (2.12) where λ = H0 R0 pcr = Pcr (EA)0 ψ1 = C1 (EI)0 H0 ψ2 = C2 (EI)0 H0 ψ3 = C3 (EI)0 H30 (2.13) and λ denotes the slenderness ratio of the spring, pcr is the dimensionless critical force and ψi means the dimensionless compliance of supports. Equation (2.12) allows one to determine the critical force in terms of dimensionless compliances of supports. In (2.12), the rotational compliances ψ1 and ψ2 occur in pairs. It means that this equation is symmetrical with respect to these quantities and it also means that the replacement of ψ1 by ψ2 and vice versa does not cause change in the critical loading. The critical loading depends also on the slenderness ratio λ; for larger values of λ the critical force is smaller. If the slenderness ratio is small enough, the buckling of the spring does not occur. Itmeans that the relation pcr = pcr(λ) has its own return point which corresponds to the limiting slenderness ratio λlim below which no buckling occurs. Equation (2.12) also allows one to determine such limiting slenderness ratios for any mode of supports (for any values of compliances). 3. Support compliances vs. stability of helical springs We analyze here relationships between modes of the support expressed by support compliances and stability of compressed helical springs.We limit our considerations to some chosen variants of the supports for which the classical fittings of ends of the springs constitute particular cases of the supports. Analysis is based on general solution (2.12) obtained via application of the equivalent column concept. 3.1. Rotational compliances In this case,we analyze the influenceof the rotational compliances ψ1 and ψ2 on the stability and on the limiting slenderness ratio of a helical compression spring. Since equation (2.12) is symmetrical with respect to ψ1 and ψ2, the dependence of the limiting slenderness ratio on these compliances should be common for ψ1 and ψ2. To find appropriate relationships, we assumed the values of ψ3, namely ψ3 = ∞ or ψ3 = 0 which refer to free movement and to completely blocked movement of the upper end of the spring, respectively, whereas ψ1 and ψ2 varies between the limiting values, ∞ ¬ ψi ¬ 0, where i = 1 or 2. That leads to analysis of three systems of supports for which three pairs of the particular (limiting) cases of supports are presented in Fig. 2. 86 J. Krużelecki, D. Szubartowski Fig. 2. Particular cases of the supports for analysis of the influence of ψ1 or ψ2 In the first case, it is assumed that only ψ1 can vary (∞ ¬ ψ1 ¬ 0) whereas ψ2 = 0 and ψ3 =∞. The particular systems of supports for the limiting values of compliances referring to this case are presented in Fig. 2a. In the second case, we examine the influence of ψ2 on pcr and λlim. It means that only ψ2 can vary (0 ¬ ψ2 ¬ ∞) whereas ψ1 = 0 and ψ3 = ∞. The limiting systems of supports are shown in Fig. 2b. In the third case, presented in Fig. 2c, we analyze the influence of ψ1 and ψ2 on the assumption that ψ3 =0. As in the first and second case, the same ψ3 =∞ is assumed, and we can consider these two cases together. Substituting ψ2 = 0 and ψ3 = ∞ or substituting ψ1 = 0 and ψ3 = ∞ into (2.12) and assuming that only the first mode of buckling is considered, equation (2.12) can be rewritten as follows λ √ pcr 2+ν 2+2ν ( 1− 1+2ν 2+2ν pcr ) = π−arctan ( ψiλ √ pcr 1+2ν 2+2ν ( 1− 1+2ν 2+2ν pcr ) ) (3.1) where i =1 or 2. In the limit case ψi =∞, we obtain pcr = 1+ν 1+2ν ( 1± √ 1− 1+2ν 2+ν π2 λ2 ) λlim = π √ 1+2ν 2+ν (3.2) whereas for ψi =0we have pcr = 1+ν 1+2ν ( 1± √ 1−4 1+2ν 2+ν π2 λ2 ) λlim =2π √ 1+2ν 2+ν (3.3) In Fig. 3a, curves pcr = pcr(λ) for the considered particular cases of supports are presented where the upper parts of the curves, marked by dashed lines, refer to the regions above the return points (marked with circles in Fig. 3a). The upper parts of the curves do not describe the real buckling loadings. For 0¬ ψi ¬∞, the curves can be obtained numerically from (3.1) and they are located between the limiting cases. Such a curve obtained for ψi = 0.8 is also presented in Fig. 3a. All these curves have the return points for the same value of pcr, namely for pcr = (1+ν)/(1+2ν) but for different values of λlim which depend on values of ψi. The relationship λlim = λlim(ψi) can be obtained numerically fromequation (3.1) and it is presented in Fig. 3b (full line). That curve can be approximated with a very good accuracy by λlim = 0.9+0.56ν −0.16ν2 0.4+ψi +π √ 1+2ν 2+ν i =1,2 (3.4) and is shown in Fig. 3b by the dashed line (both lines overlap). Influence of support compliances on stability and limiting slenderness ratio... 87 Fig. 3. Critical force vs. slenderness ratio (a), limiting slenderness ratio vs. rotational compliance ψi for ν =0.3 and ψ3 =∞ The third case, presented in Fig. 2c, shows a study of the impact ψ1 and ψ2 on the stability and the limiting slenderness ratio of the spring on the assumption that ψ3 = 0. Substituting ψ3 = 0 into (2.12) and assuming that only the first mode of buckling is considered, equation (2.12) can be rewritten as follows λ √ pcr 2+ν 2+2ν ( 1− 1+2ν 2+2ν pcr ) { (1−pcr) [ λ2pcrψ1ψ2 2+ν 2+2ν − ( 1− 1+2ν 2+2ν pcr )−1] +ψ1+ψ2 } sin ( λ √ pcr 2+ν 2+2ν ( 1− 1+2ν 2+2ν pcr ) ) − [ 2+λ2pcr(ψ1+ψ2)(1−pcr) 2+ν 2+2ν ] cos ( λ √ pcr 2+ν 2+2ν ( 1− 1+2ν 2+2ν pcr ) ) +2=0 (3.5) For the limiting values of ψi (ψi = 0 and ψi = ∞), closed solutions (3.4) can be written as follows: — for ψ1 = ψ2 =0 pcr = 1+ν 1+2ν ( 1± √ 1−16 1+2ν 2+ν π2 λ2 ) λlim =4π √ 1+2ν 2+ν (3.6) — for ψ1 = ψ2 =∞ pcr = 1+ν 1+2ν ( 1± √ 1−4 1+2ν 2+ν π2 λ2 ) λlim =2π √ 1+2ν 2+ν (3.7) For 0 < ψi < ∞, only numerical solutions to (3.5) can be obtained. Such a sample solution for ψi =0.8 together with solutions (3.6) and (3.7) for the limiting cases is presented in Fig. 4a. All curves in Fig. 4a have the return points for the same value of pcr, namely for pcr = (1+ ν)/(1+ 2ν), but for different values of λlim which depend on values of ψi. The return points are marked with circles in Fig. 4a. The relation λlim = λlim(ψi) was obtained numerically from (3.5) and is presented in Fig. 4b (full line). That curve is approximated with a very good accuracy by λlim = 0.89+0.53ν −0.11ν2 0.2+ψi +2π √ 1+2ν 2+ν i =1,2 (3.8) and is shown in Fig. 4b by dashed line (both lines overlap). 88 J. Krużelecki, D. Szubartowski Fig. 4. Critical force vs. slenderness ratio (a), limiting slenderness ratio vs. rotational compliance ψi for ν =0.3 and ψ3 =0 (b) 3.2. Lateral compliance In this case, we analyze the influence of the lateral compliance ψ3 (0 ¬ ψ3 ¬ ∞) on the stability and the limiting slenderness ratio of the spring on the assumption that ψ1 = 0 and ψ2 =∞ or ψ1 =∞ and ψ2 =0. It refers to analysis of two systems of supports for which two pairs of the limiting cases of the supports are presented in Fig. 5. Fig. 5. Particular cases of the supports for analysis of the influence of ψ3 Because general equation (2.12) is symmetrical with respect to ψ1 and ψ2, both cases can be considered together. Substituting into (2.12) ψ1 = 0, ψ2 = ∞ or ψ1 = ∞, ψ2 = 0 for the first mode of buckling, one obtains λ √ pcr 2+ν 2+2ν ( 1− 1+2ν 2+2ν pcr ) = π+arctan ( λ 1− (1+ψ3)pcr 1− 1+2ν 2+2ν pcr √ pcr 1+2ν 2+2ν ( 1− 1+2ν 2+2ν pcr ) ) (3.9) The numerical solutions to (3.9) are presented in Fig. 6 for different values of ψ3 (0 ¬ ψ3 ¬ ∞) and ν = 0.3. These curves, in general, are different from those presented for the previous cases. Decreasing the compliance ψ3 from infinity to a finite value, the systems of supports presented in Fig. 5a become more stiff. It causes that the critical force increases, especially for smaller values of ψ3 and any value of λ. Simultaneously, the return points can move towards larger values of pcr and λ. On the other hand, the shear force, which becomes larger for smaller values of ψ3, has destabilizing influence on the stability of the spring and it causes that the critical force can decrease for any value of the slenderness ratio. It can result Influence of support compliances on stability and limiting slenderness ratio... 