2017 2nd International Conference on Test, Measurement and Computational Method (TMCM 2017) ISBN: 978-1-60595-465-3

Analysis Method of Stiffness and Load Capacity of Euler Spring with High Static Stiffness and Low Dynamic Stiffness

Chun-yu QU1, Xiao-ming WANG1 and Yu-lin MEI2,*

1School of Mechanical Engineering, Dalian University of Technology, China

2School of Automotive Engineering, Dalian University of Technology, China

*Corresponding author

Keywords: Euler spring, Variable stiffness, Nonlinear Geometric Deformation, Structure design.

Abstract. A large number of studies have shown that nonlinear passive control devices have good

vibration attenuation and isolation effect. And vibration isolation systems with nonlinear stiffness

have more advantages than linear systems. The paper addresses the stiffness and load capacity

design problem of Euler springs with varying cross-sections. Firstly, in order to analyze nonlinear

behavior of curved beams due to large geometric deformations, a set of first-order differential

equations are established, which can be easily solved by numerical integration. Secondly, a curved

beam is taken for example to compare analytical results calculated by using the first-order

differential equations and ANSYS simulation results. Then the influence of geometric parameters of

curved beams with varying cross-sections on structural stiffness and load capacity is investigated by

using ANSYS. Finally, a design strategy for stiffness and load capacity of curved beams is summed

up.

Introduction

Euler springs have variable stiffness, especially the stiffness is very low at the post-buckling

stage, as shown in Figure 1. And the feature can be used to reduce natural frequencies of vibration

systems and get better vibration isolation effect. In general, buckling loads of Euler springs are

equal to the pressure of isolated objects. Under static loading, Euler springs are in the pre-buckled

state and have high stiffness 0K , so their deformations are very small. However, under dynamic

loading, because of the occurrence of additional inertia forces, the total loads applied to Euler

springs exceed the buckling loads. Consequently, Euler springs suddenly start to buckle and their

stiffness sharply reduces to tK , where tK is very small relative to 0K .

Figure 1. Force-displacement curve.

Virgin and Plaut et al. [1,2] study static and dynamic characteristics of a system composed of

Euler struts by experiment. The results show that the system has a small fundamental frequency and

a wide range of vibration isolation frequency; and the vibration isolation performance can improve

by reducing damping. Barber et al. [3] design a vertical vibration isolator using Euler springs. In

order to achieve effective vibration isolation at low frequency range, pre-pressure greater than

buckling loads is acted on Euler springs. Zhao et al. [4] review recent researches on curved beams,

including basic static theories, dynamic theories, modeling methods, in-plane vibration and

out-plane vibration analysis methods etc. For thin-walled mild steel columns with square and

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circular cross sections, Abramowicz et al. [5] report an experimental study on the transition from

initial global bending to progressive buckling under axially static and dynamic loads.

It is well known that Euler springs with high static stiffness and low dynamic stiffness can be

used to effectively obtain vibration attenuation and isolation. Currently, one of key difficulties is

how to design an Euler spring with limited scale based on the premise of maintaining its load

capacity in an expected range. In order to reduce dynamic stiffness of Euler springs, Kovacic et al.

[6] design a vibration isolator with a quasi-zero stiffness characteristic by parameter optimization.

The isolator consists of a vertical linear spring and two non-linear pre-stressed diagonal springs. Liu

et al. [7] propose a nonlinear isolator with zero dynamic stiffness, which is constructed by parallel

adding a negative stiffness corrector to a linear spring, where the negative stiffness corrector is

formed by Euler buckled beams. After dynamic loads are carried, the dynamic stiffness of the

isolator is zero at the equilibrium point, while the support capacity of the original linear isolator is

retained.

