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It is well known that the time period of a harmonic oscillator when mass $m$ and spring constant $k$ are constant is $T=2\pi\sqrt{m/k}$.

However, I would be interested to know what the time period is if $k$ is not constant. I have searched hours after hours for right answers from Google and came up with nothing. I am looking for an analytical solution.

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    $\begingroup$ When you say "not constant", do you mean "k depends on displacement x"? In that case, you have what's called a "non-linear oscillator" (which you can google. There is no single "analytical" solution, and period depends on amplitude). If the mass changes, you have to wonder "how"? Does the mass increase? What is its momentum when it "arrives"? Does it decrease? If so, does mass get ejected equally in all directions? Some clarifications are needed... $\endgroup$ – Floris Aug 18 '15 at 10:30
  • $\begingroup$ I would be interested of general case where k can be depend on displacement or time or any other variable. This might be impossible? Then we could think that k depends on t --> k(t). Mass is not varable but constant. $\endgroup$ – dr_mushroom Aug 19 '15 at 5:58
  • $\begingroup$ And, I forgot to mention: Amplitude of motion is known in this situation $\endgroup$ – dr_mushroom Aug 19 '15 at 6:10
  • $\begingroup$ In general, if k gets larger with displacement the frequency increases at larger amplitudes; if k get less, it decreases. The standard example is a "real" pendulum which becomes nonlinear at higher displacements. That has been well studied. I am not aware of a general approach for "any" nonlinear oscillator but these lecture notes are a start. $\endgroup$ – Floris Aug 19 '15 at 12:50
  • $\begingroup$ Thanks Floris for answer! I've been trying to understand how I could use these lecture notes (I have been trying to read many notes similar to those you send me) but I have to say they go much over my understanding even I have some education in math (I'm engineer). This question is a kind of sideproject for for my real workproject. $\endgroup$ – dr_mushroom Aug 19 '15 at 13:18
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Here is a solution for a spring force that varies directly with displacement. It thus varies with time implicitly, but has no explicit dependence on time or any other variable.

Givens and Assumptions

  • oscillator with mass $m$
  • amplitude of oscillation $A$
  • oscillator displacement, $x$, varies with time, but $x(t)$ is unknown
  • spring applies force varying with displacement, $F(x)$
  • The function $F(x)$ is an odd function, that is $F(-x) = -F(x)$ (otherwise the amplitude could be different in the positive and negative directions - see below for what to do in this case)
  • equilibrium position is $x=0$, that is $F(0) = 0$ (for convenience only)

Objective

Find the period of oscillation, $T$

Solution

Starting from conservation of energy, the sum of the kinetic and potential energy of the mass must be equal to the total energy, which is constant. $$KE(x)+PE(x)=E$$ $$KE(x)=\frac{1}{2}mv^2(x)$$ $$PE(x)=\intop_0^xdx'\,F(x')$$ So $PE(x)$ is the potential energy stored in the spring, with $x'$ as just an integration variable.
We can think of $PE(x)$ as another way of defining the force-displacement relationship of the spring. We can define the potential energy versus displacement or the force versus displacement, and getting the other one is fairly easy.

Now, at $x=A$, $KE(x=A)=0$, so $PE(A)=E$ is known.

And so we have $$\frac{1}{2}mv^2(x)=PE(A)-PE(x)$$ Solving for $v(x)$, $$v(x)=\sqrt{\frac{2}{m}\left(PE(A)-PE(x)\right)}$$

Because $v=\frac{dx}{dt}$, we can also write $$dt=\frac{dx}{v(x)}$$ Integrating both sides, the time to go from a position $x_0$ to $x_1$ is $$\Delta t = \intop_{x_0}^{x_1}\frac{dx}{v(x)}$$ In particular, we know the time required to go from $x=0$ to $x=A$ is $T/4$, so $$T=4\intop_0^A\frac{dx}{v(x)}$$ $$T=4\intop_0^A\frac{dx}{\sqrt{\frac{2}{m}}\sqrt{PE(A)-PE(x)}}$$ which further simplifies to...

