Types of ibuprofen available.
\r\n\t
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Diagnoses of upper extremity MSDs include muscle strain injuries, carpal and cubital tunnel syndromes, muscle myalgia/hyperalgesia, dorsal wrist tendinosis, lateral and medial epicondylopathies, rotator cuff tendinopathies, and more. These disorders often occur as a consequence of daily activities (both occupational and not), sports or military activities, and are a leading cause of pain and physical disability [1–4]. Some cases become so severe that simple personal tasks, such as buttoning a shirt, become difficult to impossible. Acute trauma may be a causal factor in some WMSDs. Yet, many result from cumulative small amplitude forces occurring with overtraining, overexertion, repetitive activities, forceful actions, and prolonged static positioning [5–8].Prevention is hampered by many problems [9, 10]. There remains a call for effective treatments for these often debilitating disorders [9, 11].
The first line of treatment for workers in pain usually entails a prescription of non
Ibuprofen was introduced to the US market as a prescription drug to treat arthritic conditions in 1974, and subsequently became available over the counter in the United States in 1984. Despite its relatively short history as an over-the-counter medicine, it has quickly achieved popularity as a treatment for musculoskeletal and peripheral nerve pain, capturing up to a third of the over-the-counter analgesics market of the US by 2002. According to background information supplied by Wyeth Pharmaceuticals (a manufacturer of Advil, a brand name ibuprofen), this occurred principally because of its strong gastrointestinal (GI) safety profile that ibuprofen was approved for over-the-counter use. There are a wide variety of ibuprofen drugs available on the market as indicated in Table 1. Tablet, caplet, injectable, and topical forms are available. Negative side effects and major concerns will be discussed later in this chapter, although it should be noted that topical ibuprofen formulas are absorbed less into blood stream than oral forms, avoiding several side effects. However, as topical NSAID drugs are not systemic, they will not reduce inflammatory responses other than at the site of application.
Types of ibuprofen available.
Ibuprofen works by inhibiting both the constitutive cyclooxygenase (COX)-1 and the more inducible COX-2 enzyme. These enzymes catalyze the generation of prostanoids (prostaglandins PGE2 and PGF2a), prostacyclins, and thromboxanes [25, 26]. Inhibition of these enzymes by ibuprofen prevents the conversion of arachidonic acid to prostaglandin H2, and in doing so blocks the prostaglandin-signaling pathway. Prostaglandins play an important role in pain and inflammatory signaling, as well as have roles in maintaining kidney function (mainly by regulating blood flow in the glomerular capsule) and the gut mucosa, and cardiovascular physiological processes [26].
\nA steady dose of ibuprofen is considered necessary to attenuate the increase in inflammation, rather than just analgesic. The dose used should be lower than the maximum limit for gastrointestinal toxicity. Those suggested maximum limits are indicated in Table 1.
Several animal models have been developed to study WMSDs and have shown that repetitive hand activities induce sensorimotor dysfunction [27–33]. A model developed in our laboratory is a unique operant rat model of voluntary reaching and grasping (Figure 1; [7, 34]). Using this model, we are able to examine the effects of voluntary performance of repetitive low or high demand tasks on sensorimotor performance and musculoskeletal tissues [7, 30, 35]. This model is nonsurgical and involves performance of voluntary repetitive tasks to induce mechanical loading of forearm tissues. Specifically, adult rats are required to voluntarily and repetitively reach for, grasp, and isometrically pull a handle with one forelimb to obtain a food reward at various reach rates and force levels determined from studies on risk exposure for WMSDs to humans [7, 34]. Additionally, several functional outcomes are tested that are similar to those tested in patients, including forepaw (hand) sensitivity, grip strength, and median nerve conduction velocity.
Rat performing HRHF repetitive reaching task. (A) Rat awaits auditory stimulus with snout in portal. (B and C) Rat reaches for force handle with right forepaw; left forepaw used for postural support. (D) Closer view, rat grasps and isometrically pulls force handle attached to force transducer (FT), until predetermined force threshold is reached and held for at least 50 ms. (E) Rat retrieves foot pellet reward by mouth from food trough.
Using this rat model, we have observed early exposure-dependent changes (duration and task level) in inflammatory responses in the form of increased macrophages and inflammatory cytokines in soft tissues involved in performing the repetitive task [7, 30, 32, 35, 36]. The greatest responses were observed in rats performing a high-repetition high-force (HRHF) task for 6–12 weeks, compared to lower demand tasks. Therefore, we picked this HRHF task regimen for experiments in which we tested the effectiveness of ibuprofen.
