Solution of Nonlinear Hardening and Softening type Oscillators by Adomian’s Decomposition Method
محورهای موضوعی : فصلنامه شبیه سازی و تحلیل تکنولوژی های نوین در مهندسی مکانیکبهرام گل محمدی 1 , قاسم اسعدی کردشولی 2 , علیرضا وحیدی 3
1 - مربی، دانشکده فنی مهندسی، دانشگاه آزاد اسلامی واحد سلماس
2 - مربی، دانشکده علوم پایه، دانشگاه آزاد اسلامی واحد شهر ری
3 - استادیار، دانشکده علوم پایه، دانشگاه آزاد اسلامی واحد شهر ری
کلید واژه: Hardening and Softening Oscillator, Nonlinear, Adomian’s Method,
چکیده مقاله :
A type of nonlinearity in vibrational engineering systems emerges when the restoring force is a nonlinear function of displacement. The derivative of this function is known as stiffness. If the stiffness increases by increasing the value of displacement from the equilibrium position, then the system is known as hardening type oscillator and if the stiffness decreases by increasing the value of displacement, then the system is known as softening type oscillator. The restoring force as a nonlinear polynomial function of order three, can describe a wide variety of practical nonlinear situations by proper choosing of constant multipliers. In this paper, a spring-mass system is considered by the restoring force of the introduced type. Choosing suitable values for a, b and n, a hardening and softening type oscillators are constructed and related equations of motion are introduced as second order nonlinear differential equations. The equations are solved directly, using the Adomian’s decomposition method (ADM). In another approach, the equations are converted to systems of first order differential equations and then solved using the same method. The results show that the ADM gives accurate results in both approaches, beside it shows that converting the equation to a system of equations of lower order, tends to more accurate solutions when ADM applies.
یکی از عوامل ایجاد اثرات غیرخطی در سیستمهای نوسانی، غیرخطی بودن تابع نیروی بازگرداننده است که طیف وسیعی از آنها با تابع چندجملهای درجه سه و انتخاب ضرایب مناسب مدلسازی میشوند. در این مقاله یک سیستم نوسانگر جرم - فنر تحت تاثیر چنین نیروی بازگردانندهای درنظر گرفته شده و دو دسته پارامتر طوری انتخاب شدهاند که معادلات حرکت دو نوسانگر سخت شونده و نرم شونده از کاربرد قانون دوم نیوتن بهدست آیند. این معادلات دیفرانسیل غیرخطی مرتبه دو ابتدا با استفاده از روش تجزیهی آدومین حل شدهاند. در مرحلهی بعد ابتدا معادلات به دستگاه معادلات مرتبه یک تبدیل و سپس مجددا با روش آدومین حل شدهاند. با توجه به اینکه طرف دوم معادلات دیفرانسیل حل شده، برابر با صفر است، برای مقایسهی نتایج، پاسخها در معادله قرارگرفته و انحراف آنها از صفر به عنوان خطا درنظر گرفته شدهاست. مقایسهی نتایج نشان میدهد که روش بهکار رفته برای هر دو مساله از دقت مناسب برخوردار است و همچنین تبدیل معادله به دستگاه معادلات مرتبهی پایینتر منجر به حصول پاسخهای دقیقتر می شود.
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