Analytical Approximate Methods For Solving Strongly Nonlinear Oscillation Problems And Their Applications

Vibration is a common phenomenon in engineering technology and natural science.All kinds of vibration processes are unified by the method of Mathematical Physics.Theory of vibration can be classified into the theory of linear vibration and the theory of nonlinear vibration.Because of the principle of superposition,the study on the linear vibration theory is well-developed.In contrast,the general techniques of solutions for nonlinear oscillation systems are less well known,since the superposition principle is no 1 onger set up in them.T herefore,the analysis and calculation method of nonlinear oscillation systems is particularly important.Analytical approximations are interesting because such solutions provide accurate explicit expressions and they allow direct discussion of the effects of related parameters in the solution.One of the most common approaches is the perturbation method.Almost all perturbation methods are based on the presence of a small parameter in the nonlinear governing equation.Such requirement of a small parameter assumption greatly restricts the applications of perturbation techniques,and it is particularly influential for strongly nonlinear problems without a small parameter.Although the harmonic balance(HB)method can be applied to nonlinear oscillatory systems with non-small parameters or even without a small parameter,it is nevertheless very difficult to construct higher-order analytical approximations which require explicit analytical solutions of a s et of complicated nonlinear algebraic equations.In this dissertation,some methods for constructing analytical approximate periodic solutions to strongly nonlinear oscillation systems are presented.Consider the following nonlinear oscillation system governed by du2/dt2+f(u)= 0,u(0)= A,du/dt(0)=0(1)The restoring force function-f(u)is odd nonl inear function of u[i.e.f(-u)=-f(u)],and the restoring force function satisfies uf(u)>0 for u#0.Obviously,u = 0 is an equilibrium position and the system oscillates within a symmetric bound[-A,A].A new independent variable ?=?t is introduced.In terms of this new variable,Eq.(1)becomes Qu + f(u)= 0,u(0)= A,u(0)= 0(2)where ?=?2 and a dot denotes differentiation with respect to ?.The new independent variable ensures the solution of Eq.(2)is a periodic function of ? of period 2?.T he corresponding frequency is given by ?=(?).The periodic solution u(?)and frequency ? are dependent on A.1 The second order Newton-HB MethodBased on the single term HB method,the initial approximation is set as u1(?)= Acos?(3)which satisfies the initial conditions in Eq.(2).U sing the odd f unction assumption f(-u)=-f(u),one can expand f(u1(?))into a Fourier series f(u1(?))=(?)aa2i1cos[(2j-1)?].(4)Substituting Eqs.(3)and(4)into Eq.(2)and setting the coefficient of cos ? to zero,which can be solved for ?(A)as?1(A)=a1/A.(5)The periodic solution of Eq.(2)and its frequency squared can be formulated as?=?1+??1,u(?)+ u1(?)+ ?u1(?)(6)Substituting Eq.(6)into Eq.(2),neglecting the third-order and higher-order power degree terms in ?u1 and ??1,lead to where subscript u denotes derivative of f(u)with respect to u.Here,?u1 is a periodic function of ? of period 2?,and both ?u1 and ??1 are undetermined quantities.Because solving the resulting nonlinear equation in ?u1 and ??1 is difficult,its solution is separated into the following two steps.First,linearizing Eq.(7)with respect to ?u1 and ??1 yields The approximate solution to Eq.(8)can be developed by setting ?u10(?)as Substituting Eq.(9)into Eq.(8),using the HB method for ?u10 and ??10,yieldNext,replacing ??1 in ??1?u1 with ??210 and one of two ?u1 terms in 0.5fuu(u1)(?u1)2 with ?u10 in Eq.(7),respectively,yields a linear equation in ?u1 and ??1 as Further letting ?u1 in Eq.(12)be y1,y2 and ??1 will be solved by using the HB method,and the solutions areFinally,more accurate analytical approximations can be constructed by using the last approximations u3 and ?3 in place of u1 and ?1,respectively,and implementing the similar steps as before.2 The predictor-corrector-HB MethodBased on the single term HB method,the initial approximation is set as Using the odd function assumption f(-u)=-f(u),one can expand f(u0(?))into a Fourier series Substituting Eqs.