Finding time period of oscillations in a multiple spring system attached to a solid cylinder A solid cylinder of mass $m$ and radius $R$ is kept in equilibrium on horizontal rough surface. Three unstretched springs of spring constant $k$, $2k$, $3k$ are attached to cylinder as shown in the figure. Find the period of small oscillations. Given that surface is rough enough to prevent slipping of cylinder. 

Now, I'm very, very stuck. Probably because I'm rather lost. 

Attempt 1:
I tried finding the restoring torques and forces at a displacement $x$ not the extreme or mean. Once I found these, I tried to equate the acceleration  and find the effective constant '$k$'. But that didn't work because I don't understand how to use the $2k$ spring, or for that matter whether torques or forces should both be considered and how...

Attempt 2:
I tried the energy method, I added spring potential energies. But I got stuck at the rotational kinetic energies. How are the kinetic  energies considered?

Now, I know my attempts look very weak because my understanding of Rotational mechanics is rather shaky, so any help is appreciated. Specifically with regard to What is considered a restoring force here, both torques? torques and which linear restoring force? 
Answer is known if that is needed
 A: First, let's review the basic ideas of simple harmonic motion (I'm assuming an early university level). Starting with Newton's equation:
$$F=ma$$
and using Hooke's law
$$ma=-kx$$
then recognizing that acceleration is the second derivative of position x
$$mx''= -kx$$
We know that simple harmonic motion is sinusoidal, so we substitute $x=\sin(\omega t)$ into the above expression (the choice of sin or cos is arbitrary, and doesn't affect any results).
$$m \frac{d^2}{dt^2}\sin(\omega t) = -k\sin(\omega t)$$
After evaluating the derivative, the sin cancels out, leaving a relation between the spring constant, mass, and angular frequency.
$$m \omega^2=k$$
This can be rearranged to solve for angular frequency.
$$\omega = \sqrt{\frac{k}{m}}$$
Recalling some of the relations between frequency and period
$$\omega = 2\pi f = 2\pi/T$$
Finally, solving for the period:
$$T = 2\pi \sqrt{\frac{m}{k}}$$
Now, time to apply these ideas to your problem. The idea is to find an effective mass and effective spring constant we can use in the above formula.
In this kind of problem where there's a constraint (the no-slip condition) it's usually much easier to apply energy ideas, because they entirely ignore the constraining force. Additionally, it's relatively easy to write energy expressions for rotation, rather than do a sum of forces.
First, we write an expression for the potential energy
$$E_p = \frac{1}{2}(k_1{x_1}^2 + k_2{x_2}^2 + k_3{x_3}^2)$$
I'm using the x variables to represent the displacements on each spring, with the corresponding spring constants.
Notice that your problem has only one degree of freedom. In other words, if you know the amount of horizontal displacement, you also know the amount of rotation, and can figure out the displacement for any of the springs.
To find the effective spring constant, choose a primary position variable, and rewrite the other positions in terms of this one. You should be able to rewrite the potential energy equation into this form:
$$E_p = \frac{1}{2}k_{eff}x^2$$
This gives you your effective spring constant.
Now to find the effective mass, write an expression for the kinetic energy, making sure to include both rotational and translational terms
$$E_k = \frac{1}{2}(I{\omega_{rot}}^2 + m{v_{cm}}^2)$$
The variables I'm using here are $I$ for moment of inertia, $\omega_{rot}$ for angular velocity of the rotation (Note: this is different from the angular frequency used above), and $v_{cm}$ for the velocity of the center of mass, which is also the center of the circle.
To find the effective mass, you must rewrite the angular velocity and center of mass velocity in terms of the velocity corresponding to the same position variable you chose when finding the effective spring constant. Doing this, you get:
$$E_k = \frac{1}{2}m_{eff}v^2$$
Finally, substitute the effective mass and spring constant into the expression for the period of the simple harmonic motion.