Final Result

$$T=\sqrt{8m}\intop_0^A\frac{dx}{\sqrt{PE(A)-PE(x)}}$$

Check of Result

For the linear case, $F(x)=kx$, so $PE(x)=\frac{1}{2}kx^2$ and $PE(A)=\frac{1}{2}kA^2$, which gives $$T=\sqrt{8m}\intop_0^A\frac{dx}{\sqrt{\frac{k}{2}}\sqrt{A^2-x^2}} =4\sqrt{\frac{m}{k}}\intop_0^A\frac{dx}{\sqrt{A^2-x^2}}$$ This integral can be looked up in a table, to obtain $$T=4\sqrt{\frac{m}{k}}\left(\sin^{-1}(1)-sin^{-1}(0)\right)=4\sqrt{\frac{m}{k}}\frac{\pi}{2}$$ $$T=2\pi\sqrt{\frac{m}{k}}$$ as expected. (QED)


We can dispense with the assumption that $F(x)$ is odd if we define two amplitude values: $A_+ > 0$ for the amplitude in the positive direction and $A_- < 0$ for the amplitude in the negative direction.

The total oscillator energy, $E = PE(A_+) = PE(A_-)$, so we can still call it $PE(A)$ as long as we remember what that means now.

Then, the period is twice the time required to go from $A_-$ to $A_+$, and so $$T=2\intop_{A_-}^{A_+}\frac{dx}{\sqrt{\frac{2}{m}}\sqrt{PE(A)-PE(x)}}$$ $$T=\sqrt{2m}\intop_{A_-}^{A_+}\frac{dx}{\sqrt{PE(A)-PE(x)}}$$

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  • $\begingroup$ Great sketch of a general framework. More useful than the Frobenius method in the other answer and also dispenses with the question of convergence of the series (which would need to be fully checked in the other answer). $\endgroup$ – WetSavannaAnimal Aug 20 '15 at 4:28
  • $\begingroup$ +1--definitely a lot more practical than the Frobenius method, haha. The only advantage I can see of the Frobenius method is that you don't have to take that $F\left(x\right)$ is an odd function. $\endgroup$ – Eric R. Anschuetz Aug 20 '15 at 5:39
  • $\begingroup$ For you also: Ho** s**t thanks! This is simple and awesome $\endgroup$ – dr_mushroom Aug 20 '15 at 11:47
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    $\begingroup$ I still think the Frobenius method is the way to go for explicit dependence on time, and could be extended to other cases as well. I've added a supplement that explains what to do if the function is not odd. I just think the original explanation is simpler if "amplitude" still means what we're familiar with from the linear oscillator. $\endgroup$ – Captain Numerical Aug 20 '15 at 11:52
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From Newton's second law we have (whether $k$ is constant or not) that: \begin{equation} m\ddot{x}+kx=0 \end{equation} The only difference is whether or not $k$ is a function of $t$ or not. If it is a function of $t$, the only general way to solve this differential equation is by using Taylor expansions. Let us take: \begin{equation} x\left(t\right)=\sum_{n=0}^\infty a_nt^n \end{equation} and: \begin{equation} k\left(t\right)=\sum_{n=0}^\infty b_nt^n \end{equation} Our differential equation then becomes: \begin{equation} \begin{aligned} m\ddot{x}+kx&=0\\ \implies\sum_{n=2}^\infty mn\left(n-1\right)a_nt^{n-2}+\left(\sum_{n=0}^\infty b_nt^n\right)\left(\sum_{n=0}^\infty a_nt^n\right)&=0\\ \implies\sum_{n=0}^\infty\left[m\left(n+2\right)\left(n+1\right)a_{n+2}+\sum_{i=0}^na_ib_{n-i}\right]t^n&=0\\ \implies m\left(n+2\right)\left(n+1\right)a_{n+2}+\sum_{i=0}^na_ib_{n-i}&=0\forall n\\ \implies a_{n+2}&=-\frac{\sum_{i=0}^na_ib_{n-i}}{m\left(n+2\right)\left(n+1\right)}\forall n \end{aligned} \end{equation} As the $k\left(t\right)$ is known all of the $b_n$ are known, and if we know two of our initial conditions two of the $a_n$ are known (let us say $a_0$ and $a_1$). Using this recurrence relation, one can read off all of the $a_n$--that is, one knows all of the coefficients of the Taylor series for $x$. You can't really see too much more analytically in this super general case (to find a period, one would have to find a $k\left(t\right)$ that generated $a_n$ such that $x\left(t\right)$ was periodic, and read off the period from that function), but a good sanity check is to check if we recover our same answer when $k$ is a constant $k_c$; that is, when $b_0=k_c$ and $b_n=0$ for all $n>0$. In this case we find that: \begin{equation} \begin{aligned} a_2&=-\frac{a_0k_c}{2m}\\ a_3&=-\frac{a_1k_c}{6m}\\ a_4&=\frac{a_0k_c^2}{24m^2}\\ &\vdots \end{aligned} \end{equation} Following the pattern, we notice that the $a_n$ for even $n$ give the Taylor series for $a_0\cos\left(\sqrt{\frac{k_c}{m}}t\right)$ and the $a_n$ for odd $n$ give the Taylor series for $a_1\sin\left(\sqrt{\frac{k_c}{m}}t\right)$, yielding an angular frequency of $\sqrt{\frac{k_c}{m}}$ and therefore a period of $2\pi\sqrt{\frac{m}{k_c}}$.

