How do I calculate the effective motion for several excited harmonic oscillators?

Let’s say I have several harmonic oscillators, e.g. springs, where each has a displacement $x(t)$ and velocity $v(t)$ with total energy $E = frac{1}{2} m v^2 + frac{1}{2} k x^2 $.

Now these oscillators are subject to a force $F(t)$. This force could be anything, but it is the same for each oscillator. The relative phases of the oscillators are a free parameter.

I am only interested in the center-of-mass motion of all oscillators combined. My question is: Is there a way to calculate the center-of-mass motion without having to calculate each oscillator individually and then summing up the motion?

I feel like this is a very basic question, but I don’t know what to google.