"Optimum" values depends on what you are trying to optimize for. Do you want to minimize displacement of one or more components, or minimize reaction force between any two components, or stay away from certain frequencies that may shake the operators organs or eyeballs, or maximize force between drill and object being drilled, or....? Your specification of constant c and k values makes the problem potentially easier to solve, but only after declaring what you want to optimize for.

You should write the governing dynamic equation based on F = M*A, and from that you can solve for equations of motion, which will then allow for the calculation of value(s) individual optimization parameter based on the system definition. Its a harmonically forced, damped oscillator so there will be a transient and a steady state response component. If you only care about one vs the other it may simplify what you have to do.

Then differentiate the governing dynamic equation and solve for maxima or minima in the solution equation of motion (solution to governing equation defined by sys config) depending on system definition values. Choose each optimization parameter, one at a time, and solve to see what constants of sys config are optimal for that parameter, then compare and contrast to see which parameters are more or less sensitive to variation of the others. This should give a good feel for the overall response of the system and its inter-related sensitivities, always a good idea when designing dynamic systems.

In general, when frequencies are of concern, adding stiffness will (typically) drive up response frequency and (typically) drive down response amplitude. Adding mass will typically have the opposite effect. One must take care when near the fundamental or resonance frequencies of the system, where slight changes in mass or stiffness my drive the system response closer to resonance which can lead to higher amplitude and associated higher resultant contact forces between components.

To avoid resonance, drive the stiffness higher and the mass lower in order to place the fundamental frequency above any possible excitation frequency of the system. If you are looking for resonance to increase drill effectiveness, for example, the variable system definition variables may be chosen to match the system fundamental frequency to the forcing frequency or a low harmonic thereof. But be careful to check that forces, amplitudes or any other critical value of the physical system is not exceeded (i.e. you can't pop out the eyes of the operator just to drill faster, so you have to know what constraints may exist practically for operational limitations).

TimH

If the springs and dampers are linear, the dynamic equilibrium conditions for the forces (equations of motion) give a system of two coupled ordinary differential equations. These equations can be solved analytically. The solution can also be found in literature. With these solutions an optimization should be possible.

https://scholar.google.com/scholar?q=source+and+propagation+of+nonlinearity+in+a+vibration+isolator+with+geometrically+nonlinear+damping&hl=en&as_sdt=0&as_vis=1&oi=scholart#d=gs_qabs&p=&u=%23p%3DQPdNkwtv98UJ