**2. The properties of protein and peptide**

A peptide is made up of short polymers of ⍺-amino acid, which is around 20 to 50 amino acids. The function of small peptides depends on the functional group of various amino acids. Examples of active peptides are glutathione, bradykinin, angiotensin, vasopressin and oxytocin [4].

Protein is a macromolecular and high molecular weight polypeptide, which is made up of long-chain amino acids (more than 50) arranged in a linear chain through peptide bonds [50]. It can exist in four different structural conformations such as primary, secondary, tertiary, and quaternary. The formation of these structures is dependent on the intermolecular interaction between functional groups of amino acids [51], through covalent bonds or non-covalent bonds.

The covalent bonds are strong bonds which include peptide bonds and disulfide bonds [51]. Peptide bonds are interactions between two consecutive amino acids through amino and carboxyl groups. Meanwhile, disulfide bonds link two cysteine residues through sulphhydryl groups [52].

On the other hand, non-covalent bonds are weak bonds that include hydrogen, electrostatic and hydrophobic bonds. Hydrogen bonds link two different peptides with the hydrogen atom of the N-H group and oxygen of the carboxylic group. Hydrophobic bonds will occur if the hydrophobic nature between non-polar side chains of amino acid interacts with each other [51].


**Author details**

Arnab Maity Department of Chemistry, Akal University, Bathinda, Punjab, India

\*Address all correspondence to<EMAIL_ADDRESS>
© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


**1. Introduction**

The theoretical concepts of political leadership creates an in-depth understanding of the societal consciousness of the political processes and the outcomes of the quality of leadership; retrospectively classical writers like; Niccolo Machiavelli, Robert Michels and Max Weber explored theoretical ideas, arguments, and definitions of the authority of political leadership. Political leadership has always had a great influence and an impact on the empowerment of the nation-building process of countries. The never-ending challenges and struggles that political leaders are faced with is the economic reality of their nations which have always been the question that dictates the achievements of the leadership quality of a political leader.

Political Leadership is the patriotic state of consciousness that ignites the spirit of nationalism, and the pride of citizenship that unifies ethnic diversity.

Political Leadership is the act of governance that distinguishes: leadership from politics, integrity from loyalty, and humanity from prosperity.

Political Leadership centers on the duties of social responsibilities, nationbuilding, cultural capital, and the welfare-betterment of humanity.

Political Leadership does not symbolize nor define left wing - right wing political competition against ideologies, positions, and party dominance on the political spectrum. It represents, justifies and validates the essence and advancement of togetherness through communal systems in the presence and acceptance of diversity.

The accountability demands, credibility measures and aspirational heights of political leadership ought to be grounded in their profound values and beliefs on humanity, capacity development, life and security. Conversely more often than not; these measures of expectations are solely directed to economics, financial gains, wealth creation and individual wealth appropriation which clearly never answers the deeper questions to the efficiency of political leadership.

Political Leadership avails a fundamental social responsibility for the unification of social cohesion in the presence of diversity, divides and conflicts in order to achieve a collective human-capital force that assures economic growth, prosperity, survival and protection. Pragmatically, these assertions do not resonate with the political realities of most developing countries; it is conversely on the opposite, state of affairs were political leadership disregards, excludes humanity from governance through the tactics of politics which hardly represents the ideals of leadership but utilizes the rustiness of power through authority.

The theoretical understanding of Political Leadership is not as elusive and fluid as presented overtime it is the trust-given responsible to serve, lead and advance the betterment of humanity harmoniously through prosperity, humanity capacity development and cooperation.

There a question in the developing countries that everyone dares and struggles to get an answer to which is:

"Why are the offices of political appointees, and leaders synonymous to the bank of wealth? ".

I believe pensively, that this question may never be answered as it references and represents a norm, a culture and a tradition that is systematically instituted to picture a reality. The inhibitions of political Leadership is found in the sensitivity and in the powerful pieces of decision making process and the execution of action that are constantly motivated by visionary leadership and inspired by the enthusiasm and compassion for humanity.

Political leaders are faced with the battle of consciousness between "Efficiency and Effectiveness". Confidently ensuring that their tasks and responsibilities are executed perfectly right apt enough to better welfare and ensuring that they are

#### *Political Leadership and Financial Emoluments: A Case of Developing Countries DOI: http://dx.doi.org/10.5772/intechopen.97524*

standing, endorsing and motivating the right actions that will justify the path of humanity towards peace, betterment and prosperity.

Political Leadership is not the absence of the quality, compassion and ideals of the other forms of leadership. But rather; it is indeed the collective accumulation of all the other forms of leadership aimed for a collective and mutual betterment and good for humanity. The stereotypical negative connotations of political leadership been associated and linked to corruption and wealth embezzlement has indeed created a social construct, a message of deceit and a norm which has hindered leadership accountability, credibility, selfless responsibilities, and expectations from the political leader.

Political Leadership is the given-responsibilities and authority by trust, belief, and assurance to an individual with the identification of patriotic loyalty, recognition of leadership, and compassion for humanity. Political leadership is not a game of politics, a display of wealth nor is it a competitive showcase of power and authority, it is the real business of putting humanity, first and above, policies, interests, ideological bents, beliefs and traditions. Political leadership is the combination of power, intelligence, security, wealth and authority into a force for enthusiasm and compassion for humanity.

Political leadership owes the people no greater speech, no greater promise, no greater policies, no greater agenda, no greater army, no greater bridges, no greater roads than the effective and efficient building of better, stronger and greater tomorrow that ensures and insures the harmonious -collective - consciousness of purpose, dedication to duty and the validation of the worthiness of humanity.

There will be no greater oath of office, affirmation of a strong-will dedication to office, than the one that specifically centers on the core fundamental values, principles and compassion for the greater good, advancement and prosperity of humanity in its worthiness and validation.

The swearing in to office of any appointed political leadership opportunity ought to be the-one-and-only social consciousness to duty that should exhilarate, justify and empower the interests, ambition and intentions to govern with political authority a leader can have.

The fear, anxiety and bravery to serve humanity through political leadership should be an oath of sacrifice, an allegiance to dedication of self and a pledge to better humanity with all the resources there is within or outside the sovereignty of nation states.

Political leadership creates the means and justifies the means to govern humanity towards the path of betterment by the people for the people and with the people. This does not make it a complicated mission to accomplish, it rather eliminates the space for self-interests, personal gains, personal agendas and competition of selfgreatness. It reaffirms the truth bearing of faith and allegiance to serve humanity for a greater, better and a more harmonious life of creation that humanity is worthy of and validated for.

Political leadership is an authoritative responsibility that creates more for the people than it takes from the people. It is an office that benefits all, an opportunity that creates sustainable growth, progressive advancements and a harmonious culture for productivity. There is an endless path to the progressive advancement of political leadership in developing countries that ought to journeyed on, that path can only be traveled through with the human- capacity- development- investment- and- advancement as the only policies of change that should be demanded and gotten.

The theoretical concept of Political leadership is asserted to be an authority symbolic to parenthood. It carries on, upon it the responsibilities, the never ending

#### **Figure 1.**

*Schematic Presentation of the title's core Components. Authors Compilation (2020).*

duty of ensuring and insuring the betterment of humanity with which all the other means and agendas of prosperity is assured.

Political leadership in developing countries owes its humanity more than just the legal obligations of citizenry by means of identity but to the embracing of what it truly means, symbolizes, represents deeply, compassionately and emotionally to be a citizen of a country regardless of its political, socio-economic and moral status. Until the betterment of humanity as the only capital source of wealth is assured it is impossible to enrich or discover the wealth in any nation in the absence of slavery and exploitation.

Political Leadership is patriotically the social-political consciousness of the authority to utilize power to the ultimate advantage of the betterment of its humanity regardless of political realities, international regimes or competitive economic markets.

It is in the authenticity and intentions of purpose and the protection of citizenry that nation's competitive advantages are born. No nation could hardly excel productively, economically and intellectually without its citizenry. It is the putting first and above of its citizenry by political Leadership that puts a nation above other nations on the global map of relevance and power. No nation could journey without a strong-willed political Leadership that assures and insures the existence of its citizenry in a proud state of humanity (**Figure 1**).

#### **1.1 Worthy of notice**

The consciousness of individuality, the awareness of self worth, the embracing of nationality and the privileges of leadership opportunities are fundamental to the understanding and recognition of the influential authority of political leadership.


*3.8. Adsorption Mechanism*

Two routes can be proposed for the adsorption of AB40 on the surface of PANI salt and PANI/Fe3O4 composite. In the first one, electrostatic interaction may occur between the molecules of AB40 and PANI. The second one involves the formation of an H-bond between the dye and –NH group of PANI. H-bond formation is also possible between AB40 and –OH group present on the surface of Fe3O4.

The electrostatic interactions are based on the fact that when dye is dissolved in water it splits into positively and negatively charged ions (Dye-SO3 −). These negatively charged anions (Dye-SO3 −) interact with positively charged sites (–+NH–) on PANI surface. The enhancement of dye adsorption in acidic medium is good evidence of electrostatic interaction expressed in Section 3.4. Existence of physical forces (H-bond) is also supported by FTIR spectra shown in Figure 4B. After adsorption of AB40, all peaks in the spectra of PANI and PANI/Fe3O4 composite are shifted towards low-frequency values. Moreover, appearance of peak at 2356.7 cm−1 shows existence of AB40 adheres to the surface of PANI and PANI/Fe3O4 composites [72].

### *3.9. Thermodynamics of Adsorption*

The nature of adsorption can be described well with thermodynamic parameters like Gibbs free energy, change in enthalpy and change in entropy. Values of Gibbs free energy were calculated by the equation shown below (Equation (10));

$$
\Delta \mathbf{G} = -\mathbf{R} \mathbf{T} \ln \mathbf{K}\_{\mathbf{e}} \tag{10}
$$

where Ke is the equilibrium constant, R is the gas constant having the value of 8.314 J K−1 mol−1 and T represents the Kelvin temperature. The negative sign of ΔG values shows that the adsorption of AB40 onto Fe3O4, PANI and PANI/ Fe3O4 composites are spontaneous (Table 5). The values of Δ G which range from −20 to zero kJ mol−1 show physical adsorption [47]. The values of ΔH and ΔS were calculated from the slope and intercept of van't Ho ff equation respectively by plotting lnke vs. 1/T (Figure 12b). The van't Ho ff equation is expressed as below;

$$
\Delta \mathbf{K}\_{\mathbf{e}} = -\frac{\Delta \mathbf{H}}{\mathbf{R} \mathbf{T}} + \frac{\Delta \mathbf{S}}{\mathbf{R}} \tag{11}
$$

$$\mathbf{K\_{e}} = \frac{\mathbf{q\_{e}}}{\mathbf{c\_{e}}} \tag{11a}$$

where qe (mg g−1) is the adsorption maximum and Ce (mg L−1) is the concentration of dye at equilibrium. The negative values of ΔH and ΔS shown in Table 4 show that adsorption is exothermic and correlate to the e ffect of temperature on adsorption expressed in Section 3.3. Activation energy also expresses the nature of adsorption. Its values are calculated from the slope of Arrhenius equation by plotting lnK vs. 1/T shown in Figure 12b. The Arrhenius equation is expressed as below;

$$\text{lnk} = \text{lnA} - \frac{\text{Ea}}{\text{RT}} \tag{12}$$

where K is the rate constant, A is Arrhenius constant, Ea is the activation energy, R is the general gas constant and T is kelvin temperature. The activation energy of adsorption of AB40 onto Fe3O4, PANI and PANI/Fe3O4 composites are 30.12, 22.09 and 26.13 kJ mol−1 showing physical adsorption. Ozcan and co-workers have demonstrated that physical adsorption is characterized by the activation energy values range from 5 to 40 kJ mol−1 and its higher values (40–800) kJ mol−1 express chemical adsorption [85].