We hypothesized that an underlying inflammatory mechanism is driving many of the sensorimotor declines, as are inflammation-linked fibrotic and degenerative/degradative tissue changes [37]. We explored this hypothesis by treating rats with systemic ibuprofen (i.e., oral) at anti-inflammatory doses. The design of these experiments is shown in Figure 2, and included normal controls (termed NC rats) and food-restricted-only controls (termed FRC rats). Rats were food-restricted to body weights of 5% less than age-matched normal controls to motivate them to work. Subsets of food-restricted rats were trained to high-force levels to determine the effects of training (10 min/day, 5 days/week, for 5 weeks) in which they learn to pull at high-force levels (1.25 Newton’s which is approximately 60% of their maximum voluntary force) [7]. The trained-only rats are termed TRHF rats. Subsets of TRHF rats went on to perform a high-repetition high-force task regimen for 2 h/day, 3 days/week for up to 12 weeks. Task requirements were a reach rate of 8 reaches/min and a target force of 60 ± 5% of their mean maximum pulling force. HRHF rats had to grasp the force lever bar and exert an isometric pull at the target level for at least 50 ms to receive a food reward. Half of each group was administered ibuprofen (Children’s Motrin Grape Flavored, Johnson & Johnson) in drinking water daily (a dose of 45 mg/kg body weight was used). This dose was lower than the maximum limit for gastrointestinal toxicity in rats, yet effective in reducing chronic inflammation [38]. The results of these experiments and the effectiveness (or lack thereof) are discussed below.
Experimental design. (A) Food restriction began after a 1-week period of daily handling. All rats but normal control (NC ± ibuprofen treatment) were food restricted to 5% less than weights of age-matched NC rats. NC and food-restricted control (FRC) rats rested until euthanasia at matched time points as HRHF rats. Trained and task rats underwent a 5-week training period (rats reached the HRHF level by last week of training). These trained-only rats (TRHF) were euthanized after training. Task rats performed a high-repetition high-force (HRHF) task for 12 weeks. NC+IBU and FRC+IBU rats received daily ibuprofen (IBU) treatment of 45 mg/kg of body weight in drinking water in the final 8 weeks, as did HRHF+IBU rats (arrow indicates the onset of ibuprofen treatment). TRHF+IBU rats received ibuprofen treatment prophylactically during training. The number of rats per group is shown at the far right.
Some mechanisms examined to date in our rat model include task-induced tissue injury, inflammation, and fibrosis, each of which contributed to declines in grip strength by producing discomfort or affecting biomechanical strength. Evidence of tissue injury was paralleled by inflammatory responses, such as increased pro-inflammatory cytokines in flexor digitorum muscles and tendons [30–32, 39], and increased macrophages in the median nerve at the level of the wrist (Figure 3A, B). Elevated levels of key pro-inflammatory cytokines, IL-1beta, and TNF-alpha were also observed in serum of rats that had performed a HRHF task for 12 weeks (12-week HRHF rats) (Figure 3D, E).
Median nerve inflammatory and fibrotic responses as well as systemic cytokine responses. (A) Photomicrographs showing increased activated macrophages in the median nerve of HRHF rats (detected immunohistochemically and denoted with arrowheads), and width of epineurial connective tissues (CT; (double arrows)) around the median nerve (N) at the wrist level. Eosin counterstain. (B) Mean number of activated macrophage in the median nerve decreased with ibuprofen treatment provided daily in task weeks 5–12. (C) Nerve conduction velocity (NCV) in meters/second (m/sec) declined in HRHF rats and was rescued by ibuprofen treatment that began after task week 4 (arrow) and that continued though task week 12. (D and E) IL-1-beta and TNF-alpha increased systemically (in serum) in untreated HRHF rats. These increases were ameliorated with 8 weeks of ibuprofen treatment. Symbols: *
Treatment of rats performing a high-repetition high-force task with oral ibuprofen in weeks 5–12 of a 12-week task regimen significantly reduced macrophage numbers and inflammatory cytokines in tissues and serum (Figure 3A, B, D). Ibuprofen treatment also improved HRHF-induced declines in several voluntary work parameters, including reach rate, voluntary pulling force, and duration of voluntary performance (Figure 4A, B) [40, 41]. Similarly, the treatment of human subjects with ibuprofen before unaccustomed exercise improves muscle strength [42, 43]. The attenuation of voluntary reach abilities in HRHF+IBU rats in parallel with reduced numbers of macrophages in the median nerve (Figure 3A) [40] indicates that a task-induced neuralgia is contributing to voluntary motor declines seen in Figure 4A and B.
