Does the Period of a Harmonic Oscillator Change under Uniform Translation?

High speed trains in China in uniform translation with identical clocks.

The most basic of all clocks is a harmonic oscillator consisting of a body of unit mass connected by a linear spring to a fixed point, described by the differential equation expressing Newton's 2nd Law:

• $\frac{d^2x}{dt^2}=(x_0-x)$   (1)

where $x(t)$ is the position of the body at time $t$ on an $x-axis$ and $x_0$ is the fixed point with $x_0-x$ the spring force. The solution $x(t)$ is periodic in time with period $2\pi$.

Suppose now we introduce another coordinate axis with coordinate $x^\prime = x-vt$, where $v$ is a given constant velocity, expressing that the $x^\prime$-axis moves with the constant velocity $v$ with respect to the $x$-axis in uniform translation. Substituting $x=x^\prime +vt$ into (1), we get

• $\frac{d^2x^\prime}{dt^2}=(x_0^\prime-x^\prime )$   (2),

because $\frac{d^2(vt)}{dt^2}=0$. We see that (1) reads identically the same as (2) sharing the same time $t$. We thus see that the motion of a harmonic oscillator is Galilean invariant since the equation describing the motion reads the same in the two space coordinate systems connected by the Galilean transformation $x^\prime = x -vt$ of uniform translation.

One way to express our experience is to say that the motion of a harmonic oscillator including the period is independent of uniform translation. This is what we expect: It is unthinkable that the physics including the period of a harmonic oscillator, could be influenced by uniform translation. Your clock must tick the same rate waiting for the train in the train station and in the train in uniform translation. Anything else is unthinkable, based on the basic Newtonian mechanics of a harmonic oscillator. 

But the special theory of relativity SR says that the clock in motion ticks at a slower rate. 

What is your conclusion? Evidence that SR is correct or false from a physical point of view?

PS1 Note that (1) has the same form in all inertial systems under uniform translation and thus satisfies Einstein's Relativity Postulate. What is then wrong with (1) from the view of SR? Well, SR is obsessed with the speed of light, but a harmonic oscillator has nothing to do with light because it is a mass-spring system. So what is wrong with the harmonic oscillator form the point of view of SR is that it is mechanical, but is it anything wrong with being a mechanical system?  Isn't a harmonic oscillator an example of basic physics?  If not, what physics is then SR?

PS2 To see that (1) is not Lorentz invariant as required in SR, recall this post. It means that (1) from the point of view of SR is not a law of physics. Einstein thus claims that a harmonic oscillator is not physics.  Do you agree?

PS3 Things like this appears to be impossible to discuss with physicists seemingly brain-washed by special relativity.