In this paper, the analysis method of stiffness and load capacity of Euler springs with high static

stiffness and low dynamic stiffness is studied. Because curved beams are very typical of a type of

Euler springs, we mainly focus on the influence of geometric parameters of curved beams on their

stiffness and load capacity. Firstly, in order to analyze nonlinear behavior of curved beams due to

large geometric deformations, a set of first-order differential equations are established, which can be

easily solved by numerical integration. Secondly, a curved beam with uniform cross-section is taken

for example to compare analytical results calculated by using the first-order differential equations

and ANSYS simulation results. And then we investigate the influence of geometric parameters of

curved beams with varying cross-sections on their stiffness and load capacity by using ANSYS

finite element programs. In the end, a design strategy for stiffness and load capacity of curved

beams is summed up.

Stiffness Analysis Method of Curved Beams with Varying Cross-sections

The deformation of a curved beam is shown in Figure 2, where ( )0

sγ

and ( )sγ describe the

curved beam before and after deformation, respectively; { }1 2 3=F F F F， ， and

{ }1 2 3=M M M M， ， are force vectors and bending moment vectors applying to ( )γ s ; ( )q s is

an externally distributed load along ( )γ s . According to balance of forces and balance of moments,

equilibrium equations for the curved beam can be written as

0, ( ) 0∂ ∂

+ × = + =∂ ∂

M dr FF q s

s ds s. (1)

Figure 2. Deformation diagram of a curved beam. Figure 3. Curved beam subjected to an external force.

For a 2D problem as shown in Figure 3, a curved beam is subjected to an external force P at an

end. Assuming that the Frenet frame of ( )sγ is { }, ,nα γ , where α , n and γ are tangent

vectors, normal vectors and binormal vectors, respectively, then we have

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( ) ( )( ), ( )

αα= = −r r

d s dn sk n s k s

ds ds. (2)

Here γk stands for the relative curvature. If the vector P is parallel to the X axis of Cartesian

coordinate system { }, ,X Y Z and the included angle between X and α is θ , then the scalar

components of P in the directions of α and n , 1F and 2F , can be calculated by

1 2cos , sinθ θ= − = −F P F P . (3)

In the case of neglecting axial deformations, Eq. (1) can be reduced to

32 0

MF

s

∂+ =

∂. (4)

Where 3M is a scalar component of M in the direction of γ , which subscript is omitted later in

order to write conveniently. And the constitutive relation between bending moments and curvatures

is given by 0

( )= −r rM EI k k , where 0r

k and rk are used to express relative curvatures of the

beam before and after deformation, E and I are the modulus of elasticity and the area moment

of inertia, respectively. Substituting M into Eq.(4), the following transformation is yielded

22

0

2 2sin

dd P

ds EI ds

θθθ= − + . (5)

Considering / cosθ=dX ds and / sinθ=dY ds , a set of first-order differential equations are

obtained

2

0

2sin , , cos , sin

ξ θ θθ ξ θ θ= − + = = =

d P d d dX dY

ds EI ds ds ds ds (6)

The above formulation can be solved by numerical integration methods. And after initial boundary

conditions are given, the deformations of curved beams can be easily gotten.

When the initial curvature of the beam is constant, Eq. (6) can be simplified as

2

2sin

d P

ds EI

θθ= − . (7)

Multiplying by θdds

at both sides of Eq.(7), and after performing the integration, we have

2 2

0( ) 2 (cos )d

k Cds

θθ= − . (8)

Where 2=k P EI , ( )' 2 2

0 0 0cos / 2θ θ = − C k , 0θ and 0 'θ denote function values and derivative

values calculated at s=0. As shown in Fig.3, 0θ is the included angle between X and α at the

end of the curved beam, and 0 'θ is the corresponding curvature. Supposing that / 0θ <d ds , the

positive square root of Eq.(8) is

02 cosd

k Cds

θθ= − − . (9)

Substituting sin sin2

θϕ= b and ( )

1/2

01 / 2= − b C into Eq. (9), we have / 2 cosθ ϕ= −d ds kb ,

and then the following equation is obtained

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2 22 cos 1 sin

d dds

kb k b

θ ϕ

ϕ ϕ= = −

− −. (10)

