• Cylindrical compression spring

Springs

Characteristics & design

CDIM Team Translation – Marion Fourmeau

Lecture plan

• Generalities

• Spring functions

• Types of springs & material

• Spring assembly

• Cylindrical compression spring

• Geometry, load, stress, deflection & rigidity

• Other design criteria

• Graphical and organigramme design

• Example

• Other springs

• Torsion springs

• Belleville bushers

• Leaf springs

Generalities on springs

• The design of springs is based on :

• Material resistance (most often in fatigue)

• Dynamics

• Know-how

• However, these relationships are often not sufficiant : several springs may fit a given application.

Iterations based on technological considerations are often required.

Calibration spring: provides a treashold of mechanical action

Regulation spring: provide a linear action

Return spring: stores and returns a mechanical energy

Play adjustment: provides a compensating displacement Typical spring functions

Typical spring functions

Assemble 2 parts while enabling a relative and « controlled » displacement

The « force-displacement » relationship depends on the spring geometry Typical spring functions

Different types of springs

Sorting as a function of external actions Compression

SpiresBushersElasticblocs

Tension

Spires

Torsion

SpiresSpirale

Torsion rods

Bending

À Lames

Material Cost T (°C) σe(MPa) σr(MPa) Observations Piano string C65

C70 C80

2,5 120 275

275 275

700 800 900

Small resistant springs

Steel chrome vanadium

3 220 700 850 Endurance, impact resistance

Steel silicon 45Si7 55Si7

4 250 510-780

620-880

640-980 780-1080

Quenched Steel chrome silicon

45SiCrMo6

4 250 780-870 950-1050 Idem + life time Stainless steel 7 to 11 350 Suivant nuances Corrosion resistance Copper alloys CuNi25

CuSn7P CuNi26Zn27

40

200

500 900 650

Corrosion + low temperature resistance

For steels : τe=0,8.σe; σ-1=0,5.σr and τ-1=0,8.σ-1

The shear modulus depends also on temperature : G = G_ref– (θ-20)*2,2783 Spring material

Spring assembly

• In parallel

P P

k1

k2

k3

δ kn

Therefore, an element with larger rigidity than others takes over the assembly rigidity, and transmit most of the actions

The relative displacement d is similar for all springs The action transmitted Pi transmis for a spring i is :

P

= K

. δ

The equivalent rigidity is _i

i

K=

∑

Spring assembly

Loaded

k₁ k₂ k₃

No load

L1

L2

P

δ

δ P1

P

k₃ k2

k1

P2

1

1 1. 1

Keq K P K L

=

= δ< L₁

L₁< δ< L₂

1 K2

Keq= K +

2 1 2 1

1 1 1 2 2 1

1 2 2 2 1

( )

K . ( ).( )

K . ( )

P P Keq L L

L K K L L

L K L L

= + −

= + + −

= + −

1 K 2 K3

Keq=K + + L₂< δ

• In parallel : example

Spring assembly

• In series

P k1 k2 k3 P

The action transmitted by each spring is similar:

P

= P

The relative displacement is different:

P

= k

. δ

The equivalent rigidity is : 1

equi 1

i i

K

K

=

∑

A softer element than others takes over the softness of the system (opposite to parallel)

Spring assembly

P

δ

No load Loaded

k₁ k2

k₃

L

1

L₂

L₂< δ: Keq=K3

P _k

3

k₂

k₁

1

1 2 3

1

1 1 1

Keq

K K K

= + +

δ< L₁:

P₁= K_eq1.L₁

P₁

L₁ P₂

L₂

L₁< δ< L₂: 2

2 3

1

1 1

Keq

K K

= +

P₂= P₁+K_eq2.(L₂-L₁)

