Multi-degree-of-freedom Systems

The numerical simulations show strong agreement with the theory for a SDOF oscillator so attention is now turned to two MDOF scenarios; two masses each defined by one generalised coordinate and a single mass able to move in two directions thus defined by two generalised coordinates. For the former case a two mass system of the form of Figure 3.5 is taken with equations of motion is compared to this amount. A system is analysed with parameter values of m1= 1.5kg, m₂= 0.7kg, c₁= 0.4kg/s, c₂= 0.2kg/s, c₃= 0.05kg/s, k₁= 15N/m, k₂= 15N/m, k₃= 5N/m, ε = 2000N/m³and narrowband base acceleration with spectrum given by Eq. (3.66) where ω0= 6rad/s, ζ0= 0.2 and S₀= 2W/kg and the time and real part of the frequency domain first kernels are plotted in Figure 3.6.

As might be expected since it contains a summation of the velocities of the two masses, the Wiener kernel contains two dominant frequencies: the resonant frequency of each mass.

Interestingly, the first resonance has very little noise, whereas at frequencies above it the kernel is noisier presumably due to effects from the stiffening nonlinearity interacting with the second resonance. The integral over the frequency kernel is calculated as 1.09kg which is very close to the theoretically predicted result using Eq. (3.63) of (m₁+ m₂)/2 = 1.1kg and

Fig. 3.5 Two degree-of-freedom system with two masses connected by a linear spring and one mass connected to the base via a nonlinear spring where F_NL= k₁x+ εx³.

-2 0 2 4 6 8 10 12 14

τ -1

-0.5 0 0.5 1 1.5 2

k1(τ)

(a)

0 5 10 15 20 25 30 35

ω 0

0.5 1 1.5 2

Re[K1(ω)]

(b)

Fig. 3.6 a) Time and b) frequency domain first Wiener kernels under narrowband noise for the 2 mass system of Figure 3.5.

-1 0 1 2 3 4 5 6 7 8 9

Fig. 3.7 a) Time and b) frequency domain first Wiener kernels under narrowband noise for the inverted pendulum system of Eq. (3.59) with k₂= 50 (blue), 100 (red) and 200N/m (yellow).

the power dissipated has been calculated directly from the simulations as 7.87W which is again very close to the power calculated from Eq. (3.62) of 7.89W.

The second MDOF case considered is the inverted pendulum described by Eq. (3.59) where Ef^TM⁻¹f = m. The first kernel is shown in Figure 3.7 for three different values of rod stiffness, k₂= 50, 100 and 200N/m and other parameters m = 1kg, L = 0.75m, c₁= 0.1kgm²/s/rad, c₂= 0.5kg/s, k₁= 5Nm/rad, g = 9.81m/s²and narrowband base acceleration with spectrum given by Eq. (3.66) where ω₀= 1rad/s, ζ0= 0.5 and S₀= 2W/kg. As before the kernel has two dominant frequency components, approximately from oscillating in the θ direction at p

k₁/L²m= 3.0 rad/s and in the r direction at p

k₂/m rad/s. The power calculated directly from the simulations is 94, 81 and 76mW for k₂= 50, 100 and 200N/m and again shows close resemblance to the power calculated from Eq. (3.62) as 95, 81 and 76mW. The integral over the frequency kernel is calculated as 0.49, 0.50, and 0.49kg which is very close to the theoretically predicted result using Eq. (3.63) of 0.5kg.

As the rod stiffness is increased, Figure 3.7 shows that the frequency of the second resonance increases and also the magnitude of the peak decreases. With greater rod stiffness, eventually the axial resonant frequency of the rod will be higher than the maximum frequency provided by the noise which is a computational limitation dictated by the minimum time-step used for the simulations and therefore the computation time of the simulations. From the discussions in Section 3.4.1 this is similar to the case where the axial displacement of the rod is constrained and the system becomes SDOF with equations of motion given by Eq. (3.60) and is discussed in the following section.

-2 0 2 4 6 8 10

Fig. 3.8 a) Time and b) frequency domain first Wiener kernels under narrowband noise for the SDOF inverted pendulum system in Eq. (3.60).