By taking the integral of both sides of Eq.(10), we get

0

2 2

1

1 sin

ds

k b

ϕ

ϕ

ϕ

ϕ=

−∫ . (11)

Where ( ){ }0 0arcsin sin / 2 /ϕ θ= b . And Eq.(11) can be transformed into

( ) ( )( )0

1, ,s F b F b

kϕ ϕ= − . (12)

Substituting Eq.(12) into Eq.(9) yields

( ) ( ) ( ) ( )0 0 0

2 2 1, , , (cos cos ), , ,ϕ ϕ ϕ ϕ ϕ ϕ= − − = − = −

bx E b E b s y s F b F b

k k k. (13)

Where ( ),F bϕ

and ( ),E bϕ are called the incomplete elliptic integrals of the first and second

kind [8], respectively.

For a curved beam hinged at both ends, considering the symmetry, only a half is taken into

account, which length is l . At one end of the half beam, 0s = , the bending moment is zero, 0 'θ

is constant; at another end, =s l , 0θ =l , 0ϕ =l , ( )0 , /ϕ=l F b k , ( )02 , /ϕ= −x E b k l . During the

analyses and calculation procedure, material and structure parameters as well as initial values are

given as follows, 35.236 10−= ×l m , 112.1 10= ×E Pa , 16 41 10−

= ×I m , 0

10θ =� , 0 ' 100 / 3θ = , the

initial distance between the two ends 2

01.042 10X m

−= × . With the increase of forces, the

deformations of the beam become larger and larger and 0θ constantly changes. Here, an iterative

algorithm is utilized by adjusting 0θ to make =s l . In this way, the solutions of 0θ are figured

out under different forces. Subsequently, k , 0C , b , 0ϕ are gotten. The overall amount of

compression between the two ends of the whole curved beam is:

( )( )0 0 02 2 4 , /x X x X l E b kϕ∆ = − = + − . (14)

In Eq.(14), ∆x is a function of the force P . In order to compare errors between analytical results

based on Eq.(14) and ANSYS simulation results based on FEM, two load-displacement curves are

given in Figure 4, where x∆ is displayed along the horizontal axis and P on the vertical axis,

the blue line is the result based on Eq.(14), the red one is the ANSYS simulation result. A part of

Figure 4(a) is scaled up as shown in Figure 4(b). By comparison, it can be found that errors between

two curves are less than 0.4%, which means that the ANSYS analysis method may be used to

accurately simulate nonlinear geometric deformations of curved beams.

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(a) Error comparisons (b) Drawing of partial enlargement

Figure 4. Load-displacement curves.

Influence of Geometric Parameters on Stiffness and Load Capacity of Curved Beams

Compared with curved beams with uniform cross-sections, curved beams with varying

cross-sections can further improve stress distribution, increase static stiffness and reduce dynamic

stiffness. In this section, aiming at the curved beams with varying cross-sections, we will

investigate the influence of geometric parameters on stiffness and load capacity. It should be noted

that Eq. (6) applies to curved beams with uniform cross-sections or varying cross-sections. However,

for convenience, ANSYS software is employed here.

We take a structure in Figure 5(a) for example, which consists of the upper and lower rigid bodies

as well as curved beams on each side. The material of curved beams is ultra-high-strength steel with

Elastic Modulus 52.1 10× MPa , Poisson ratio 0.3 and Density 3 37.8 10 /× kg m .Because of the

symmetry of the structure, in order to reduce redundant computation, we only analyze a quarter

model subject to symmetric boundary condition, as shown in Figure 5(b). During the simulation

process, all degrees of freedom of the lower surface are constrained, and a displacement of 0.2mm

is applied to the upper surface in the negative direction of Z axis. The curved beam model is shown

in Figure 6(a) or Figure 6(b). In Figure 6(a), the curved beam only has one minimum in its

cross-sectional area; and In Figure 6(b), the curved beam has two minimums in its cross-sectional

area. Where H is the height of the curved beam; v is the initial deflection of the curved beam; t

is the width of a cross-section in the symmetry plane of the curved beam; a is the width of end

section of the curved beam; h is the height of a cross-section with minimum area, which is

defined as the distance from the cross-section to the symmetry plane of the curved beam; b is the

width of a cross-section with minimum area.