• In series : example

Lecture plan

• Generalities

• Spring functions

• Types of springs & material

• Spring assembly

• Cylindrical compression spring

• Geometry, load, stress, deflection & rigidity

• Other design criteria

• Graphical and organigramme design

• Example

• Other springs

• Torsion springs

• Belleville bushers

• Leaf springs

Cylindrical compression spring

• Geometrical definition

D p

D_e D_i

D_i: internal diameter D_e: external diameter

D : average diameter, D=0,5.(D_i+D_e) d : string diameter

p : step : distance between 2 turns with no load. Practical value : p=0.3D

n : number of active turns (= Na) n_T: total number of turns

: coiling ratio or spring index. Providers advice ranges between 8 and 10 to ease the manufacturing process.

c D

=d

d

• Load & stress

Static analysis says :

• Normal load P₁= Pcosα

• Transverse load P₂= Psinα

• Bending moment M_f= P₂D/2

• Torsion moment M_t= P₁D/2 By considering torsion only (main load whenαsmall*) :

τ = 8PD / π d ³

• D : average diameter

• d : string diameter

• P : axial load

*For α < 5°, Mf = 8,7% de Mt

D P

d

Cylindrical compression spring

• Load & stress

Cylindrical compression spring

F

F

F

F Mt

Shear stresses (F)

τ(F)

Resulting stresses

τ Torsion stresses (Mt)

τ(Mt)

max 3

Mt

8. . F D τ d

= π

max 2

F

4. F

τ d

= π

max 3

8. . 1 0.5*

Mt F

F D d

d D

τ

π

 

=  + 

 

Cylindrical compression spring

• Stresses : resistance criteria

. 3

E

F u

σ

τ τ

< ≈

α

max 3

. 8. .

D

k F D τ d

= π

Practically:

Correction coefficient

Linear beam model : shear ratio Neglect curvature effect Satisfying approach in static

(1 0.5 ) k

= + c

Often limites to 4/5

Curved beam model : Wahl factor Account for string curvature Necessary approach in fatigue

4 1 0.615

4 4

D

k c

c c

−

 

=   − +  

• Stresses : resistance criteria

4 1 0,615 The Wahl coefficient

4 1

4 4

0,615 is an approximation of the

« real » stress distibution inside the spring, which can be obtained experimentally or by numerical simualtion

0 1 2 3 4 5 6 7 8 9 10

0 5 10 15 20 25 30

diamètre de fil Ød

rapport d'enroulement c

Wahl coefficient k_D= 4c-1/(4c-4) + 0,615/c

c D

=d Coiling ratio:

Cylindrical compression spring

Cylindrical compression spring

Relationship « force - deflection »

F = K . ∆ H

Stored energy

1 2

2

.

U = K ∆ H

Circular section spring

Rigidity : K Deflection : ∆H

Tension spring (preload F₀) Compression spring

3 4

8. . . . H n D F

∆ = G d 8. .

( )

.

H n D F F

∆ = G d −

4 0

3 3

. .

8. . 8. .

F F G d G d

K H n D n c

= − = =

∆

4

3 3

. .

8. . 8. .

F G d G d

K = H = n D = n c

∆

RdM

Rigidity deflection

• Deflection & rigidity

τ= k_D.8Pc / πd² Deflection =

8Pc³N_a/ Gd Rigidity =

dG / 8c³N_a

∆τ= k_D.8∆P.c / πd²

No load Calibration

Use

N_a: number of active turns G : shear elasticity modulus c : coiling ratio

d : string diameter P : axial load K_D: Wahl coefficient Loaded characteristics

Spring length

Spring deflection in compression

Minimal Joint coils Or «to the limit »

Springs often work in fatigue, so the alternating stress should be considered.

The limit of alternating stress is provide by Goodman diagram, dedicated to springs

∆τ= 450 Mpa for τ1= 200 MPa

∆τ= 200 Mpa for τ1= 700 MPa

Contraintes minimales

C o n tr a in te s m ax im a le s

Alternating stress limit : Goodman diagram (DIN 17 223-1 C spring)

D + d < D_A D – d > D_B H > H_mini

Minimal turn distance

H < H_flambage

norme DIN 2076

D_A

D_B

Manufacturing constrains:

• c is ranging from 4 (difficult to process) to 10 (too soft).

