Dtip57.doc

Packaging solutions for MEMS/MOEMS using thin films as mechanical components

P. Boyle a, R.R.A. Syms b and D.F. Moore a

a Cambridge University Engineering Department, Trumpington Street, Cambridge CB2 1PZ, UK

b Department of Electrical Engineering, Imperial College of Science and Technology and Medicine, Exhibition Road, London SW7 2BT, UK

ABSTRACT

Optoelectronic subsystems for are becoming increasingly important to reduce the costs of assembly and packaging. The mechanical properties of vapour-deposited thin films can be used to advantage; for example three micrometer thick silicon nitride microclips to hold single mode optic fibres in place in silicon V-shaped grooves. This paper describes the proposed use of pairs of thin film microcantilevers to precisely locate an optic component such as a filter or a mirror in an optical bench. In this configuration the precision of the lithographic process for the cantilevers determines the exact location of the component in the package, and to first order the etched shape in the substrate is unimportant. Simulation software based on variational principles has been developed to examine the behaviour of structures undergoing large scale elastic deflections. The design software consists of spreadsheet front end to enter parameters, and then Visual Basic ( VBA ) code and Frontline 'solver' software to run simulations. The fabrication process is described for 5 µm thick silicon carbide beams which are then tested by bending them using a surface profiler ( such as a Dektak ) to deflect the fifty micrometer wide silicon nitride cantilevers through large angles. The possible consequences for more efficient optoelectronic packaging are briefly assessed. Keywords: microsystem, simulation, packaging, SiC, beams

I. INTRODUCTION

Microelectromechanical systems ( MEMS ) are of growing importance1-21. Many microsystems involve small elastic deflections of cantilevers and membranes e.g. accelerometers and pressure sensors1. Small angle deflections are considerably more amenable to analysis than large-scale deflections. Many proposed micropackages involve elastic deflections large enough to make the point of application of a force change hence making the system geometrically non-linear9.

Conventional analytical analysis of such a system is an arduous task resulting in a set of problem specific equations whose solution requires numerical methods. Many finite element packages can only deal with quite specific geometrically non-linear problems. Even if they can the packages themselves are expensive, the problems difficult to set up and solutions are computationally intensive. This lengthens the period and may even make infeasible the optimisation of microsystems based on large elastic deflections.

We report a simple method, which is applicable to a variety of problems. It can deal with large deflections as for example the packaging proposal sketched in Figure 1, and with some coupled physics problems where an electric field may be applied. It works on Variational principles, finding shape functions of elastic structures, which minimise the system energy given a set of user defined displacement constraints. Energy methods considerably simplify non-linear geometry problems. The encoding is done in a spreadsheet environment making full use of ready built macros and functions to create a user friendly and highly flexible design tool that can simulate non-linear deflections of variable cross-section and composite microcantilevers. The specific analysis outlined produces results for a configuration of microcantilevers that can be arranged to hold optical components in a vertical position.

Lumped mass analysis involves splitting a continuous section into a series of discrete elements whose properties approximate those of the original bar. It is used widely in vibration analysis to investigate system resonances. The continuous beam is split into a series of hinged torsional springs joined by rigid members of zero mass as in Figure 2. When the beam undergoes deformation the rigid members can rotate relative to each other and elastic energy is stored in the torsional springs. The total elastic energy stored in the bar is simply the sum of spring energies.

Si Substrate

Inserted Component

Clip

Clip Length, L

Clearance α

Tip Force

Figure 1. Cross section of the proposed packaging structure.

