A comparison of robustness and performance of linear and nonlinear Lanchester dampers

Abstract

In this paper, we study and compare performance and robustness of linear and nonlinear Lanchester dampers. The linear Lanchester damper consists of a small mass attached to a primary system through a linear dashpot, whereas the nonlinear Lanchester damper is linked to the primary mass through dry friction forces. In each case, we propose a semi-analytical method for computing the frequency response, for different values of the design parameters, in order to evaluate the performance and robustness of the two kinds of damper. Overall, it is shown that linear Lanchester dampers perform better than nonlinear damper both in terms of attenuation and robustness. Moreover, the nonlinear frequency response curves, that include the intrinsic non-smooth nature of the friction force, may serve as reference curve for further numerical studies.

Appendix A: Symmetry in the response curves

We consider the following harmonically forced dynamical system:

$$\begin{aligned} m\ddot{{ x}} + c\dot{{ x}} + kx + { f}_\mathrm{nl}({ {\dot{x}}}) = { f}_\mathrm{e}\cos \Omega t \end{aligned}$$

(A.1)

where x(t) is a function of time t, ${\dot{\circ }}$ is the time derivative of $\circ$, m, c, k are mass, damping and stiffness constants, ${ f}_\mathrm{nl}({ {\dot{x}}})$ is the nonlinear internal force and ${ f}_\mathrm{e}$ is the external force.

We assume that ${ f}_\mathrm{nl}({ \dot{x}})$ is an odd function of ${ \dot{x}}$, which means that for all ${ \dot{x}}$:

$$\begin{aligned} { f}_\mathrm{nl}(-{ \dot{x}})=-{ f}_\mathrm{nl}({ \dot{x}}). \end{aligned}$$

(A.2)

As a consequence, its Taylor series expansion around 0 contents only odd terms:

$$\begin{aligned} { f}_\mathrm{nl}({ \dot{x}})=a_1{ \dot{x}}+a_3{ \dot{x}}^3+a_5\dot{x}^5+\cdots =\sum _{i=1,3,5}^{+\infty } a_{i} \dot{x}^{i}, \end{aligned}$$

(A.3)

where the $a_i$, $i\in \mathbb {N}$ are the Taylor coefficients of $f_\mathrm{nl}$. If periodic solutions of Eq. (A.1), in the steady state, are under concern, they can be written as the following Fourier series:

$$\begin{aligned} x(t)=x_0+\sum _{h=1}^{+\infty }\left( x_h^c\cos h\Omega t + x_h^s\sin h\Omega t\right) \end{aligned}$$

(A.4)

where $\Omega =2\pi /T$ is the frequency of motion, T its period and $(x_h^c, x_h^s)$ are the Fourier coefficients of x(t). In Eq. (A.1), the harmonics content of x(t) is created by the only nonlinear term of the equation, the function $f_\mathrm{nl}$. Since it is odd, it creates only odd harmonics in x(t). For instance, if $x(t)=\cos (\Omega t+\varphi )=\cos \phi$, $x^3=(3\cos \phi + \cos 3\phi )/4$, $x^5=(10\cos \phi + 5\cos 3\phi +\cos 5\phi )/16$... so that:

$$\begin{aligned}&x(t)=\cos (\Omega t+\varphi )\quad \Rightarrow \quad f_\mathrm{nl}(\dot{x})\nonumber \\&\quad =\sum _{h=1,3,5}^{+\infty } \left( f^c_{h} \cos h\Omega t+ f^s_{h}\sin h\Omega t\right) , \end{aligned}$$

(A.5)

As a consequence, the simplest solution x(t) of Eq. (A.1) is composed only by odd harmonics:

$$\begin{aligned} x(t)=\sum _{h=1,3,5}^{+\infty } \left( x^c_{h} \cos h\Omega t+ x^s_{h}\sin h\Omega t\right) , \end{aligned}$$

(A.6)

Even harmonics can still be created after a symmetry breaking bifurcation, a case out of the scope of the present text. Now, we consider the following symmetry property of the T-periodic time function x(t):

$$\begin{aligned} \forall t,\qquad x(t+T/2)=-x(t) \end{aligned}$$

(A.7)

so that one half of the period is the inverse mirror of the other half-period. Inserting Eq. (A.7) into Eq. (A.4), one shows that necessarily all even harmonics of x(t) are zero, so that x(t) has an odd harmonics content [Eq. (A.6)]. The reciprocal rule is also verified: any time periodic function with an odd harmonics content verifies the symmetry property (A.7).

As a conclusion, if $f_\mathrm{nl}(\dot{x})$ is an odd function of $\dot{x}$, the harmonics content of the basic (without considering symmetry breaking bifurcations) nonlinear solution x(t) of Eq. (A.1) is necessarily odd and the symmetry property (A.7) is verified.