• The minimal number of active turns Na is 2 (to spread the load)

• Ød >1,2mm for granulated strings

Minimal turn distance

Most design criteria enable to provide criteria on Ød and c, represented in a [c; Ød] graph Geometrical limitations

D minimal

D maximal c minimal

c maximal

∆τmaximal τmaximal

Na minimal Na donné

ØD donné

For a given material (G)

Coiling ratio

St ri ng di a m e te r

Graphical representation of spring design

Coiling ratio limit

Coiling ratio

St ri ng di a m e te r

Graphical representation of spring design

For a maximal spring diameter D_B

B

B B

D d D c.d d D d D

+ ≤ → + ≤ → ≤c 1 +

Coiling ratio

St ri ng di a m e te r d

Graphical representation of spring design

For a minimal diameter D_B

B

B B

D d D c.d d D d D

− ≥ → − ≥ → ≤c 1

−

St ri ng di a m e te r

Graphical representation of spring design

For a load P_max

D max

max 2 u

K (c).8.P .c

d f (c)

d ^τ

τ = ≤ τ → ≥

π

Coiling ratio

St ri ng di a m e te r

Graphical representation of spring design

For an alternating load∆Pmax

D max

max 2 L

K (c).8. P .c

d f (c)

d ^∆τ

∆τ = ∆ ≤ ∆τ → ≥

π

Coiling ratio

St ri ng di a m e te r

Graphical representation of spring design

K > K_min Na_min=2

3

L L

3

dG 16c

K d .K

8c .2≥ → ≥ G

Coiling ratio

St ri ng di a m e te r

Graphical representation of spring design

D minimal

D maximal c minimal

c maximal

∆τmaximal τmaximal

Na minimal Na fixed

ØD fixed

St ri ng di a m e te r

Admissible area Graphical representation of spring design

D minimal

D maximal c minimal

c maximal

∆τmaximal τmaximal

Na minimal Na fixed

ØD fixed

Rapport d’enroulement

D ia m è tr e du fi l

D is fixed :

D c.d d D

= → =c

Graphical representation of spring design

D minimal

D maximal c minimal

c maximal

∆τmaximal τmaximal

Na minimal Na fixed

ØD fixed

Rapport d’enroulement

D ia m è tr e du fi l

Na and K are fixed:

3 3

dG K8.c .Na

K d

8.c .Na G

= → =

Graphical representation of spring design

Buckling equations are applied to the spring and depend on :

• the material (elastic properties : Young modulus E and shear modulus G),

• Excentricitiy coefficient χaccounting for how the load is applied This provides a critical length for buckling Hf

2 1

1 1 1

0,5

χ= 2 χ= 1 χ= 0,7 χ= 0,5

Buckling (DIN 2076 norm)

1,5 0,0015 0,1 !

The critical length depends on how srping extermities are used. We define Hj , the length when turns are joints, and the minimal length Hmini = Hj+e with the minimal turn distance 1,5 0,0015 ^" 0,1 !

• Non-closed, non-milled : Hj = (Na+2,5)d N_total= Na+1,5

• Non-closed, milled : Hj = (Na+1)d N_total= Na+1,5

• Closed, non-milled : Hj=(Na+2,5)d N_total= Na+1,5

• Closed, milled : (Nt, terminale turns) : Hj=(Na+Nt-0,5)d N_total= Na+2Nt-0,5 Extremities and spring shape

Input data

external load, ØDAet ØD_B, extremity

shape…

Material choice

Diameter ØD choice

Start with largest, if ØDmini is reached, take a

harder material

Diameter Ød choice

Start with smallest, if Ødmaxi is reached,

decrease ØD

Compute c = D/d

Check that c > 4 or decrease Ød

Computeτ2, τ1,∆τ Check thatτ2 max and∆τmax,

or increaseØd

Compute Na

Check Na>2 or increase Ød

Compute Hj

Check that Hj>H2or increase Ød

Compute minimal turn distance e

If not, increase Ød

Compute D_miniand D_maxi

Or increase Ød

Compute Hf

Check Hf>H2or increase Ød

OK ! Organigram for spring design (B5435 norm)

Hydaulic engine commande Design issue :

The life time of the spring is lower than expected

Example : regulating spring

But the spring is in admissible zone!