**Table 5.** Activation energy and thermodynamic parameters of AB40 adsorption.

**Figure 12.** (**a**) Arrhenius plot and (**b**) van't Hoff plot for calculation of activation energy and thermodynamic parameters.


**1.2 Jungian archetypes and use of fairy tales**

Cultural myths emerge as artifacts of society in form such as stories handed down from generation to generation, often then being converted into dramatic and artistic multimedia artforms such as plays, movies, ballet, music, and film. These stories may include many elements which can be interpreted metaphorically. The universality of fairy tales in crossing cultural boundaries and addressing social and generational issues within the context of child and adolescent psychiatry is noted by Wallace and colleagues [36]. From a psychological point of view, fairy tales provide "grist for the mill" in human experiences, and have fascinated writers and practitioners in the therapeutic fields such as C.G. Jung. Myths and attitudes about mothers and motherhood are pervasive and foundational across all societies, and Jung describes the archetype of the mother within an array of archetypes [37–39]. An archetype may simply be described as a universal pattern in the psyche, in a similar manner to instincts [37]. Archetypes comprise key elements often in heightened imaginal form, providing the ingredients to understand human attitudes and behaviors which in turn assist the person with changes experienced in everyday life. Such archetypal patterns typically follow a distinct goal, "the meaning of which becomes more and more clear as they unfold in their actual expressions or images" [40]. Broadly from a Jungian stance, myths are seen as both particular and universal, needing translation into the language of the audience [41]. Archetypes typically address personal complexes linked to relationship issues, thereby enlarging awareness and empowering the living of an authentic life [20].

Archetypes are important in therapy, potentially echoing themes in the life experience of a person and bringing to consciousness what was previously unconscious or preconscious, in turn leading to transformation. Beebe lists a wide range of archetypal complexes which serve as patterns for internal dialogs within the self, including the hero, good father, trickster, divine child, good mother, and bad witch [42]. This internal process of development and transformation, terms individuation by Jung, is seen as a naturally occurring process within the human psyche where archetypal complexes are typically explored and ultimately resolved [20, 37, 40]. Therefore, the purpose of an archetype is to bring materials to consciousness and in doing so enlarge experience and make meaning for the person, contributing to the overall process of psychological growth and development. Clearly, issues from the primary family and their own experiences of being mothered affect progress toward individuation for any person. Additionally, significant issues occur for "replacement children" (after the loss of another child) as they grow and develop during adulthood, in turn affect their own attachment, identity, grief, and guilt [21].

The archetypal information carried by images can be worked with during progress toward individuation. Such images are found in the form of dreams and active


2. Governing equation of the AFG beam

This studied free vibration of the axially functionally graded beam, which is a nonuniform and nonhomogeneous structure because of the variable width and height, as shown in Figure 1. Based on Euler-Bernoulli beam theory, the governing differential equation of the beam can be written as

$$\frac{\partial^2}{\partial \mathbf{x}^2} \left[ E(\mathbf{x}) I(\mathbf{x}) \frac{\partial^2 w(\mathbf{x}, t)}{\partial \mathbf{x}^2} \right] + \rho(\mathbf{x}) A(\mathbf{x}) \frac{\partial^2 w(\mathbf{x}, t)}{\partial t^2} = \mathbf{0}, \qquad \mathbf{0} \le \mathbf{x} \le L \tag{1}$$

where w xð Þ ; t is the transverse deflection at position x and time t; L is the length of the beams; E xð ÞI xð Þ is the bending stiffness, which is determined by Young's modulus E xð Þ and the area moment of inertia I xð Þ; and ρð Þ x A xð Þ is the unit mass

Free Vibration of Axially Functionally Graded Beam DOI: http://dx.doi.org/10.5772/intechopen.85835

Figure 1. The geometry and coordinate system of an AFG beam.

length of beam, which is determined by volume mass density ρð Þ x and variable cross-sectional area A xð Þ.

Because of the particularity of AFG beam, bending stiffness E xð ÞI xð Þ and unit mass ρð Þ x A xð Þ will change with the axis coordinates, which makes the original constant coefficient differential equation become variable coefficient differential equation and to some extent increases the difficulty of solving. In order to facilitate calculation, we simplify the calculation process by introducing dimensionless parameters. Reference flexural stiffness EI0 and reference mass ρA0 are introduced, and the above two dimensionless parameters are invariant. Suppose E xð ÞI xð Þ¼ EI0 þ E xð ÞI xð Þ and ρð Þ x A xð Þ¼ ρA0 þ ρð Þ x A xð Þ, where EI0 and ρA0 are the invariant parts and E xð ÞI xð Þ and ρð Þ x A xð Þ represent flexural stiffness and mass per unit length, respectively, and vary with the axial coordinates. Here, we introduce a dimensionless space variable ξ ¼ x=L and a dimensionless time variable τ ¼ t L2 ffiffiffiffiffiffi EI0 ρA0 q ; Eq. (1) can be rewritten in the dimensionless form:

$$\frac{\partial^2}{\partial \xi^2} \left\{ \left[ \mathbf{1} + f\_1(\xi) \right] \frac{\partial^2 w(\xi, \tau)}{\partial \xi^2} \right\} + \left[ \mathbf{1} + f\_2(\xi) \right] \frac{\partial^2 w(\xi, \tau)}{\partial \tau^2} = \mathbf{0}, \qquad \mathbf{0} \le \xi \le \mathbf{1} \tag{2}$$

where

frequency response curves of a nonuniform beam undergoing nonlinear oscillations by the multiple time scale method. Wang [69] proposed the differential quadrature element method for the natural frequencies of multiple-stepped beams with an aligned neutral axis. Abdelghany [70] utilized the differential transformation

The asymptotic development method, which is a kind of perturbation analysis method, is always used to solve nonlinear vibration equations. For example, Chen et al. [71, 72] studied the nonlinear dynamic behavior of axially accelerated viscoelastic beams and strings based on the asymptotic perturbation method. Ding et al. [73, 74] studied the influence of natural frequency of transverse vibration of axially moving viscoelastic beams and the steady-state periodic response of forced vibration of dynamic viscoelastic beams based on the multi-scale method. Chen [75] used

method to examine the free vibration of nonuniform circular beam.

Mechanics of Functionally Graded Materials and Structures

the asymptotic perturbation method to analyze the finite deformation of

analysis.

conclusions are presented.

∂2

4

∂x2 E xð ÞI xð Þ ∂2

2. Governing equation of the AFG beam

differential equation of the beam can be written as

w xð Þ ; t ∂x2

prestressing hyperelastic compression plates. Hao et al. [76] employed the asymptotic perturbation method to obtain the nonlinear dynamic responses of a cantilever FGM rectangular plate subjected to the transversal excitation in thermal environment. Andrianov and Danishevs'kyy [77] proposed an asymptotic method for solving periodic solutions of nonlinear vibration problems of continuous structures. Based on the asymptotic expansion method of Poincaré-Lindstedt version [78], the longitudinal vibration of a bar and the transverse vibration of a beam under the action of a nonlinear restoring force are studied. The asymptotic development method is applied to obtain an approximate analytical expression of the natural frequencies of nonuniform cables and beams [79, 80]. Cao et al. [81, 82] applied the asymptotic development method to analyze the free vibration of nonuniform axially functionally graded (AFG) beams. Tarnopolskaya et al. [83] gave the first comprehensive study of the mode transition phenomenon in vibration of beams with arbitrarily varying curvature and cross section on the basis of asymptotic

The present topic focus on the free vibration of AFG beams with uniform

perturbation method (APM) and Meijer G-function. First, the governing differential equation for free vibration of nonuniform AFG beam is summarized and rewritten in a form of a dimensionless equation based on Euler-Bernoulli beam theory. Second, the analytic equations are then derived in detail in Sections 3 and 4, respectively, where the nature frequencies are obtained and compared with the results of the finite element method and the published references. Finally, the

This studied free vibration of the axially functionally graded beam, which is a nonuniform and nonhomogeneous structure because of the variable width and height, as shown in Figure 1. Based on Euler-Bernoulli beam theory, the governing

w xð Þ ; t

where w xð Þ ; t is the transverse deflection at position x and time t; L is the length of the beams; E xð ÞI xð Þ is the bending stiffness, which is determined by Young's modulus E xð Þ and the area moment of inertia I xð Þ; and ρð Þ x A xð Þ is the unit mass

∂t2 ¼ 0, 0≤x≤L (1)

þ ρð Þ x A xð Þ ∂2

or nonuniform cross sections using analytical method: the asymptotic

$$f\_1(\xi) = \frac{\overline{E(\xi)I(\xi)}}{EI\_0} \text{ and} \\ f\_2(\xi) = \frac{\overline{\rho(\xi)A(\xi)}}{\rho A\_0} \tag{3}$$

are the nondimensional varying parts of the flexural stiffness and of the mass per unit length, respectively.

## 3. Asymptotic perturbation method

#### 3.1 Equation deriving

In this section, the APM is introduced to obtain a simple proximate formula for the nature frequency of the AFG beam. Firstly, we assume that

$$w(\xi,\tau) = W(\xi)\sin(a\tau) \tag{4}$$

where Wð Þξ is the amplitude of vibration and ω is the circular frequency of vibration. We obtain the following equation by substituting Eq. (4) with Eq. (2): Mechanics of Functionally Graded Materials and Structures

$$\frac{d^2}{d\xi^2} \left\{ \left[ \mathbf{1} + f\_1(\xi) \right] \frac{d^2 W}{d\xi^2} \right\} - \alpha^2 \left[ \mathbf{1} + f\_2(\xi) \right] W = \mathbf{0}, \qquad \mathbf{0} \le \xi \le \mathbf{1} \tag{5}$$

To use the APM, a small perturbation parameter ε is introduced:

$$f\_1(\xi) \to \mathfrak{e}f\_1(\xi),\ f\_2(\xi) \to \mathfrak{e}f\_2(\xi) \tag{6}$$

and the frequency equation is

is satisfied. As a result,

Integrating by parts, we obtain

dξ d2 W0 dξ2 W0 þ f 1

d3 W0 dξ3 W0 � f 1

d2 W0 dξ2 W0 þ

dW0 dξ � d3

!�

W0 dξ3 W0

þ ð1 0 f 1 d2 W0 dξ2 !2

2 4

f 1ð Þ¼ ξ Eð Þξ Ið Þξ

d2 W0 dξ2

> d3 W0

> > ð1 0

Eð Þξ Ið Þξ EI0

)� � � � �

1

0 þ 1 EI0

h1ð Þξ W0dξ ¼ df 1

By definition we have

geneous beam.

