\r\n\tCongenital hearing loss means hearing loss that is present at birth. I have managed children with hearing loss for many years, and the most touching thing is the light that blooms on the face while the hearing-impaired child heard his mother's voice at first time. The scene of "happy tears" impressed me so much. To hear the voice that has not been heard is so pleasant, as if this ordinary listening experience is a supreme listening enjoyment.
\r\n\r\n\tAge-related hearing loss means a progressive loss of ability to hear high frequencies with aging, also known as presbycusis. Among them are the influence of internal and external factors such as genes, drugs and noise exposure. The studies pointed out that the brain stimulation of the hearing-impaired person is greatly reduced compared with subjects with normal hearing. The connection of auditory cortex and other brain areas has declined a lot, which is probably one of the important causes of dementia or even depression in the elderly.
\r\n\r\n\tNoise-induced hearing loss is hearing impairment resulting from exposure to loud sound. There is actually continuous and endless noise in many workplaces, which may cause chronic and cumulative damage. Some young people often work hard but easily neglect to protect themselves. In addition, in recent years, entertainment noise (such as nightclubs, concerts, and personal listening devices) has caused hearing impairment in young people. These should be avoidable and preventable.
\r\n\r\n\tHearing Science is the study of impaired auditory perception, the technologies and other rehabilitation strategies for persons with hearing loss. Public health has been defined as "the science and art of preventing disease", improving quality of life through organized efforts. To avoid the “epidemic” of hearing loss, it is necessary to promote early screening, use hearing protection, and change public attitudes toward noise.
\r\n\r\n\tBased on these concepts, the book incorporates updated developments as well as future perspectives in the ever-expanding field of hearing loss. Besides, it is also a great reference for audiologists, otolaryngologists, neurologists, specialists in public health, basic and clinical researchers.
",isbn:"978-1-83968-678-8",printIsbn:"978-1-83968-677-1",pdfIsbn:"978-1-83968-679-5",doi:null,price:0,priceEur:0,priceUsd:0,slug:null,numberOfPages:0,isOpenForSubmission:!1,hash:"a4b7dbb02ba00e7412422cd5dbffa029",bookSignature:"Dr. Tang-Chuan Wang",publishedDate:null,coverURL:"https://cdn.intechopen.com/books/images_new/10529.jpg",keywords:"Hidden Hearing Loss, Plasticity, Electrophysiology, Otoacoustic Emission, Newborn Hearing Screening, Genetics, Aging, Hearing Aids, Noise Exposure, Occupational Hearing Loss, Epidemiology, Prevention",numberOfDownloads:null,numberOfWosCitations:0,numberOfCrossrefCitations:null,numberOfDimensionsCitations:null,numberOfTotalCitations:null,isAvailableForWebshopOrdering:!0,dateEndFirstStepPublish:"September 3rd 2020",dateEndSecondStepPublish:"October 1st 2020",dateEndThirdStepPublish:"November 30th 2020",dateEndFourthStepPublish:"February 18th 2021",dateEndFifthStepPublish:"April 19th 2021",remainingDaysToSecondStep:"3 months",secondStepPassed:!0,currentStepOfPublishingProcess:4,editedByType:null,kuFlag:!1,biosketch:"Dr. Tang-Chuan Wang is an excellent otolaryngologist-head and neck surgeon in Taiwan; a research scholar of Harvard Medical School and University of Iowa Hospitals. He worked in the Hospital of the University of Pennsylvania, Boston Children's Hospital, and Massachusetts Eye and Ear. Due to his contribution to biomedical engineering, he was invited into the executive committee of HIWIN-CMU Joint R & D Center in Taiwan.",coeditorOneBiosketch:null,coeditorTwoBiosketch:null,coeditorThreeBiosketch:null,coeditorFourBiosketch:null,coeditorFiveBiosketch:null,editors:[{id:"201262",title:"Dr.",name:"Tang-Chuan",middleName:null,surname:"Wang",slug:"tang-chuan-wang",fullName:"Tang-Chuan Wang",profilePictureURL:"https://mts.intechopen.com/storage/users/201262/images/system/201262.gif",biography:'Dr. Tang-Chuan Wang is an excellent otolaryngologist – head and neck surgeon in Taiwan. He is also a research scholar of Harvard Medical School and University of Iowa Hospitals. During his substantial experience, he worked in Hospital of the University of Pennsylvania, Boston Children\'s Hospital and Massachusetts Eye and Ear. Besides, he is not only working hard on clinical & basic medicine but also launching out into public health in Taiwan. In recent years, he devotes himself to innovation. He always says that "in theoretical or practical aspects, no innovation is a step backward". 