89 in changing of locations of the return points, namely they can move towards smaller values of pcr and λ. These two different effects cause the curves have the inflexion points for the lateral compliance ψ3 whose values are closer to ψ3 =0.Moreover, the returnpoints,marked by circles in Fig. 6, move towards larger values of pcr and λ for smaller values of ψ3. So, in this case not only the limiting slenderness ratio but also the critical force of the return points depends on the lateral compliance ψ3. Fig. 6. Critical force vs. slenderness ratio for ν =0.3, ψ1 =0, ψ2 =∞ or ψ1 =∞, ψ2 =0 The largest dimensionless critical force, definedby (2.13)2, andwhose physical interpretation is the critical compressive strain, theoretically (diameter of the wire is equal to zero) can reach a value of 1, pmax cr =1. For larger values of pcr than 1, the coils of the spring are in contact one with another (spring is closed). However, taking into account that diameter thewire is of a finite dimension, then the maximum pcr is lower, namely pmaxcr = 1−nd/H0, where n denotes the number of coils and d is the diameter of thewire. This horizontal line, shown inFig. 6, separates the admissible solutions (lines below) from those which are not possible to obtain (lines above). This line shows also that the return points can be reached only for large enough values of ψ3, ψ3 ­ ψ3. The critical force for the return points vs. the lateral compliance ψ3 can be approximated with a very good accuraccy by pcr = 0.4 0.9+ψ3 + 1+ν 1+2ν for ψ3 ­ ψ3 (3.10) whereas the limiting slenderness ratio for the return points can be described by λlim = 0.47+0.49ν +0.13ν2 0.61−0.66ν +0.63ν2+ψ3 +π √ 1+2ν 2+ν for ψ3 ­ ψ3 (3.11) where ψ3 = 0.4(1+2ν) ν − (1+2v)nd H0 −0.9 (3.12) and the above approximations are valid for ν ­ 0.3. For the lateral compliance ψ3 < ψ3, neither the critical force nor the limiting slenderness ratio can be specified because the neighbouring coils settle one on another and the stability of the fully compressed spring cannot be described in this way. In Fig. 7a, the critical force for the return points vs. the lateral compliance ψ3 is presented, whereas in Fig. 7b, the limiting slenderness ratio is shown for ν = 0.3 on the assumption that d =0. 90 J. Krużelecki, D. Szubartowski Fig. 7. Critical force (a), limiting slenderness ratio (b) vs. compliance ψ3 for the return points, ν =0.3, d =0 4. Final remarks Analysis of the influence of support compliances on the stability and the limiting slenderness ratio showed that rotational compliances of the upper and lower ends of a compression helical spring have identical effects on the investigated quantities – algebraic equation (2.12) describing the critical load is symmetrical with respect to the rotational compliances ψ1 and ψ2. Each curve describing the critical force by both rotational compliances is located between the curves obtained for particular modes of the supports which were investigated in details. The values of the critical forces for the return points are the same for any value of the rotational compliances and it depends on the Poisson ratio. The values of the limiting slenderness ratio, below which the spring never buckles, also depend on the rotational compliances. 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