ð1 0

so that

(

þ d2 W0 dξ2

7

¼ d E½ � ð Þξ Ið Þξ EI0dξ

df 1 dξ d2 W0 dξ2 W0 þ f 1

The spatial mode shape can be obtained as

Free Vibration of Axially Functionally Graded Beam DOI: http://dx.doi.org/10.5772/intechopen.85835

> ð1 0

> > ω1 ¼

Ð 1

2ω0 Ð 1 0 W2 0dξ

0 h1ð Þξ W0dξ

Because h1ð Þξ is linearly correlated with W0, the former equations indicate that the arbitrary amplitude of W0 does not impact ω1. This finding yields the first-order correction of the natural frequency ω0 corresponding to a nonuniform and homo-

> d3 W0 dξ3 W0 � f 1

!�

� ω2

E0I ¼ Eð Þξ Ið Þ� ξ EI0

dW0 dξ

EI0

� � � �

1

0 þ ð1 0 f 1 d2 W0 dξ2 !2

dξ3 W0 � Eð Þξ Ið Þξ

Eð Þξ Ið Þξ d2

EI0

W0 dξ2 !2

d2 W0 dξ2

> dξ � ð1 0

0 f 2W2 0

W0 ¼ cosh ð Þ� kξ cosð Þþ kξ C

Now, the solution of the first-order equation is analyzed. In Eq. (9), both h1ð Þξ and W1 are linearly correlated with W0. Based on the theory of ordinary differential equations [84], the solvability conditions of linear differential equations can be expressed by the orthogonality of solutions of homogeneous systems of equations. At the same time, according to the orthogonality of modal vibration theory, the solution of Eq. (9) exists under the condition of the solvability of differential equation:

cos k cosh k þ 1 ¼ 0 (16)

½ � h1ð Þ� ξ 2ω1ω0W0 W0dξ ¼ 0 (18)

d2 W0 dξ2

> 3 5dξ

dW0 dξ

� � � �

1

0

dξ

dW0 dξ

> d2 W0 dξ2 !2

D ½ � sinh ð Þ� kξ sin ð Þ kξ (17)

(19)

(20)

(21)

dξ

(22)

According to the Poincaré-Lindstedt method [78–82], we assume the expansion for ω and Wð Þξ as

$$\begin{aligned} \omega &= \omega\_0 + \varepsilon \alpha\_1 + \varepsilon^2 \alpha\_2 + \dots \\ \mathcal{W}(\xi) &= \mathcal{W}\_0(\xi) + \varepsilon \mathcal{W}\_1(\xi) + \varepsilon^2 \mathcal{W}\_2(\xi) + \dots \end{aligned} \tag{7}$$

Substituting these expressions with governing Eq. (5) and then expanding the expressions into a ε-series, Eqs. (8) and (9) are obtained by equating the coefficients of ε0 and ε1 to zero, yielding a sequence of problems for the unknowns ωi and Wið Þξ :

$$a\frac{d^4W\_0}{d\xi^4} - a\_0^2W\_0 = 0\tag{8}$$

$$\frac{d^4W\_1}{d\xi^4} - a\_0^2W\_1 + h\_1(\xi) - 2a\_1a\_0W\_0 = 0\tag{9}$$

where

$$h\_1(\xi) = 2\frac{d\mathcal{f}\_1(\xi)}{d\xi}\frac{d^3\mathcal{W}\_0}{d\xi^3} + \frac{d^2\mathcal{f}\_1(\xi)}{d\xi^2}\frac{d^2\mathcal{W}\_0}{d\xi^2} + w\_0^2[\mathcal{f}\_1(\xi) - \mathcal{f}\_2(\xi)]\mathcal{W}\_0 \tag{10}$$

For Eq. (8), the following general solution can be obtained:

$$W\_0 = A\sin\left(k\xi\right) + B\cos\left(k\xi\right) + C\sinh\left(k\xi\right) + D\cosh\left(k\xi\right) \tag{11}$$

where

$$k = \sqrt{a\_0} \tag{12}$$

For simplicity, we consider clamped-free (C-F) beams, and the corresponding boundary conditions are

$$\mathcal{W}\_0 = \frac{dW\_0}{d\xi} = \mathbf{0}, \ \xi = \mathbf{0} \tag{13}$$

$$\frac{d^2W\_0}{d\xi^2} = \frac{d^3W\_0}{d\xi^3} = 0, \ \xi = 1\tag{14}$$

Then, the following equations from equation can be obtained:

$$\begin{aligned} A + C &= 0\\ B + D &= 0\\ \frac{C}{D} &= \frac{\sin k - \sinh k}{\cos k + \cosh k} \end{aligned} \tag{15}$$

Free Vibration of Axially Functionally Graded Beam DOI: http://dx.doi.org/10.5772/intechopen.85835

and the frequency equation is

d2

for ω and Wð Þξ as

Wið Þξ :

where

where

6

h1ð Þ¼ ξ 2

boundary conditions are

dξ2 1 þ f 1ð Þξ � � d2

( )

Mechanics of Functionally Graded Materials and Structures

W dξ2

To use the APM, a small perturbation parameter ε is introduced:

ω ¼ ω0 þ εω1 þ ε2ω2 þ …,

d4 W0 dξ4 � ω2

d2 f 1ð Þξ dξ2

For Eq. (8), the following general solution can be obtained:

d2 W0 dξ2 þ ω2

k ¼ ffiffiffiffiffiffi ω0

For simplicity, we consider clamped-free (C-F) beams, and the corresponding

W0

D ¼ sin k � sinh k cos k þ cosh k

W0 ¼ dW0

Then, the following equations from equation can be obtained:

C

A þ C ¼ 0 B þ D ¼ 0

d2 W0 dξ2 ¼ d3

W0 ¼ A sin ð Þþ kξ B cosð Þþ kξ Csinh ð Þþ kξ D cosh ð Þ kξ (11)

d4 W1 dξ4 � ω2

df 1ð Þξ dξ

d3 W0 dξ3 þ

According to the Poincaré-Lindstedt method [78–82], we assume the expansion

Substituting these expressions with governing Eq. (5) and then expanding the expressions into a ε-series, Eqs. (8) and (9) are obtained by equating the coefficients of ε0 and ε1 to zero, yielding a sequence of problems for the unknowns ωi and

� ω2 1 þ f 2ð Þξ � �W ¼ 0, 0≤ξ≤ 1 (5)

0W0 ¼ 0 (8)

0 f 1ð Þ� ξ f 2ð Þξ � �W0 (10)

p (12)

dξ ¼ 0, ξ ¼ 0 (13)

dξ3 ¼ 0, ξ ¼ 1 (14)

(15)

0W1 þ h1ð Þ� ξ 2ω1ω0W0 ¼ 0 (9)

f 1ð Þ!ξ εf 1ð Þξ , f 2ð Þ!ξ εf 2ð Þξ (6)

Wð Þ¼ ξ W0ð Þþ ξ εW1ð Þþ ξ ε2W2ð Þþ ξ …: (7)

$$k\cos k\cosh k + 1 = 0\tag{16}$$

The spatial mode shape can be obtained as

$$W\_0 = \cosh\left(k\xi\right) - \cos\left(k\xi\right) + \frac{C}{D} [\sinh\left(k\xi\right) - \sin\left(k\xi\right)] \tag{17}$$

Now, the solution of the first-order equation is analyzed. In Eq. (9), both h1ð Þξ and W1 are linearly correlated with W0. Based on the theory of ordinary differential equations [84], the solvability conditions of linear differential equations can be expressed by the orthogonality of solutions of homogeneous systems of equations. At the same time, according to the orthogonality of modal vibration theory, the solution of Eq. (9) exists under the condition of the solvability of differential equation:

$$\int\_{0}^{1} [h\_1(\xi) - 2\alpha\_1 \alpha\_0 \mathcal{W}\_0] \mathcal{W}\_0 d\xi = 0 \tag{18}$$

is satisfied. As a result,

$$\rho\_1 = \frac{\int\_0^1 h\_1(\xi) \, W\_0 d\xi}{2\alpha\_0 \int\_0^1 W\_0^2 d\xi} \tag{19}$$

Because h1ð Þξ is linearly correlated with W0, the former equations indicate that the arbitrary amplitude of W0 does not impact ω1. This finding yields the first-order correction of the natural frequency ω0 corresponding to a nonuniform and homogeneous beam.