Example : regulating spring

e

Hj

H2 < Hmini Hmini

Example : regulating spring

H2 > Hmini☺ Hmini

Example : regulating spring

Lecture plan

• Generalities

• Spring functions

• Types of springs & material

• Spring assembly

• Cylindrical compression spring

• Geometry, load, stress, deflection & rigidity

• Other design criteria

• Graphical and organigramme design

• Example

• Other springs

• Torsion springs

• Belleville bushers

• Leaf springs

Torsion springs

F D

r d Material is loaded in bending

1,2 3

32. . . . k F r

σ

=

π

4 1 4 1

max ,

4 ( 1) 4 ( 1)

c c c c

k c c c c

 − − + − 

=  

− +

 

The deformation is measured by an angle

4

64 . . . . F r D n θ = E d

(simplified relationship)

.

64 . K E d

θ

= D n

Branching influence (norm DIN)

.

(N/mm) 3667. .

K E d

n D Sa Sb

θ

=

+ +

Branching geometry & spring geometry

Torsion rod

P d

P d

L

o This is one of the simplest spring possible o One extremity is fixed, the other is connected

to external load and displacement.

. .

t o

M L

α = G I

.

o

M r

τ = I

• Conical shape

• Lie on their basis diameter

• Load applied on their small diameter

d

D ho

• Reduced spacing

• Enable to build a column of spings in series. Or in parallel

• Important load for reduced deflection Belleville bushers

Belleville bushers

• Almen and Lazlo formula

d

D h_o

« No load »

∆h

F

« Loaded »

( )

2 2

4. . .

. . .( )

1 ^{o o}

E C e h

F h h h h e

ν D^{∆  }

= −  − ∆ − ∆ +  ^Avec

2 1 2

1 . 1 ln

C

π δ δ

δ δ

δ

   

=    − 

− −

   

et D

δ= d

!

( )

2

.( )

. ^{o o} 1 . _a

o

h h h h

F h F

h e

− ∆ − ∆

 

=∆  + 

 

3

2 2

. 4. . . 1

o a

e h F E C

ν D

= −

Load for flat configuration

Belleville bushers

• Characteristic load

2

12

. ^o . 1 . 1 1

a o o o

h

F h h h

F h e h h

     

∆   ∆ ∆

=    −    − + 

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

-0.5 0 0.5 1 1.5 2 a 2.5

F F

o

h h

∆

e 1 h_o

=

e 2 h_o

=

e 2 h_o

= 2

e 2 h_o

=

Belleville bushers

• Stresses

d_o

Neutral line :

1

. ln d

d δ

δ

= −

Max stress : . ⁴ ₂. ₂ ._{1 2}

1 2

M o

E h h

C C h C e

σ D

ν

∆   ∆  

=− −   − + 

1

6 1

ln ln 1

C δ

π δ ⁻ δ

 

=  − 

 

3( 1) C δ ln

π δ

= −

Leaf spring

• Generalities

F F

2F L_t

L_c L_e

Consist of parallel leafs (b x a) :

• Np full leaves

• Ng graduated leavesenabling a homogeneous bending stress

F b_e

L_e

1 1

2 3

e T b

L = L − L

( ).

e p g

b = N +N b

1 _p.

b =N b a

Leaf spring

• Deflection & stress

F

2

6. .

( ). .

e

p g

L F N N b a σ =

+

3 3

4. . .

. .( ).

e

p g

h Q L F

E a N N b

∆ = +

2 3

3 1 3

. 2 ( ln )

(1 ) 2 2

Q λ λ λ λ

 

= −  − + − 

1 e

b λ =b