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    $\begingroup$ +1 See user89613's answer as a complement to yours. $\endgroup$ – WetSavannaAnimal Aug 20 '15 at 4:29
  • $\begingroup$ Ho** s**t thanks Captain Numerical and Eric R. Anschuetz! $\endgroup$ – dr_mushroom Aug 20 '15 at 11:44
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To obtain some kind of practical answer, you have to determine how k varies. For example, if k varies with temperature, I would determine its value at -50, 0, and 50 degrees, then use those values and calculate T (which varies inversely as the square root of k). I would Use more points if a higher accuracy is required. If a formula is required, I would use the "best fit curve" through the points, to generate it.

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  • $\begingroup$ I gladly show you all what I am doing after I have some preminilary results. The problem had something to do with cylinder full of liquid. Cylinder has also piston. This piston is pushed down with weight that is dropped from above. This whole process can be descibed quite accurately with equation mx''(t)+k(p)x(t)=0 in which k is dependent of pressure. This pressure dependence of k is because liquid has pressure dependent stifness (yes this really is true, pressure is thousands of bars). So liquid acts like a spring. Well then question is that how to get k(p)-->k(x).. $\endgroup$ – dr_mushroom Aug 20 '15 at 13:48
  • $\begingroup$ The trick here is that I also know that because I have measured x(t) and corresponding p(t). From this knowledge I think I can calculate k(x). Why I am so interested about this? Because I would like to measure stifness of that spring from period of oscillation. And I just might be able to do it now when you have helped me $\endgroup$ – dr_mushroom Aug 20 '15 at 13:52
  • $\begingroup$ And I almost forgot to say: dropped the weight from different heights, so I know period of oscillation with different pressure "pulses".. $\endgroup$ – dr_mushroom Aug 20 '15 at 13:57
  • $\begingroup$ And what could be the practical outcome here? Just to know how this kind of device works theoretically, not just practically $\endgroup$ – dr_mushroom Aug 20 '15 at 14:01
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Alright, according to my knowledge there are some cases with a time dependent spring constant where a closed form solution is known. One of my favorites is the following where the spring constant is a power function. Assume that $k/m = \omega^2/t^\beta$ where $\omega \in {\mathbb R}$ and $\beta \ge 0$. Then the ODE in question take a form: \begin{equation} \ddot{x}_t + \frac{\omega^2}{t^\beta} x_t = 0 \end{equation} and it has the following linearly independent solutions: \begin{equation} \sqrt{t} J_{\pm \frac{1}{2 \beta-2}}\left[\frac{\omega}{\beta-1} t^{1-\beta} \right] \end{equation} where $J_\cdot[]$ is a Bessel function.

The result is proven in several different ways in https://math.stackexchange.com/questions/673737/solution-to-a-second-order-ordinary-differential-equation .

A quick and dirty "proof" is given by the following Mathematica code:

In[166]:= w =.; b =.; t =.;
Table[FullSimplify[(D[#, {t, 2}] + w^2/t^(2 b) #) & /@ {Sqrt[
      t] BesselJ[eps/(2 b - 2), w/(b - 1) t^(1 - b)]}], {eps, -1, 1, 
  2}]

Out[167]= {{0}, {0}}

Now, having said all this it would be natural to generalize our ODE by by replacing the power function by a linear combination of two power functions. In other words we seek solutions to the following ODE:

\begin{equation} \ddot{x}_t + \left(\frac{\omega_1^2}{t^\beta_1} + \frac{\omega_2^2}{t^\beta_2}\right)x_t = 0 \end{equation}

I have long been struggling to solve this problem without any considerable success (however see this https://math.stackexchange.com/questions/1018897/an-ordinary-differential-equation-with-time-varying-coefficients). The usual way "plug the result into Mathematica" or "ask a mathematician" has led to nothing. As it seems mathematics as usual lags several decades behind physics and it will never catch up...;-).

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