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CraigH
#1
Dec10-12, 02:53 PM
P: 199
What is the general solution to the differential equation describing a mass-spring-damper?

t=time
x= extension of spring
M=Mass
K=Spring Constant
C=Damping Constant
g= acceleration due to gravity

Spring has 0 length under 0 tension
Spring has 0 extension at t = 0

If the Force downwards due to the mass equals:

Force downwards = M*g

And this is opposed by tension in the spring:

Force upwards = - K*x

And there is a 3rd force acting on the system caused by the damper. This allways acts in the opposite direction as velocity.

Force = - C*(dx/dt) //because damping is proportional to velocity.

So the overall force acting on the mass equals:

Net Force = M*g - K*x - C*(dx/dt)

This net force causes an acceleration in the direction of the force:
F=Ma so...

M*(d2x/dt2) = M*g - K*x - C*(dx/dt)

\\you can see that this will oscillate, because when x is 0 the acceleration will be positive, when M*g= K*x + C*(dx/dt), the acceleration will be 0, and then become negative, and then the velocity will become 0, and then negative, and then the x will become 0 again, and it will repeat.

So the differential equation describing this system is:

M*(d2x/dt2) + C*(dx/dt) + K*x = Mg

Mx''+Cx'+Kx=Mg

This is a second order linear non homogeneous differential equation.
Mx''+Cx'+Kx=g(t)
where g(t) is a constant.

Now my question is, how do I solve it for x?

I tried using the undetermined coefficients method, but I do not get an equation that implies that x is oscillating. Shouldn't I get a sin or cos somewhere in the answer?

Here is what I tried:

corresponding Homogeneous Equation:

Mx''+Cx'+Kx=0

characteristic equation = M*(r^2) + C*r + K

General solution to the homogeneous equation:

C1(e^pt)+C2(e^qt)
where
p=(-C+SQRT((C^2)-4*M*K))/2M
q=(-C-SQRT((C^2)-4*M*K))/2M //from the quadratic equation

Now to calculate the particular solution:

Mx''+Cx'+Kx=Mg

The right hand side is a polynomial of degree 0. try a polynomial as the solution

x=Ax^2 + Bx + C
x'=2Ax +B
x''=2A

Sub these in to the differential equation and you end up with:

x=Mg/K

So the general solution to the differential equation is:

x(t) = C1(e^pt)+C2(e^qt) + Mg/K

This can't be correct, because x should oscillate over time.
C1, C2, p, q, ang Mg/K are all constants so this equation does not describe an oscillation.

Where did I go wrong?

Thanks!
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