Natural Frequency of a Unimorph Cantilever

48 3-11 Figure 3-2: A cross-sectional view of a unimorph structure. The bending modulus per unit width, D of the unimorph cantilever is given by [91], 3-12 where e i is the elastic modulus for the particular layer e e denotes elastic modulus for electrode layer and e p denotes elastic modulus for piezoelectric layer, h is the thickness of a particular layer of the structure and h N is the neutral axis from the reference point, “ ”. For simplification to estimate the natural frequency of a symmetrical unimorph cantilever, the neutral axis is assumed to be coincident with the centroid of the PZT layer. Therefore, the bending modulus per unit width for a unimorph structure as shown in Figure 3-2 is, 3-13 h e + ½ h p -h e + ½ h p ½ h p -½ h p e e p p w h h m   2     dh h h e D n i N i      1 2           2 2 3 3 2 3 4 3 3 2 12 1 e p e p e e p p h h h h h e h e                                             3 8 1 3 2 1 2 3 8 1 2 3 3 3 p e p e p p h h h e h e            p p p e p p p e h h h h h e p h h h e unimor ph dz z e dz z e dz z e D 2 1 2 1 2 1 2 1 2 2 2 1 2 1 2 p e p p p h h h e h h p z e z e 2 1 2 1 3 2 1 2 1 3 3 2 3                 49 The first mode natural frequency of the unimorph structure can be calculated by substituting equations 3-11 and 3-13 into 3-9, 3-14 Lower resonant frequency is desirable for miniature integrated system. However, as size scales down, resonant frequency scales up, therefore additional proof masses are needed to be attached at the end of the cantilever to further reduce the resonant frequency of the cantilever. The natural frequency for a cantilever with proof mass, f M can be obtained by comparing with resonant frequency for a cantilever without proof mass as, 3-15 where M m is the additional proof mass and m eff is the effective mass at the tip of the cantilever, which is given by [92], 3-16 where  b , w b , h b and l b are density, width, total thickness and total length of the beam and m b is the total beam mass, 3-17 The total mass at the tip of a cantilever when attached with a proof mass, M m is therefore, 3-18 e e p p e p e p e e p p b N h h h h h h h e h e l f   2 2 3 4 3 8 1615 . 2 2 3 3 2            m eff eff N M M m m f f   b b b b b eff m l h w m 236 . 236 .        p p e e e b b b h h h l w m      2 1                 b b m b b eff m eff l w M h h h l w m M M 236 . 236 .   50

3.4.2 Location of Neutral Axis of a Unimorph Cantilever

Figure 3-3: a Side view of Bending beam with bending moment and radius of curvature, bTransformed cross-section of a composite unimorph beam, with PZT layer width, w p and transformed electrode width of n ep w p . A bending beam is subjected to tension and compression proportional to the distance above and below the neutral axis respectively as shown in Figure 3-3a. There is no resultant force acting on the cross section at the neutral axis and the stress, σ x is the multiplication of elastic modulus, e , curvature, κ and the distance from the neutral axis, y . Since E and κ are nonzero, therefore, 3-19 A composite beam can be analysed with the transformed-section method [93], where the cross section of a composite beam is transformed into an equivalent cross section of an imaginary beam that is composed of only one material, with elastic modular ratio, 3-20        A A A dA h dA h e dA   p e ep e e n  h ue h le h p h N Neutral axis a b h h Bending Moment Ref d ½h p w p n e w p Radius of curvature Bending Moment 51 Therefore the distance of the neutral axis from a reference point as shown in Figure 3-3b can be derived as, 3-21 where h i is the distance from the reference point to the centre of each layer of the material and A i is the area of the i -th layer of the structure. The distance from the centroid of PZT layer to the neutral axis is therefore, 3-22 We can see from Equation 3-22 that, if the thickness of the upper electrode is similar to the lower electrode, h e1 = h e2 , the neutral axis is located at the centre of the PZT layer, therefore, d = 0. This will give a zero resultant stress, which will be discussed in the following section.

3.4.3 Maximum Allowed Stress

Figure 3-4: Bending beam of unimorph structure. l h d Neutral axis Centroid plane of PZT Layer Upper Electrode Lower Electrode PZT layer Rigidly Clamped      n i i n i i i N A A h h 1 1       1 2 1 2 2 2 1 2 2 2 2 1 e e ep p e p e e ep p p h h n h h h h h n h h d       