Shape Design of Cantilever Springs

Shape Design of Cantilever Springs

Gokula Krishnan Suresh, Hong Zhou Department of Mechanical Engineering Texas A&M University-Kingsville Kingsville, Texas, USA

AbstractCantilever springs are simple flat springs in which one end is fixed and the other end is loaded. There are various applications for cantilever springs that include automobiles, medical devices and consumer products. Cantilever springs commonly have straight, slender and uniform beam configurations and are designed for small deflections that are perpendicular to the beam axis. Because of the small transverse deflection, the longitudinal deflection is usually ignored, the

In Equation (1), E is Youngs modulus of the beam material. F is the force that is perpendicular to the beam axis and applied at the free end of the cantilever beam.

When the longitudinal deflection at the loading end of the beam is ignored, the bending stress ( ) along the beam axis (y) can be calculated by the following formula [4].

spring stiffness is regarded as constant and the maximum stress is considered to be proportional to the deflection. With the

( y) 6Fy

bt 2

(2)

increase of the transverse deflection of a cantilever spring, the longitudinal deflection gradually becomes large that cannot be ignored, the spring stiffness can no longer be regarded as constant, the maximum stress does not have linear relationship with the transverse deflection. It is not trivial to design nonlinear cantilever beam springs. In this paper, the longitudinal deflection is derived for a nonlinear cantilever spring. The shape of a cantilever spring is designed to reduce its longitudinal deflection. The results of the paper provides a roadmap for designing nonlinear cantilever beam springs.

KeywordsCantilever Spring; Shape Design; Deflection; Analysis.

The maximum bending stress ( max bt 2 ) occurs at the fixed end of the beam. Bending stress is tensile on one side of the beam and compressive on the other side.

1. INTRODUCTION

   Flat springs usually refers to springs that are made of sheet, strip or plate. Although term flat is used, the shape of a flat spring is not necessarily flat. It may contain bends and other complicated forms [1]. The main purpose of using term flat is to distinguish the shapes of flat springs from those of helical, spiral, washer or power springs. Cantilever springs are simple flat springs in which one end is fixed and the other free end is loaded. They are designed to generate desired force and deflection relationships. There are various applications for cantilever springs that include automobiles, medical devices and consumer products [2].

   Cantilever springs commonly have straight, slender and uniform beam configurations and are designed for small deflections that are perpendicular to the beam axis. Because of the small transverse deflection, the longitudinal deflection is usually ignored, the spring stiffness is regarded as constant and the maximum stress is considered to be proportional to the deflection.

   Figure 1 shows a cantilever beam with uniform rectangular cross section. The in-plane thickness and out-of-plane width of

   Fig. 1 A cantilever beam with straight and uniform undeformed shape.

   The material of the cantilever beam is considered as homogeneous and isotropic in the paper. The slender beam is assumed to be inextensible. The strain ( ) of the beam remains small and is within its linear elastic range. The cantilever beam in the paper is an Euler-Bernoulli beam. Plane hypothesis holds for the cantilever beam, i.e., a plane cross section that is perpendicular to the neutral axis of the beam before deformation remains plane and perpendicular to the neutral axis after deformation. The bending moment (M) of an Euler- Bernoulli beam is proportional to its curvature ( ). The relationship can be written as follows [5].

   the cantilever beam are t and b, respectively. The beam has

   (s) 1

   d M (s)

   (3)

   length of L. When the transverse deflection ( x ) at the loading

   (s)

   ds EI

   end is small, it can be calculated as follows [3].

   4FL3

   x Ebt 3

   (1)

   In Equation (3), s is the arc length along the deflection curve, is the radius of curvature, is the slope of the

   deflection curve, I is the moment of inertia of the cross section of the beam.

   In the rectangular coordinate system as shown in Figure 1, the curvature of the deflection curve of the cantilever beam can be written as:

2. BEAM DEFLECTION ANALYSIS

   For the cantilever beam shown in Figure 1, we assume x

   (s)

   1

   (s)

   d 2 x

   dy 2

   dx

   2 3 2

   M ( y) EI

   (4)

   is given together with known beams cross sectional sizes (t and

   b) and Youngs modus (E) of the beam material. We are trying to find out y and F.

   Rearranging Equation (3) yields the following equation.