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From chapter submission and review, to approval and revision, copyediting and design, until final publication, I work closely with authors and editors to ensure a simple and easy publishing process. I maintain constant and effective communication with authors, editors and reviewers, which allows for a level of personal support that enables contributors to fully commit and concentrate on the chapters they are writing, editing, or reviewing. I assist authors in the preparation of their full chapter submissions and track important deadlines and ensure they are met. I help to coordinate internal processes such as linguistic review, and monitor the technical aspects of the process. As an ASM I am also involved in the acquisition of editors. 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Venkateswarlu",coverURL:"https://cdn.intechopen.com/books/images_new/371.jpg",editedByType:"Edited by",editors:[{id:"58592",title:"Dr.",name:"Arun",surname:"Shanker",slug:"arun-shanker",fullName:"Arun Shanker"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}}]},chapter:{item:{type:"chapter",id:"70295",title:"Periodic Solution of Nonlinear Conservative Systems",doi:"10.5772/intechopen.90282",slug:"periodic-solution-of-nonlinear-conservative-systems",body:'Conservative systems can be defined as oscillating or vibrating systems in which the total energy content of the system remains constant. In order words, the total energy in the system is conserved. Ideally, such a system will continue to be in periodic oscillatory motion ad infinitum because the energy content of the system does not diminish due to the absence of dissipative force or increase due to additional energy input. However, for real cases where dissipative mechanisms such as friction or viscous damping cannot be completely eliminated, a conservative system can be thought of as one in which the energy dissipated is negligible during the time range under consideration. For example, the first few seconds of the oscillation of a simple pendulum may be considered conservative since the effect of air friction is negligible, but in the long run, the initial energy content is gradually dissipated until the pendulum comes to a halt. Other examples of practical conservative systems include mass-spring oscillator, structural elements (i.e., beams, plates, and shells), slider-crank mechanism [1], human eardrum [2], relativistic oscillator [3], planetary orbits around the sun [3], and current-carrying conductor in the electric field of an infinite rod [4]. Hence, a large number of oscillating physical systems can be studied as conservative systems.
At any point in time, the energy of a conservative system is composed of kinetic (
Exact analytical solutions for the nonlinear vibration models of conservative systems can be derived only in very few situations, and the solutions are usually derived in terms of special functions. Alternatively, highly accurate numerical solutions can be obtained for the nonlinear vibration model of any conservative system. However, as it is well recognized among the nonlinear science community, numerical solutions often have the limitations of lack of physical insight and convergence issues. Furthermore, there is the possibility of obtaining inaccurate convergent solutions for a nonlinear ordinary differential equation (ODE) [6], thus necessitating the independent verification of the convergent numerical solution by another numerical or analytical method. These limitations have driven the search for approximate analytical schemes capable of providing periodic solutions to nonlinear conservative oscillators. It can be rightly concluded that this search has been very fruitful considering the many approximate analytical schemes that now appear in the nonlinear science literature. The purpose of this section is to provide a brief survey of some of the notable achievements in the development of approximate analytical schemes for the periodic solution of nonlinear conservative oscillators. It should be noted that an approximate analytic method for nonlinear oscillators is considered adequate if it gives accurate predictions for the frequency-amplitude response and the oscillation history as well [7, 8].
