Can we shake an atom that hard so the nucleus flies out? Take a single atom. A collection of hadrons (nucleons) sitting at the center rather inert, surrounded by orbitals of leptons (electrons).
Somehow we let the atom move slowly oscillatory. Then we increase the frequency of the oscillation steadily. Will the motion of the whole be such that the nucleus lags behind the motion of the orbitals? Will the nucleus fly out through the orbitals at some moment? Or will the orbitals adjus or will the whole atom just be torn apart?
 A: You imagine incorrectly - a molecule or atom that is gripped firmly is a solid, full stop. You can't just go and grab an atom without doing chemistry to it to get it into solid form: they're almost perfectly bouncy and perfectly slippery. We can modify the question so that it's possible-ish by putting the atom in a solid. Now the atom is in fact gripped firmly on all sides and we can mechanically shake it.
Let's use hydrogen, since it's simple.*
The electron has a quantized set of energy levels. These energy levels correspond to the size and shape of the electron orbitals. The electrons can't be continuously displaced from their energy levels, but skip discontinuously between them. Therefore it is impossible for the nucleus to shift around continuously inside of the orbital.
It is possible for the hydrogen atom (or larger atoms with more complex systems) to be hit hard enough that the electron is temporarily excited to a higher energy level. The electron will then decay back down to a lower level after a probabilistically determined (but very short) amount of time, emitting a photon.
It is even possible to knock the atom so hard that its electron flies free entirely. (Or in the case of heavier elements, one or more electrons.) This is called ionization because the result is an ion - an atom with a nonstandard charge.
So, could we take our hydrogen atom in a solid - call it a piece of plastic - and vibrate it hard enough to make it glow blue? After all, each direction reversal in the oscillation imparts energy to each atom in the solid, so maybe we could impart enough to pop the electron up an energy level - the quantum-mechanical equivalent of hitting it hard enough to make the nucleus shift relative to the electron/s.
No. The first reason is that the bonds that make the solid structurally stiff enough to vibrate are far weaker per unit volume than the energy density required to excite our electron. Either the sample or the machine we used to vibrate it would break.
What if we constructed some sort of device that would vibrate the sample using something that couldn't break, for instance by blasting it with lasers or something?
Still no - the second reason is that the bonds holding the solid together are themselves much weaker per unit volume than the energy density required to excite our electron. The sample will probably glow because of blackbody radiation, but it will vaporize before it emits photons from electron energy level decay.
Okay, now we've vaporized the sample. We can't grip it any more, but we can still blast it with lasers or fling other atoms at it to bounce it back and forth violently. Now can we shake it hard enough?
Yes. Now since there aren't any material constraints keeping our energy density from getting too high, there's nothing stopping us from blasting the atom hard enough to make its electrons skip up to higher energy levels.

*Hydrogen is much less simple if we put it into a compound, so maybe the choice of atom doesn't matter much. Fortunately we can get by with order-of-magnitude estimates for energy densities - I think it should be order of $10^{27} \;\text{eV}$ per kg to start affecting electron energy levels, order of $10^{25} \;\text{eV}$ per kg to vaporize the sample, if it's a plastic.
A: For the question about whether the atom will be torn apart, yes. Heat can shake an atom violently at extremely high temperatures, causing nuclei to disintegrate, with even higher and more incomprehensible temperatures allowing the disintegration of protons and neutrons into their fundamental particles (quarks). This is in fact what happened in the Big Bang, the immense heat allowed for a short period of time in which it was so hot that nuclei couldn't form, and neither could quarks themselves assemble themselves into protons and neutrons.
A: The main handle that I'm familiar with for grabbing atoms is using an electric field. We can apply force to atoms using the electric field from laser beams. These electric fields put forces on both the electron cloud and the nucleus. However, a single proton is about 2000 times heavier than the electron. This means that the electric fields predominantly put forces on and cause motion of the electrons rather than the nucleus.
If we put on large forces to "shake" the ion we cause the electron cloud to oscillate back and forth. If our laser is tuned to resonant frequencies of the electron-nucleus system then we can efficiently drive energy into the electrons and cause larger and larger charge oscillations. We can also drive energy just by off-resonantly putting huge electric fields (high intensity, many photons).
Yes, if shake hard enough we can drive the electrons into "continuum" or "unbound" states, as opposed to the "bound" states that constitute an atom. As others have pointed out, this is called ionization. For many-electron atoms we typically strip electrons off one at a time, requiring more and more energy for every subsequent ionization.
So the answer to your question is yes, we can rip apart atoms by shaking them hard enough. Though it's not quite right to think of it as shaking so hard that the nucleus falls out because (in the center of mass frame) it is actually the electrons flying away from nucleus which doesn't move much.
A: Neutrons can hit a nucleus and give it such a momentum that it will leave the atom very quickly. See my paper on arXiv.
