3.4 Power from the Extended Series
3.4.1 Integral of the First Extended Wiener Kernel
In what follows, a useful property of the first kernel, namely 1
that influences understanding of power dissipation will be derived. By using Eq. (3.26), Eq.
(3.42) can be rewritten as
k1(0) =EfTM−1f
2 (3.43)
and can be derived by thinking of the base excitation as a series of impulses of magnitudes given by ξ (t). A small change in z(t) due to an impulse ξ (T ) at time T , termed z(t)|ξ (T ) to differentiate it from the change in z due to previous impulses, can be assessed both from physical reasoning and from the Wiener series.
Physically, the effect of an impulse will instantaneously only change the acceleration term, M(y)¨y, and have no effect on the nonlinear term g(y, ˙y,t) so from Eq. (3.33)
lim
δ t→0
M(y)δ ˙y|ξ (T )
δ t = f(y)ξ (T ) (3.44)
where δ ˙y|ξ (T )
δ t represents the accelerations of the generalised coordinates only due to the excitation at time T . This equation can be described physically as a unit impulse giving a unit change in momentum.
Rearranging Eq. (3.44), a small change in ˙y, δ ˙y|ξ (T ), from the excitation is therefore lim
δ t→0
δ ˙y|ξ (T )= M−1fξ (T )δ t. (3.45) The small time-step, δ t, here is from a time just before the impulse, T−, to a time just after the impulse, T+, where T+− T− = δ t and the magnitude of the impulse is ξ (T )δ t.
The change in the velocity vector from excitation ξ (T ) at time T , δ ˙y|ξ (T ) can also be calculated from the Wiener series. From Eq. (3.36) the change in the velocity vector provides a change in the z(t) variable
δ z|ξ (T )= f(y)Tδ ˙y|ξ (T ) (3.46) where the Wiener series of Eq. (3.37) can be rewritten in a variational form
δ z|ξ (T )=
∞
∑
n=0
∆gn[kn; ξ (T )]. (3.47)
and the ∆gnterms represent the change in nth g-functional due to the excitation at time T , ξ (T ).
The zeroth order ∆g-functional is simply
∆g0[k0; ξ (T )] = 0 (3.48)
since it does not depend on the excitation. The first order ∆g-functional is
∆g1[k1; ξ (T )] = lim
δ t→0
Z δ t/2
−δt/2
k1(τ)ξ (T − τ)dτ. (3.49)
Since the instantaneous response to a single impulse of magnitude ξ (T )δ t is being assessed, the integral of Eq. (3.49) disappears and the first ∆g-functional becomes
∆g1[k1; ξ (T )] = lim
δ t→0
k1(0+)ξ (T )δ t (3.50)
meaning that immediately after the impulse the first ∆g-functional contributes k1(0+)ξ (T )δ t to the response where the argument 0+ in k1(τ) represents the time instantaneously after τ1= 0. This is analogous to the impulse response of a linear system where immediately after an impulse, the response would be given by the value of the impulse response just after the impulse multiplied by the magnitude of the impulse. However, whilst in the linear case k1(τ) would completely define the response to the impulse, in the nonlinear case the higher order functionals may also contribute to this value so must also be assessed.
The second order ∆g-functional is subtraction of the double integral between −δ t/2 and δ t/2 on the first line accounts for the counting of the −δ t/2 ≤ τ1≤ δt/2, −δt/2 ≤ τ2≤ δt/2 region in both of the first two terms on the right hand side. The value of this integral clearly goes to zero as δ t → 0 since it is of order δ t2. For all higher order ∆g-functionals, this region will also be negligible since its value will be of order δ tnwhere n is the order of the functional. For the same reason as the first functional, one argument of the second kernel is evaluated at τ1= 0+.
Similarly, the nth ∆g-functional is
∆gn[kn; ξ (T )] = lim where the limit of the summation is different from that of Eq. (3.11) because only terms that involve at least one excitation term can affect δ z.
Combining Eqs. (3.45), (3.46), (3.47) and (3.52) yields expected value taken (noting that depending on the system both f and M(y) can be random variables) giving It is clear that for any even n, the nth term will be zero since it will include an ensemble average of an odd number of zero mean excitation terms. The limit of the sum over m therefore becomes (n − 1)/2 and the nth term can be investigated in a similar manner as Section 3.2.
The ensemble average containing n − 2m − 1 terms can be rewritten as a sum of products of autocorrelation functions. There are (n − 2m − 1)!/((n − 2m − 1)/2)!2(n−2m−1)/2distinct ways of pairing the terms so there will be this many terms in the sum and the integral over each term will be the same due to the symmetry property of the kernels. The nth term in Eq.
(3.54) therefore becomes where the integral is the same for every value of m. Analysis of this term shows that it is equal to zero since the sum becomes
(n−1)/2
where M = (n − 1)/2 and a binomial expansion around the point −1 has been used. Eq.
(3.54) therefore becomes
k1(0+) = EfTM−1f . (3.56)
The value of the first kernel, k1(τ), around the τ = 0 point will now be explored. When τ = 0−, instantaneously before any excitation, the response must be zero because the system is causal so k1(0−) = 0. Combining with Eq. (3.56) suggests that at τ = 0 the value of the kernel is halfway between k1(0) and k1(0+), so k1(0) = EfTM−1f /2 as required by Eq.