   1 dy

   EI d M (s)

   (8)

   Equation (4) is a second order nonlinear differential equation. It is difficult to have an analytical closed-form solution to the equation. When beam deflection is small, dy

   is small and (dx dy)2 approaches zero. When the denominator

   ds

   Differentiating both sides of Equation (8) with respect to s, we have

   2

   of Equation (4) is approximated as one, the nonlinear

   EI d dM (s)

   (9)

   differential equation is simplified as follows.

   ds2 ds

   d 2 x M ( y)

   M (s) is calculated by Equation (6) as FL y y. We

   dy 2 EI

   (5)

   have dM (s) ds F dy ds . Since dy ds cos , we have

   M ( y) in Equation (4) can be derived from Figure 1 as:

   dM (s) ds F cos . Substituting the expression of

   M ( y) FL y

   • y

     (6)

     dM ds into Equation (9) and moving the term on the right hand side to the left yields the following equation.

     When y is ignored as zero for small deflection, M ( y) can be approximately simplified as M ( y) FL y. Equation (5)

     d 2

     EI

     ds2

     • F cos 0

       (10)

       becomes a second order linear differential equation with the simplified M ( y) . Its analytical closed-form solution can be

       From Equation (10), we have

       solved as:

       d 1 d 2

       Fy2

       EI

       ds 2 ds

       • F sin 0

         (11)

         x( y)

         6EI

         3L y

         (7)

         When beam deflection is not small, the denominator in Equation (4) cannot be simplified as one, and the longitudinal

         Equation (11) can be solved with its solution as

         1 d 2

         deflection y in M ( y) can no longer be ignored. Although it

         EI

         2 ds

       • F sin C

     (12)

     is difficult to have an analytical closed-form solution to the

     second order nonlinear differential equation of Equation (4), many different numerical approaches have been proposed and published [6], which include elliptic integral approach [7],

     C in Equation (12) is an arbitrary constant. It can be decided by the boundary condition of the deflected cantilever beam. At the loading end of the beam, we have s L , M (L) 0 , and

     power series approach [8], Runge-Kutta approach [9], finite element approach [10], equivalent system approach [11], and others [12]. In most existing approaches, the problem is focused

     d

     ds sL

     0 . Assume the slope of the beam at the loading end

     on solving the large deflection cure of a cantilever beam or a frame under a given loading such as concentrated, distributed or combined. However, we are more interested in the reverse problem for cantilever beam springs, i.e., solving the needed reaction force (F) and corresponding longitudinal deflection (

     y ) under the given transverse deflection ( x ). For a cantilever

     is m . Substituting the expressions of d ds and at the

     loading end into Equation (12) yields the following equation.

     F sinm C (13)

     Substituting Equation (13) into Equation (12), we have

     beam spring, y is usually an undesired axial deviation that can

     d 2

     ds

     2F (sin sin )

     EI m

     (14)

     be reduced by designing the shape of the cantilever beam. The

     authors of the paper are motivated by the challenges facing cantilever beam springs. The research objective of the paper is to provide a guideline and systematic approach for the analysis

     Taking square root on both sides of Equation (14) yields the following equation.

     and shape design of cantilever beam springs.

     The remainder of the paper is organized as follows. The beam deflection analysis is presented in section II. The analysis

     d

     ds

     2F sin

     EI m

   • sin

     (15)

     on cantilever beam springs is provided in section III. Section IV is on the shape design of cantilever beam springs. Conclusions are drawn in section V.

     Rearranging Equation (15) yields the following equation.

     ds

     EI d

     (16)

     2F sinm sin

     Integrating Equation (16) from the fixed end to the loading end of the cantilever beam yields the following equation.

     L EI m d

     (17)

     2F 0

     sinm sin

     Substituting Equation (16) into following equation.

     dx cos ds

     yields the

     dx

     EI cos d

     (18)

     2F sinm sin

     Integrating Equation (18) from the fixed end to the loading end of the cantilever beam yields the following equation.

     Fig. 2 The solid model of a cantilever beam spring with straight undeformed shape.

     The solid model created in ANSYS Design Modeler is then analysed in ANSYS Mechanical [17] that is also an application

     EI m cos d

     (19)

     of ANSYS Workbench. The lower end of the beam is fixed and

     x 2F 0 sin sin

     its upper end is for loading. A x of 50 mm is applied at the

     m

     m and F are unknowns now. They can be solved by Equations (17) and (19). Combining Equations (17) and (19) and Eliminating F from them yields the following equation.

     upper end of the beam. The deformed and undeformed shapes of the beam are shown in Figure 3.

     The deformation numbers and their corresponding colors shown in Figure 3 are for the transverse deformation of the

     d

     cos d

     cantilever beam that is directional deformation in ANSYS

     x m L m

     (20)

     along x axis. Because of the large deformation of the beam, its

     0 sinm sin 0 sinm sin

     m is the only unknown in Equation (20). It can be solved numerically.

