In the following problems, when I request "equations of motions" I simply mean the force balance equations from the free body diagrams.

The three dashpots in the drawing on the left can be replaced by a single dashpot as shown on the right.

a) Find an expression for b_{eq} in terms of b_{1}, b_{2},
and b_{3}.

b) Find the ratio of v_{2} to v_{1}. (v_{2}
is the velocity of the point labeled x_{2}...).

The task in this problem is to analyze the system shown.

a) Before starting, coordinates must be defined for the free body diagrams. Define appropriate coordinates on the diagram. Make all coordinates positive to the right.

b) Draw a free body diagram for each coordinate

c) Write equations of motion for each free body diagram. The left
side of each equation should have x_{1} variables, starting with
highest order derivatives, followed by x_{2} variables, ....
The right hand side should have system inputs.

d) If you were to define the system with a single differential equation, what order system would you expect? Why?

a) Three independent coordinates are needed to acco7unt for all positions in system.

b)

c)

d) Five energy storage elements (3 springs, 2 masses), therefore we
expect a 5^{th} order system.

Consider the system shown. A mass, m, is connected to a beam of
length L (and moment of inertia, I, made of material with Young's modulus,
E). A force, f_{a}, is applied at the other end of the beam,
as shown. We take motion of the mass to be purely horizontal (i.e.,
deviations from the beam being purely vertical are small).

Recall that the tip deflection of a cantilevered beam is related to the force by

so we can replace the beam with an equivalent spring:

a) Draw free body diagrams for the system shown below.
Note x_{1} and x_{2} point in different directions.

b) Write equations of motion (one for m_{1} and one for m_{2})
for the system.

a)

b)

Consider the system shown. Two masses (m_{1} and m_{2})
are connected to a fixed reference by beams of length
L, moment of inertia, I, with Young's modulus, E.
The masses are suspended vertically (gravity acts downwards). A force f_{a}, is applied
to one of the masses as shown. Note that x_{1} and x_{2}
are in the same direction.

b

a) Draw free body diagrams if the beams are undeflected at the zero position (i.e.,at the position without gravity acting).

b) Write equations of motion.

c) Solve for the equilibrium position under the influence of gravity (with no force applied).

d) Now assume x_{1} and x_{2} are zero at the equilibrium
position under the influence of gravity. Draw free body diagrams and
solve for the equations of motion.

a)

b)

c)

>> syms k keq m1g m2g >> inv([keq+k -k;-k keq+k])*[m1g; m2g] ans = (m1g*(k + keq))/(keq*(2*k + keq)) + (k*m2g)/(keq*(2*k + keq)) (m2g*(k + keq))/(keq*(2*k + keq)) + (k*m1g)/(keq*(2*k + keq)) >> pretty(simple(ans)) +- -+ | k m1g + k m2g + keq m1g | | ----------------------- | | keq (2 k + keq) | | | | k m1g + k m2g + keq m2g | | ----------------------- | | keq (2 k + keq) | +- -+

d)

Consider the system shown. Two masses
are connected to a fixed reference by springs of value k_{1}, and to
each other with a spring of value k_{2}. There is viscous
friction between the masses, b. A force f_{a}, is applied
to one of the masses as shown. Note that x_{1} and x_{2}
are in the same direction.

a) Draw free body diagrams if the springs are unstretched (or compressed) when the positions are zero (i.e., at the position without gravity acting).

b) Write equations of motion. The left side of each equation should
have x_{1} variables, starting with highest order derivatives,
followed by x_{2} variables. The right hand side should have
system inputs.

c) Solve for the equilibrium position under the influence of gravity
(with no force applied, f_{a}(t)=0).

d) Now assume x_{1} and x_{2} are zero at the equilibrium
position under the influence of gravity. Draw free body diagrams and
write the equations of motion. This should be quite easy by reusing
the work done previously.

a)

b)

c) At equilibrium derivatives (and input) are equal to zero

Add the equations and solve

d) Same as before without gravitational terms

The diagram below shows a MEMS (MicroElectroMechanical System) accelerometer

from:
http://isites.harvard.edu/fs/docs/icb.topic499289.files/Lecture%209-%20Applications%20of%20MEMS.pdf

A micrograph of such a system is shown below. The mass "m" in the model represents the large block of material in the middle. The spring, "k", models the four "legs" coming off the middle block, and damping comes from material properties...

from:
http://www.palomartechnologies.com/applications/microelectronic-mechanical-systems/

The output of the device is measured by the displacement of the mass relative to the support structure.

