Equivalent spring constants

To further design and configure the geometry and number of the load-bearing pillars and sensing structures, a model can be made in which each component is represented by its spring constant for normal and shear forces. This will give the resulting forces and displacements in vertical and horizontal directions under applied load for each structure, from which the stress and strain of the material can be calculated. Four types of springs can be identified from the design: 1, the load-bearing pillars, connecting the top wafer to the bottom wafer; 2, the displacement- transmitting pillars, that connect the top wafer to the cross-beam structures measuring the vertical displacement; 3, the cross-beam structure itself; 4, the axial springs measuring the horizontal displacements, as shown in Figure 3.4. As the axial springs are directly connected to the bulk of the top wafer, no spring is placed in between. While the axial springs may only be used to measure one component of horizontal displacements, their structure does work as a spring for normal forces and shear forces perpendicular the the beam, influencing the total spring constant of the sensor and the displacement of the top wafer. When the all the spring constants of the structures of the sensor are combined, two total spring constants are found, one for forces in the vertical directions,

kz,totaland one for the forces in the horizontal directions,kx,total.

Horizontal spring constant

To calculate kx,total, the spring constants in the horizontal directions are determined for the

and multiplying by the total number of these pillars on the sensor: kx,1= 3n1Eπr14 l3 1 (3.26) where n1 is the number of load-bearing pillars. To find the spring constant of the displacement-

transmitting pillars (3.15) is used again, this time multiplied by the number of these pillars:

kx,2=

3n2Eπr24

l3 2

(3.27) withn2 the number of displacement-transmitting pillars.

The cross-beam structure normally has arms on four sides, but each side can have multiple arm parallel. For a horizontal force in either the xor they direction this means that on two of the sides beams are being axially loaded, while the beams on the other two sides are bend. To find the horizontal spring constant for the cross-beam (3.5) and (3.14) are combined:

kx,3= ( 2Eh3w3 l3 +2Eh3w 3 3 l3 3 )n3nc= 2n3ncEh3w3 l3 (1 +w 2 3 l2 3 ) (3.28)

where nc is the number of beams on each side of the cross-beam and n3 is the number of cross-

beam structures on the sensor. Notice that since the beams are bend in the horizontal direction, it is the widthwthat is taken to the third power.

The axial springs measuring the horizontal displacements consist of two axial beams, one of which elongates and the other shortens as a force is applied. The spring constant of this combination is found by the taking the spring constant of two axially loaded beams in parallel. To measure the horizontal displacements in both the xand y direction, two axial spring structures are placed orthogonal to each other. This means that the axial springs also add a spring constant in the direction perpendicular to the beams. The horizontal spring constant of the axial springs is therefore similar to the spring constant of the cross-beam, with two beams axially loaded and two beams bend:

kx,4= n4 2 ( 2Eh4w4 l4 +2Eh4w 3 4 l3 4 ) = 2n4Eh4w4 l4 (1 + w 2 4 l2 4 ) (3.29)

whitn4 the total number of axial springs of the device.

When these spring constants are combined, the equivalent horizontal spring constant is found. As the force-transmitting pillars are placed upon the cross-beam structure, their spring constants are taken in series. This combined spring constant is then placed parallel to the load-bearing pillars and axial springs:

kx,total= 1 1 kx,2 + 1 kx,3 +kx,1+kx,4 (3.30)

Vertical spring constant

To findkz,total, the spring constants of the separate structures in the vertical direction are used.

The spring constant of an axially loaded pillar is given by (3.6). To find the spring constant of all the load-bearing pillars, the formula is multiplied by the number of pillars:

kz,1=

n1Eπr12

l1

(3.31) withn1the number of load-bearing pillars. The spring constant of the displacement-transmitting

pillars is based on the same formula but now multiplied with the number of these pillars:

kz,2=

n2Eπr22

l2

3.3. WIRING AND READOUT 33

withn2 the number of displacement-transmitting pillars.

The spring constant of the cross-beam structure can be found using the formula for a bending beam (3.14) and multiplying it for the number of arms of the cross. The vertical spring constant of all cross-beam structures is found by multiplying by the number of these structures on the sensor:

kz,3=

4n3ncEw3h33

l3 3

(3.33) with n3 the number of cross-beam structures and nc the number of arms on each side of the

structure. The axial spring consists of two beams and the vertical spring constant is found by using (3.14) again, times two and multiplying by the total of axial springs on the device:

kz,4=

2n4Ew4h34

l3 4

(3.34) withn4 the number of axial springs.

The equivalent vertical spring constant is composed from the previously found equations. First, the displacement-transmitting pillars and cross-beam structures are taken in series. Then, these are placed parallel to the load-bearing pillars and axial springs:

kz,total= 1 1 kz,2 + 1 kz,3 +kz,1+kz,4 (3.35)

With these two spring constants, it is possible to relate the displacements to the applied force. Using the formulas for stress and strain, the structures can be dimensioned such that maximum strain is accomplished but none of them exceed their maximally allowed stress.