Approximate analytical techniques to solve the nonlinear ODE governing the oscillations of a conservative system have been formulated for at least 100 years and can be classified as perturbation and nonperturbation methods. The first attempts were based on perturbation theory and are referred to as classical perturbation methods. The perturbation methods are formulated based on the concept that an unknown nonlinear system can be studied by introducing a small disturbance to a known linear system in equilibrium. For this reason, the classical perturbation methods (see Nayfeh [9] for a comprehensive treatment of classical perturbation methods) depend on the assumption of a small parameter. The problem with the small parameter assumption is that it has a small range of validity and only produces reliable solutions for cases of small-amplitude oscillations and weak nonlinearity. Nevertheless, the classical perturbation methods are still very relevant today for introducing and investigating various nonlinear concepts.
More recently, in the last four decades, a number of approximate analytical schemes have been proposed. Most of these recent schemes are nonperturbation methods, but some recent perturbation methods that attempt to improve on their classical counterparts have been formulated too. The recent perturbation methods include
In contrast to the perturbation methods, the nonperturbation methods do not use any small or artificial parameter. Examples include Adomian decomposition method [29], Homotopy analysis method [30], Variational iteration method [31], Energy balance method [2] and its modifications [32, 33, 34], He Chengtian’s interpolation method [27] also called max-min approach, amplitude-frequency formulation [35], Hamiltonian approach [36], global error minimization method [37], Harmonic balance method [4] and its modifications [38, 39, 40, 41, 42], cubication methods [43, 44, 45, 46], variational methods [47, 48, 49], differential transform method [50], and continuous piecewise linearization method [8]. Nonperturbation methods also have various limitations. For instance, a study [51] showed that the Adomian decomposition method does not converge to the correct solution in some cases, and the study proposed an optimal convergence acceleration parameter to deal with this issue. Also, methods that rely on a simple harmonic approximation of the oscillation history, such as the energy balance method, amplitude-frequency formulation, Hamiltonian approach, max-min approach, and variational methods, can only give reliable estimate of the frequency-amplitude response. Sometimes these methods perform poorly in predicting the oscillation history during large-amplitude and/or strong nonlinear vibrations. Other methods that usually require high-order approximations, such as Adomian decomposition method, harmonic balance method, and variational iteration method, present algebraic complexities in their determination of higher order solutions and may be impractical for oscillators with highly complex nonlinearities such as the slider-crank mechanism [1] and the bifilar pendulum [52]. Furthermore, it has been observed that higher order estimates do not always improve the solution of the oscillation history [27]. Finally, some nonperturbation methods are heuristic in nature (e.g., energy balance method and variational methods) and require experience to choose the initial trial function and the condition for error minimization [8].
The continuous piecewise linearization method (CPLM) is an iterative analytic algorithm that was formulated to overcome most of the above challenges by providing simple and accurate solutions for the oscillation history and frequency-amplitude response of Duffing-type oscillators. In another study [53], the CPLM was modified in order to generalize it so that it can handle more complicated nonlinear conservative oscillators. Interestingly, the CPLM does not require higher order approximations or any small, artificial, or embedded parameter. Also, the algorithm is inherently stable, straightforward, and based on closed-form analytical approximations. This chapter is aimed at presenting the generalized CPLM algorithm as a veritable approach for accurate periodic solution of Hamiltonian and non-Hamiltonian conservative oscillators with complex nonlinearity. As is shown later, the CPLM retains the same simplicity in its implementation irrespective of the complexity of the nonlinear conservative system.
The main idea of the CPLM is based on the piecewise linearization of the nonlinear restoring force of a conservative oscillator. The linearization technique used by the CPLM was first applied in another algorithm [6, 54] for the solution of half-space impact models called force indentation linearization method (FILM). The FILM has been applied to formulate theoretical solutions for rigid body motions and local compliance response during nonlinear elastoplastic impact of dissimilar spheres [55]. However, because the FILM is limited to impact excitations that are nonoscillatory, it cannot be applied to nonlinear conservative oscillators. Hence, the CPLM applies the piecewise linearization technique of the FILM to provide a periodic solution for nonlinear conservative oscillators.