2. ANALYSIS OF GEOMETRICALLY NON-LINEAR ELASTIC DEFLECTIONS

The springs are such as to give the same bending rigidity per unit length as the continuous bar. We can then write the torsional spring constant as EIk

s=

∆ M kθ= E.I - Flexural rigidity of bar

We can develop expressions for the energy in the bar by looking at a small section. The energy stored in a single torsional spring is E Mdθ= ∫ 2E 1

2kθ=

At node n the stored energy is given by

21( )

2n nEI

nEs

θ θ −= −∆

The total energy is the sum of the separate energy terms

2

0 2

N

tot nn

EIEs

θ=

= ∆∆∑

In a given system the bar assumes the shape corresponding to the lowest energy, so by minimising the above function given a set of geometric constraints we can find the shape of a grossly deformed bar. Finding the minimum energy is not the simple problem of minimizing a several variable constrained function. We are interested in bringing about a particular condition for a given expression by varying the functions on which the expression depends. Specifically the shape function of a fixed length bar so as to minimize elastic energy given a set of geometric constraints. The bar shape function can be specified in terms of Cartesian coordinates, y(x). In order to obtain an explicit solution y(x), the problem must first be expressed in integral form. The stored elastic energy can be expressed as an integral involving y(x).

Minimize

0

( , , )L dyE k F y x dx

dx= ∫

To solve analytically and find an explicit solution for y(x) we w use the Euler-Lagrange equation. The working of such a solution is long and difficult but changes be made to make the problem more amenable. Previously the shape function was given in terms of Camenable to solution if the shape function is given in intrinsic cothe angle changes along the length of the bar. We now look at tM is the moment and κ is the local curvature of the section. evaluated over the whole bar. totE Mκ= ∫ ds

Making use of the Euler constitutive relationship for bending beam

d Mds EIθκ = =

Approximating the integral expression as a series expression give

2

0 2

N

tot nn

EIEs

θ=

= ∆∆∑

[Un, Vn]

[Un-1 , Vn-1 ] n

n-1

θn

θn-1

∆s

Figure 2. Discretisation of cantilever.

3. MODELLING SILICON CARBID

The primary function of the microclips is to precisely locate a bench as sketched in Figure 1. The problem essentially comprigap α , which is smaller than L. We wish to find the resulting shtip of the cantilever. Intrinsic coordinates are used to describe relative to some datum. In the discrete intrinsic coordinate modlength segments. The geometric constraint that the bar must fit intrinsic coordinates; it is necessary to make a switch to Carteperfectly rigid so, given the set of bar angles, the node points carecursive expressions for node points are obtained

1.n nV s Sin Vnθ −= ∆ +

1.n nU s Cos Unθ −= ∆ +

Given the location of the node points we can impose displacementhe shape function. In the microclip example the end of the barsize α. UN = α.

ould can

artesian coordinates. The problem is more ordinates, θ(s). This function describes how he internal virtual work of a bar in bending. The stored energy is an integral function

s.

s

∆s

n+1

E MICROCLIPS

variety of optical components on an optical ses of fitting a cantilever of length L into a ape function θ(s) and the normal force at the the angle of the bar at every point along it el the bar is divided into a number of equal into the gap α cannot be easily described in sian coordinates. The bars are taken to be n be plotted on the xy plane. The following

t constraints and minimize energy to obtain , i.e. the last node, must be equal to the gap

4. IMPLEMENTING THE VARIATIONAL MODEL

Excel includes an optimisation plugin called Solver developed by Frontline software19. It allows you to minimize an objective function (Energy) by changing a set of variables ( bar angles ) subject to constraints. Actual implementation is straightforward. (i) Changing variables – Bar angles θ0 …..θN (ii) Objective function – this is a function of the changing variables. Minimize Etot subject to constraints: Utip = α ; tip of bar fits into gap; θn <= 90 ; and angle of segments cannot exceed 90o. Solver finds the solution to this non-linear optimisation problem by using a reduced gradient method. Effective use of Solver and the results it produces requires knowledge of some the subtleties of optimisation. The Frontline Software website19 contains a detailed user manual for Solver outlining what problems Solver can tackle, how to set them up for fast solution, effective constraint specification, interpreting Solver messages etc.

The geometry of the bent cantilever is independent of the Young’s modulus and the second moment of area. These two parameters do, however, affect the resulting normal force acting at the tip of the cantilever. In order to calculate the force either static equilibrium conditions or a work argument can be used. Looking at a free body diagram for the resulting bent cantilever, it is clear a force acts at the tip at a perpendicular distance δ from the root. The resulting moment must be balanced at the root to maintain static equilibrium. The moment of a single torsional spring in the discrete model is given by.

