27 Cantilever

Dr. Ayushi Paliwal and Dr. Monika Tomar

Introduction

Major challenge of the twenty first century is to develop self powered systems, especially those which can be operated using energy harvester as electrical power sources [Jones et al. (2001)]. Self powered devices are used in a wide range of wireless applications ranging from encapsulated implants to industrial process monitoring. Furthermore, the increasing demands of wireless sensor networks in mobile devices and the recent advent of extremely low power operated electrical and mechanical devices make such energy harvester sources very attractive. Traditional power sources such as batteries have various limitations in current wireless remote sensor systems which include their large volume, limited lifetime, contribution to environmental pollution and huge maintenance requirements etc. Exploring the possibilities of renewable, sustainable and green energy sources, to replace fossil fuels, is one of the most significant and challenging issues in the energy research. A number of ambient sources such as sunlight, heat, magnetic energy and mechanical vibrations have been studied for generating useful electrical voltage. Among them, mechanical vibrations are independent of weather conditions and offer great potential in various applications as energy harvesters. So, for harvesting mechanical energy, piezoelectric cantilevers are the promising candidates. Let us discuss about the piezoelectric cantilevers in detail in this module.

2. Cantilever

A structural element anchored at one end to a support and subjected to load transverse to its axis at the other end is known as a cantilever. A cantilever is classified under a broader category of beam. In general, a beam can be either free from any axial force or the effect of this force may be negligible. Usually, a beam is considered in horizontal direction and load in vertical direction. The load can be of two types, 1) Concentrated load and 2) Distributed load. The concentrated load is assumed to act at a particular point, though in practice it may be distributed over a small area. On the other hand, distributed load is the one which is spread over the length of the cantilever. However, the rate of loading may be uniform or may vary from one point to another. There are different types of supports for beam which are as follows:

1. Roller support: In case of roller support, a beam rests on a sliding surface like a roller or a flat surface (Figure 1). The roller support can sustain a force normal to its surface as the possible movement on the supporting surface does not allow any resistance in that direction. Therefore, the reaction (R) along the rolling surface is zero and it is present only normal to the surface.

Figure 1: Schematic of roller support

2. Hinged Support: In case of hinged support, the possibility of translational displacement of the beam is zero, however, rotation is possible. In this case, there can be reactions in vertical (R) as well as in horizontal direction (H) (Figure 2).

Figure 2: Schematic of Hinged support

3. Fixed or encastre or built-in support: A built-in rigid support which does not allow any type of movement or rotation is known as fixed or encastre or built-in support. A fixed support exerts a fixed moment (M) and a reaction (R) on the beam (Figure 3).

Figure 3: Schematic of built-in support

A beam with one end fixed and the other end free is called cantilever (Figure 3). There is a vertical reaction(R) and a moment (M) at the fixed end and is called fixed moment. In the present chapter, cantilever beam is made which is supported from one end (fixed support) and free from other ends. In this case, cantilever beam transfers the load to the rigid support where it manages the moment of force and shear stress [Duan et al. (2014)].

Shear force

Shear force is one of the most important parameters in case of a cantilever. It is an unbalanced vertical force on one side (other than fixed support) of the cantilever beam and is the sum of all the normal forces [Duan et al. (2014)]. In other words, it represents the tendency of either portion of the cantilever to slide or shear laterally relative to the other. Shear force is considered positive when the resultant of the forces to the left of a section is upwards or to the right downwards.

Bending moment

Bending moment is another parameter of interest while understanding the theory behind cantilever. Bending moment at some section of a beam is defined as the algebraic sum of the moments about the section of all the forces on one side of the section [Duan et al. (2014)].

Natural frequency of cantilever

In this section, the natural frequency of the cantilever having tip mass at the free end has been calculated, followed by the frequency calculation of the cantilever with distributed mass. Subsequently, the frequency of the cantilever with distributed mass and tip mass has also been shown.