     After m is solved through Equation (20), F can then be solved from either Equation (17) or (19).

     loading end of the beam has a significant longitudinal deformation, which is shown in Figure 4. The longitudinal deformation is directional deformation in ANSYS along y axis. The y of the cantilever beam is 10.41 mm, which is large that

     cannot be ignored. In Figure 4, the longitudinal deformation has negative sign. That is because it is downward and the positive y axis direction is upward.

     Substituting Equation (16) into following equation.

     dy sin ds

     yields the

     dy

     EI sin d

     (21)

     2F sinm sin

     Integrating Equation (21) from the fixed end to the loading end of the cantilever beam yields the following equation.

     EI m sin d

     (22)

     y 2F 0

     sinm

   • sin

   Equation (22) leads to the solution of y .

3. CANTILEVER SPRING ANALYSIS

   The deflection and reaction force of a cantilever beam spring can be directly analyzed by finite element analysis software ANSYS [13-15]. The stress of the deflected beam can also be directly obtained during the analysis process.

   Figure 2 shows the solid model of an initially straight and uniform cantilever beam spring. The height of the beam is 150 mm. The thickness (t) and width (b) of the beam are 0.25 mm and 10 mm, respectively. The material of the cantilever spring is structural steel with Youngs modules (E) of MPa, Poissons ratio () of 0.3, yield strength (y) of 250 MPa. The Design Modeler [16] of ANSYS is used to create the solid model. ANSYS Design Modeler is an ANSYS Workbench application that provides modeling tool for the creation and modification of geometries.

   Fig. 3 The transverse deformation of the cantilever beam spring with straight undeformed shape.

   The maximum stress within the deformed cantilever beam is 175.51 MPa, which is below the yield strength of the beam material. The stress distribution in the deformed beam is shown in Figure 5. To have the transverse deformation of 50 mm at the loading end of the cantilever beam, an input force of 0.1315 N is needed. The input force is called reaction force in ANSYS Mechanical. The reaction force is shown in Figure 6. The input force is small because the cantilever beam is slim.

   The first shape change is to make the vertically straight undeformed shape become slantingly straight undeformed shape. Figure 7 shows the solid model of the cantilever spring. The slanted beam has vertical height of 150 mm that is along y axis. Its lower end is fixed. Its upper end is now away from its

   lower end horizontally by 50 mm. When a x

   of 50 mm is

   applied at the free loading end of the slanted cantilever spring, its horizontal and vertical deformations along the beam are

   shown in Figures 8 and 9, respectively. The y

   of the slanted

   Fig. 4 The longitudinal deformation of the cantilever beam spring with straight undeformed shape.

   Fig. 5 The stress of the cantilever beam spring with straight undeformed shape.

   Fig. 6 The input force of the cantilever beam spring with straight undeformed shape.

4. SHAPE DESIGN OF CANTILEVER SPRINGS Shape design of a cantilever beam spring is to improve its

   performance and better meet its needs and requirements by changing its shape. The vertical cantilever beam spring analyzed in the preceding section has uniform cross section. Its height, width and thickness are 150 mm, 10 mm and 0.25 mm, respectively. In this section, we use the same spring material, beam cross section, beam height along y axis, but change the beam shape to see the performance difference under the same

   x of 50 mm.

   cantilever beam is now 6.4863 mm, which is smaller than that of 10.41 mm from the vertical cantilever beam. As shown in Figure 9, the vertical deformation has positive sign, which means its direction is upward along positive y axis.

   Fig. 7 The solid model of a cantilever beam spring with slanted undeformed shape.

   Fig. 8 The horizontal deformation of the cantilever beam spring with slanted undeformed shape.

   The stress distribution in the deformed cantilever beam spring is shown in Figure 10. The maximum stress is 152.01 MPa, which is below that of 175.51 MPa from the vertical cantilever beam. To have the horizontal deflection of 50 mm at the loading end of the slanted cantilever beam, an input horizontal force of 0.1013 N is needed, which is smaller than that of 0.1315 N from the vertical cantilever beam.

   The undeformed beam shape does not have to be straight. It can be curved. The solid model shown in Figure 11 has a circular undeformed shape. Both the fixed and loading ends of the cantilever beam are on the y axis. The arc length of the beam is set as the same as the length of the slanted straight beam shown in Figue 7, which is 158.11 mm.

   Fig. 9 The vertical deformation of the cantilever beam spring with slanted undeformed shape.

   Fig. 10 The stress of the cantilever beam spring with slanted undeformed shape.