Power for Constrained Systems

For the constrained inverted pendulum system of Eq. (3.60) with an infinitely stiff rod the triple product and therefore the integral over the first kernel are no longer equal to the mass since Ef^TM⁻¹f = mE sin²θ. First, this result and its subsequent implications for power will be verified numerically and then the disparity between Ef^TM⁻¹f for a two degree-of-freedom (2DOF) inverted pendulum with a stiff rod and the SDOF will be discussed.

An inverted pendulum with m = 1kg, L = 0.75m, c₁= 0.1kgm²/s/rad, k₁= 5Nm/rad and g = 9.81m/s² is excited by narrowband base acceleration with spectrum given by Eq.

(3.67) where ω₀= 1rad/s, ζ0= 0.5 and S₀= 2W/kg and the resulting first kernel is plotted in Figure 3.8. For this SDOF case, the response has one resonant frequency and due to the constraint of the infinite rod stiffness, the integral over the kernel in Eq. (3.63) is less than the mass, equalling 0.40kg which is close to the value of Ef^TM⁻¹f calculated directly from simulations as 0.39kg and the decrease can be observed by a reduction in the initial jump of the time kernel from 1 to approximately 0.8 when compared to the unconstrained 2DOF case.

The power dissipated by the damper calculated directly from the simulations is 70mW and from Eq. (3.62) is 70mW suggesting that the equation is still valid for the constrained case.

The transition from a two to a one degree of freedom system as rod stiffness is increased is interesting because of its impact on the integral over the first kernel and therefore triple product Ef^TM⁻¹f and power dissipation. In theory for a finite, but very high, stiffness the system is always 2DOF since the first kernel is valid for frequencies higher than the axial resonant frequency of the rod. The inclusion of the second resonance in the kernel

0 1 2 3 4 5 6 7 8 9

Fig. 3.9 a) Time and b) frequency domain first Wiener kernels under band-limited noise for the 2DOF (blue) and SDOF (red) inverted pendulum systems where the maximum frequency in the simulations is greater than the second resonance of the 2DOF system.

adds to the integral over it and the triple product is equal to the mass. The difference in the kernels between the constrained SDOF and unconstrained 2DOF cases can be seen in Figure 3.9 where for the first resonance the kernels are almost identical, whereas the 2DOF contains the second resonance. The parameters selected are the same as in Figure 3.7, but with k₂= 6000N/m providing an axial natural frequency of 77.5rad/s.

The result on the Ef^TM⁻¹f term is apparent from the integral over the frequency domain kernel. For the 2DOF system the integral contains the second resonance and is 0.48kg, close to the expected theoretical value of Ef^TM⁻¹f /2 = m/2 = 0.5kg. Conversely, the integral over the frequency domain kernel for the SDOF system is Ef^TM⁻¹f /2 = Em sin²(θ ) /2 = 0.34kg and is less than the 2DOF system because the second resonance is not included.

The consequence of modelling constrained systems carefully is illuminated when analysing the power dissipated by such systems. Applying band-limited noise with a flat spectrum of magnitude 2W/kg in the range 0-50rad/s and therefore not including the second resonance, the power dissipated by the SDOF and 2DOF systems respectively directly from simulations is 0.21 and 0.21W. This is very close to the values calculated using Eq. (3.62) of 0.21 and 0.23W. However, when white noise of magnitude 2W/kg is applied to the two systems the power dissipated by the SDOF and 2DOF systems respectively from simulations is very different: 0.22 and 0.29W. Here, the 2DOF has dissipated more power than the SDOF due to its second resonance. Whilst Eq. (3.62) can accurately calculate the power dissipated by constrained and unconstrained systems, failure to include all resonances of a system in the frequency range of interest will produce underestimates of the power.

In summary, the theory of Eqs. (3.62) and (3.63) has been validated within the bounds of reasonable noise error provided the frequency range used includes all significant frequency responses of the system. Where this is not the case, either the maximum frequency content of the simulations should be increased or a constraint should be added to the system such that the higher resonance is removed and Ef^TM⁻¹f is no longer equal to M_Tot.