EIMs

θ= ∆∆

This equation is identical to the Euler linear constitutive relationship for beams

Ms EIθ∆

=∆

With a given shape function (found by Solver) the bending moment can be evaluated at node points using the above relationship. Evaluation at node 0 corresponds to the root moment.

0

rr

EIMs

θ= ∆∆

Tip force is simply the root moment divided by the perpendicular distance from root to tip. For large deflections the point of application of the force does not act at the tip of the cantilever but at the point at which the cantilever goes flat against the inserted component. To calculate this distance it is necessary to introduce an extra column in the spreadsheet to check where the cantilever makes its first perpendicular contact. Details are given in Appendix A and Appendix B. The previous method of statical analysis works well for this particular case. Using a work argument allows for a more generalised method to find applied forces. A bar constrained to fit into a gap α1 will store elastic energy E1. If the gap size is reduced by a small amount ∆d the potential energy in the bar will increase to E2. This increase in internal energy must equal the work done by the external load F1 acting over a distance ∆d. Force is the spatial derivative of energy: hence the tip force can be approximated for different gap sizes by numerically differentiating the energy profile as seen in Figure 4. Using the central difference approximation, the force is given by

21 1 ( )2

i iE EF Od

+ − d−= +

∆∆

Solver can be run via a macro, which automates the solution procedure. In a simulation the gap size is reduced in small increments and automatically solved to get the corresponding stored energy. energy array can then be numerically differentiated to obtain a graph of how tip force varies with gap size. The advantage of this method is its flexibility for use in other problems. We can effectively apply a force at any point on the bar and in any direction by moving the corresponding node point where the force is applied in the direction of the force. This again produces an energy array, which can be numerically differentiated to get the force.

Figure 3. Cross sect

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0

Ener

gy (J

)

Figure 4. Simulated

However with this particular procedure is to obtain the forforce. It is necessary to ensuvalidation involves solving thof the excel simulation. Thererrors of less than one perceappreciably increase accuracy

5. PRO

The functional requirement ofabrication there may be promisalignment and variation inthe misalignment to these varrequire a probabilistic modeoptimised for cost, applied fo

We can abstract the system tessentially replace the bendinprofile were established previforce can be completely descmodulus, and the second mom

ion through a bent microbeam.

Energy/Force Profile

.2 0.4 0.6 0.8 1 1.2

Gap Size

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Forc

e (N)

Energy

Force

Force and Energy Profile.

model it is not possible to apply a force and find the resulting deflection. The ce-distance profile, and then determine the deflection corresponding to a given re that the results concerning energy and force have acceptable accuracy. Full e equations of the elastica for a particular case and comparing it with the results e is very close agreement between Timoshenko and the numerical model, with nt. A 30 node model provides accurate results, finer node refinement does not .

BABILISTIC DESIGN AND PROCESS ERRORS f the microclip is to hold the inserted component in a vertical position. During cess variations, which lead to vertical misalignment. For example, front-back beam geometry will cause the inserted component to skew. The sensitivity of

iations will also depend on other parameters such as the substrate thickness. We l that will give bounds on the misalignment. The microclip design is then rce, breaking stress, etc. keeping within the tolerance bounds for the alignment. o obtain a model where a component is simply acted on by force units. We g cantilevers with non-linear springs, whose spring constant and force-distance ously. Firstly we need to derive a useful expression for the force function. The ribed with three parameters; the ratio of gap size to beam length, the Young’s

ent of area F(x) = f { α/L , E , I , 1/L2 }. The dependence on α/L can be

described by another function, which effectively changes the coefficient in front of other terms F(x) = K α/L . E . I / L2 , where K α/L = f(x). Using the spreadsheet model we can obtain this function. All the geometric parameters and the Young’s modulus are set to one and a simulation is performed. We obtain a plot of the function coefficients. The data are in the form of discrete data points so we need to curve fit the data. There are two distinct regimes in the coefficient graph, tip angle less than 90o and when the cantilever is flat against the component. Initial attempts to find high order polynomial curve fits did not adequately cover the full range. Separate curve fits were performed for each of the regions in turn. While providing close agreement with the data there are problems with this procedure. Solver will later use this function, which has a small discontinuity in it. This can sometimes affect the performance of Solver both in terms of speed of operation and reliability of results.