Voluntary and reflexive motor abilities. (A) Mean voluntary pulling force on the handle (percent of maximum pulling force) in grams. Across weeks of task performance, the mean voluntary pulling force was lower than target levels in untreated HRHF rats, yet met target levels in ibuprofen-treated rats (ibuprofen was provided in task weeks 5–12, with onset indicated by arrow). (B) Across the weeks, the duration of voluntary task performance decreased in untreated HRHF rats. By contrast, the duration was near target levels in HRHF+IBU rats in weeks 9 and 12. (C) Grip strength (maximum reflexive grip strength in grams) in the preferred reach limb decreased in both groups, compared to baseline naïve levels. Ibuprofen treatment only partially rescued this nocifensive motor behavior. *
However, reflexive grip strength was not rescued in 9- and 12-week HRHF+IBU rats (Figure 4C) [41]. This type of nocifensive motor behavior has been termed muscle hyperalgesia [44] and is a type of chronic pain. We postulate that ibuprofen did not rescue reflexive grip strength declines because it did not prevent inflammation-associated changes in the central nervous system. We stained cervical regions of the spinal cord for pro-inflammatory cytokine IL-1-beta levels using immunohistochemical methods and found that both untreated HRHF and HRHF+IBU animals expressed this cytokine in neurons and some glial cells at roughly the same frequency and intensity (Figure 5A, B, D). This was in sharp contrast to IL-1-beta immunoexpression in spinal cords of normal control rats, which showed an almost absence of IL-1-beta immunoexpression (Figure 5C). We postulate that ibuprofen, or other anti-inflammatory drug, would have to be provided earlier than week 4 prior to the onset of pain behaviors in order to be fully effective. Future studies need to consider these negative central nervous system changes to successfully treat chronic pain behaviors in subjects with WMSDs.
Inflammatory cytokine (IL-1-beta) immunoexpression was increased in neurons of spinal cords of both HRHF and HRHF+IBU rats, compared to NC rats, indicative of central sensitization (
Muscles undergo repetitive strain-induced fibrosis. Stauber and colleagues have shown that repeated muscle strains at fast velocities resulted in fibrotic myopathy with increased collagen content, collagen cross-links, and non-contractile tissues [45–48]. Factors and mechanisms of repetitive strain-induced fibrosis are still under investigation. They appear to involve transforming growth factor beta-1 (TGFB-1) and connective tissue growth factor (CTGF), a key downstream mediator of TGFB-1’s effects on matrix protein production [49–53]. Strong links between mechanical loading and increased TGFB-1 and CTGF protein levels in muscles and tendons
CTGF production also appears to be regulated by pro-inflammatory cytokines, IL-1-beta, and TNF-alpha, which are also thought of as pro-fibrogenic cytokines [37, 56, 57]. Since we have observed that task-induced tissue inflammation precedes tissue fibrotic responses, including increased CTGF and collagen type 1 production [58–60], we next examined the effects of secondary ibuprofen treatment on fibrogenic processes in our rat model [40, 61]. In addition to successful reductions of tissue and serum inflammatory responses after ibuprofen treatment, we observed significant reductions in TGFB-1 and CTGF protein expression as well as collagen deposition in median nerves (Figure 3A) and flexor digitorum muscles of 6-week and 12-week HRHF+IBU rats (Figure 6) [40, 61]. These findings support an underlying inflammatory drive on at least some fibrogenic processes. This reduction in collagen deposition within and around tissue components of the upper extremity may also aid the return of function, such as the return of median nerve conduction velocity in median nerves of 12-week HRHF rats as shown in Figure 3C.
Fibrogenic protein levels (TGFB1, CTGF, and collagen type 1 (Col I)) were increased in forearm muscles of 6-week HRHF rats, increases that were reduced after a 2-week treatment with ibuprofen provided in task weeks 5 and 6. Cross sections of flexor digitorum muscle are shown. (A–C, G) TGFB1 staining was absent in muscles of normal control (NC) rats shown in panel A, high in muscles of untreated 6-week HRHF rats (visible as red staining at the edges of the myofibers in panel B), and reduced in muscles of 6-week HRHF rats treated with ibuprofen (panel C). (D–F, H) A small number of CTGF-immunostained cells (red in color) were present around myofibers in NC rats as shown in panel D, increased in muscles of untreated 6-week HRHF rats as shown in panel E, but reduced back to control levels in muscles of 6-week HRHF rats treated with ibuprofen as shown in panel F. (G&F) Quantification of percentage area of muscle with TGFB1 and CTGF staining. *
Joint degeneration may occur for a number of reasons including joint trauma from increased repetition of joint loading, high impact joint loading, increased inflammatory processes (e.g., autoimmune), or pathological metabolic processes [62–64]. Radiocarpal and intercarpal joints of the wrist and hand, respectively, can show signs of increased incidence of hand osteoarthritis in individuals involved in intense (defined as long duration, high repetition, and/or high force) occupationally related physical activities [65–67]. A high incidence of radiographic of hand osteoarthritis has been identified in middle-aged female dentists and teachers [66, 67]. Several studies report that increasing radiographic severity of hand osteoarthritis is associated with reduced hand function and increased pain [66, 68, 69]. Therefore, the impact of hand osteoarthritis is considerable [68, 69].