Integrating by parts, we obtain

$$\begin{split} \int\_{0}^{1} h\_{1}(\xi) \mathcal{W}\_{0} d\xi &= \left( \frac{df\_{1}}{d\xi} \frac{d^{2} \mathcal{W}\_{0}}{d\xi^{2}} \mathcal{W}\_{0} + f\_{1} \frac{d^{3} \mathcal{W}\_{0}}{d\xi^{3}} \mathcal{W}\_{0} - f\_{1} \frac{d^{2} \mathcal{W}\_{0}}{d\xi^{2}} \frac{d \mathcal{W}\_{0}}{d\xi} \right) \bigg|\_{0}^{1} \\ &+ \int\_{0}^{1} \left[ f\_{1} \left( \frac{d^{2} \mathcal{W}\_{0}}{d\xi^{2}} \right)^{2} - a\_{0}^{2} f\_{2} \mathcal{W}\_{0}^{2} \right] d\xi \end{split} \tag{20}$$

By definition we have

$$f\_1(\xi) = \frac{\overline{E(\xi)I(\xi)}}{E\_0 I} = \frac{E(\xi)I(\xi) - EI\_0}{EI\_0} \tag{21}$$

so that

$$
\begin{split}
&\left(\frac{d\mathcal{f}\_{1}}{d\tilde{\xi}}\frac{d^{2}W\_{0}}{d\xi^{2}}W\_{0} + f\_{1}\frac{d^{3}W\_{0}}{d\xi^{3}}W\_{0} - f\_{1}\frac{d^{2}W\_{0}}{d\xi^{2}}\frac{dW\_{0}}{d\xi}\right)\bigg|\_{0}^{1} + \int\_{0}^{1}f\_{1}\left(\frac{d^{2}W\_{0}}{d\xi^{2}}\right)^{2}d\xi\\&= \left\{\frac{d[E(\xi)I(\xi)]}{EI\_{0}d\xi}\frac{d^{2}W\_{0}}{d\xi^{2}}W\_{0} + \frac{E(\xi)I(\xi)}{EI\_{0}}\frac{d^{3}W\_{0}}{d\xi^{3}}W\_{0} - \frac{E(\xi)I(\xi)}{EI\_{0}}\frac{d^{2}W\_{0}}{d\xi^{2}}\frac{dW\_{0}}{d\xi}\right.\\&\left.+\frac{d^{2}W\_{0}}{d\xi^{2}}\frac{dW\_{0}}{d\xi} - \frac{d^{3}W\_{0}}{d\xi^{3}}W\_{0}\right\}\bigg|\_{0}^{1} + \frac{1}{EI\_{0}}\int\_{0}^{1}E(\xi)I(\xi)\left(\frac{d^{2}W\_{0}}{d\xi^{2}}\right)^{2}d\xi - \int\_{0}^{1}\left(\frac{d^{2}W\_{0}}{d\xi^{2}}\right)^{2}d\xi.\tag{22}
\end{split}
$$

Choosing the reference bending stiffness

$$EI\_{0} = \frac{\left. \frac{d[E(\xi)I(\xi)]}{d\xi} \frac{d^{3}W\_{0}}{d\xi^{2}} \mathcal{W}\_{0} + E(\xi)I(\xi) \frac{d^{3}W\_{0}}{d\xi^{3}} \mathcal{W}\_{0} - E(\xi)I(\xi) \frac{d^{2}W\_{0}}{d\xi^{2}} \frac{dW\_{0}}{d\xi} \right|\_{0}^{1} + \int\_{0}^{1} E(\xi)I(\xi) \left(\frac{d^{3}W\_{0}}{d\xi^{2}}\right)^{2} d\xi}{\left(\frac{d^{3}W\_{0}}{d\xi} \mathcal{W}\_{0} - \frac{d^{2}W\_{0}}{d\xi} \frac{dW\_{0}}{d\xi}\right) \Big|\_{0}^{1} + \int\_{0}^{1} \left(\frac{d^{3}W\_{0}}{d\xi^{2}}\right)^{2} d\xi} \tag{23}$$

$$\text{we have } \left(\frac{d\xi\_1}{d\xi}\frac{d^2W\_0}{d\xi^2}W\_0 + f\_1\frac{d^3W\_0}{d\xi^3}W\_0 - f\_1\frac{d^2W\_0}{d\xi^2}\frac{dW\_0}{d\xi}\right)\Big|\_{0}^{1} + \int\_{0}^{1} f\_1\left(\frac{d^2W\_0}{d\xi^2}\right)^2 d\xi = 0.$$

Analogously, we choose

$$
\rho A\_0 = \frac{\int\_0^1 \rho(\xi) A(\xi) \mathcal{W}\_0^2 d\xi}{\int\_0^1 \mathcal{W}\_0^2 d\xi} \tag{24}
$$

giving Ð 1 0 f 2W2 0dξ ¼ 0. Then, we obtain ω1 ¼ 0. These values are the properties of the equivalent homogeneous beam having the same frequency (at least up to the first order) as the given nonuniform AFG beam.

The nth natural circular frequency of the AFG beam can be derived as

$$
\lambda\_n = \frac{1}{L^2} \sqrt{\frac{EI\_0}{\rho A\_0}} \rho\_0 \tag{25}
$$

cross-sectional area A xð Þ and moment of inertia I xð Þ along the beam axis can be

Geometry and coordinate system of an AFG beam for different taper ratios: (a) case 1, cb ¼ ch ¼ 0; (b) case 2,

� �, Ixð Þ¼ IL 1 � cb

respectively. AL and IL are cross-sectional area and area moment of inertia of the beam left sides, respectively. It is instructive to remember that if cb ¼ ch ¼ 0, the beam would be uniform; if ch ¼ 0, cb 6¼ 0, the beam would be tapered with constant height; if cb ¼ 0, ch 6¼ 0, the beam would be tapered with constant width; and if

corresponding to Figure 2(a)–(d), respectively. Moreover, the material properties such as Young's modulus E xð Þ and mass density ρð Þ x along the beam axis are

� �, ρð Þ¼ x ρL 1 þ

where EL and ρL are Young's modulus and mass density of the beam left sides,

Table 2 shows the first third-order natural frequencies of the AFG beam, case of Figure 2(a), which is uniform but nonhomogeneous. It can be clearly seen that the analytical results obtained from the asymptotic development method are in good

nonuniform AFG beams with different boundary configurations were obtained. The results were listed in Tables 2–7, respectively, and it also was compared with

x L � � 1 � ch

HL are the breadth and height taper ratios,

x L þ

ρLAL=ELIL

x L

� �2 � � (27)

p ) of the four cases of

x L � �3

(26)

x L

cb 6¼ 0, ch 6¼ 0, the beam would be double tapered. These four cases are

x L

Based on the introduced analytical equation, the first third-order

nondimensional natural frequencies (Ωn ¼ λnL2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

clearly expressed as follows:

A xð Þ¼ AL 1 � cb

where cb ¼ 1 � BR

assumed as

Figure 2.

respectively.

9

x L � � 1 � ch

BL and ch ¼ 1 � HR

ch ¼ 0, cb 6¼ 0; (c) case 3, cb ¼ 0, ch 6¼ 0; and (d) case 4, cb 6¼ 0, ch 6¼ 0.

Free Vibration of Axially Functionally Graded Beam DOI: http://dx.doi.org/10.5772/intechopen.85835

E xð Þ¼ EL 1 þ

those of published work by Shahba et al. [38].

agreement with those given by Ref. [38].

Each order of frequency of ω0 can be determined by Eq. (16) (in turn, positive numbers from small to large). The required variables have been computed by the above expression. Eq. (25) is an approximate formula for the natural frequencies of variable cross-sectional AFG beams.

In order to show the applicability of this method, we study other supporting conditions, and we can easily get the corresponding boundary conditions of Eqs. (13) and (14). Due to the limited space, the detailed derivation process is omitted, and the final results are shown in Table 1.

#### 3.2 Numerical results and discussion

Based on the above analysis, four kinds of AFG beams with various taper ratios are considered, as shown in Figure 2. The numerical simulations are carried out, and the results are compared with the published literature results to verify the validity of the proposed method.

In Figure 2, BL and BR are the width of the left and right ends of the beams, respectively, and HL and HR are the height of the left and right ends of the beams, respectively. Here, we assume that the geometric characteristics of AFG beams vary linearly along the longitudinal direction. Therefore, the variation of

Table 1.

Frequency equations and mode shapes for various beams.

Free Vibration of Axially Functionally Graded Beam DOI: http://dx.doi.org/10.5772/intechopen.85835

Figure 2.

Choosing the reference bending stiffness

dξ2 W0 þ Eð Þξ Ið Þξ d3

Mechanics of Functionally Graded Materials and Structures

first order) as the given nonuniform AFG beam.

omitted, and the final results are shown in Table 1.

equation

Simply supported (S-S) sin k ¼ 0 W0 ¼ sin ð Þ kξ

Frequency equations and mode shapes for various beams.

d3 W0 dξ3 W0 � d2

> d3 W0 dξ3 W0 � f 1

� ��

ρA0 ¼

Ð 1

W0

� ��

n o�

dξ3 W0 � Eð Þξ Ið Þξ d2

d2 W0 dξ2 dW0 dξ

0 ρ ξð ÞAð Þξ W2

ffiffiffiffiffiffiffiffi EI0 ρA0

Ð 1 0 W2 0dξ

of the equivalent homogeneous beam having the same frequency (at least up to the

s

Each order of frequency of ω0 can be determined by Eq. (16) (in turn, positive numbers from small to large). The required variables have been computed by the above expression. Eq. (25) is an approximate formula for the natural frequencies of

In order to show the applicability of this method, we study other supporting conditions, and we can easily get the corresponding boundary conditions of Eqs. (13) and (14). Due to the limited space, the detailed derivation process is

Based on the above analysis, four kinds of AFG beams with various taper ratios are considered, as shown in Figure 2. The numerical simulations are carried out, and the results are compared with the published literature results to verify the

In Figure 2, BL and BR are the width of the left and right ends of the beams, respectively, and HL and HR are the height of the left and right ends of the beams, respectively. Here, we assume that the geometric characteristics of AFG beams vary linearly along the longitudinal direction. Therefore, the variation of

Mode shape

Clamped-pinned (C-P) tan k � tanhk ¼ 0 W0 ¼ cosh ð Þ� kξ cosð Þ� kξ cosh k� cos k sinh k� sin k ½ � sinh ð Þ� kξ sin ð Þ kξ

Clamped-clamped (C-C) cos k cosh k � 1 ¼ 0 W0 ¼ cosh ð Þ� kξ cosð Þþ kξ sin kþ sinh k

The nth natural circular frequency of the AFG beam can be derived as

λn ¼ 1 L2 � � 1 0 þ Ð 1 0 d2 W0 dξ2 � �2

W0 dξ2 dW0 dξ

W0 dξ2 dW0 dξ

� � 1 0 þ ð1 0 f 1 d2 W0 dξ2 !2

0dξ

0dξ ¼ 0. Then, we obtain ω1 ¼ 0. These values are the properties

� � 1 0 þ Ð 1

dξ

ω0 (25)

cos k� cosh k ½ � sinh ð Þ� kξ sin ð Þ kξ

0 Eð Þξ Ið Þξ d2

W0 dξ2 � �2

(23)

(24)

dξ ¼ 0.

dξ

EI0 ¼

d E½ � ð Þξ Ið Þξ dξ

we have df 1

giving Ð 1

d2 W0

dξ d2 W0 dξ2 W0 þ f 1

Analogously, we choose

0 f 2W2

variable cross-sectional AFG beams.

3.2 Numerical results and discussion

validity of the proposed method.

Boundary conditions Frequency

Table 1.

