# ODE Applications - Unforced Mechanical Vibrations

## Homework Statement

A spring and dashpot system is to be designed for a 32lb weight so that the overall system is critically damped.

(a) How must the damping constant γ and spring constant k be related?
(b) Assume the system is to be designed so that the mass, when given an initial velocity of 4 ft/sec from its rest position, will have a maximum displacement of 6 inches. What values of damping constant γ and spring constant k required?

## The Attempt at a Solution

For the system to be critically damped, γ^(2) = 4km

(a) γ = sqrt(4km)
k = γ^(2)/4m

(b) the IVP is... my'' + λγ' + ky = 0, y(0) = 6, y'(0) = 4

Since γ^2 = 4km,

The solution of the IVP is... y(t) = c1e^(λt) + c2te^(λt)

Imposing the initial conditions, I got c1 = 6
and c2 = 4 + 12 k^(1/2)*m^(-1/2)

Now, how do I solve for the damping constant γ and the spring constant k?

rock.freak667
Homework Helper
The first part is correct. But since your damping ratio is 1, I think your solution for 'y' should be
of this form:

y(t)=(A+Bt)ert

Your form shows that it will be over damped.

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The first part is correct. But since your damping ratio is 1, I think your solution for 'y' should be
of this form:

y(t)=(A+B)ert

Your form shows that it will be over damped.

In my text, I have y(t) = c1e^(λ1t) + c2e^(λ2t) as an over-damped case,

y(t) = c1e^(λt) + c2te^(λt) as a critically-damped case,

and y(t) = (A+B)e^(rt) as an underdamped case.

I'm wondering if I made a mistake somewhere with the initial conditions.
It says in the question, that it will have a max. displacement of 6 in.
I assumed that the max. displacement will occur at t=0. Is this correct?

Also, I have trouble understanding how the initial velocity can be 4 ft/sec.
Don't you always start at zero?

rock.freak667
Homework Helper
In my text, I have y(t) = c1e^(λ1t) + c2e^(λ2t) as an over-damped case,

y(t) = c1e^(λt) + c2te^(λt) as a critically-damped case,

and y(t) = (A+B)e^(rt) as an underdamped case.

I'm wondering if I made a mistake somewhere with the initial conditions.
It says in the question, that it will have a max. displacement of 6 in.
I assumed that the max. displacement will occur at t=0. Is this correct?

Also, I have trouble understanding how the initial velocity can be 4 ft/sec.
Don't you always start at zero?

Well you can have a zero initial position such that y(0)=0, but you give it an initial velocity. This is what that meant, so your other condition is that y'(0)=4

Okay, so my equation and the initial conditions are correct.
How do I go about getting γ and k?
Would I get constants (numbers) or would it be a function of something?

With only knowing C1 and C2, I have no idea how to proceed.

rock.freak667
Homework Helper
It said also that the maximum displacement is 6 inches, meaning that the velocity at the time to reach 6 inches is zero.