\nAfter 12 weeks of performing the HRHF task, untreated task animals demonstrated evidence of joint inflammation (loss of proteoglycan staining as shown in Figure 7B as compared to controls in Figure 7A) [70]. This loss of proteoglycan staining in untreated 12-week HRHF rats is captured in the form of elevated Mankin histopathological scores (Figure 7E), a scoring system that also reflects a development of pannus and apoptotic cells in the joint cartilage. Each of these changes was indicative of task-induced joint degeneration. Serum biomarker testing revealed increased levels of a serum biomarker of collagen degradation, C1,2C (a marker of collagen type 1 and 2 degradation fragments produced by collagenase cleavage of type II collagen) in untreated 12-week HRHF rats [70]. Increased activated macrophages, cyclooxygenase immunopositive cells, and inflammatory cytokine levels were detected in the distal radius, ulna, and carpal bones (the latter shown in Figure 7F, G), supporting our hypothesis of an underlying inflammatory mechanism.
HRHF-induced degeneration of radiocarpal joint cartilage was attenuated by ibuprofen treatment. (A–D) Distal radii articular cartilage stained with safranin O and fast green from (A) untreated TRHF rat, (B) TRHF+IBU rat (trained controls receiving ibuprofen treatment prophylactically), (D) HRHF rats that performed the task for 12 weeks show dramatically reduced proteoglycan staining in the articular cartilage (red-pink safranin O staining), and (E) HRHF+IBU rats that performed the task for 12 weeks while receiving ibuprofen treatment (45 mg/kg body wt, daily, oral) in the last 8 weeks. (E) Histopathological Mankin scores for distal radius articular cartilage of the reach limb in TRHF, TR+IBU, HRHF, and HRHF+IBU rats. (F&G) Cytokine concentrations in wrist joint (distal radius, ulna, and carpal bones) and in diaphysis of the radius and ulna bones, tested using ELISA. Levels of (F) IL-1-alpha and (G) IL-1-beta are shown for each group. *
Eight weeks of ibuprofen administration reduced all of these changes, despite continued task performance (Figure 7). This latter finding indicates that the joint degenerative changes observed were a consequence of the inflammatory response induced by this high-repetition high-force task that was 12 weeks in duration. Each of these changes were attenuated by ibuprofen treatment, suggesting that such treatment is chondroprotective, at least during the early phases of cumulative loading-induced inflammation and degeneration in hand and wrist joints.
Cyclical loading and high-force loads are known to affect bone quality [71–74]. However, only a few studies have examined changes occurring in upper extremity bones as a consequence of prolonged performance of occupational tasks. Bone scan studies of patients with upper extremity MSDs show increased blood flow and pooling (suggestive of inflammation) in affected bones, although the sensitivity and accuracy of the results were variable across studies [75–77]. We found that the performance of a HRHF task for 12 weeks reduced trabecular bone (Figure 8A, B) and cortical bone thinning in the radius and ulna in untreated HRHF rats (Figure 8B) [39, 40]. Bone levels of IL-1-beta, an inflammatory cytokine known to stimulate osteoclastogenesis and activity [78, 79], increased in involved distal forelimb bones (Figure 8C). This increase was matched by increased osteoclasts (Figure 8C) and increases in two serum biomarkers of bone degradation (Trap5b, band 5 tartrate-resistant acid phosphatase, and a biomarker of osteoclast activity and bone resorption, and CTX1, the C-terminal telopeptide of collagen type I cleaved by osteoclasts during bone resorption). Thus, a 12-week task at high-repetition high-force levels leads to a net loss of trabecular bone volume in the radius and ulna.
Microcomputed tomography (MicroCT), bone cytokines, and osteoclast numbers in distal radial trabecular region. (A) Representative transaxial microCT slices of the metaphysis of the radius and ulna (at 166 slices, 1.5 mm from the distal edge of their respective growth plate) from an FRC, 12-week HRHF, and 12-week HRHF+IBU rat. (B) MicroCT analysis of trabeculae of distal radius showing reduced trabecular bone volume (BV/TV) in HRHF rats that was rescued by ibuprofen treatment in task weeks 5–12. (C) IL-1-beta in forelimb bones (radius and ulna), tested using ELISA. *
Fortunately, systemic anti-inflammatory treatment with ibuprofen prevented these bone catabolic changes (Figure 8) [40, 70]. Eight weeks of continual ibuprofen treatment reduced bone inflammatory cytokine levels, and osteoclast numbers and activity, despite continued task performance. These results suggest that bone catabolism in the untreated HRHF rats was the result of increased inflammatory cytokines and their activating effects on osteoclasts. In summary, forearm bone osteopenia can be one consequence of prolonged high-intensity hand and wrist tasks. This increase in osteopenia and perhaps even fracture risk of workers performing this type of task is under-investigated in human and should be the focus of future studies.