8

Geometry and coordinate system of an AFG beam for different taper ratios: (a) case 1, cb ¼ ch ¼ 0; (b) case 2, ch ¼ 0, cb 6¼ 0; (c) case 3, cb ¼ 0, ch 6¼ 0; and (d) case 4, cb 6¼ 0, ch 6¼ 0.

cross-sectional area A xð Þ and moment of inertia I xð Þ along the beam axis can be clearly expressed as follows:

$$A(\boldsymbol{\kappa}) = A\_L \left(\mathbf{1} - c\_b \frac{\boldsymbol{\kappa}}{L}\right) \left(\mathbf{1} - c\_h \frac{\boldsymbol{\kappa}}{L}\right), \quad I(\boldsymbol{\kappa}) = I\_L \left(\mathbf{1} - c\_b \frac{\boldsymbol{\kappa}}{L}\right) \left(\mathbf{1} - c\_h \frac{\boldsymbol{\kappa}}{L}\right)^3 \tag{26}$$

where cb ¼ 1 � BR BL and ch ¼ 1 � HR HL are the breadth and height taper ratios, respectively. AL and IL are cross-sectional area and area moment of inertia of the beam left sides, respectively. It is instructive to remember that if cb ¼ ch ¼ 0, the beam would be uniform; if ch ¼ 0, cb 6¼ 0, the beam would be tapered with constant height; if cb ¼ 0, ch 6¼ 0, the beam would be tapered with constant width; and if cb 6¼ 0, ch 6¼ 0, the beam would be double tapered. These four cases are corresponding to Figure 2(a)–(d), respectively. Moreover, the material properties such as Young's modulus E xð Þ and mass density ρð Þ x along the beam axis are assumed as

$$E(\mathbf{x}) = E\_L \left( \mathbf{1} + \frac{\mathbf{x}}{L} \right), \quad \rho(\mathbf{x}) = \rho\_L \left[ \mathbf{1} + \frac{\mathbf{x}}{L} + \left( \frac{\mathbf{x}}{L} \right)^2 \right] \tag{27}$$

where EL and ρL are Young's modulus and mass density of the beam left sides, respectively.

Based on the introduced analytical equation, the first third-order nondimensional natural frequencies (Ωn ¼ λnL2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ρLAL=ELIL p ) of the four cases of nonuniform AFG beams with different boundary configurations were obtained. The results were listed in Tables 2–7, respectively, and it also was compared with those of published work by Shahba et al. [38].

Table 2 shows the first third-order natural frequencies of the AFG beam, case of Figure 2(a), which is uniform but nonhomogeneous. It can be clearly seen that the analytical results obtained from the asymptotic development method are in good agreement with those given by Ref. [38].

#### Table 2.

Nondimensional natural frequencies of the AFG uniform beam (case 1) with different boundary conditions.

#### Table 3.

Nondimensional natural frequencies of the AFG-tapered beam with constant height (case 2) and different boundary conditions.

As can be seen from Tables 3 and 4, the first third-order dimensionless natural frequencies of AFG conical beams with only varying width or height are studied, respectively. It is easy to find the following conclusions. This method has higher accuracy on the equal height AFG-tapered beam. When the height changes, there is

cb 0.2 0.4 0.6 0.8

0.2 Present 2.6873 2.9380 3.3113 3.9455

0.4 Present 2.8226 3.0877 3.4796 4.1377

0.6 Present 3.0640 3.3506 3.7700 4.4625

0.8 Present 3.5271 3.8475 4.3081 5.0458

0.2 Present 17.7225 18.3289 19.1598 20.3725

0.4 Present 16.7822 17.4061 18.2458 19.4418

0.6 Present 16.1771 16.8214 17.6687 18.8380

0.8 Present 16.0947 16.7493 17.5836 18.6877

0.2 Present 50.2194 51.1534 52.4114 54.1995

0.4 Present 46.1970 47.2734 48.6925 50.6520

0.6 Present 43.2042 44.4117 45.9613 48.0269

0.8 Present 41.7065 42.9817 44.5636 46.5828

Nondimensional natural frequencies of the AFG double-tapered beam (case 4); boundary conditions: C-F.

Ref. [38] 2.6863 2.9336 3.2993 3.9219

Ref. [38] 2.7987 3.0486 3.4181 4.0471

Ref. [38] 2.9699 3.2237 3.5985 4.2355

Ref. [38] 3.2794 3.5401 3.9232 4.5695

Ref. [38] 17.7501 18.2379 18.9501 20.2432

Ref. [38] 16.4092 16.8571 17.5139 18.7164

Ref. [38] 14.9567 15.3627 15.9616 17.0694

Ref. [38] 13.3850 13.7466 14.2848 15.2955

Ref. [38] 50.3934 50.8645 51.6029 53.1332

Ref. [38] 44.9504 45.4003 46.0957 47.5129

Ref. [38] 39.0605 39.4844 40.1304 41.4236

Ref. [38] 32.4229 32.8123 33.3986 34.5521

ch First mode

Free Vibration of Axially Functionally Graded Beam DOI: http://dx.doi.org/10.5772/intechopen.85835

ch Second mode

ch Third mode

According to Figure 2(d), when the height and width of AFG beams change simultaneously, we can see that AFG beams are not uniform. The

found that the natural frequencies of AFG beams at low order are in good

natural frequencies of three boundary conditions (free clamping, simply supported, and clamping) are studied in Tables 5–7. From the data in the table, it can be clearly

agreement with Ref. [38], while at high order, there are some errors in the natural

a certain fractional error in the AFG-tapered beam.

frequencies.

11

Table 5.

#### Table 4.

Nondimensional natural frequencies of the AFG-tapered beam with constant width (case 3) and different boundary conditions.

Free Vibration of Axially Functionally Graded Beam DOI: http://dx.doi.org/10.5772/intechopen.85835

#### Table 5.

Boundary condition First mode Second mode Third mode C-F Present 2.439 18.437 54.339

S-S Present 9.053 35.834 80.470

C-C Present 20.585 56.251 109.869

Nondimensional natural frequencies of the AFG uniform beam (case 1) with different boundary conditions.

cb C-F S-S C-C

First mode

0.2 Present 2.613 18.887 54.951 9.068 35.957 80.772 20.457 56.196 110.003 Ref. [38] 2.605 19.004 55.534 9.060 36.342 81.685 20.415 56.472 110.862 0.4 Present 2.854 19.483 55.753 9.088 36.117 81.165 20.294 56.124 110.177 Ref. [38] 2.851 19.530 56.023 9.087 36.315 81.645 20.288 56.298 110.671 0.6 Present 3.214 20.311 56.853 9.113 36.332 81.697 20.079 56.028 110.411 Ref. [38] 3.214 20.296 56.800 9.099 36.297 81.624 20.019 55.921 110.250 0.8 Present 3.832 21.542 58.453 9.147 36.638 82.456 19.783 55.892 110.743 Ref. [38] 3.831 21.676 58.435 9.069 36.277 81.639 19.385 54.971 109.142

Nondimensional natural frequencies of the AFG-tapered beam with constant height (case 2) and different

ch C-F S-S C-C

First mode

0.2 Present 2.5054 17.2596 49.4982 8.1416 32.1888 72.2680 18.2420 50.1851 98.2992 Ref. [38] 2.5051 17.3802 50.0491 8.1341 32.5236 73.1138 18.2170 50.4801 99.1734 0.4 Present 2.6293 16.2995 45.3519 7.2793 28.9717 65.1203 16.3027 45.0600 88.4345 Ref. [38] 2.6155 16.0705 44.6181 7.1531 28.4747 63.9942 15.8282 44.0246 86.6272 0.6 Present 2.8535 15.6697 42.2358 6.4872 26.3694 59.4850 14.9152 41.2502 80.9747 Ref. [38] 2.7835 14.6508 38.7446 6.0357 24.1101 54.0921 13.2293 36.9653 72.8740 0.8 Present 3.2889 15.5662 40.6554 5.7966 24.6371 55.9734 14.2233 39.1823 76.7690 Ref. [38] 3.0871 13.1142 32.1309 4.6520 19.1314 42.6954 10.2235 28.7492 56.8109

Nondimensional natural frequencies of the AFG-tapered beam with constant width (case 3) and different

Second mode

Third mode

First mode Second mode

Third mode

Third mode Second mode

Third mode

First mode Second mode

Third mode

Third mode

Table 2.

Table 3.

Table 4.

10

boundary conditions.

boundary conditions.

First mode

First mode Second mode

Second mode

Mechanics of Functionally Graded Materials and Structures

Ref. [38] 2.426 18.604 55.180

Ref. [38] 9.029 36.372 81.732

Ref. [38] 20.472 56.549 110.947

Nondimensional natural frequencies of the AFG double-tapered beam (case 4); boundary conditions: C-F.

As can be seen from Tables 3 and 4, the first third-order dimensionless natural frequencies of AFG conical beams with only varying width or height are studied, respectively. It is easy to find the following conclusions. This method has higher accuracy on the equal height AFG-tapered beam. When the height changes, there is a certain fractional error in the AFG-tapered beam.

According to Figure 2(d), when the height and width of AFG beams change simultaneously, we can see that AFG beams are not uniform. The natural frequencies of three boundary conditions (free clamping, simply supported, and clamping) are studied in Tables 5–7. From the data in the table, it can be clearly found that the natural frequencies of AFG beams at low order are in good agreement with Ref. [38], while at high order, there are some errors in the natural frequencies.

### Mechanics of Functionally Graded Materials and Structures

#### Table 6.

Nondimensional natural frequencies of the AFG double-tapered beam (case 4); boundary conditions: S-S.

E xð Þ¼ EL 1 � f E

, ρE ¼ 1 � ρR

ρL

1 � f Ex w00 00

2 t and βn

where f E ¼ 1 � ER

Table 7.

ch First mode

Free Vibration of Axially Functionally Graded Beam DOI: http://dx.doi.org/10.5772/intechopen.85835

ch Second mode

ch Third mode

qnðÞ¼ t An cos βn

13

EL

beam. Eq. (2) is then rewritten as

2

t þ Bn sin βn

The governing equation is then derived as

x L

cb 0.2 0.4 0.6 0.8

0.2 Present 18.2779 18.3231 18.3818 18.4612

0.4 Present 16.4975 16.7396 17.0484 17.4563

0.6 Present 15.2512 15.6622 16.1771 16.8423

0.8 Present 14.6662 15.2004 15.8587 16.6925

0.2 Present 50.4430 50.7713 51.2035 51.7981

0.4 Present 45.6257 46.3346 47.2495 48.4763

0.6 Present 42.0890 43.1214 44.4245 46.1234

0.8 Present 40.2151 41.4614 42.9975 44.9420

0.2 Present 98.2992 99.7466 100.8219 102.313

0.4 Present 88.4345 90.9806 92.8200 95.3023

0.6 Present 80.9747 84.4598 86.8967 90.0855

0.8 Present 76.7690 80.8426 83.5867 87.0576

Nondimensional natural frequencies of the AFG double-tapered beam (case 4); boundary conditions: C-C.