Essentially, the linearization technique of the CPLM involves
Discretization of the restoring force of a typical nonlinear oscillator.
The standard form for representation of a nonlinear conservative oscillator moving in the
where
where
Since
Eq. (10) is a nonhomogeneous linear ODE and its solution depends on whether the sign is positive or negative.
If
where
For the oscillation stage that moves from
The time at the end of each discretization is
If
where
The sign before the square root in Eq. (8) is negative for the oscillation stage that moves from
Therefore,
In very rare situations, we may have
where
From the above presentation of the CPLM formulation, it is obvious that the CPLM algorithm is simple and can be implemented by undergrads without difficulty.
The CPLM is inherently stable and does not have convergence issues [8].
For few discretization, say
When dealing with conservative oscillators with odd nonlinearity, which are symmetrical about the origin, discretization of the restoring force is only required for 0 to
The CPLM algorithm retains the same simplicity in implementation irrespective of the complexity of the restoring force. Only the
The CPLM relies on the explicit expression of restoring force, which means that the model for the oscillator must be expressed in the form of Eq. (1). For Hamiltonian systems, the oscillator model is formulated naturally in the form of Eq. (1). For non-Hamiltonian conservative systems, the oscillator model is not formulated naturally in the form of Eq. (1). Therefore, there is a need to transform the model of non-Hamiltonian conservative systems into the form of Eq. (1) before applying the CPLM algorithm. Fortunately, this transformation only requires a simple algebraic manipulation as demonstrated in Section 4.
The phase equation gives the relationship between the state variables (displacement and velocity) and can be derived exactly in closed-form for all conservative systems. The CPLM extends this bilateral relationship into a tripartite one by finding the value of the corresponding independent variable (i.e., time) that matches the values of the state variables in each discretization.
For few discretization, say
The usual initial conditions investigated for nonlinear conservative oscillators are nonzero displacement and zero velocity. However, the CPLM algorithm can comfortably handle nonzero initial conditions for displacement and velocity.
The simple pendulum is arguably the most investigated physical system and provides very interesting insights into nonlinear phenomena. Butikov [57] calls it “an antique but evergreen physical model.” The undamped oscillation of a simple pendulum is a Hamiltonian system governed by the well-known nonlinear ODE as shown:
where
Where
From Eq. (13), the restoring force for the pendulum is
(a) Frequency-amplitude response of the simple pendulum for 0°<A<180°. (b) CPLM error analysis.
Oscillation history of the simple pendulum for (a) A=45° and (b) A=135°. CPLM estimate—Lines; exact solution—Markers.
Consider the motion of a satellite along a path that is equidistant from two identical massive stars with mutually interacting gravitational fields. If the distance between the two stars is
where
Restoring force for Eq. (18): M=105kg; d=100m; A=500.
The main CPLM constant is
The input values used for simulation are
(a) Frequency-amplitude response for satellite. (b) CPLM error analysis.
Oscillation history of satellite for (a) A=50 and (b) A=1500. CPLM estimate—Lines; exact solution—Markers.
On the other hand, Figure 6 shows the oscillation history of the satellite during small-amplitude (
An interesting oscillator that has been the subject of several studies [60, 61, 62, 63, 64, 65] is the Hamiltonian oscillator with odd fractional nonlinearity. For the purpose of the present investigation, we consider an oscillator that is characterized by a general fractional nonlinearity as follows [65]:
where the restoring force,
while the exact oscillation history was obtained by the numerical solution of Eq. (18) using the NDSolve function in Mathematica™. The CPLM solution demonstrates an excellent agreement with the exact solution.
(a) Frequency-amplitude response for u1/3 oscillator. (b) CPLM error analysis.
Oscillation history of u1/3 oscillator for (a) A=0.01 and (b) A=10. CPLM estimate—lines; exact solution—markers.