Accelerometers are used in cars to sense decelerations (and allow for deployment in airbags). Typically there are three of them to measure acceleration in the x, y and z directions. They are also used in smart phones, tablets, the Nintendo Wii and other devices, both to measure acceleration (e.g., in games) but also to detect "up" so that the screen can rotate to the proper orientation.

Call the the input of the system z_{in}, and set it to be the position
of the outer box (in the lumped parameter model) and the output, z_{out}, to
be the position of the mass. Also assume all positions and velocity
are zero at equilibrium.

- Draw a free body diagram without gravity,
- write a differential equations with output variables on the left, and input variables on the right,
- redraw the free body diagram if gravity acts downward,
- find the equilibrium position of the mass if
gravity is acting (and z
_{in}=0).

a)

b)

c)

d) Rewrite differential equation and set derivatives to zero to find equilibrium

A company (Kurt Kinetic) makes an indoor trainer for a bicycle (it turns your bicycle into a stationary bicycle). The rotation of the rear bicycle wheel is resisted by a frictional force in the device. This company also sells a “power meter” that calculates how much power you are generating based on the speed of rotation of the wheel. The curve relating speed to power is shown below (Kinetic Fluid Resistance Unit). Note the vertical axis reads wattage – just read it as Watts (and make a mental note not to let marketing people near a graph with numbers)

from https://kurtkinetic.com/files/kurtkinetic/ckfinder/images/Kinetic_FluidPCurve.jpg

A fit to the curve is given on their web page as Power=5.244·velocity + 0.019·velocity³. Determine which mode(s) of friction (static, kinetic, viscous, drag) are present, or is it impossible to tell? Briefly, (one or two sentences) justify your answer.

Since Power=Force·velocity the two force terms must be a constant (5.244) and a squared term (0.19·velocity²), the sources of friction are predominantly modeled by kinetic and aerodynamic (or drag) friction.

I was riding my bike down a long hill with a constant at an angle θ with respect to horizontal. I put on the brakes to maintain a constant speed, v_{0}.

Artist's rendition

https://patriciadorothy.files.wordpress.com/2010/10/patricia-dorothy-bicycle.jpg

a) Derive an expression for the power being dissipated in the brakes in terms of m, g, v_{0} and θ (mass, gravity, velocity and angle of decline of hill).

b) If the hill was Mont Ventous (a mountain in France) with a slope of about 6 degrees, and assuming my bike and I weigh 100 kg, and that France is on earth (so g=9.8 m/s²), and that my final velocity was v_{0}=40 kph, determine how many Watts of power are being dissipated. You should assume that the only thing slowing me down is the brakes, so you can ignore wind resistance, rolling resistance of tires.... Be careful with units.

Note: Mont Ventoux is an environmental cautionary tale. Though it was once covered by forest, the mountain was clearcut for wood beginning in the 12th century. In the 19th and 20th centuries efforts were made to reforest the lower slopes, but the top of the mountain is still barren with no top soil.

https://teamwildside.files.wordpress.com/2011/07/ventoux-top.jpg

**Force argument**: The only forces affecting the speed of the bike are gravity and the brakes. Since the force of gravity is m·g·sin(θ)=102N. The velocity of the bike is v_{0}=40kph=11.1m/s. Therefor the power being dissipated is 102*11.1=1.1kW (the same as eleven 100W light bulbs).

**Energy argument:** Potential energy=m·g·h=m·g·x·sin(θ). Velocity is not changing, so kinetic energy is constant. Power is derivative of energy with respect to time, this gives m·g·v_{0}·sin(θ). This is the same as the previous answer.