Ref. [38] 18.1996 18.1286 17.9437 17.4566

Ref. [38] 15.8498 15.8350 15.7367 15.4025

Ref. [38] 13.2896 13.3319 13.3238 13.1529

Ref. [38] 10.3229 10.4255 10.5168 10.5339

Ref. [38] 50.4565 50.3599 50.1017 49.3728

Ref. [38] 44.0553 44.0370 43.9027 43.4066

Ref. [38] 37.0509 37.1137 37.1104 36.8678

Ref. [38] 28.8912 29.0409 29.1842 29.2402

Ref. [38] 99.1474 99.0414 98.7543 97.9046

Ref. [38] 86.6608 86.6414 86.4932 85.9176

Ref. [38] 72.9681 73.0382 73.0375 72.7615

Ref. [38] 56.9674 57.1341 57.2991 57.3787

end of the beam, and ρL and ρR are the mass density at the left/right end of the

Based on the vibration theory, we assume w xð Þ¼ ; t ϕð Þ x q tð Þ, where

, ρð Þ¼ x ρL 1 � f ρ

þ 1 � f ρx

x L

w€ ¼ 0 (29)

2 is the modal frequency for dimensionless.

. EL and ER are Young's modulus at the left/right

(28)


**Author details**

the model with the behavioral parameter sets shows a reduction in P-factor and R-factor values (**Table 4**). **Figure 7** shows that while the uncertainty band has been reduced, the model is undersimulating both high and low flows, and does not bracket the moderate to low flows. This could be associated with input data uncertainties, or some processes are not well represented in the model. ALPHA\_BF was the most sensitive parameter identified through the sensitivity analysis, and apart from a lack of observed ground water information, difficulties of SWAT in simulating

This study assessed model uncertainty using a combined uncertainty approach that assumes all sources of uncertainty have been considered within the model. In such an approach it is hard to separate the sources of uncertainty, and therefore a follow-up analysis of uncertainty should be undertaken by determining how erroneous input data influence model results. Although not assessed within the research questions of this study, the results highlight potential uncertainties in the input rainfall and evaporation data. The use of these data was justified and used in the simulations but could potentially have influenced the overall model perfor-

The uncertainty analysis was carried out using 20 sensitive parameters, which is a large number considering the interactions between them. Therefore, some less sensitive parameters should be fixed and allow only the most sensitive parameters to vary. This will reduce the effect of parameter interactions and hence the none-uniqueness problem. Although this model has been shown to generate reasonable results, it is worthwhile to consider the challenges associated with setting up a distributed model. In this research, large-scale spatial datasets have been used, and a homogenous model was assumed because the spatial data resolution was insufficient to represent large numbers of hydrological response units. However, even when the resolution was sufficient, attribute values for most of the parameters are lacking. Because of difficulties associated with parameter representation across spatial scales, it is better to use a homogenous set up because biases and uncertainty can be added by the modeler when trying to parameterize values within the hydrological response unit at a size larger than

• The SUFI-2 approach has capabilities of identifying behavioral parameter. However, the

• The scatter plots of the parameter values against objective functions obtained after simulation provided an initial qualitative overview of the uncertainties involved in the represen-

• The 95% of the predictive uncertainty (95 PPU) for stream flow computed using SUFI-2 using the Latin Hypercube sampling with 2000 runs, did not bracket all simulations, indicating that some processes are not represented in the model. Hence additional information

• It is also important to emphasize that the prediction limits obtained with SUFI-2 are highly dependent on the threshold selected to separate behavioral from non-behavioral parameter sets and that the subjective choice of the threshold value and objective function can lead to

ground water flow [12] might have contributed to the negative aspects of these results.

mance and uncertainties that cannot be explained.

64 Achievements and Challenges of Integrated River Basin Management

its coverage. The overall conclusions from this assessment include;

results are influenced by large equifinality.

additional uncertainty in the simulation results.

tation of basin's behavior.

is needed to improve the results.

#### Madaka Harold Tumbo

Address all correspondence to<EMAIL_ADDRESS>
Institute of Resource Assessment, University of Dar es Salaam, Tanzania


**3. Demultiplexing employing thin-film interference filters**

The technology based on thin-film interference filters is mature and one of the most commonly applied technologies for realization of WDM demultiplexers in single-mode glass fiber communication. The demultiplexers for coarse WDM applications cascade the interference filters to provide up to 16 flattop channels between 1271 and 1611 nm, with 20 nm minimum channel spacing [19]. The typical parameters of commercial 4-, 8-, and 16-channel demultiplexers with IL <1.6, 2.7, and 3.7 dB, respectively, can be found in [20]. The thin-film filter-based demultiplexers for dense WDM applications are commercially available with up to 40 channels in 1550 nm region and <8 dB IL. Instead of simply cascading the filters, those devices usually employ a modular configuration described in by Dutta et al. [21]. The same reference provides a typical transfer function of the 40-channel demultiplexer with 3–6 dB IL and 100 GHz (0.8 nm) channel spacing.

In the visible spectrum, and thus within the application range of SI-POF, a vast variety of thin-film interference filters is available from various manufacturers. Even though not particularly intended for POF applications, the visible interference filters represent an attractive solution for POF demultiplexers, where wavelength selectivity, low IL, and high isolation are required.

A dichroic mirror is a special type of interference filter intended for the spatial separation or combination of light at different wavelengths. It is designed to operate at 45° AOI, such that a certain spectral range is transmitted, whereas the rejected wavelength range is reflected at 90° angle with respect to the incident optical axis. A commercial visible spectrum dichroic mirror has a transition slope between the transmission and reflection band of typically 30–40 nm (see **Figures 3** and **4**). This is significantly less steep compared to the standard interference filters designed for

#### **Figure 4.**

*Principle of separation of two collimated wavelength channels employing thin-film interference filters: (1) dichroic mirror (45° AOI); (2) interference filter (0° AOI).*

*Optoelectronic Key Elements for Polymeric Fiber Transmission Systems DOI: http://dx.doi.org/10.5772/intechopen.86423*

the normal incidence. Unlike an interference filter, e.g., a long-pass mirror must be not only highly transmissive above the cutoff wavelength but also highly reflective below it. Therefore, producing steeper slopes would require increased complexity of the coating and, accordingly, a significant rise in production costs.

The interference filters show significant angular dependence of their transmission characteristic measured by Lee et al. [22]. To be applicable for SI-POF, the highly divergent beam from the fiber must be transformed into a bundle of parallel rays prior to the incidence. To increase the channel isolation, an additional bandpass filtering in each of the output channels should be implemented prior to the focusing of light. As an example, a selection of the dichroic mirror and interference filters for demultiplexing two wavelength channels centered around λ1 = 450 nm and λ2 = 525 nm is shown in **Figure 4**.

#### **3.1 Data transmission techniques for increasing the channel capacity**

Two data transmission techniques were used to overcome the bandwidth limitation of a POF WDM channel, which is primarily caused by the intermodal dispersion of SI-POF. Those were:

In the single-channel POF systems with intensity modulation and direct detection (IM/DD), those are well-known techniques for increasing the channel capacity. The next two subsections briefly introduce the two techniques: non-return-to-zero modulation and feed-forward equalization.

According to the Nyquist theorem for two-level signaling, the maximum bit rate (in bits per second) for a noiseless channel of the bandwidth *B* equals 2*B*. Since the real systems encounter noise, this theoretical limit cannot be achieved. Therefore, for 100 m SI-POF link, the data rates less than 100 Mb/s are possible. If the data rate further increases, the eye diagram of the received signal becomes partially or completely closed due to the combination of the ISI and additive noise (introduced, e.g., by receiver's transimpedance amplifier).

The equalization techniques are used to open the eye diagram at the receiver for clock and data recovery (CDR). In its principle, the equalizer compensates for ISI, which is deterministic (unlike the random noise) and determined by the low-pass frequency response of a POF channel. In the work a simple linear FFE equalization technique was employed to correct the distorted signal waveforms at the receiver.

An FFE equalizer is realized as a discrete-time finite impulse response filter with adjustable coefficients. The output of the equalizer is obtained as the weighted sum of the delayed samples of the input signal as

$$V\_{\epsilon}(t) = \sum\_{k=0}^{N-1} c\_k \, V\_r(t - k \, T\_D),\tag{6}$$

where *Ve*(*t*) is the equalized voltage sample at the time *t*, *ck* is the equalizer coefficient (weighting factor), *TD* is the tap delay, and *Vr*(*t* − *kTD*) is the *k* times delayed uncorrected received voltage sample. A block diagram of the FFE equalizer is shown in **Figure 5**.

Through its coefficients the equalizer may synthesize a transfer function corresponding to the inverted channel frequency response, thus eliminating the ISI. In

**Figure 5.** *Basic structure of an FFE equalizer.*

**Figure 6.**

*Calculation of the Qy-factor from the eye diagram: P(1) and P(0), probability distributions of received logic 1 and 0 levels, respectively.*

a noisy POF channel, and due to inverting the channel frequency response, this would lead to great noise amplification at higher frequencies where the channel frequency response is small in magnitude. Typically, to minimize the probability of the decision error, the weighting factors are calculated to minimize the noise power at the cost of a certain amount of residual ISI after equalization. The descriptions of different algorithms for optimizing the equalizer coefficients can be found in Loquai et al. [23].

The BER performance of the system was estimated based on the Q-factor of the equalized eye diagram (**Figure 6**)

$$\mathbf{Q} = \frac{V\_1 - V\_0}{\sigma\_1 - \sigma\_0},\tag{7}$$

where *V*1 and *V*0 are the mean values and σ1 and σ0 are the standard deviations of the equalized signal voltages associated with logic 1 and 0 levels, respectively, as shown in **Figure 6**. The corresponding BER was obtained as

$$BER(Q) = \frac{1}{2} erfc\left(\frac{Q}{\sqrt{2}}\right),\tag{8}$$

where *erfc*(*x*) = (2/√ \_\_ *π*) ∫*x* ∞ *exp*(−τ 2 )*d* is the complementary error function.

### **3.2 DMT: discrete multitone modulation**

To provide high spectral efficiency of the signals transmitted within POF WDM channels, a DMT modulation technique was used. The DMT is a multicarrier modulation format and represents a baseband version of a better-known


**Field Performance Test of an Air-Cleaner with Photocatalysis-Plasma Synergistic Reactors for Practical and Long-Term Use**

### **Tsuyoshi Ochiai, Erina Ichihashi, Naoki Nishida, Tadashi Machida, Yoshitsugu Uchida, Yuji Hayashi, Yuko Morito and Akira Fujishima**

**Abstract:** A practical and long-term usable air-cleaner based on the synergy of photocatalysis and plasma treatments has been developed. A field test of the air-cleaner was carried out in an office smoking room. The results were compared to previously reported laboratory test results. Even after a treatment of 12,000 cigarettes-worth of tobacco smoke, the air-cleaner maintained high-level air-purification activity (98.9% ± 0.1% and 88% ± 1% removal of the total suspended particulate (TSP) and total volatile organic compound (TVOC) concentrations, respectively) at single-pass conditions. Although the removal ratio of TSP concentrations was 98.6% ± 0.2%, the ratio of TVOC concentrations was 43.8% after a treatment of 21,900 cigarettes-worth of tobacco smoke in the field test. These results indicate the importance of suitable maintenance of the reactors in the air-cleaner during field use.