(3.43). This is equivalent to saying at k1(0), only half of the impulse from the excitation has occurred so the response is only half the magnitude. The importance of this point is observed when examining the relationship between the time and frequency domain kernels, k1(τ) and K1(ω) connected via Eqs. (3.25) and (3.26). The causality of k1(τ) implies that the real and imaginary parts of the frequency domain kernel are even and odd respectively.
When an inverse Fourier transform is taken of the frequency kernel, the real and imaginary parts generate even, k1E(τ), and odd, k1O(τ), time functions respectively. For τ < 0 the two functions are equal and opposite, k1E(τ < 0) = −k1O(τ < 0), so cancel to ensure a causal response whereas for τ > 0 the two functions are equal, k1E(τ > 0) = k1O(τ > 0), and add to generate the time kernel. However, when τ = 0 the odd function must be zero, k1O(0) = 0, but the even function has a finite value given by the integral over the real part of the frequency kernel. Assuming k1E(τ) is smooth, this value will be k1(0+)/2. The factor of
two is therefore important for calculating power dissipation according to Eq. (3.41). This is similar to analysis of a point at a discontinuity in Fourier analysis where the calculated value at the discontinuity is the mean of the values to either side of it.
Although the triple product term EfTM−1f in Eq. (3.56) seems physically unintuitive, it has been discussed in detail in [48] where it has been shown that for a system with a total oscillating mass, MTot,
EfTM−1f ≤ MTot. (3.57)
In [48] Langley shows that if a system is constrained to reduce its number of degrees of freedom then EfTM−1f < MTot. However, if there is no constraint on the system, the inequality of Eq. (3.57) becomes an equality so
EfTM−1f = MTot. (3.58)
An illustrative example is given where, quoting directly from the text, ‘a planar inverted pendulum, consisting of a lumped mass m mounted on a massless rod of length L. The rotation of the pendulum from the vertical is θ and the elastic extension of the rod is r. The elastic stiffness of the rod is k2and a rotational spring of stiffness k1is attached to the base of the rod; dampers of rate c2and c1are attached in parallel with the springs.’ The equations of motion for this system are
m(L + r)2 0
0 m
! ¨θ
¨r
!
+ 2m(L + r)˙r ˙θ + c1θ + k˙ 1θ − mg(L + r) sin θ
−m(L + r) ˙θ2+ c2˙r + k2r+ mg cos θ
!
= m(L + r) sin θ
−m cos θ
!
¨b(t) (3.59)
where g is the acceleration due to gravity and ¨b(t) is the acceleration of the base of the rod in the vertical direction. Here, the mass matrix and force vector provide EfTM−1f = m since the mass is free to move anywhere on the plane. If, however, the system is constrained such that the rod becomes infinitely stiff, a single-degree-of-freedom system results with equation of motion
mL2θ + c¨ 1θ + k˙ 1θ − mgL sin θ = mL sin θ ¨b(t). (3.60) In this case EfTM−1f = mE sin2θ so the triple product is less than the total oscillating mass because motion has been constrained to one generalised coordinate, θ , and motion in the entire plane is not possible.
In practice the two systems of Eq. (3.59) and (3.60) could be expected to give almost identical responses when a very high rod stiffness is used in the unconstrained case. It seems
counter-intuitive therefore that the constrained system should provide such a different triple product and thus first Wiener kernel from Eq. (3.56) (which has consequences for power dissipation) from that of the unconstrained system. Langley provides physical reasoning for this effect arguing that any constrained system such as the one of Eq. (3.60) is in reality still an unconstrained system like Eq. (3.59), but with components or couplings of infinite stiffness that reduce the number of generalised coordinates.
For white noise excitation as used in [48], which covers all frequencies, the system must still include the infinite frequency components in order to account for all resonances and be a good model across the entire frequency range. According to Langley ‘the use of [Eq.
(3.60)] is equivalent to the assumption that the excitation does not extend to the second linear resonance, so that the system is essentially excited by band limited noise, leading to a lower result for the power input by the base excitation.’
The inverted pendulum systems of Eqs. (3.59) and (3.60) are simulated and used to illustrate this discussion in Section 3.5.2. The case of band-limited noise is interesting since the kernel is undefined for any frequency range that contains no excitation and this will be discussed separately in Section 3.6.
To summarise the results so far: the power dissipated by a general nonlinear oscillator, Eq. (3.33), under random excitation of a general spectrum, Sξ ξ(ω), can be calculated using Eq. (3.41). To do this, the first Wiener kernel must be calculated from either simulations or experimentally using Eq. (3.24), but crucially for a designer of a system desiring a preliminary estimate of power dissipation, the first Wiener kernel has the property of Eq.
(3.42) where the triple product term is simply equal to the oscillating mass provided the system is unconstrained.
It should be noted that this approach extends the results for white noise excitation of [47] and [48] to general random excitation. The white noise results are retrieved by using Sξ ξ(ω) = πS0 where the factor of π remains in the spectrum since the Fourier transform formula used is Eq. (3.27) with an autocorrelation function Rbb(τ) = πS0δ (τ ). This is substituted into Eq. (3.41) and combining with Eq. (3.42) and (3.57) the power dissipated by white noise is
PW hite=π S0EfTM−1f
2 ≤ π S0MTot
2 . (3.61)
The value of this approach for non-white noise, particularly when applied to energy harvesting, will be illustrated in the following sections. Additionally, for a system with linear damping, the approach is not limited to calculation of power, but also provides the mean square velocity of the response since this is proportional to power.