(16)and(17)into Eq.(2)and setting the coefficient of cos ? to zero,which can be solved for ?(A)asNext,the combination of the predictor-corrector technique and the HB method is formulated to solve Eq.(2).T he first step is the linearization of Eq.(2).The periodic solution and the square of frequency of Eq.(2)can be formulated as follows:Substituting Eq.(19)into Eq.(2)and linearizing with respect to the correction terms?u10 and ??10 lead to The approximate solution to Eq.(20)can be developed by setting ?u10(?)as By using the HB method work out ?u10 and ??10,then obtain the periodic solution and the square of frequencyBased on the prediction above,the periodic solution and the square of frequency of Eq.(2)can be further expressed as Substituting Eq.(24)into Eq.(2)and linearizing with respect to the correction terms Au20 and ??20,still at u = u0 and ?=?0 yield The approximate solution to Eq.(25)can be developed by setting Au20(?)as Eqs.(25)will be solved by using the HB method,and the solutions areFinally,more accurate analytical approximations can be constructed by using the last approximations uc and ?c in place of u,and ?1,respectively,and implementing the similar steps as before.Applying the methods mentioned above,we have studied a series of strongly nonlinear vibration systems,including Duffing oscillators,antisymmetric,constant force oscillator,oscillator with fractional-power restoring force,the singular oscillator and Duffing-Harmornic oscillators.T heir high accurate analytical approximate periodic solutions have been constructed.The methods previously above also can be applied to more complex nonlinear systems,such as MEMS/NEMS.3 Oscillations in MEMS/NEMSA doubly-clamped undamped micro-/nanobeam made of an elastic material and actuated electrostatically by a full-length electrode on only one side of the beam is considered.The gap is assumed significantly smaller comparing with the length and the residual stress is considered as uniform while the effect of residual stress gradients is neglected.H ere b,h(b>h)and L are the width,thickness and length of beam,respectively.The nominal gap is g,the tensile or compressive axial load is N,and the fixed electrode potential is V.Ignore damping,the dimensionless nonlinear integral-differential governing equation for the electromechanical system can be expressed asDeflection function expressed as And the Galerkin procedure is applied to derive a reduced order model with one degree-of-freedom.Set the time derivatives in Eq.(31)to zero the potential V can be solved and expressed in terms of the normalized static beamcenter deflection as.The deflection can be expressed as Substitute Eq.(32)into(31),and using Taylor series expansion of F(as +u,V)about u at a = as,eliminating the terms representing equilibrium and keeping the linear part of increment u only yields the fundamental natural frequency of beams,asThe equation of motion for ROM can be expressed as The system will thus oscillate in an asymmetric interval[-B,A]where both-B(B>0)and A have the same energy level,i.e.,?(-B,V)= ?(A,V).The denominator of electrostatic force is expended into a Taylor series at u = 0 up to the third order.Eq.(34)is approximated byIntroducing a new independent variable ?=?t,Eq.(35)can be expressed as Where ???2 and(')denotes differentiation with respect to ?.Two new nonlinear systems which oscillate between the symmetric bounds[-H,H]is introduced here as where ?= ±1 where H=A for ?=+1,and H=B for ??-1,respectively.With the method mentioned above,yields Using these analytical approximate solutions,the corresponding n th(n = 1,2)analytical approximate period and periodic solution for Eq.(36)can be constructedThe methods presented in this dissertation are simple in principle and easy to be applied.They do not require either non-zero linear part of restoring force or small parameters.These approaches yield brief and exact accurate analytical approximate solutions for a wide range of oscillation amplitudes,including the limiting case of infinite oscillation amplitude.