Reprinted from *Molecules.* Cite as: Ochiai, T.; Ichihashi, E.; Nishida, N.; Machida, T.; Uchida, Y.; Hayashi, Y.; Morito, Y.; Fujishima, A. Field Performance Test of an Air-Cleaner with Photocatalysis-Plasma Synergistic Reactors for Practical and Long-Term Use. *Molecules* **2014**, *19*, 17424-17434.

#### **1. Introduction**

Photocatalysis-plasma synergistic reactors have been recently proposed for use in air-cleaners [1–4]. The synergistic effects of photocatalysis and plasma excitation achieve significant oxidative decomposition of gaseous compounds in laboratory tests. The coil-shaped reactor (Figure 1), using plasma-assisted catalytic technology (PACT) [5] and a TiO2 impregnated Ti-mesh filter (TMiPTM) [6], shows long-term capability of removing tobacco smoke compounds. High-level air-purification activity was maintained in the air-cleaner with the coil-shaped reactor (98, 98.9% ± 0.1%, and 88% ± 1% removal of the odour concentration, total suspended particulate (TSP), and total volatile organic compound (TVOC) concentrations, respectively) even after the treatment of 12,000 cigarettes-worth of tobacco smoke, which is the equivalent of using the air-cleaner in the smoking room for 6 months [4]. In this study, a field test of the air-cleaner using the coil-shaped reactor was carried out for 84 days in a functioning smoking room (Figure 2) of a typical office building. The air-purification ability and the long-term usability of the air-cleaner in the smoking room were discussed by comparison with previously reported laboratory test results.

**Figure 1.** Image and schematic illustration of the coil-shaped PACT-TMiP synergistic reactor (**left**) and the air-cleaner (**right**). Reproduced from Ochiai *et al.* [4], published by Scientific Research Publishing Inc., 2014.

#### **2. Results and Discussion**

#### *2.1. Evaluations of the Photocatalysis-Plasma Synergistic Air-Cleaner*

Figure 3 shows the temperature distributions of the coil-shaped reactors in the air-cleaner 1, 5, and 10 min after the device is switched on without air flow. The temperature of the reactors reached an almost steady state at around 100 °C for 10 min (Figure 3c). This temperature is lower than the anatase-to-rutile transformation temperature of TiO2 [7]. Therefore, the TiO2 photocatalyst on the TMiP surface cannot be affected by the air-plasma. Conversely, ozone and NOx concentrations near the reactors in the air-purifier were 8–9 and 0.7–0.8 ppm, respectively under a 1200 m3 /h flow rate. These species can accelerate the decomposition of TVOCs [1]. However, ozone and NOx concentrations at the air-outlet of the air-cleaner were below the detection limit. Thus, the ozone-cut and activated carbon filters shown in Figure 2 can reduce excess ozone and NOx.

#### *2.2. Usage of the Smoking Room*

The field test was carried out in a smoking room with a volume of approximately 31 m3 (5.0 m × 2.5 m × 2.5 m), which was used by several smokers (Figure 2). The number of cigarettes burned every 30 min, from 11:00 to 17:00 during the working day, was counted and the results summarised in Figure 4. On average, 32 ± 11 and 350 cigarettes are burned on an hourly and daily basis, respectively. Generally, an estimated 12,000 cigarettes are burned in the smoking room every six months [4]. Therefore, the number of cigarettes burned in the smoking room in this study is around six times higher than in an average smoking room.

**Figure 2.** Schematic illustration of the test method in the smoking room for evaluating the air-purification ability of the air-cleaner (unit: cm). Sampling point (1) is near the air inlet of the air-cleaner, and (2) is inside the duct. Detailed experimental procedures are included in Section 3.

**Figure 3.** Temperature distribution of the coil-shaped reactors in the air-cleaner (**a**) 1; (**b**) 5; and (**c**) 10 min after the device is switched on without air flow.

**Figure 4.** Summary of the number of cigarettes burned every 30 min during the working day (11:00–17:00, 1 May 2014).

#### *2.3. TSP Removal*

Figure 5a shows TSP concentrations at sampling points 1 (near the air inlet of the air-cleaner, Figure 2) and 2 (inside the duct, Figure 2) after the treatment of 2300 cigarettes-worth of tobacco smoke. TSP concentrations at sampling point 1 fluctuated from 0.026 to 1.24 mg/m3 with changes in the number of cigarettes burned. The TSP concentrations at sampling point 2 fluctuated from below the detection limit (0.0008 mg/m3 ) to 0.014 mg/m3 . Thus, the average removal ratio of TSPs was 98.7% ± 0.4%. Moreover, the removal ratio maintained high-levels (98.6% ± 0.2%) after the treatment of 21,900 cigarettes (Figure 5b). These values indicate that the air-cleaner is able to remove TSPs efficiently, in agreement with previously reported air-cleaner trends observed in laboratory test [4].

**Figure 5.** TSP concentrations at sampling points 1 (black filled diamonds, Figure 2) and 2 (black open diamonds, Figure 2) and the removal ratios of TSPs (red filled diamonds) after the treatment of (**a**) 2300 and (**b**) 21,900 cigarettes-worth of tobacco smoke.

#### *2.4. TVOC Removal*

Figure 6a shows the normalised GC-MS chromatograms of the air samples at sampling points 1 and 2 after the treatment of 2300 cigarettes-worth of tobacco smoke. Many distinctive VOC peaks were observed in the chromatogram of the air sample at sampling point 1, which had almost disappeared at point 2. TVOC concentrations were calculated for all peaks between *n*-hexane (16.1 min) and *n*-hexadecane (54.7 min); they were then calibrated and converted to toluene peak (28.5 min) equivalents. In Figure 6a, the TVOC concentrations at sampling points 1 and 2 were 128.1 and 3.4 ȝg/m3 , respectively (corresponding to a 97.3% removal ratio of TVOCs). However, several peaks remained or were amplified in the chromatogram of the sample at point 2 following the treatment of 21,900 cigarettes-worth of tobacco smoke (Figure 6b), especially between the *n*-hexane (16.1 min) and toluene (28.5 min) peaks. The TVOC removal ratio calculated from Figure 6b dramatically decreased to 43.8%, while the ratio for TSPs did not decrease (Figure 5). These data indicate that the TVOC removal efficiency of the air-cleaner under the present conditions is more easily affected than TSP removal efficiency by catalyst poisoning and the adsorption/desorption of VOCs on the filters during long-term use [8–11].

**Figure 6.** Normalised GC-MS chromatograms of the air samples from the Figure 2 sampling points 1 (black) and 2 (blue) after the treatment of (**a**) 2300 and (**b**) 21,900 cigarettes-worth of tobacco smoke.

*2.5. Comparison of the Field and Laboratory Tests; The Problems and Future Directions*

The removal ratios of the TSPs and TVOCs from tobacco smoke by the air-cleaner in the field laboratory tests are summarised in Figure 7. In both the field and the laboratory tests, TSP removal ratios continued at high-levels (around 98.5%) throughout the experimental period. However, the TVOC removal ratios decreased with increases in the number of cigarettes. Intriguingly, the TVOC removal ratios in the field test decreased sooner than in the laboratory test. In this case, TSPs were removed by the HEPA filter and electrostatic precipitation in the plasma treatment [12,13]. However, as mentioned in Section 2.4, TVOC removal was easily affected by catalyst poisoning and adsorption/desorption of VOCs on the filters during long-term field use. To improve long-term usability, suitable maintenance methods must be developed such as plasma ashing of the reactor surfaces.

**Figure 7.** Removal ratios of TSPs (blue) and TVOCs (red) in tobacco smoke by the air-cleaner (**a**) in the field and (**b**) in laboratory tests. Panel (b) is reproduced from Ochiai *et al.*, [4], published by Scientific Research Publishing Inc., 2014.

Another issue that must be considered is CO removal. Figure 8 shows CO and CO2 concentrations at sampling points 1 and 2. The concentrations fluctuated with changes in the number of burning cigarettes, as was observed with the TSP concentrations (Figure 5). However, there are no clear differences between the concentrations at points 1 and 2 after the treatment of 2300 and 21,900 cigarettes-worth of tobacco smoke. These data indicate that the present experimental conditions of the photocatalysis-plasma synergistic reactor were not adequate for CO removal, despite the success with the TSP and TVOC removal. Several studies have investigated the synergistic effects of catalysis-plasma or photocatalysis-plasma systems. They found that synergism existed extensively but only under specific conditions [14–18]. Hence, a number of factors have been suggested that can affect efficiency such as catalyst loading level, input power, temperature, adsorption process, *etc*. These factors can be easily influenced by the smoking room usage in this study. This may be the biggest drawback for developing a versatile and effective air-cleaner with a photocatalysis-plasma synergistic reactor. Currently, the oxidation of CO to CO2 in the presence of noble metals is being studied for the development of effective catalytic converters and fuel cells [19–21]. In this study, the causes of the decrease in TVOC removal efficiency and the poor CO removal efficiency are still unclear. However, more suitable experimental conditions and combinations of catalysis, photocatalysis, and the plasma treatment for effective TVOC and CO removal in the field are being tested.

**Figure 8.** CO (black) and CO2 (red) concentrations from the Figure 2 sampling points 1 (filled diamonds) and 2 (open diamonds) after the treatment of (**a**) 2300 and (**b**) 21,900 cigarettes-worth of tobacco smoke.

#### **3. Experimental Section**

*3.1. Fabrication and Evaluation of the Photocatalysis-Plasma Synergistic Reactor and the Air-Cleaner*

The image and schematic of the coil-shape PACT-TMiP synergistic reactor and air-cleaner are shown in Figure 1. The basic design and fabrication methods of the reactor and the air-cleaner have been previously described [4]. In this study, a voltage of 10 kV (peak-to-peak), a frequency of 25 kHz, and a power of 45 W were used. Air can be drawn through the gaps of the reactor while maintaining high surface contact with TMiP and air-plasma. A high efficiency particulate air (HEPA) filter, ten coil-shape reactors, two ozone-cut filters, an activated carbon filter, and a fan were arranged inside the air-cleaner. When the fan is turned on, air flow is generated inside the casing from the air inlet towards the air outlet, passing through the filters and the PACT-TMiP reactor. Temperature distributions of the coil-shaped reactors in the air-cleaner were measured by thermography using a Handy Thermo TVS-200EX (Nippon Avionics Co., Ltd., Tokyo, Japan). Ozone and NOx concentrations were monitored using a Model 106-L ozone monitor (2B Technologies, Boulder, CO, USA) and MODEL42 I NOx Analyser (Thermo Fisher Scientific, Waltham, MA, USA), respectively.