\nA loss of bone mineral density has been reported in metacarpal bones and distal radius and ulna of patients with long-term carpal tunnel syndrome [80]. Surgical release treatment for carpal tunnel syndrome rescues this decline in distal forearm bone mineral density [81]. Those authors hypothesized that nerve-compression-induced muscle weakness led to bone loss as a consequence of reduced muscular loading on the bone [80], since the muscles involved in performing hand-grip actions produce forces on forearm bones [82, 83]. In our model, ibuprofen may be sparing bone volume by reducing osteoclastogenesis and activity as well as by reducing fibrotic nerve compression, thus sparing muscle activity and muscle-pulling forces on bones (refer to Figure 3C and 4A again).
Ibuprofen treatment is inexpensive and readily available over the counter. Yet, its use should be limited to short-term treatments (we have tested only up to 8 weeks). Ibuprofen medication may inhibit skeletal muscle hypertrophy and adaptation [42, 84–86], although a more recent study shows no effect of ibuprofen on muscle hypertrophy [43]. Long-term use of ibuprofen-related NSIADs could increase gastrointestinal bleeding, renal toxicity, risk of myocardial infarction, and hypertension [87, 88]. NSAIDs are also not always successful for long-term treatment of pain and dysfunction [16], similar to our results with reflexive grip strength.
It is unlikely that a single drug will be effective in treating all WMSDs since their development is multi-factorial. Multipronged treatment should be developed that are individualized to the subject for complete reversal of WMSD-induced tissue inflammation/fibrosis/degeneration and recovery of function. Figure 9 shows various points of interventional treatment, indicating that early treatment is needed to alter acute inflammatory responses, while chronic inflammatory responses are accompanied by several signs and symptoms of chronic pain and should be treated with secondary anti-inflammatory drugs such as ibuprofen or anti-tumor necrosis factor alpha drugs [89]. The latter drugs have yet to be tested in subjects with WMSDs, but have been tested in our animal model and show fair to strong efficacy [39, 61]. In subjects with chronic or persistent pain, negative neuroplasticity in the CNS, termed central sensitization, may have occurred. Treatment options of such central sensitization should be explored carefully in future studies to reduce chronic pain. At the right side of this figure, we show the onset of fibrosis, which may compress and damage axons (such as in carpal tunnel syndrome), and tether tissues. We are currently exploring options of blocking fibrogenic-signaling pathways in our rat model.
Summary of results and possible points of intervention indicated. Repetitive tasks can cause local injury and acute inflammation that could be prevented with reduced loads. Acute and chronic inflammation can be treated by prophylactic and secondary ibuprofen treatment at anti-inflammatory doses. Means to treat fibrosis are still under investigation as are most effective ways to rescue persistent sensorimotor declines. Abbreviations: CTGF = connective tissue growth factor; IL-1 = interleukin 1; Macs = macrophages; TNF = tumor necrosis factor; TGFB1 = transforming growth factor beta 1. Modified and used by permission from Barr and Barbe [
One new non-pharmaceutical direction may be modeled manual therapy. A recent review examined the effectiveness of exercise versus several types of mobilization methods for the treatment of carpal tunnel syndrome and concluded that there was only poor support [90]. However, two recent pilot studies examined massage therapy methods specifically and observed reduced symptoms of discomfort and increased strength post treatment in patients being treated for carpal tunnel syndrome [91, 92]. Another type of massage termed “sports massage’ has been used to treat post-exertional muscle soreness, which is also known as delayed onset muscle soreness (DOMS). While the clinical utility of sports massage for DOMS is supported overall, a comprehensive review of the literature by Moraska in 2005 shows its effectiveness in some studies and a lack thereof in others [93]. Perhaps, this is because sports massage therapy treatment is typically short term. With regard to the use of massage therapies for individuals with repetitive motion disorders, clinicians should be aware that these disorders are not acute in nature. Instead, repetitive motion disorders are the consequence of underlying tissue changes that take weeks to years. It is unlikely that a single, short-term treatment will be effective.
