Applying the method of normal forms to second-order nonlinear vibration problems

Abstract

Vibration problems are naturally formulated with second-order equations of motion. When the vibration problem is nonlinear in nature, using normal form analysis currently requires that the second-order equations of motion be put into first-order form. In this paper, we demonstrate that normal form analysis can be carried out on the second-order equations of motion. In addition, for forced, damped, nonlinear vibration problems, we show that the invariance properties of the first-and second-order transforms differ. As a result, using the second-order approach leads to a simplified formulation for forced, damped, nonlinear vibration problems.

A generalized frequency detuning method for multidegree-of-freedom oscillators with nonlinear stiffness

“…Therefore as the nth diagonal elements of matrices Λ γ and Υ 2 are ω 2 γ and −ω 2 rn respectively, it can be seen that these matrices are similar (but opposite sign). Hence we can write (18). It is this detuning approximation that we will discuss in this paper.…”

“…Please see [18,19,21] for more details of the derivation. Here we use the already defined ω a parameter such that, either ω an = ω γn if no detuning is applied or ω an = ω rn for the detuning case.…”

“…The ε 0 equation is satisfied by setting P u = P v . To satisfy the ε 1 equation, (18), the form of the response needs to be considered. Since the near-identity transform removes non-resonant nonlinear terms from the equations of motion, the response for each state u 1 , u 2 .…”

“…Normal forms is usually applied to first-order nonlinear oscillator equations and an assessment of their accuracy is given in [17]. Here we consider a recently developed formulation that can be applied directly to second-order nonlinear oscillators directly, termed second-order normal forms [18,19]. The second-order normal form technique has the useful property that the nonlinear transform removes non-resonant terms for each mode, rather than for each state, as is the case using the first-order formulation.…”

“…In Section 2 the generalised detuning method is derived based on the second-order normal form technique from [18], and the effect of the selec-tion of linearised frequency has on the predicted response is analysed. Then in Section 3, the first example we give is of a single degree-of-freedom Duffing oscillator, which is used to shown that the detuning approximation is equivalent to linearising the system using the resonant frequency.…”