### *3.2. The Evaluation Method of the Air-Purification Activity by the Air-Cleaner in the Smoking Room*

The schematic of the test method for evaluating the air-purification activity of the air-cleaner is shown in Figure 2. Air flow is generated inside the smoking room from the door and air inlet to the duct, passing through the air-cleaner. Under these conditions, the smoking room was filled with tobacco smoke from the sequential burning of cigarettes by the smokers (32 ± 11 cigarettes/h, 350 cigarettes/d). The concentrations of TSPs, TVOCs, carbon monoxide (CO), and carbon dioxide (CO2) were measured after the treatments of 2300, 6500, 14,900, and 21,900 cigarettes-worth of tobacco smoke at sampling points 1 (near the air inlet) and 2 (inside the duct) defined in Figure 2. TSP concentrations were monitored using a digital real-time LD-3K2 dust monitor (Sibata Scientific Technology Ltd., Saitama, Japan) every minute for an hour. TVOC concentrations were calculated by qualitative and quantitative analysis using GC-MS analysis. A GC-17A-GCMS-QP5050A combination (Shimadzu, Kyoto, Japan) was used at an ionization voltage of 70 eV and a mass range of 35–200. The system was equipped with a 60 m × 0.25 mm internal diameter × 1.4 ȝm DB-624 fused silica capillary column (Agilent Technologies, Santa Clara, CA, USA) with split injection (split ratio 11:1). The oven was programmed to start at 35 °C (for 15 min) reaching 240 °C (for 8 min) at a rate of 6 °C/min. Samples were collected by drawing 60 L (0.6 L/min) of air through a charcoal tube, desorbed with 1 mL of carbon disulphide, and analysed by GC-MS. TVOC concentrations were calculated for all compounds eluted between *n*-hexane and *n*-hexadecane, they were then calibrated and converted to toluene equivalents. The removal ratios were calculated using the formula (A1 í A2)/A1, where A1 and A2 are the amounts at sampling points 1 and 2, respectively. An important point to note is that the tobacco smoke was treated by the air-cleaner once, *i.e*., this was a single-pass system. CO and CO2 concentrations were also measured using a COX-3 CO/CO2 analyser (Sibata Scientific Technology Ltd., Saitama, Japan) every minute for an hour.

#### **4. Conclusions**

The photocatalysis-plasma synergistic air-cleaner and its long-term usability in the field were investigated. Compared with previously reported laboratory test results for the air-cleaner, TSP removal ratios remained at high-levels (around 98.5%) throughout the experimental period in both the field and laboratory tests. However, the TVOC removal ratios in the field test decreased three times sooner than in the laboratory test. Additionally, the CO removal ability of the air-cleaner was almost negligible. In conclusion, these results indicate that the photocatalysis-plasma synergistic air-cleaner was effective in the long-term removal of TSPs, given the tobacco smoke conditions in the smoking room investigated. Suitable maintenance methods for the reactor surfaces would improve the long-term TVOC removal ability of the air-cleaner in the field.

#### **Acknowledgments**

We are grateful to N. Uchiyama, H. Inoue, H. Kasagi (I'm PACT World Co., Ltd.), Y. Abe, I. Kurihara, R. Yokoyama, H. Shigeno, and S. Arai (TANASHIN DENKI Co. Ltd.) for the experiments and discussions. We would also like to thank the staff at the SGS, Inc. offices for providing the opportunity for field testing and their support.

#### **Author Contributions**

Tsuyoshi Ochiai, Tadashi Machida, and Yoshitsugu Uchida participated in study design. Tadashi Machida and Yoshitsugu Uchida conducted the study. Data was collected and analysed by Tadashi Machida and Erina Ichihashi. The manuscript was written by Tsuyoshi Ochiai, Naoki Nishida, Tadashi Machida, and Yoshitsugu Uchida. Yuko Morito designed and made the TMiP. Yuji Hayashi, Yuko Morito, and Akira Fujishima provided valuable discussions and advice on the manuscript.

#### **Conflicts of Interest**

The authors declare no conflicts of interest.

#### **References**

*Sample Availability*: Not available.


**7. Ischemic Stroke**

Epidemiological studies also suggest an association between periodontitis and ischemic stroke. Stroke is the second most common cause of mortality worldwide and approximately 80% of strokes are caused by focal cerebral ischemia [76]. A recent meta-analysis [76] has shown a significant association between periodontitis and ischemic stroke based on three cohort studies [77–79] (pooled RR 2.52, 95% CI 1.77–3.58) and five case-control studies (pooled RR 3.04, 95% CI 1.10–8.43) [80–84]. Chi et al. (2019) examined mice with both experimental periodontitis induced by periodontal injection of LPS and photothrombotic ischemia. The study has demonstrated that chronic periodontitis exacerbates ischemic stroke through increasing the activation of microglia/astrocytes and the expression of nod-like receptor protein 3 inflammasome and IL-1β [85], suggesting that chronic periodontitis is a driving force for neuroinflammation associated with ischemia.


**1. Introduction**

Food is essential for life, and the main sources of food for animals especially for humans are different parts of the plant such as leaf, stem, fruits, seeds, flowers, and roots. Plants and their parts are not only used as a food but are also used as medicines. This is due to the presence of active chemical components in parts of the plant. Nowadays, various research and study applications have focused on the active chemical constituents of the plants and their functions due to the effectiveness, less side effects, synthesized easily, and most importantly economically feasible. Traditional uses of plants and plant parts also demonstrate the direction to use plants for the specific type of disease. *A. marmelos* (L.) Correa (**Figure 1**) is a very important indigenous plant to the Indian subcontinent for over 5000 years because of its economically feasible dietery and medicinal uses. The fruits of bael are of

**Figure 1.** *Leaves and fruits of Aegle marmelos.*

dietetic use, and the fruit pulp is used to make delights such as murabba, puddings, and juice. The subtropical fruit *A. marmelos* or commonly known as bael belongs to the Rutaceae family [1]. *A. marmelos* L*.*, known by different names in different languages, such as Bel, Beli, and Belgiri in Hindi, Bilva, Shivadrums, Shivaphala, and Vilva in Sanskrit and Bael, Bengal quince, Indian quince, holy fruit, and golden apple in English and also known as Japanese bitter orange, stone apple, or wood apple (**Table 1**) [2]. In India, it is cultivated as a temple garden plant, and the leaves of *A. marmelos* are used to pray to Lord Shiva.

In different fields of science and technology, every part of this plant has a great utilization to cure many diseases. The variety of fruits is smaller in size than the other variety in the similar cultivated type [1]. Most common cultivar varieties of *A. marmelos*/Bael are Kagzi Gonda, Gonda no 1, Gonda no 2, Kagzi Etawah, Mirzapuri, and Baghel [1]. The potential effects of bael in different medicinal systems such as Ayurvedic, Unani, and Siddha have been already in use, and it

#### **Table 1.**

*Various names of Aegle marmelos in different languages.*

*Medicinal Uses with Immense Economic Potential and Nutritional Properties… DOI: http://dx.doi.org/10.5772/intechopen.102876*

states the functionality and feasibility of bael [3]. Many researches studies on bael approved that it consists of many nutritional elements such as a large group of phytochemicals, which includes alkaloids, flavonoids, tannins, phenolic acids, and coumarins. Except the above chemicals, other class of beneficial compounds such as amino acids, vitamins, a wide range of organic acids, carbohydrates, fatty acids, fibers, minerals make bael a highly nutritious fruit with immense health benefits, and the natural availability makes it economically feasible [4]. *A. marmelos* may cure gastrointestinal diseases such as diabetes, cardiac issues, and inflammationrelated problems. It can acquire protective effects against the wound, microbes, radiation, free radical generation, and depression. The natural healing power of *A. marmelos* is proved by many research studies and applications [5]. *A. marmelos* fruit is a natural source of antioxidants and functional foods. It can be used in its dehydrated form such as jam, slab, bael powder, which makes it more useful for any kinds of season. *A. marmelos* and its various products and its various uses and applications make it extremely profitable, and the availability provides greater advantage over other similar kind [6, 7]. All seeds of *A. marmelos* are surrounded by gum having adhesive, waterproofing, and oil emulsion coating properties [8]. Due to the digestive and stomachic properties of bael, it is used in the treatment of diarrhea and dysentery [9]. It acquires tannins, reducing sugar, high amount of pectin, and very small amount of nonreducing sugar [10]. Due to presoluble and insoluble mucilaginous dietary fiber and pectin, it has great nutritional values. Many minerals and vitamins such as riboflavin, ascorbic acid, thiamine, niacin, vitamin A, calcium, and phosphorus are found in bael, which makes it healthy food [11]. Focus of this review is to highlight the composition of beneficial nutrients, formulation, and their medicinal uses with availability, quantification of phytochemicals, and economic view for production. This article will also be helpful for the industries for the production of medicines, bael products, food supplements and also helpful for researchers to make it more fruitful.

### **2. Botanical description**

The growth of the *A. marmelos* tree is slow, and the height is nearly about 762 cm and 90–120 cm in girth [12], with a somewhat spiny branches of stem are observed in the bael tree [13]. Branches are armed with straight sharp axillary thrones, 2.5 cm long. Leaves alternate, 3 – foliate, rarely 5 – foliate; petiole 2.5–6.3 cm long, terete, 2.5 cm long. Leaflets 5–10 by 2.5–6.3 cm, ovate or ovate – lanceolates, crenate, acuminate, membranous, pellucid-punctate, the lateral opposite, subsessile, the terminal long petioluled. **Table 2** shows the botanical classification of bael.

The color of the leaves is different in different phase of the leaves; it is very green and fresh to see in the early stages, but later on it looks like dark green in color [13]. On observing the bark of the tree, it has been found that it is thick in nature, and

**Table 2.**

*Botanical classification of Aegle marmelos/bael.*

after slightly thrashing the bark, then gum secretion is also observed, and it becomes thicker with the presence of air [13]. *A. marmelos* flower is hypogynous and bisexual in nature, and it smells sweet. The greenish-white bael flowers are collectively held by some lateral panicles, which are held by leaf axil [1, 14]. *A. marmelos* fruit has a hard, woody exocarp, and the fruit color is mainly yellowish green, having a diameter of 5.3–7.2 cm, with an approximate weight of 77.2 g, volume of 73.7 mL, and sphericity of 93.72 ± 2.78% [15]. The color of the fruit pulp may vary from bright orange to sunset yellow, and the pulp contains seeds that are present in grooves and is surrounded by thick, clear mucilage. It contains some dots on the outer surface and also contains numerous seeds, which are hard and having thread-like hairs over their outer surface, which is white. The fruit is greenish when unripe, and upon ripening, it turns into yellowish color. Both ripe and unripe fruits are used for their medicinal values [16, 17].