Figure 7 compares the CPLM estimates of the frequency-amplitude response with Eq. (19), and the maximum error of the CPLM solution is 0.22% for
The non-Hamiltonian conservative systems are generally more complex and difficult to solve compared with the Hamiltonian systems. In order to demonstrate the application of the CPLM algorithm to non-Hamiltonian conservative systems, we consider the motion of a particle on a rotating parabola. This system consists of a frictionless mass sliding on a vertical parabolic wire described by
Hence, the Lagrangian is:
(a) Schematic of particle on a rotating parabola. (b) Restoring force when q=1.0; Λ=10.0; A=2.0.
Next, we substitute Eq. (21) into the Euler–Lagrange equation to derive the equation of motion. The Euler–Lagrange equation can be written as:
Therefore, using Eqs. (21) and (22), the motion of a particle on a rotating parabola is governed by:
where
To solve Eq. (23) using the CPLM, it must be recast in the form of Eq. (1). The conservation of energy for Eq. (23) is given as [4]:
where
Therefore, the restoring force is
where
A comparison of CPLM frequency estimate and the exact frequency is shown in Figure 10, while the oscillation histories for
(a) Frequency-amplitude response for particle on a rotating parabola. (b) CPLM error analysis.
Oscillation history of particle on a rotating parabola for (a) A=0.50 and (b) A=2.0. CPLM estimate—Lines; exact solution—Markers.
Conservative oscillators generally exhibit nonlinear response, and they form a large class of natural and artificially vibrating systems. Hence, the study of the dynamic response of nonlinear conservative systems is important for understanding many physical phenomena and the design of systems. The main challenge in the theoretical analysis of nonlinear conservative systems is that exact solutions are normally not available except for a few special cases where exact solutions are derived in terms of special functions.
To date, many approximate analytical methods have been formulated for the periodic solution of nonlinear conservative oscillators. This chapter provides a brief survey of the recent advances in the formulation of approximate analytical schemes and then introduced a recent approximate analytical algorithm called the continuous piecewise linearization method. The CPLM has been shown to overcome the challenges of solution accuracy and simplicity usually encountered in using most of the existing approximate analytical methods. The CPLM combines major desirable features of solution schemes such as inherent stability, accuracy, and simplicity. It is simple enough to be introduced at the undergraduate level and is capable of handling conservative oscillators with very complex nonlinearity. Conservative systems of broad interest were used to demonstrate the wide applicability of the CPLM algorithm. As demonstrated above, an accuracy of less than 1.0% relative error can be achieved for most oscillators using few discretizations, say
The authors are grateful to the Vice-Chancellor of the University of Port Harcourt, Prof. Ndowa E.S. Lale and the Dean of Engineering, Prof. Ogbonna Joel, for approving the publication grant for this chapter. The OAPF was funded by the African Centre of Excellence in Oilfield Chemicals Research (ACE-FOR), University of Port Harcourt.
Conservative systems can be defined as oscillating or vibrating systems in which the total energy content of the system remains constant. In order words, the total energy in the system is conserved. Ideally, such a system will continue to be in periodic oscillatory motion ad infinitum because the energy content of the system does not diminish due to the absence of dissipative force or increase due to additional energy input. However, for real cases where dissipative mechanisms such as friction or viscous damping cannot be completely eliminated, a conservative system can be thought of as one in which the energy dissipated is negligible during the time range under consideration. For example, the first few seconds of the oscillation of a simple pendulum may be considered conservative since the effect of air friction is negligible, but in the long run, the initial energy content is gradually dissipated until the pendulum comes to a halt. Other examples of practical conservative systems include mass-spring oscillator, structural elements (i.e., beams, plates, and shells), slider-crank mechanism [1], human eardrum [2], relativistic oscillator [3], planetary orbits around the sun [3], and current-carrying conductor in the electric field of an infinite rod [4]. Hence, a large number of oscillating physical systems can be studied as conservative systems.
