The simple spring system consists of a mass m attached to the lower end of a spring that is itself suspended vertically from a mounting. The system is in its equilibrium position when it is at rest.
The mass is set in motion by one or more of the following means: displacing the mass from its equilibrium position, providing it with an initial velocity, or subjecting it to an external force F(t).
Hooke's Law: The restoring force F of a spring is equal and opposite to the forces applied to the spring and is proportional to the extension (contraction) l of the spring as a result of the applied force; that is, F = —kl, where k denotes the constant of proportionality, generally called the spring constant.
Example 1. A steel ball weighing 128 lb is suspended from a spring, whereupon the spring is stretched 2 ft from its natural length. The applied force responsible for the 2-ft displacement is the weight of the ball, 128 lb. Thus, F = -128 lb. Hooke's law then gives -128 = -k(2), or k = 64 lb/ft.
For convenience, we choose the downward direction as the positive direction and take the origin to be the center of gravity of the mass in the equilibrium position. We assume that the mass of the spring is negligible and can be neglected and that air resistance, when present, is proportional to the velocity of the mass. Thus, at any time t, there are three forces acting on the system: F(t), measured in the positive direction; a restoring force given by Hooke's law as `F_s=-kx` , k > 0; and a force due to air resistance given by `F_a=-a (dx)/(dt)'`, a > 0, where a is the constant of proportionality. Note that the restoring force `F_s` always acts in a direction that will tend to return the system to the equilibrium position: if the mass is below the equilibrium position, then x is positive and -kx is negative; whereas if the mass is above the equilibrium position, then x is negative and -kx is positive. Also note that because a > 0 the force `F_a` due to air resistance acts in the opposite direction of the velocity and thus tends to retard, or damp, the motion of the mass.
It now follows from Newton's second law that `m(d^2x)/(dt^2)=-kx-a (dx)/(dt)+F(t)` or `(d^2x)/(dt^2)+a/m (dx)/(dt)+k/m x=(F(t))/m` .
If the system starts at t = 0 with an initial velocity `v_0` and from an initial position `x_0` , we also have the initial
conditions `x(0)=x_0,\ x'(0)=v_0` .
The force of gravity does not explicitly appear in differential equation, but it is present nonetheless. We automatically compensated for this force by measuring distance from the equilibrium position of the spring. If one wishes to exhibit gravity explicitly, then distance must be measured from the bottom end of the natural length of the spring. That is, the motion of a vibrating spring can be given by `(d^2x)/(dt^2)+a/m (dx)/(dt)+k/m x=g+(F(t))/m` if the origin, x = 0, is the terminal point of the unstretched spring before the mass m is attached.
Example 2. A mass of 2 kg is suspended from a spring with a known spring constant of 10 N/m and allowed to come to rest. It is then set in motion by giving it an initial velocity of 150 cm/sec. Find an expression for the motion of the mass, assuming no air resistance.
Here, externally applied force on the mass F(t) = 0, and no resistance from the surrounding medium, a = 0. The mass and the spring constant are given as m = 2 kg and k = 10 N/m, respectively, so differential equation becomes `(d^2x)/(dt^2)+5x=0` . The roots of its characteristic equation `r^2+5=0` are purely imaginary, so its solution is
At t=0, the position of the ball is at the equilibrium position `x_0=0` m. Applying this initial condition we find that `x(0)=0=c_1cos(sqrt(5)*0)+c_2sin(sqrt(5)*0)` or `c_1=0` .
So, `x(t)=c_2sin(sqrt(5)t)` . Thus, `x'=sqrt(5)c_2cos(sqrt(5)t)` .
Applying second initial condition `x'(0)=150 cm/s=1.5 m/s` gives that `x'(0)=1.5=sqrt(5)c_2cos(sqrt(5)*0)` or `c_2=1.5/sqrt(5)` .
Finally, `x(t)=1.5/sqrt(5) sin(sqrt(5)t)` .
Example 3. A 128-lb weight is attached to a spring whereupon the spring is stretched 2 ft and allowed to come to rest. The weight is set into motion from rest by displacing the spring 6 in. (which is equal to 0.5 ft.) above its equilibrium position and also by applying an external force `F(t)=8sin(4t)` . Find the subsequent motion of the weight if the surrounding medium offers a negligible resistance.
Here `W=128` . We need to find m using W. Since `W=mg` and `g=32 ft/s^2` then `m=W/(g)=128/32=4` . From Hooke's law: `W=-kl` where `l=2` .
So, `k=-W/(l)=-128/2=64 (lb)/(ft)` .
There is no air resistance, so a=0.
Thus, given differential eqaution is `(d^2x)/(dt^2)+16x=2sin(4t)` .
Solution of the corresponding homogeneous equation is `x_h=c_1cos(4t)+c_2sin(4t)` .
To find particular solution use method of undetermined coefficients. We can't assume that `x_p=Acos(4t)+Bsin(4t)` because these terms are already in the homogeneous solution, so assume that `x_p=Atcos(4t)+Btsin(4t)` .
Then `x_p'=Acos(4t)-4Atsin(4t)+Bsin(4t)+4Btcos(4t)` and `x_p''=-4Asin(4t)-4Asin(4t)-16Atcos(4t)+4Bcos(4t)+4Bcos(4t)-16Btsin(4t)` .
Plugging these values into equation gives:
Equating like terms with `8sin(4t)` gives
which has solution `A=-1/4,\ B=0` . So, `x_p=-1/4 tcos(4t)` .
Therefore, `x=x_h+x_p=c_1cos(4t)+c_2sin(4t)-1/4 tcos(4t)` .
Now, find x'(t): `x'(t)=-4c_1sin(4t)+4c_2cos(4t)-1/4 cos(4t)+tsin(4t)` .
Now applying initial conditions `x(0)=-0.5` , `x'(0)=0` gives
Which has solution `c_1=-1/2,\ c_2=1/16` .
Finally, `x(t)=-1/2 cos(4t)+1/16 sin(4t)-1/4 tcos(4t)` .
Note that `|x|\ ->oo` as `t->oo` . This phenomenon is called pure resonance. It is due to the forcing function F(t) having the same circular frequency as that of the associated free undamped system.