Effect of Uncertainty on the support’s dynamics on the isolation platform dynamics

If we write the equation of motion of the system in Figure 1, we obtain:

After eliminating \(x^\prime\), we obtain:

Let the support have:

• a nominal mass of \(m^\prime = 1000\ [kg]\)
• a nominal stiffness of \(k^\prime = 10^8\ [N/m]\)
• a nominal damping of \(c^\prime = 10^5\ [N/(m/s)]\)

mpi = 1e3;
cpi = 5e4;
kpi = 1e8;

Let’s also consider some uncertainty in those parameters:

mp = ureal('m', mpi, 'Percentage', 30);
cp = ureal('c', cpi, 'Percentage', 30);
kp = ureal('k', kpi, 'Percentage', 30);

The compliance of the support without the isolation platform is \(\frac{1}{m^\prime s^2 + c^\prime s + k^\prime}\) and its bode plot is shown in Figure 2.

One can see that support has a resonance frequency of \(\omega_0^\prime = 50\ Hz\).

Let’s first fix the mass of the payload to be isolated:

m = 100;

And we generate three isolation platforms:

• A soft one with \(\omega_0 = 0.1 \omega_0^\prime = 5\ Hz\)
• A medium stiff one with \(\omega_0 = \omega_0^\prime = 50\ Hz\)
• A stiff one with \(\omega_0 = 10 \omega_0^\prime = 500\ Hz\)

The obtained dynamics from \(F\) to \(x\) for the three isolation platform are shown in Figure 3.

Let’s now consider a general support described by its compliance \(G^\prime(s) = \frac{x^\prime}{F^\prime}\) as shown in Figure 4.

Now let’s consider the system consisting of a mass-spring-system (the isolation platform) on top of a general support as shown in Figure 5.

We have to following equations of motion:

And by eliminating \(F^\prime\) and \(x^\prime\), we find the plant dynamics \(G(s) = \frac{x}{F}\).

In order to verify that the formula is correct, let’s take the same mass-spring-damper system used in the system shown in Figure 1: \[ \frac{x^\prime}{F^\prime} = \frac{1}{m^\prime s^2 + c^\prime s + k^\prime} \]

And we obtain \[ \frac{x}{F} = \frac{m^\prime s^2 + c^\prime s + k^\prime}{(ms^2 + cs + k)(m^\prime s^2 + c^\prime s + k^\prime) + ms^2(cs + k)} \] Which is the same transfer function that was obtained in section 1 (Eq. \eqref{eq:plant_simple_system}).

We model the support by a mass-spring-damper model with some uncertainty.

The nominal compliance of the support is corresponding to the compliance of a mass-spring-damper system with a mass of \(1000\ kg\) and a stiffness of \(10^8\ [N/m]\). The main resonance of the support is then \(\omega^\prime = \sqrt{\frac{m^\prime}{k^\prime}} \approx 50\ Hz\).

m0 = 1e3;
c0 = 5e4;
k0 = 1e8;

Gp0 = 1/(m0*s^2 + c0*s + k0);

Let’s represent the uncertainty on the compliance of the support by a multiplicative uncertainty (Figure 6): \[ G^\prime(s) = G_0^\prime(s)(1 + w_I^\prime(s)\Delta_I(s)) \quad |\Delta_I(j\omega)| < 1\ \forall \omega \]

This could represent unmodelled dynamics or unknown parameters of the support.

We choose a simple uncertainty weight: \[ w_I(s) = \frac{\tau s + r_0}{(\tau/r_\infty) s + 1} \] where \(r_0\) is the relative uncertainty at steady-state, \(1/\tau\) is the frequency at which the relative uncertainty reaches \(100\ \%\), and \(r_\infty\) is the magnitude of the weight at high frequency.

The parameters are defined below.

r0 = 0.5;
tau = 1/(100*2*pi);
rinf = 10;

wI = (tau*s + r0)/((tau/rinf)*s + 1);

We then generate a complex \(\Delta\).

DeltaI = ucomplex('A',0);

We generate the uncertain plant \(G^\prime(s)\).

Gp = Gp0*(1+wI*DeltaI);

A set of uncertainty support’s compliance transfer functions is shown in Figure 7.

Let’s express the uncertainty of the plant \(x/F\) as a function of the parameters as well as of the uncertainty on the platform’s compliance:

Let’s first fix the mass of the payload to be isolated:

m = 100;

And we generate three isolation platforms:

• A soft one with \(\omega_0 = 5\ Hz\)
• A medium stiff one with \(\omega_0 = 50\ Hz\)
• A stiff one with \(\omega_0 = 500\ Hz\)

Soft Isolation Platform:

k_soft = m*(2*pi*5)^2;
c_soft = 0.1*sqrt(m*k_soft);

G_soft = 1/(m*s^2 + c_soft*s + k_soft + m*s^2*(c_soft*s + k_soft)*Gp);
G0_soft = 1/(m*s^2 + c_soft*s + k_soft + m*s^2*(c_soft*s + k_soft)*Gp0);
wiI_soft = Gp0*m*s^2*(c_soft*s + k_soft)*G0_soft*wI;

Mid Isolation Platform

k_mid = m*(2*pi*50)^2;
c_mid = 0.1*sqrt(m*k_mid);

G_mid = 1/(m*s^2 + c_mid*s + k_mid + m*s^2*(c_mid*s + k_mid)*Gp);
G0_mid = 1/(m*s^2 + c_mid*s + k_mid + m*s^2*(c_mid*s + k_mid)*Gp0);
wiI_mid = Gp0*m*s^2*(c_mid*s + k_mid)*G0_mid*wI;

Stiff Isolation Platform

k_stiff = m*(2*pi*500)^2;
c_stiff = 0.1*sqrt(m*k_stiff);

G_stiff = 1/(m*s^2 + c_stiff*s + k_stiff + m*s^2*(c_stiff*s + k_stiff)*Gp);
G0_stiff = 1/(m*s^2 + c_stiff*s + k_stiff + m*s^2*(c_stiff*s + k_stiff)*Gp0);
wiI_stiff = Gp0*m*s^2*(c_stiff*s + k_stiff)*G0_stiff*wI;

The obtained transfer functions \(x/F\) for each of the three platforms are shown in Figure 9.

The obtain result is very similar to the one obtain in section 1, except for the stiff isolation that experience lot’s of uncertainty at high frequency. This is due to the fact that with the current model, at high frequency, the support’s compliance uncertainty is much higher than the previous model.

Now that we know the expression of the uncertainty on the plant, we can wonder what parameters of the isolation platform would lower the plant uncertainty, or at least bring the uncertainty to reasonable level.

The uncertainty of the plant is described by an inverse multiplicative uncertainty with the following weight: \[ w_{iI}(s) = \frac{ms^2(cs + k)G_0^\prime(s) w_I(s)}{ms^2 + cs + k + ms^2(cs + k)G_0^\prime(s)} \]

Let’s study separately the effect of the platform’s mass, damping and stiffness.