Spring Elements

Translating Springs

A spring with free length L

Simply Supported

A beam that is supported vertically or horizontally (non-fixed) but is allowed to deflect and rotate.

Spring Types

For different types of springs, see Week 4 (Spring Elements) in OneNote.

Uses Hooke’s Law:
$F=kx$

Torsional Springs

If we apply a torque, we make a spring rotate by theta.
Here:

• d = wire diameter
• n = number of coils Coil Spring:
  $k_T=\frac{E{d}^4}{64nD}$
  Hollow Shaft:
  $k_T=\frac{\pi G\left(D^4-d^4\right)}{32L}$

General Equation

The general equation of a simple harmonic oscillator is
$m{x}^{{}^{{}^{\prime \prime }}}+kx=0$
where the natural frequency of the harmonic oscillator is
${\omega }_n=\sqrt{\frac{k}{m}}$
There could be other things in front of the k term which is brought into the natural frequency equation

Series and Parallel Springs

Springs in series are like resistors in parallel. This makes the equivalent stiffness less stiff.
$f={\frac{k_{1}k2}{k_1+k_2}}_{}x$
Where the term in front of x is an *equivalent stiffness.

Springs in parallel are like resistors in series. This makes the equivalent stiffness more stiff.

Assuming the same deflection x:
$f=\left(k_1+k_2\right)x$
Where you can see the equivalent stiffness is higher.

Examples

See OneNote (Week 5 - Spring Elements) for examples.

Parallel Springs

Say a wheel has two springs connected to it leading in two different directions, these springs would be in parallel!

Mass or Inertia Elements

$f=M{x}^{{}^{\prime \prime }}$

Mass-Spring Systems

This is a mass on wheels attached to some spring which is also attached to a fixed end.

In general (if the spring is pulling left)
$-f_k=M{x}^{{}^{\prime \prime }}$
If we bring pulleys into this, if the pulley cable does not slip then,
$y=\left({\frac{R_1}{R_2}}_{}\right)x$

Cancelling mg

Often, the weight mg is canceled by the effect of the static spring force so it doesn’t show up in the equation of motion

For Pendulum Mass Spring Questions

Take a sum about the pivot point to find equations of motion and natural frequency

Assuming small angles

This means sin and cos can be approximated:

• sin(theta) = theta
• cos(theta) = 1

Strategies

The strategies for solving spring mass systems depend on the type of question asked but in general:

• Draw FBD of mass
• Determine Spring force
  • x could be a relative displacement using angles or the non-slipping condition
  • Are the springs in parallel or series? Use the appropriate equation to solve them
• Solve for equation of motion (2nd order ODE)
• For rotational elements
• Take a sum of moments about some pivot (once again taking into account relative displacement)
  • Maybe use the natural frequency equation
  • Maybe use the x to y no slipping equation
• Solve for equation of motion (2nd order ODE for theta)

Multi-degrees of freedom mass-spring systems

Degrees of freedom correspond to directions of movements

Oops

I guess I did the practice problems too early but the kinematics portion of this lecture is the learning outcomes from the practice problems (writing equations of motion with relative displacement)

Energy Methods

For a conservative force
$P\left(x\right)={\int }_{}^{}dP=-{\int }_{}^{}f\left(x\right)dx$
The resisting force of a spring is conservative f=-kx

where you sub in the spring equation in the integral above to derive the *potential energy of a linear spring deflected by x from its free length

$P\left(x\right)=\frac{1}{2}k{x}^2$

For torsional springs
the results are basically the same using
$T=k_T\theta$ and
${\int }_0^{\theta }Td\theta$
Yielding:
$P\left(\theta \right)=\frac{1}{2}{k}_t{\theta }^2$

Conservation of Mechanical Energy

$\Delta PE+\Delta KE=0$
Example: Moving Mass-Spring system (mass on wheels connected to a spring)
You can equate initial and final energies (where the energies considered are kinetic and spring) for an equation of motion, then you can factor by grouping which yields, once again, relative displacement

$\frac{1}{2}m\left(x^{\prime }-x_0^{\prime }\right)+\frac{1}{2}k\left(x^{\prime }-x_0^{\prime }\right)=0$ Example: Mass-Spring system attached to ceiling
In this case, a mass is attached to a ceiling through a spring, and the positive y-direction is defined as downwards. Once again using energy…
$\frac{1}{2}m{y}^{{}^{\prime }2}+\frac{1}{2}k{y}^2-mgy=const$

Obtaining Equations of motion using energy

Use
$\frac{d}{dt}\left(PE+KE\right)=0$
Once again using a mass on wheels
Starting by equating the energy equation to some constant value…
We can differentiate with respect to time

$\frac{1}{2}\left(2m{x}^{\prime }{x}^{{}^{\prime \prime }}\right)+\frac{1}{2}\left(2k{x}^{\prime }x\right)=0$ Yielding $kx+m{x}^{{}^{\prime \prime }}=0$

Energy Strategies for Wheel Spring Systems

1. Write kinetic energy equations taking into account inertia
   1. Remember the no slip condition v=Rw
   2. Find the geometrical I for this given problem
2. Write potential energy equations considering ALL SPRINGS
3. Differentiate the sum of KE + PE (d/dt)
4. Organize expression for a 2nd order ODE (this is the equation of motion)