At any point in time, the energy of a conservative system is composed of kinetic (
Exact analytical solutions for the nonlinear vibration models of conservative systems can be derived only in very few situations, and the solutions are usually derived in terms of special functions. Alternatively, highly accurate numerical solutions can be obtained for the nonlinear vibration model of any conservative system. However, as it is well recognized among the nonlinear science community, numerical solutions often have the limitations of lack of physical insight and convergence issues. Furthermore, there is the possibility of obtaining inaccurate convergent solutions for a nonlinear ordinary differential equation (ODE) [6], thus necessitating the independent verification of the convergent numerical solution by another numerical or analytical method. These limitations have driven the search for approximate analytical schemes capable of providing periodic solutions to nonlinear conservative oscillators. It can be rightly concluded that this search has been very fruitful considering the many approximate analytical schemes that now appear in the nonlinear science literature. The purpose of this section is to provide a brief survey of some of the notable achievements in the development of approximate analytical schemes for the periodic solution of nonlinear conservative oscillators. It should be noted that an approximate analytic method for nonlinear oscillators is considered adequate if it gives accurate predictions for the frequency-amplitude response and the oscillation history as well [7, 8].
Approximate analytical techniques to solve the nonlinear ODE governing the oscillations of a conservative system have been formulated for at least 100 years and can be classified as perturbation and nonperturbation methods. The first attempts were based on perturbation theory and are referred to as classical perturbation methods. The perturbation methods are formulated based on the concept that an unknown nonlinear system can be studied by introducing a small disturbance to a known linear system in equilibrium. For this reason, the classical perturbation methods (see Nayfeh [9] for a comprehensive treatment of classical perturbation methods) depend on the assumption of a small parameter. The problem with the small parameter assumption is that it has a small range of validity and only produces reliable solutions for cases of small-amplitude oscillations and weak nonlinearity. Nevertheless, the classical perturbation methods are still very relevant today for introducing and investigating various nonlinear concepts.
More recently, in the last four decades, a number of approximate analytical schemes have been proposed. Most of these recent schemes are nonperturbation methods, but some recent perturbation methods that attempt to improve on their classical counterparts have been formulated too. The recent perturbation methods include
In contrast to the perturbation methods, the nonperturbation methods do not use any small or artificial parameter. Examples include Adomian decomposition method [29], Homotopy analysis method [30], Variational iteration method [31], Energy balance method [2] and its modifications [32, 33, 34], He Chengtian’s interpolation method [27] also called max-min approach, amplitude-frequency formulation [35], Hamiltonian approach [36], global error minimization method [37], Harmonic balance method [4] and its modifications [38, 39, 40, 41, 42], cubication methods [43, 44, 45, 46], variational methods [47, 48, 49], differential transform method [50], and continuous piecewise linearization method [8]. Nonperturbation methods also have various limitations. For instance, a study [51] showed that the Adomian decomposition method does not converge to the correct solution in some cases, and the study proposed an optimal convergence acceleration parameter to deal with this issue. Also, methods that rely on a simple harmonic approximation of the oscillation history, such as the energy balance method, amplitude-frequency formulation, Hamiltonian approach, max-min approach, and variational methods, can only give reliable estimate of the frequency-amplitude response. Sometimes these methods perform poorly in predicting the oscillation history during large-amplitude and/or strong nonlinear vibrations. Other methods that usually require high-order approximations, such as Adomian decomposition method, harmonic balance method, and variational iteration method, present algebraic complexities in their determination of higher order solutions and may be impractical for oscillators with highly complex nonlinearities such as the slider-crank mechanism [1] and the bifilar pendulum [52]. Furthermore, it has been observed that higher order estimates do not always improve the solution of the oscillation history [27]. Finally, some nonperturbation methods are heuristic in nature (e.g., energy balance method and variational methods) and require experience to choose the initial trial function and the condition for error minimization [8].