The graph is separated into two regimes and a curve fit is performed on each data set. When the tip angle is less than 90o the force variation can be described with a linear function. When the cantilever is flat against the component the force variation is best described by a power law.

6. EQUILIBRIUM ANALYSIS

So far we have found a function to describe the force acting on the component. It is dependent on beam parameters hence slight process variations will result in different forces. We are interested in finding the equilibrium position and misalignment angle for a system where process error is present. The force functions are nonlinear so we will use Solver to calculate equilibria. We must specify the changing variables, the objective function and constraints. The component position in space needs to be adjusted until there is a force and moment balance within the system. Therefore the changing variable is component position. There are a number of ways in which position can be specified; the simplest involves only two variables. The two gap sizes α1 α3, will completely specify all translations and rotations of the inserted component, these will be the changing variables. The other two gap sizes are automatically known because the trench width and component width are both known. The component must be in equilibrium so we must encode this as the objective function. There must be a force and moment balance from any frame of reference. As discussed the force depends on the ratio of gap size to cantilever length (this gives the coefficient whose form was found by the curve fits), Young’s modulus and second moment of area. Solver will vary the gap size until the resolved horizontal force in Figure 5 is zero and the moment about both the top and bottom of the component is zero. Resolve horizontally: F1(x) + F3(x) – F2(x) – F4(x) = 0 Moment about bottom and top: [F1(x) – F2(x)].d = 0 ; [F3(x) – F4(x)].d = 0 All the equations above must be satisfied simultaneously. This can be done by setting the sum of the squares of the above equations to zero. When solver is run it finds the gap sizes that satisfy all the equilibrium constraints. The component alignment is found from simple trigonometry on the gap sizes.

d

F3(x) F4(x)

F2(x) F1(x)

Figure 5. Forces acting on a component and representation of component displacement and rotation.

7. SPREADSHEET FOR PROCESS ERRORS AND OPERATING TOLERANCES

We now have a model that will give a value for vertical orientation. The next step is to simulate many process scenarios in which the parameters can vary. We assume that most of the parameters follow a normal distribution with a given mean and variance. For example when we deposit a film of thickness 3 µm there may be a process error of plus or minus 0.1 µm. The procedure is then quite simple; we draw a set of values for each parameter from a given distribution and solve to find the equilibrium position and the vertical misalignment. When the procedure is run hundreds of times we can build up a distribution function for the vertical misalignment as seen in Figure 6. This table contains parameter values drawn from a normal distribution for a particular process scenario. With these values the equilibrium position of the component is calculated. When many simulations are performed we build a misalignment distribution. In the following example we can confidently state that for this system the misalignment angle will be less a 0.1 degree 75% of the time. We can now begin to optimise design while seeing what effect particular changes will have on the distribution. The parameters that introduce uncertainty are photolithographic error, variation in thin film deposition, front-back misalignment and inserted component tolerances. The model allows these parameters to vary individually or all together in order to obtain the cumulative error. The main driver for vertical misalignment is the front-to-back misalignment tolerance.

0

20

40

60

80

100

120

-0.2 -0.12 -0.04 0.04 0.12 0.2

Misalignment Angle

Freq

uenc

y

0%

20%

40%

60%

80%

100%

120%

0

20

40

60

80

100

120

140

160

-1 -0.6 -0.2 0.2 0.6 1

Misalignment Angle

Freq

uenc

y

0%

20%

40%

60%

80%

100%

120%

Figure 6. Histograms of simulated component misalignments measured in degrees. The histogram to the right is for microclips on a thinner substrate with worse front-back alignment, resulting in a wider distribution of misalignment angles.