Figure 4: (a) Cantilever beam having mass (mt) mounted at its free end (mt >> m), and (b) free body diagram of the cantilever system

Consider a cantilever beam having tip mass (mt) mounted on its free end as shown in figure 4 (a). Assume that the end-mass (mt) is much greater than the mass (m) of the cantilever. Free body diagram of the cantilever system is also shown in figure 4 (b), where, E is the modulus of elasticity of the material of cantilever, I is the moment of inertia of cantilever about the fixed support and normal to its surface, L is length of the cantilever, g is acceleration due to gravity, mt is the tip mass mounted at free end of cantilever, R is the reaction force and, MR is the reaction bending moment. Applying Newton’s law for static equilibrium, algebraic sum of rotational force (ѺF) and translational force (ҐF) must be zero [Duan et al. (2014)]. Therefore, at the free end of the cantilever (Figure 4 (b)), we have ∑(Ѻ + Ґ ) = 0 R – mg = 0 and R = mg (1) Also, the algebraic sum of the rotational moment (ѺM) and translational moment (ҐM) must be zero. Therefore, at the fixed end of the cantilever, ∑(Ѻ + Ґ ) = 0 MR – mg L = 0 (Ґ = = ) and MR = mg L

Consider the vibration of cantilever of length L about the fixed end and due to tip mass (mt) present at the free end. Under dynamic equilibrium, the cantilever is expected to vibrate with its natural frequency (fn). To determine the value of fn, consider a small segment of length x of the cantilever, starting from the fixed end as shown in figure 5.

Let the tip mass (mt) in the cantilever result in a deflection y at small distance x from the fixed end of the cantilever beam. Let M be the moment due to motion of the cantilever segment. Then the algebraic sum of the moments at a distance x from free end of the segment is

The moment M and the deflection y are related as [Duan et al. (2014)] M = EI y¢¢ (4) where y¢¢ is the acceleration of the segment at distance x from fixed end. Substituting the moment M in equation (3), we get

     EI y¢¢ = M R – Rx

EI y¢¢ = mgL – mg x        (using equation 1 and 2)

EI y¢¢ = mg(L – x)

According to Hooke’s law for a linear spring, restoring force corresponding to a displacement y from the mean position is given by

F = k y

where k is the stiffness constant. The direction of force is not considered in the relation for convenience. Since, the force at the free end (x=L) of the cantilever is mg. The stiffness constant using equation (8) is given as

4. Natural frequency of cantilever without tip mass

Let us now consider a cantilever beam having r as mass per unit length as shown in figure 6. We assume that the cantilever has a uniform cross section. The natural frequency and effective mass of the cantilever (without tip mass) is determined, where the distributed mass is represented by a discrete end-mass.

Figure 6: Schematic of cantilever beam having ρ as mass per unit length

As discussed in previous case, consider a segment of cantilever of small length x from the fixed end and having a displacement y towards normal to cantilever surface at distance x. The governing differential equation of the segment cantilever is given by [Duan et al. (2014)]

where yo is the displacement of cantilever at free end i.e. x=L. It is important to note that the given solution (equation (13)) meets all the boundary conditions except for the zero shear force at the free end of the cantilever (i.e. x=L). Despite the failure of quarter cosine wave solution (equation (13)) to satisfy zero shear force condition at x=L, it is accepted as an approximate solution in order to describe the deflection of cantilever [Eysden and Sader (2006)]. The Rayleigh method [Turner et al. (2011)] is used to find the natural frequency of the cantilever with distributed mass (without tip mass). The total potential energy P of the cantilever is given by [Turner et al. (2001)]

5. Natural frequency of practical cantilever

Consider a practical cantilever beam where both the distributed mass and the end-mass (mt) are significant. The total mass m1 for the practical cantilever having tip mass (mt) and distributed mass (m) at free end (x=L) is given by,

6. Summary:

Piezoelectric cantilevers are the sole candidates which can convert the mechanical vibrations into electrical energy efficiently. The cantilever can be defined as a structural element anchored at ne end to a support and subjected to a load transverse to its axis at the other end. Usually a beam is considered in horizontal direction and load in a vertical direction. The load can be of two types, 1) Concentrated load and 2) Distributed load. On the other hand, distributed load is the one which is spread over the length of the cantilever. However, the rate of loading may be uniform or may vary from one point to another. There are two major forces which are acting on the cantilever. One is shear force and the other one is bending moment. Shear force is an unbalanced vertical force on one side (other than fixed support) of the cantilever beam and is the sum of all the normal forces. Bending moment is another parameter of interest while understanding the theory behind cantilever. Bending moment at some section of a beam is defined as the algebraic sum of the moments about the section of all the forces on one side of the section.

Then we have derived the expression for the natural frequency of the cantilever having tip mass at the free end. The similar study has been carried for the cantilever without tip mass also.