The continuous piecewise linearization method (CPLM) is an iterative analytic algorithm that was formulated to overcome most of the above challenges by providing simple and accurate solutions for the oscillation history and frequency-amplitude response of Duffing-type oscillators. In another study [53], the CPLM was modified in order to generalize it so that it can handle more complicated nonlinear conservative oscillators. Interestingly, the CPLM does not require higher order approximations or any small, artificial, or embedded parameter. Also, the algorithm is inherently stable, straightforward, and based on closed-form analytical approximations. This chapter is aimed at presenting the generalized CPLM algorithm as a veritable approach for accurate periodic solution of Hamiltonian and non-Hamiltonian conservative oscillators with complex nonlinearity. As is shown later, the CPLM retains the same simplicity in its implementation irrespective of the complexity of the nonlinear conservative system.
The main idea of the CPLM is based on the piecewise linearization of the nonlinear restoring force of a conservative oscillator. The linearization technique used by the CPLM was first applied in another algorithm [6, 54] for the solution of half-space impact models called force indentation linearization method (FILM). The FILM has been applied to formulate theoretical solutions for rigid body motions and local compliance response during nonlinear elastoplastic impact of dissimilar spheres [55]. However, because the FILM is limited to impact excitations that are nonoscillatory, it cannot be applied to nonlinear conservative oscillators. Hence, the CPLM applies the piecewise linearization technique of the FILM to provide a periodic solution for nonlinear conservative oscillators.
Essentially, the linearization technique of the CPLM involves
Discretization of the restoring force of a typical nonlinear oscillator.
The standard form for representation of a nonlinear conservative oscillator moving in the
where
where
Since
Eq. (10) is a nonhomogeneous linear ODE and its solution depends on whether the sign is positive or negative.
If
where
For the oscillation stage that moves from
The time at the end of each discretization is
If
where
The sign before the square root in Eq. (8) is negative for the oscillation stage that moves from
Therefore,
In very rare situations, we may have
where
From the above presentation of the CPLM formulation, it is obvious that the CPLM algorithm is simple and can be implemented by undergrads without difficulty.
The CPLM is inherently stable and does not have convergence issues [8].
For few discretization, say
When dealing with conservative oscillators with odd nonlinearity, which are symmetrical about the origin, discretization of the restoring force is only required for 0 to
The CPLM algorithm retains the same simplicity in implementation irrespective of the complexity of the restoring force. Only the
The CPLM relies on the explicit expression of restoring force, which means that the model for the oscillator must be expressed in the form of Eq. (1). For Hamiltonian systems, the oscillator model is formulated naturally in the form of Eq. (1). For non-Hamiltonian conservative systems, the oscillator model is not formulated naturally in the form of Eq. (1). Therefore, there is a need to transform the model of non-Hamiltonian conservative systems into the form of Eq. (1) before applying the CPLM algorithm. Fortunately, this transformation only requires a simple algebraic manipulation as demonstrated in Section 4.
The phase equation gives the relationship between the state variables (displacement and velocity) and can be derived exactly in closed-form for all conservative systems. The CPLM extends this bilateral relationship into a tripartite one by finding the value of the corresponding independent variable (i.e., time) that matches the values of the state variables in each discretization.
For few discretization, say
The usual initial conditions investigated for nonlinear conservative oscillators are nonzero displacement and zero velocity. However, the CPLM algorithm can comfortably handle nonzero initial conditions for displacement and velocity.
The simple pendulum is arguably the most investigated physical system and provides very interesting insights into nonlinear phenomena. Butikov [57] calls it “an antique but evergreen physical model.” The undamped oscillation of a simple pendulum is a Hamiltonian system governed by the well-known nonlinear ODE as shown:
where
Where
From Eq. (13), the restoring force for the pendulum is
(a) Frequency-amplitude response of the simple pendulum for 0°<A<180°. (b) CPLM error analysis.
Oscillation history of the simple pendulum for (a) A=45° and (b) A=135°. CPLM estimate—Lines; exact solution—Markers.
Consider the motion of a satellite along a path that is equidistant from two identical massive stars with mutually interacting gravitational fields. If the distance between the two stars is
where
Restoring force for Eq. (18): M=105kg; d=100m; A=500.
