Tallest mountain

Smithsonian has this post for the tallest mountain in our solar system. Let alone the 15.5 miles high Olympus (yeah, what other names can you come up with, Mordor?)  on Mars, one thing I didn’t know is the mountain height is measure from base to peak, which makes Everest far less impressive than Mauna Loa at Hawaii.

Mauna Loa (as seen from nearby Mauna Kea) is tall enough to have snow, at least when the volcano isn't erupting (courtesy of flickr user superfluity)

via

Point in triagle test

For mortar method in non-conforming mesh FEM, this might come handy.

Harmonic vibration of a damped system

Unlike in previous post that phase change could only be or , in the damped system, as the ratio between dynamics response and static response is expressed in the same way as the undamped case

the phase lag varies in . The following Mathematica outputs the displacement response factor, phase lag and normalized time history as the function of damping ratio and frequency ratio .

Rd[w_, \[Zeta]_] := 1/Sqrt[(1 - w^2)^2 + (2 \[Zeta] w)^2]
\[Phi][w_, \[Zeta]_] := -ArcTan[1 - w^2, -2 \[Zeta] w]
ud[w_, \[Zeta]_, t_] :=
 Rd[w, \[Zeta]] Sin[2 Pi (t - \[Phi][w, \[Zeta]]/(2 Pi))]
us[t_] := Sin[2 Pi t]
Manipulate[
 GraphicsRow[{GraphicsColumn[{Plot[Rd[w, \[Zeta]], {w, 0, 3},
      AxesOrigin -> {1, 1}, PlotRange -> {{0, 3}, {0, 5}},
      AxesLabel -> {"\!\(\*FractionBox[\"\[Omega]\", SubscriptBox[\"\
\[Omega]\", \"0\"]]\)", "\!\(\*SubscriptBox[\"R\", \"d\"]\)"},
      Epilog -> {PointSize[Large], Red, Point[{w, Rd[w, \[Zeta]]}]},
      Ticks -> {{0, 1, 2, 3}, {0, 1, 2, 3, 4, 5}}],
     Plot[\[Phi][w, \[Zeta]], {w, 0, 3}, AxesOrigin -> {1, Pi/2},
      PlotRange -> {{0, 3}, {0, Pi}},
      AxesLabel -> {"\!\(\*FractionBox[\"\[Omega]\", SubscriptBox[\"\
\[Omega]\", \"0\"]]\)", "\[Phi]"},
      Epilog -> {PointSize[Large], Red,
        Point[{w, \[Phi][w, \[Zeta]]}]},
      Ticks -> {{0, 1, 2, 3}, {0, Pi/2, Pi}}]}, Frame -> All],
   Plot[{ud[w, \[Zeta], t], us[t]}, {t, 0, 2},
    PlotStyle -> {Thick, Dashed},
    AxesLabel -> {"t/T",
      "\!\(\*SubscriptBox[\"u\", \"d\"]\)/\!\(\*SubscriptBox[\"u\", \
\"st\"]\)"}]}, Frame -> All], {{w, 0.5, "Frequency ratio"}, 0,
  3}, {{\[Zeta], 0.2, "Damping ratio"}, 0, 1}]

Special cases include

1. Long period excitation, i.e., , gives a pseudo-static response. In this case, the system “waits” until it “feels” the excitation completely. is greater but very close to , and the displacement is essentially in phase with excitation force, in other words, dynamic effect is near to none.
   

   Long period/low frequency loading response

2. Short period excitation gives very small , though the phase lag is . Here the system barely reacts when the load is reversed, thus leads to small displacement.
   

   Short period/high frequency loading response

3. Resonant period, i.e., , leads to . Now is very sensitive to damping change, namely, the response is controlled by the damping: a small change of damping ratio leads to great reaction of the structure. When , we have .
   

   Resonant frequency: small damping

   

   Resonant frequency: medium damping

   

   Resonant frequency: large damping

The last case is the what’s essentially behind the viscous damping devices applied to buildings like this.

Harmonic vibration of an undamped system

This can be found in virtually every dynamics textbook. The undamped system follows the frequency of the excitation, while the displacement response factor .  Response is out of phase when excitation frequency is greater than the natural frequency:  . If there is only one thing to keep in mind of harmonic SDF vibration,  I guess it’s the plot for the response against the frequency ratio. The dynamic object in Mathematica can be built by:

Manipulate[GraphicsRow[
  {GraphicsColumn[{Plot[Rd[r], {r, 0, 3}, AxesOrigin -> {1, 0},
      Frame -> True, PlotRange -> {{0, 3}, {-5, 5}},
      AxesLabel -> {"\!\(\*FractionBox[\"\[Omega]\", SubscriptBox[\"\
\[Omega]\", \"0\"]]\)", "\!\(\*SubscriptBox[\"R\", \"d\"]\)"},
      Epilog -> {Red, PointSize[Large], Point[{w, Rd[w]}]}],
     Plot[Abs[Rd[r]], {r, 0, 3}, AxesOrigin -> {1, 1}, Frame -> True,
      PlotRange -> {{0, 3}, {0, 5}},
      AxesLabel -> {"\!\(\*FractionBox[\"\[Omega]\", SubscriptBox[\"\
\[Omega]\", \"0\"]]\)", "|\!\(\*SubscriptBox[\"R\", \"d\"]\)|"},
      Epilog -> {Red, PointSize[Large], Point[{w, Abs[Rd[w]]}]}]},
    Alignment -> Right, Frame -> All],
   Plot[{1/(1 - w^2) Sin[w t], Sin[w t]}, {t, 0, 4 Pi}, Frame -> True,
     PlotStyle -> {Thick, Dashed},
    PlotRange -> {{0, 4 Pi}, {-10, 10}}, AxesLabel -> {"t", "u"}]
   }, Frame -> All]
 , {{w, 0, "Frequency ratio"}, 0, 3}]

In the right, the dashed curve is the excitation history, while the thick one is for response, which would be out of phase to the excitation when frequency ratio is greater than 1.0 ().

New water wave patterns found

As reported in PhysOrg.com, French researches have found two new patterns of shallow water waves, through excitation of a Hele-Shaw tank. The complete paper at the Physics Review Letters can be found here. The following snapshots of the two waves are provided in PhysOrg.com.

         

Theses highly localized waves, as long as their well-known predecessors in the solitary wave family, actually are governed by very flow model, the potential flow. Their complex behaviors are rooted in the free surface motion, i.e. the change of the domain’s boundary geometries, instead of unsteady nature of the governing equations itself. The balance between the dissipation due to water viscosity and the wave dispersion characterizes such nonlinear wave motions. One feature of there two waves is the they are both steep, i.e. with high ratio between wave height and wave length, and this is the indication of high nonlinearity nature of the wave. The hysteresis we observe during the snapshots are exactly the balancing process of energy transform.