8. LASER MICROMACHINING AND DEKTAK TESTING OF SiC MICROBEAMS

Laser micromachining was used to fabricate 5 µm thick chemical-vapour deposited (cvd) silicon carbide beams by (i) cutting a track in the SiC coated Si wafer using a focused laser20 and (ii) undercutting (and removing the debris) by anisotropic silicon etching using KOH in water2. Figure 7 is a secondary electron micrograph showing the pattern etched into the silicon carbide layer after a brief silicon etch. Figure 8 is a micrograph after fully undercutting the silicon carbide beams. The central isolated region of SiC in Fig. 7 has been undercut and floated away in the wet Si etch. This approach to etch only the perimeters of shapes with the laser and allowing the silicon etch process to undercut and remove the large areas allows for efficient prototyping of micromechanical test structures. As can be seen in the micrograph in Fig. 8, Beam III has been trimmed using a focused ion beam at the base of the SiC beam where it joins the SiC bank7 . Cutting slots in the SiC removes the effect of the undercut from the Si etch to make a simple beam.

Preliminary data from the beams were obtained by testing using the force produced by a stylus profilometer (Dektak21). Figure 9 shows data from Beam III in Fig. 8 giving the height as a function of distance along the beam for an applied force of 0.4 mN. The 100 µm wide beam is deflected more than 30 µm at the end. The

Young’s modulus deduced from these data for the SiC is 360 +/- 50 GPa, which is comparable with literature values 1. Other experiments in progress include stress testing with special structures such as the four focused ion beam cut slots in Beam I. This approach to testing new materials by laser patterning followed by undercutting part of the Si substrate with wet etchants is very flexible for prototyping MEMS structures.

100µm

Laser ablated trench in SiC followed by short

SiCSiC

Silicon Substrate IV

III

II

I

Figure 7. Micrograph of a partly fabricated structure. Figure 8. Micrograph of four SiC microbeams

50µm 100µm 150µm 200µm 100µm 200µm 300µm 400µm

0µm 0µm

10 µm

20 µm

30 µm

0µm

0.5 µm

1 µm

1.5 µm

Figure 9. Experimental bending of a SiC beam. The right hand plot is a smaller region of the other plot. While SiC provides a high Young’s Modulus and resistance to harsh environmental conditions, its fabrication is more difficult than other thin film materials. Similar cantilevers were fabricated using Silicon Nitride films, whose principle advantage over SiC is ease of fabrication. Using the DEKTAK these cantilevers were subject to gross deformation. The resulting deflection was perfectly elastic and even after many runs the cantilevers remained intact. It is planned to investigate other materials such as DLC and Silicon to access whether or not they can sustain large deformation.

9. EFFECTS OF OPTICAL MISALIGNMENT

Figure 10 shows a simple packaging problem based on a micromachined silicon breadboard. A beam is coupled out from an optical fibre, collimated by a ball lens, and reflected back into the fibre by a reflective component such as a dielectric filter or mirror. Fibre and lens are mounted in anisotropically etched features

V-groove Pyramidal pit Elastic fixture

Fibre Ball lens Filter

Si substrate

Figure 10. Plan view of the proposed optical bench.

To estimate the dependence of loss on alignment accuracy, we assume that the linear misalignment ∆ of the back-reflected image resulting from an angular misalignment δθ of the filter is ∆ = 2fδθ. The coupling efficiency η back into the fibre can be found by assuming that the fibre mode is Gaussian, with a characteristic radius w, and then performing an overlap integral calculation to obtain a round-trip coupling efficiency of η = exp(-∆2/w2). The corresponding round-trip loss is then L = -10 log10(η).

Assuming a mode field radius w ~ 4 µm for standard 8/125 µm telecommunications fibre operating at 1.5 µm wavelength and a ball lens with a focal length f ~ 500 µm to generate a beam of reasonable size, we obtain the results shown in Table II.