\nBecause we could not identify any studies using manual therapies for WMSDs (other than carpal tunnel syndrome), we recently performed a study examining the effectiveness of modeled manual therapy (MMT) as a treatment for symptoms of discomfort, reduced grip strength, and increased tissue fibrosis occurring in forearms of rats performing a HRHF task for 12 weeks [33]. We began the MMT immediately post training to the high-force level, a time point when the rats began to display signs and symptoms consistent with WMSDs. Results were compared to untreated HRHF rats and to age-matched control rats. The MMT protocol included a mixture of manual therapy submodalities: gentle mobilization, skin rolling, and myofascial release (deep massage) of the forearm flexor compartment; joint mobilization of the wrist (gentle rotation and traction of the wrist); and stretching of the entire upper limb from the shoulder to the fingers. The therapy was provided 5 days per week for 12 weeks, while the animals performed the HRHF task for a food reward (as above, for 3 days/week, 2 h/day, in 30-min sessions). Compared to untreated HRHF rats, the HRHF rats receiving the MMT (called HRHF-MMT rats) showed significantly fewer behaviors suggestive of discomfort and had increased numbers of successful reaches. Grip strength had decreased significantly post training to the high-force levels, compared to the rats’ naïve levels. However, the MMT protocol improved grip strength within 2 weeks of treatment, an improvement that continued through week 12 despite continued performance of the HRHF task by the HRHF-MMT rats. An examination of tissues post euthanasia showed decreased nerve and connective tissue fibrosis, and decreased collagen and TGF-B1 in the 12-week HRHF rats, compared to the untreated HRHF rats. These observations support further investigation of manual therapy as a preventative for repetitive motion disorders.
Research reported in this publication was supported by the National Institutes of Health, the National Institute of Arthritis and Musculoskeletal and Skin Diseases, under Award Numbers AR056019 and AT009350 to MFB.
Both the ordinary and partial differential equations have an important role in explaining many phenomena that occur in nature or in medical engineering, biotechnology, economic, ocean, plasma physics, etc. [1, 2]. Duffing equation is considered one of the most important differential equations due to its ability for demonstrating the scenario and mechanism of various nonlinear phenomena that occur in nonlinear dynamic systems [3, 4, 5, 6, 7, 8, 9, 10, 11]. It is one of the most common models for analyzing and modeling many nonlinear phenomena in various fields of science such as the mechanical engineering [12], electrical engineering [13], plasma physics [14, 15], etc. Mathematically, the Duffing oscillator is a second-order ordinary differential equation with a nonlinear restoring force of odd power
where
However, these undamped models (without friction/dissipation) do not exist much in reality except under harsh conditions. In order to describe and simulate the natural phenomena that arise in many realistic physical models and dynamic systems, the friction/dissipation forces must be taken into account, as is the case in many plasma models and electronic systems. Accordingly, the following damped (non-conservative) Duffing equation will be devoted for this purpose
If the frictional force does not neglect, so that all PDEs that have “Burgers
Many authors investigated the (un)damped Duffing equation, (un)damped Helmholtz Eq. [16, 29, 30, 31], and undamped H-D equation. On the contrary, there is a few numbers of published papers about damped Duffing-Helmholtz equation [32, 33]. For example, Zúñiga [32] derived a semi-analytical solution to the damped Duffing-Helmholtz equation in the form of Jacobian elliptic functions, but he putted some restrictions on the coefficient of the linear term, and then obtained a solution that gives good results compared to numerical solutions. Also, it is noticed that Zúñiga solution [32] is very sensitive to the initial conditions. Gusso and Pimentel [33] obtained obtain improved approximate analytical solution to the forced and damped Duffing-Helmholtz in the form of a truncated Fourier series utilizing the harmonic balance method.
In this chapter, we display some novel semi-analytical (approximate analytical) solutions to the strong higher-order nonlinear damped oscillators of the following initial value problem (i.v.p)
and its family (
Our new semi-analytical solution to Eq. (3) is derived in terms of Weierstrass and Jacobian elliptic functions. Also, we will solve Eq. (3) numerically using Runge–Kutta 4
Let us consider the standard (undamping) Duffing equation in the absence both friction (
which is subjected to the following initial conditions
The general solution of Eq. (4) maybe written in terms of any of the twelve Jacobian elliptic functions.
For example, let us assume
By inserting solution (6) in Eq. (4), we get
where cn
Equating to zero the coefficients of cn
Thus, the general solution of Eq. (4) reads
The values of the constants
For
Making use of the additional formula
the solution (10) could be expressed as
where
Solution (12) is a periodic solution with period
Let us consider the i.v.p.
Using formula (10), the exact solution of the i.v.p. (15) reads
According to the relation (12)-(13), the exact solution of the i.v.p. (15) is also written as
and its periodicity is given by
In Figure 1, the comparison between the exact analytical solution (17) and the approximate numerical RK4 solution is presented. Full compatibility between the two analytical and numerical solutions is observed.
A comparison between the analytical solution
For
where
with initial conditions
Inserting ansatz (18) into Eq. (4) and taking the below relation into account
we get
Equating the coefficients of
Note that the i.v.p. (19)-(20) has a positive discriminant and it is given by
Then the problem reduces to the first case. Accordingly, the solution of the i.v.p. (4)-(5) maybe written in the form,
where
The solution (23) is unbounded and its periodicity is given by
Let us assume the following i.v.p.
The solution of the i.v.p. (26) according to the relation (23) reads
and the periodicity of this solution is given by
Solution (27) is displayed in Figure 2.