(a) Analysis structure (b) A quarter model

Figure 5. Analysis structure and its quarter model.

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(a) With one minimum (b) With two minimums

Figure 6. Curved beam models.

For the curved beam model with one minimum in cross-sectional area as shown in Figure 6(a),

the parameters are given as: 0.07=t mm , 0.14=a mm , 0.43=v mm , 10=H mm .And the

simulation results are shown in Table1 and Figure 7. In Table 1, 1F and 2F are the external loads

causing the deformation of the curved beam as much as 1% H and 2% H , respectively, and 2F

is regarded as the working load, which represents the load capacity of the structure; 0K , 1K and

2K are the structure stiffness corresponding to the deformations of 0.05% H , 1% H and 2%H ,

respectively; K % is the ratio of static and dynamic stiffness, which is defined as 0 2 0/−K K K ;

σ is the maximum stress corresponding to the deformation of 2% H . And it should be pointed

out that [ ]σ MPa in all the Tables are less than the yield stress of ultra-high-strength steel.

Table 1. Simulation results (for Figure 6(a)).

[ ]1F N [ ]2

F N [ ]0/K N mm [ ]1

/K N mm [ ]2/K N mm %K [ ]MPaσ

23.55 30.06 523.10 103.51 35.43 93.23 1194.80

The load-displacement curve and the stiffness-displacement curve are plotted in Figure 7(a) and

Figure 7(b), respectively. The results indicate that the ratio of static and dynamic stiffness of the

structure is very high, which characteristic can be used to attenuate and isolate vibration. In Figure

7(c), a color nephogram illustrates the equivalent stress distribution of the structure after

deformation, where the maximum stress value is 1194.8MPa occurring at the thinnest section of

the curved beam, corresponding to the position “1”.

(a) Load-displacement curve (b) Stiffness-displacement curve (c) Color nephogram

Figure 7. Simulation results (for Figure 6(a)).

Next, in order to study the influence of geometric parameters on structural stiffness and load

capacity, we will vary a parameter little by little, such as [ ]t mm , [ ]v mm or [ ]H mm , and meanwhile,

402

the other parameters are remained unchanged. Numerical results are shown in Table 2, Table 3,

Table 4 and Figure 8. Where, Table 2 and Figure 8(a) are the results gotten by only changing [ ]t mm ;

similarly, Table 3 and Figure 8(b) corresponding to the change of [ ]v mm , Table 4 and Figure 8(c)

corresponding to the change of [ ]H mm . Where definitions of %K and σ are the same as above.

Table 2. Influence of [ ]t mm on structural stiffness and load capacity.

[ ]t mm 0.09 0.08 0.07 0.06 0.05

%K 92.51 92.78 93.23 93.68 93.98

[ ]MPaσ 1242.73 1209.20 1194.80 1148.78 1132.72

Table 3. Influence of [ ]v mm on structural stiffness and load capacity.

[ ]v mm 1.00 0.80 0.60 0.43 0.30

%K 67.56 77.74 87.07 93.23 96.44

[ ]MPaσ 857.13 913.79 1104.45 1194.80 1492.43

Table 4. Influence of [ ]H mm on structural stiffness and load capacity.

[ ]H mm 10.00 9.50 9.00 8.50 8.00

%K 93.23 92.28 91.12 88.52 88.16

[ ]MPaσ 1194.80 1238.31 1259.55 1289.74 1431.10

(a) Changing [ ]t mm (b) Changing [ ]v mm (c) Changing [ ]H mm

Figure 8. Influence of geometric parameters on stiffness and load capacity (for Figure 6(a)).