   Fig. 12 The horizontal deformation of the cantilever beam spring with right circular undeformed shape.

   Fig. 13 The vertical deformation of the cantilever beam spring with right circular undeformed shape.

   The stress distribution in the deformed cantilever beam spring is shown in Figure 14. The maximum stress is 155.35 MPa, which is close to that of 152.01 MPa from the slanted cantilever beam. To have the horizontal deflection of 50 mm at the loading end of the right circular cantilever beam, an input horizontal force of 0.1107 N is needed, which is slightly below that of 0.1013 N from the slanted cantilever beam.

   If the right circular cantilever beam is flipped with respect to y axis, the cantilever beam becomes a left circular beam. Its

   solid model is shown in Figure 15. When a x

   of 50 mm is

   Fig. 11 The solid model of a cantilever beam spring with right circular undeformed shape.

   When a x of 50 mm is applied at the free loading end of the circular cantilever spring, its horizontal and vertical deformations along the beam are shown in Figures 12 and 13, respectively. The y of the right circular cantilever beam is

   now 6.0895 mm, which is close to that of 6.4863 mm from the slanted cantilever beam. As shown in Figure 13, the vertical deflection is downward along negative y axis.

   applied at the free loading end of the left circular cantilever spring, its horizontal and vertical deformations along the beam are shown in Figures 16 and 17, respectively. The y of the left

   circular cantilever beam is now 16.995 mm, which is well above that of 6.0895 mm from the right circular cantilever beam. As shown in Figure 17, the vertical deformation is downward along negative y axis. The stress distribution in the deformed cantilever beam spring is shown in Figure 18. The maximum stress is 181.07 MPa, which is larger than that of

   155.35 MPa from the right circular cantilever beam. To have the horizontal deflection of 50 mm at the loading end of the left circular cantilever beam, an input horizontal force of 0.1429 N is needed, which is above that of 0.1107 N from the right circular cantilever beam.

   Fig. 14 The stress of the cantilever beam spring with right circular undeformed shape.

   Fig. 15 The solid model of a cantilever beam spring with left circular undeformed shape.

   Fig. 16 The horizontal deformation of the cantilever beam spring with left circular undeformed shape.

   Fig. 17 The vertical deformation of the cantilever beam spring with left circular undeformed shape.

   Fig. 18 The stress of the cantilever beam spring with left circular undeformed shape.

   The shape design results are summarized in the following table. The cantilever beam spring in each case has the same material, same cross section, and same horizontal input deflection (50 mm).

   Table 1 Shape design results of the cantilever spring

   Cantilever Beam Spring Shape	Vertical Deflection	Maximum Stress
   Vertical Straight	-10.41 mm	175.51 MPa
   Slanted Straight	6.4863 mm	152.01 MPa
   Right Circular	-6.0895 mm	155.35 MPa
   Left Circular	-16.995 mm	181.07 MPa

   As shown in Table 1, the cantilever beam spring with right circular shape has the smallest vertical deflection. The slanted straight springs vertical deflection is slightly above that of the right circular spring. Their vertical deflections have opposite directions. The lowest maximum stress comes from the spring with slanted straight shape. The right circular springs maximum stress is a little above that of the slanted straight spring. Cantilever beam springs with vertical straight or left circular shape have larger vertical deflection and higher maximum stress than cantilever beam springs with any of the other two shapes.

5. CONCLUSIONS

A cantilever beam spring usually has straight shape and is designed for small deflection that is perpendicular to the springs straight line. When the transverse (horizontal) deflection at the free end of the cantilever beam is large, the longitudinal (vertical) deflection is also not small. To reduce the vertical deflection, the springs straight line can be slanted (which is no longer perpendicular to the horizontal deflection). A cantilever beam spring with slanted straight shape has smaller vertical deflection and lower maximum stress than its corresponding vertical straight spring. Besides slanting a vertical straight spring, the vertical straight shape of the spring can be changed to be circular to reduce its vertical deflection. There are two symmetric circular shapes among which one (called right circular in the paper) decreases vertical deflection and another (left circular) increases vertical deflection. Cantilever beam springs with slanted straight and right circular shapes have similar effects on reducing vertical deflection and lowering the maximum stress. Their vertical deflections have opposite directions.

ACKNOWLEDGMENT

The authors of this paper gratefully acknowledge the research instrument support of the US National Science Foundation under Grant No. 1337620. Any opinions, findings, recommendations or conclusions expressed in this paper are those of the authors and do not necessarily reflect the views of the US National Science Foundation.

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