The profile of solution
If the discriminant vanishes
reads
which may be verified by direct computation.
is given by
is given by
with
the solution of the i.v.p. (4)-(5) could be written in terms of the Weierstrass elliptic function
with
and
The solution (36) is periodic with period
where
where
and
Let us consider the i.v.p.
The approximate solution in trigonometric form is given by
The exact solution reads
with period
The error on the interval
The comparison between the approximate analytic solution (44) and the exact analytic solution (45) is illustrated in Figure 3.
A comparison between the approximate solution
where
and
The approximate trigonometric solution of
reads
The exact solution is
with period
The error on the interval
Figure 4 demonstrates the comparison between the approximate analytic solution (50) and the exact analytic solution.
A comparison between the approximate solution
The undamped Duffing-Helmholtz equation reads
We will give a solution to the i.v.p. (53) in terms of Weierstrass elliptic functions. For solving this problem the following ansatz is considered
where
Substituting the ansatz (54) into the ordinary differential equation (ode)
with
Equating the coefficients
The values of
We have
The number
The solution of the i.v.p.
according to the relation (54) is given by
In Figure 5, the comparison with the approximate analytic solution (61) and the approximate numerical solution using RK4 is investigated.
A comparison between solution
The periodicity of solution (61) is given by
Suppose that the physical system to be studied is under the influence of some constant external/excitation force, so the standard Duffing-Helmholtz equation can be reformulated to the following constant forced Duffing-Helmholtz i.v.p.
For solving the i.v.p. (62), the following assumption is introduced
where
Substituting Eq. (63) into the i.v.p. (62), we have
Note that the constant forced Duffing-Helmholtz Eq. (62) has been reduced to the standard Duffing-Helmholtz Eq. (65) with the following new initial conditions
Suppose that we have the following i.v.p. and we want to solve it
It is clear that the i.v.p. (67) is a constant forced Duffing-Helmholtz equation. The solution of this problem is given by
The comparison between the solution (68) and the RK4 solution is introduced in Figure 6.
A comparison between the solution
The periodicity of solution (68) is given by
Let us define the following i.v.p.
Suppose that
then the first equation in system (69) can be written as
For solving the i.v.p. (69), the following ansatz is assumed
with
where the function
Let us define the following residual
and by applying the condition
By solving this equation we can get the value of
Let
The approximate analytic solution of the i.v.p. (77) reads
The distance error as compared to the RK4 numerical solution is given by
Also, the comparison between solution (78) and RK4 solution is presented in Figure 7.
A comparison between solution
The value of
In this case, the number
Let us define the following new i.v.p.
We suppose that
Let us define the solution of i.v.p. (82) as follows
where
The function
where
The value of
Let
The approximate analytic solution of the i.v.p. (89) is given by
The distance error according to the RK4 numerical solution is calculated as
Moreover, solution (90) is compared with RK4 solution as shown in Figure 8.
A comparison between solution
Let us consider the following ordinary differential equation [36]
which is subjected to the following initial conditions
a. Suppose that
where the function
The values of the coefficients
and the value of the quantity
The solution to the the i.v.p. (95) is obtained from the formulas in the first section.
b. Suppose that
where the function
The values of the coefficients
and the value of
Note that the solution of the i.v.p. (100) could be obtained from the formulas in the first section.
Proof: case (a)
Inserting ansatz (94) into Eq. (92) taking the following equation into consideration
and using Eq. (100), we have
with
where
Equating the coefficients
Let us define the following i.v.p.
We seek approximate analytic solution in the ansatz form
with
where
Define the residual
then, the condition
Some real roots of Eq. (111) give the value of
Let
The approximate analytical solution of the i.v.p. is given by
where
The distance error as compared to the RK4 numerical solution reads
In Figure 9, we make a comparison between our solution, RK4 solution, and Zuñiga solution given in Ref. [17]. It is clear that the accuracy of our solution is better than the solution of Zuñiga [17].
A comparison between RK4 solution and (a) Solution
The above solutions could be applied to various fields of physics and engineering such as they could be used for describing the behavior of oscillations in RLC electronic circuits, plasma physics etc. In the below section, the above solution will be devoted for studying oscillations in various plasma models.
In the RLC series circuits consisting of a linear resistor with resistance
where the relation between the current the charge is given by
with
The solution of Eq. (117) can be devoted for interpreting and analyzing the oscillations that can generated in the RLC circuit.
