# ODE Applications - Unforced Mechanical Vibrations

• VeganGirl
In summary: I'm sorry, I misread your solution as the underdamped one.In summary, a spring and dashpot system is to be designed for a 32lb weight so that the overall system is critically damped. γ^(2) = 4km and k = γ^(2)/4m must be related. Assuming 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, damping constant γ and spring constant k required are γ^(2) = 4km and k = γ^(2)/4m
VeganGirl

## 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?

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.

Last edited:
rock.freak667 said:
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?

VeganGirl said:
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.

VeganGirl said:
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.

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

## 1. What is an unforced mechanical vibration?

An unforced mechanical vibration is a type of motion that occurs in a system without any external forces acting on it. It is the result of the system's natural tendency to oscillate due to its own internal properties.

## 2. What are some common examples of unforced mechanical vibrations?

Some common examples of unforced mechanical vibrations include the swinging of a pendulum, the vibration of a guitar string, and the natural movement of a bridge or building in response to wind or other disturbances.

## 3. How are unforced mechanical vibrations described mathematically?

Unforced mechanical vibrations can be described mathematically using ordinary differential equations (ODEs). These equations take into account the system's mass, stiffness, and damping properties to model the motion of the system over time.

## 4. What is the role of initial conditions in modeling unforced mechanical vibrations?

Initial conditions, such as the initial displacement and velocity of the system, are crucial in modeling unforced mechanical vibrations. These conditions determine the behavior of the system over time and are used to solve the ODEs that describe the vibrations.

## 5. How are unforced mechanical vibrations useful in real-world applications?

Unforced mechanical vibrations have several practical applications, such as in the design of buildings, bridges, and other structures to ensure their stability and safety. They are also used in musical instruments and other mechanical systems to produce specific frequencies and harmonics.

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