Loss L Efficiency η ∆/w ∆θ ∆θ 0.1 dB 0.9772 0.1517 0.6069 mrad 0.035° 0.5 dB 0.8912 0.3393 0.1357 mrad 0.077° 1.0 dB 0.7943 0.4798 0.1919 mrad 0.110°

Table I. Angular tolerances for different misalignments δθ.

From this we conclude that the packaging solution demonstrated can clearly achieve a performance level corresponding to the higher end of the loss regime. However, to reduce losses further additional advances are required.

10. CONCLUSIONS

The simulation software is a convenient design aid for micropackaging applications and this approach has good prospects for application in other microsystems. Preliminary data from silicon carbide microbeams indicate that it is a promising material for packaging applications. Even though the fabrication tolerances for the proposed packaging approach the are very demanding, our initial simulation results indicate that we will be able to achieve vertical alignments to better than 0.10. This opens up our low cost passive alignment system for use in low loss, free space optical benches.

ACKNOWLEDGEMENTS

Discussions with H. Fujita and A. Tixier on packaging; S. M. Spearing and K. Turner on silicon carbide; I. Baldwin on laser machining; M. Hopcroft and M. M. Ahmad on beam bending are gratefully acknowledged.

REFERENCES

1. Madou, M. J, Fundamentals of microfabrication, CRC Press, Boca Raton (1997) ISBN 849394511 2. Elwenspoek, M., Jansen, H., Silicon micromachining, Cambridge University Press,

(1999) ISBN 052159054X 3. Toshiyoshi, H., Mita, Y., Ogawa, M., Fujita, H., “Chip level 3-D assembling of microsystems”,

SPIE 3680, (1999) pp. 679-686 4. Moore, D. F. and Syms, R. R. A., “Recent developments in micromachined silicon”, I.E.E. Electronics

and Communication Eng. J., Vol. 11, (1999), pp. 261-70 5. Thienpont, H. et al, “Low cost MOEM interconnect modules for Tb/s.cm2 aggregate bandwidth to Si

chips”, SPIE Vol. 4408, (2001), pp. 6-18 6. Bustillo, J. M., Howe, R. T., Muller R.S., “Surface microengineering for microelectro-mechanical

systems”, IEEE Proceedings Vol. 86, (1998), pp. 1552-1574 7. Syms, R. R. A. and Moore, D. F., “FIB tuning of in-plane vibrating micromechanical resonators”,

Electronics Letters Vol. 35, (1999), pp. 1277-1278 8. Wu, M. C., “Micromachining for optical and optoelectronic systems”, IEEE Proceedings

Vol. 85, (1997), pp. 1833-1856 9. Strandman, C. and Bäcklund, Y., “Bulk silicon holding structures for mounting optical fibres in V

grooves”, IEEE J. Microelectromech. Sys. Vol. 6, (1997), pp. 35-40 10. Bostock, R. M., Collier, J. D., Jansen, R-J. E., Jones, R., Moore, D. F. Moore and Townsend, J. E.,

“Silicon nitride microclips for the kinematic location of optic fibres in silicon V-shaped grooves”, Journal Micromechanics Microengineering Vol. 8, (1998), pp. 343-360; U.S.A. patent 5961849

11. Conradie, E. H. and Moore, D. F., “SU-8 thick photoresist processing as a functional material for MEMS applications”, 12th Micromech. Europe Workshop MME2001 Cork, Ireland, 16-18 Sept. 2001 pp. 35-38

12. Spearing, S., M., “Design diagrams for reliable layered materials”, AIAA Journal Vol. 35, (1997), pp. 1638-1644

13. Beheim, G. and Salupo, C.S., “Deep reactive process for silicon carbide power electronics and MEMS”, MRS Symposium Proceedings Vol. 622, (2000), pp. 1-6

14. Daniel, J. H., Micromachining silicon for MEMS, MTP, Cambridge (1999) ISBN 095351420X 15. Tai, Y.-C, and Muller, R. S., “Measurement of Young’s modulus on microfabricated structures using a

surface profiler”, IEEE J. Microelectromechanical Systems (1990) pp. 147-152 16. Florando, J., Fujimoto, H., Ma, Q., Kraft, O., Schwaiger, R. and Nix, W. D., “Measurement of mechanical

properties in small dimensions by microbeam deflection”, MRS Symposium Proceedings Vol. 353, (1999), pp. 231-236