The main CPLM constant is
The input values used for simulation are
(a) Frequency-amplitude response for satellite. (b) CPLM error analysis.
Oscillation history of satellite for (a) A=50 and (b) A=1500. CPLM estimate—Lines; exact solution—Markers.
On the other hand, Figure 6 shows the oscillation history of the satellite during small-amplitude (
An interesting oscillator that has been the subject of several studies [60, 61, 62, 63, 64, 65] is the Hamiltonian oscillator with odd fractional nonlinearity. For the purpose of the present investigation, we consider an oscillator that is characterized by a general fractional nonlinearity as follows [65]:
where the restoring force,
while the exact oscillation history was obtained by the numerical solution of Eq. (18) using the NDSolve function in Mathematica™. The CPLM solution demonstrates an excellent agreement with the exact solution.
(a) Frequency-amplitude response for u1/3 oscillator. (b) CPLM error analysis.
Oscillation history of u1/3 oscillator for (a) A=0.01 and (b) A=10. CPLM estimate—lines; exact solution—markers.
Figure 7 compares the CPLM estimates of the frequency-amplitude response with Eq. (19), and the maximum error of the CPLM solution is 0.22% for
The non-Hamiltonian conservative systems are generally more complex and difficult to solve compared with the Hamiltonian systems. In order to demonstrate the application of the CPLM algorithm to non-Hamiltonian conservative systems, we consider the motion of a particle on a rotating parabola. This system consists of a frictionless mass sliding on a vertical parabolic wire described by
Hence, the Lagrangian is:
(a) Schematic of particle on a rotating parabola. (b) Restoring force when q=1.0; Λ=10.0; A=2.0.
Next, we substitute Eq. (21) into the Euler–Lagrange equation to derive the equation of motion. The Euler–Lagrange equation can be written as:
Therefore, using Eqs. (21) and (22), the motion of a particle on a rotating parabola is governed by:
where
To solve Eq. (23) using the CPLM, it must be recast in the form of Eq. (1). The conservation of energy for Eq. (23) is given as [4]:
where
Therefore, the restoring force is
where
A comparison of CPLM frequency estimate and the exact frequency is shown in Figure 10, while the oscillation histories for
(a) Frequency-amplitude response for particle on a rotating parabola. (b) CPLM error analysis.
Oscillation history of particle on a rotating parabola for (a) A=0.50 and (b) A=2.0. CPLM estimate—Lines; exact solution—Markers.
Conservative oscillators generally exhibit nonlinear response, and they form a large class of natural and artificially vibrating systems. Hence, the study of the dynamic response of nonlinear conservative systems is important for understanding many physical phenomena and the design of systems. The main challenge in the theoretical analysis of nonlinear conservative systems is that exact solutions are normally not available except for a few special cases where exact solutions are derived in terms of special functions.
To date, many approximate analytical methods have been formulated for the periodic solution of nonlinear conservative oscillators. This chapter provides a brief survey of the recent advances in the formulation of approximate analytical schemes and then introduced a recent approximate analytical algorithm called the continuous piecewise linearization method. The CPLM has been shown to overcome the challenges of solution accuracy and simplicity usually encountered in using most of the existing approximate analytical methods. The CPLM combines major desirable features of solution schemes such as inherent stability, accuracy, and simplicity. It is simple enough to be introduced at the undergraduate level and is capable of handling conservative oscillators with very complex nonlinearity. Conservative systems of broad interest were used to demonstrate the wide applicability of the CPLM algorithm. As demonstrated above, an accuracy of less than 1.0% relative error can be achieved for most oscillators using few discretizations, say
The authors are grateful to the Vice-Chancellor of the University of Port Harcourt, Prof. Ndowa E.S. Lale and the Dean of Engineering, Prof. Ogbonna Joel, for approving the publication grant for this chapter. The OAPF was funded by the African Centre of Excellence in Oilfield Chemicals Research (ACE-FOR), University of Port Harcourt.
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