For studying the plasma oscillations using fluid theory, the basic equations of plasma particles using the reductive perturbation method (RPM) will be reduced to some evolution equations such as KdV equation and its family [37, 38, 39, 40, 41]. Let us consider a collisionaless and unmagnetized electronegative complex plasma, consisting of inertialess cold positive and negative ion species, inertia non-Maxwellian electrons in addition to stationary negative dust impurities [42]. Thus, the quasi-neutrality condition reads:
Now, the RPM is introduced to reduce the fluid plasma equations to the evolution equation. According to the RPM, the independent quantities
with the coefficients of the quadratic nonlinear, cubic nonlinear, dispersion, and dissipation terms
where
It is shown that the coefficients
To convert EKdVB Eq. (118) to the damped H-D Eq. (4), the traveling wave transformation
where
Note that the coefficient
The analytical and semi-analytical solutions for nonlinear oscillator integrable and non-integrable equations have been investiagted. First, the standard integrable Duffing equation has been analyized and its solutions have been obtained depending on the sign of its discriminant
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His studies in robotics lead him not only to a PhD degree but also inspired him to co-found and build the International Journal of Advanced Robotic Systems - world's first Open Access journal in the field of robotics.",institutionString:null,institution:{name:"TU Wien",country:{name:"Austria"}}},{id:"441",title:"Ph.D.",name:"Jaekyu",middleName:null,surname:"Park",slug:"jaekyu-park",fullName:"Jaekyu Park",position:null,profilePictureURL:"https://mts.intechopen.com/storage/users/441/images/1881_n.jpg",biography:null,institutionString:null,institution:{name:"LG Corporation (South Korea)",country:{name:"Korea, South"}}},{id:"465",title:"Dr",name:"Christian",middleName:null,surname:"Martens",slug:"christian-martens",fullName:"Christian Martens",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null},{id:"479",title:"Dr.",name:"Valentina",middleName:null,surname:"Colla",slug:"valentina-colla",fullName:"Valentina Colla",position:null,profilePictureURL:"https://mts.intechopen.com/storage/users/479/images/358_n.jpg",biography:null,institutionString:null,institution:{name:"Sant'Anna School of Advanced Studies",country:{name:"Italy"}}},{id:"494",title:"PhD",name:"Loris",middleName:null,surname:"Nanni",slug:"loris-nanni",fullName:"Loris Nanni",position:null,profilePictureURL:"https://mts.intechopen.com/storage/users/494/images/system/494.jpg",biography:"Loris Nanni received his Master Degree cum laude on June-2002 from the University of Bologna, and the April 26th 2006 he received his Ph.D. in Computer Engineering at DEIS, University of Bologna. On September, 29th 2006 he has won a post PhD fellowship from the university of Bologna (from October 2006 to October 2008), at the competitive examination he was ranked first in the industrial engineering area. He extensively served as referee for several international journals. He is author/coauthor of more than 100 research papers. He has been involved in some projects supported by MURST and European Community. 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Delac received his B.Sc.E.E. degree in 2003 and is currentlypursuing a Ph.D. degree at the University of Zagreb, Faculty of Electrical Engineering andComputing. His current research interests are digital image analysis, pattern recognition andbiometrics.",institutionString:null,institution:{name:"University of Zagreb",country:{name:"Croatia"}}},{id:"557",title:"Dr.",name:"Andon",middleName:"Venelinov",surname:"Topalov",slug:"andon-topalov",fullName:"Andon Topalov",position:null,profilePictureURL:"https://mts.intechopen.com/storage/users/557/images/1927_n.jpg",biography:"Dr. Andon V. Topalov received the MSc degree in Control Engineering from the Faculty of Information Systems, Technologies, and Automation at Moscow State University of Civil Engineering (MGGU) in 1979. He then received his PhD degree in Control Engineering from the Department of Automation and Remote Control at Moscow State Mining University (MGSU), Moscow, in 1984. From 1985 to 1986, he was a Research Fellow in the Research Institute for Electronic Equipment, ZZU AD, Plovdiv, Bulgaria. In 1986, he joined the Department of Control Systems, Technical University of Sofia at the Plovdiv campus, where he is presently a Full Professor. He has held long-term visiting Professor/Scholar positions at various institutions in South Korea, Turkey, Mexico, Greece, Belgium, UK, and Germany. And he has coauthored one book and authored or coauthored more than 80 research papers in conference proceedings and journals. His current research interests are in the fields of intelligent control and robotics.",institutionString:null,institution:{name:"Technical University of Sofia",country:{name:"Bulgaria"}}},{id:"585",title:"Prof.",name:"Munir",middleName:null,surname:"Merdan",slug:"munir-merdan",fullName:"Munir Merdan",position:null,profilePictureURL:"https://mts.intechopen.com/storage/users/585/images/system/585.jpg",biography:"Munir Merdan received the M.Sc. degree in mechanical engineering from the Technical University of Sarajevo, Bosnia and Herzegovina, in 2001, and the Ph.D. degree in electrical engineering from the Vienna University of Technology, Vienna, Austria, in 2009.Since 2005, he has been at the Automation and Control Institute, Vienna University of Technology, where he is currently a Senior Researcher. 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