Seeing from Table 3 and Figure 8(b), we can find that the initial deflection has a great influence

on the ratio of static and dynamic stiffness and the load capacity of the curved beam, where with the

reduction of v mm from 1.00 mm to 0.30 mm , both %K and the working load 2F increase

greatly. According to Table 2,Table 4, Figure 8(a) and Figure 8(c), an alternative way to make sure a

high load capacity of the curved beam is by increasing [ ]t mm or decreasing [ ]H mm in a certain

range, however, the influence of the two parameters on %K is not obvious.

In the same way, we take the curved beam model with two minimums in cross-sectional area for

another example, as shown in Figure 6(b), to explore the influence of the parameters, such as

[ ]h mm and [ ]b mm , on structural stiffness and load capacity. In one case, we set 0.07=t mm ,

0.14=a mm , 0.43=v mm , 10=H mm 1.5=h mm and change [ ]b mm from 0.06mm to

0.03mm . The simulation results are shown in Table 5 and Figure 9(a). In another case, we set

0.07=t mm , 0.14=a mm , 0.43=v mm , 10=H mm , 0.04=b mm and change [ ]h mm from

2.50mm to 1.00mm . The simulation results are shown in Table 6 and Figure 9(b).

Table 5. Influence of [ ]b mm on structural stiffness and load capacity.

[ ]b mm 0.06 0.05 0.04 0.03

%K 93.65 94.21 94.90 95.85

[ ]MPaσ 1198.33 1143.71 1121.65 1110.49

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Table 6. Influence of [ ]h mm on structural stiffness and load capacity.

[ ]h mm 2.50 2.00 1.50 1.00

%K 94.88 94.27 94.90 94.22

[ ]MPaσ 868.45 921.05 1121.65 1399.35

The simulation results show that the variation of [ ]b mm and [ ]h mm makes a lot of difference to

the structural load capacity, but almost does not work on the ratio of static and dynamic stiffness. In

Figure 9, it is obviously noted that 2F multiplies with the increase in [ ]b mm or the reduction in

[ ]h mm . And in Table 5 and Table 6, it can be found that the maximum stress [ ]σ MPa is greatly

affected by [ ]h mm .

(a) Changing [ ]b mm (b) Changing [ ]h mm

Figure 9. Influence of geometric parameters on stiffness and load capacity (for Figure 6(b)).

To sum up, compared with the curved beam with one minimum in cross-sectional area, the

curved beam with two minimums in cross-sectional area has more potential for vibration

attenuation and isolation. Because it has the higher ratio of static and dynamic stiffness and its load

capacity can be guaranteed by adjusting geometric parameters of the structure, including the initial

deflection of the curved beam, the height of the curved beam, the width or the height of a

cross-section with minimum area. Moreover, for the curved beams, a promising way of increasing

the ratio of static and dynamic stiffness is to decrease the initial deflection in a certain range.

Conclusions

This paper studies the analysis method of stiffness and load capacity of curved beams, mainly

focuses on the influence of geometric parameters of curved beams on their stiffness and load

capacity. And some conclusions are as follows:

1. By comparing the analysis and simulation results of the curved beams proposed in this

paper, it can be found that the curved beam with two minimums in cross-sectional area has more

potential for vibration attenuation and isolation.

2. Initial deflections of the curved beams have a significant influence on the ratio of static and

dynamic stiffness. And with the reduction in initial deflections in a certain range, the ratios of static

and dynamic stiffness increase greatly.

3. The load capacity can be guaranteed by adjusting the geometric parameters of the curved

beam, such as initial deflections, the height, the width or the height of a cross-section with

minimum area and so on.

Acknowledgement

This research was financially supported by the National Science Foundation No.11372059 and

No.11272073.

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