17. Ayon, A. A., Nagle, S., Frechette, L., Epstein, A. and Schmidt, M. A., “Tailoring etch directionality in a deep reactive etching tool”, J. Vacuum Science Technology Vol. B18, (2000), pp.1412-16

18. Gartner, C., Blumel, V., Hofer, B., Kraplin, A., Possner, T., and Schreiber, “Flexible microassembly setup for optical beam tramsformation systems for high power diode laser bars and stacks”, Microtec 2000, VDE Hannover September 2000, pp. 39-42

19. www.solver.com 20. www.new-wave.com 21. www.veeco.com

Implementation

To solve the problem we use Excel’s ‘Solver’ plugin. From the ‘Tools’ menu select ‘Solver’. If it is not there select ‘Add-Ins’ from the ‘Tools’ menu and click the check box beside Solver. The following pop up window appears:

4) Zero rotation at root When solver has completed its iterations the bar angles will have changed into a configuration that minimises energy. Look at the

xy graph of the bar shape to ensure the solution is reasonable and the tip satisfies displacement constraints.

Appendix A

Problem

Plot an x-y graph with range C18:D28

Problem Formulation

Target Cell Changing Cells

● Enter the relevant beam parameters in cells B3 to B6 ● Cell B8 enter =B4*POWER(B5,3)/12 ● Cell B9 enter =B3/10 (The denominator 10 equals no nodes) ● Cells B18 to B28 = 0 ● Cells C18 and D18 = 0 ● Cell C19 enter =C18 + $B$9*COS(B19) (Fill Down to C28) ● Cell D19 enter =D14 + $B$9*SIN(B19) (Fill Down to D28) ● Cell E19 enter =(POWER(B19-B18,2)+$B$6*$B$8)/(2*$B$9) (Fill Down to E28) ● Cell F19 enter =((B19-B18)/$B$9)*$B$6*$B$8

7) Click ‘Solve’ to obtain solution 6) This constrains the horizontal tipdisplacement to a specified value. In thiscase we are constraining the 0.5m bar to fitinto a gap of 0.45m

5) Bar Angles arenegative relative tohorizontal datum.

3) Bar Angles

2) Set to minimise energy 1) Sum Energy Cell

dialog box.

To add code enter the VBA editor by pressing Alt+F11 or selecting Tools>Macro>Visual Basic Editor. Copy the code below into the editor. To create a buttonto run the subroutine open the forms menu and add a button to the worksheet. Assign the AutoSolve macro to the new button in the

Sub AutoSolve() Dim EnergyArray(500) As Double Dim ClearanceArray(500) As Double

' Reset to the start tip displacement Range("I21").Value = Range("I17").Value

'delta defines the step size. It is the start distance minus the end distance divided by the number of steps delta = (Range("I17").Value - Range("I18").Value) / (Range("I19").Value)

'The following loop finds the beam shapes in the specified range For i = 1 To Range("I19").Value SolverOk SetCell:="$E$30", MaxMinVal:=2, ValueOf:="0", ByChange:="$B$18:$B$28" SolverSolve True EnergyArray(i) = Range("E30").Value ClearanceArray(i) = Range("I21").Value

'Increment to the next clearance value Range("I21").Value = Range("I21").Value - delta Next i

'The next loop writes the stored simulation data to the spreadsheet For i = 1 To Range("I19").Value Range("k2").Offset(i, 0).Range("A1").Value = ClearanceArray(i) Range("l2").Offset(i, 0).Range("A1").Value = EnergyArray(i) Next i

End Sub

Automating the Solution Procedure

The screenshot below shows the results of a simulation run. In order to automate the procedure and build an energy / force profile we need to add some extra cells and VBA code. Copy the contents of cells H17 to I23 into your spreadsheet. These specify the range over which the tip of the beam moves and how many steps there are between.

Appendix B