Vibrations due to Flow-Driven Repeated Impacts

<table class="table-group" width="800px"><tr class="fig-image" id="a"><td><img alt="760939.fig.0010a" src="https://static.hindawi.com/articles/mpe/volume-2013/760939/figures/760939.fig.0010a.jpg"/></td></tr><tr class="fig-caption"><td align="center"><table><tr><td><b>(a) </b>Flow velocity <svg height="14.85" id="M158" style="vertical-align:-2.22495pt" version="1.1" viewbox="0 0 50.3125 14.85" width="50.3125" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
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q0 -76 10 -138.5t37 -107.5t69 -45q32 0 55.5 25t35.5 68.5t17.5 91.5t5.5 105t-5.5 105.5t-18 92t-36 68t-56.5 24.5z" id="x30"></path></g><g transform="matrix(.017,-0,0,-.017,8.222,12.012)"><path d="M113 -12q-24 0 -39.5 16t-15.5 42q0 24 16 40.5t40 16.5t40 -16.5t16 -40.5q0 -26 -16 -42t-41 -16z" id="x2E"></path></g><g transform="matrix(.017,-0,0,-.017,12.098,12.012)"><path d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z" id="x31"></path></g><g transform="matrix(.017,-0,0,-.017,23.079,12.012)"><path d="M797 0h-214v26q46 5 58 17.5t12 63.5v167q0 116 -88 116q-48 0 -112 -52q3 -18 3 -42v-195q0 -46 11 -58t53 -17v-26h-214v26q48 5 59.5 17t11.5 60v172q0 115 -88 115q-47 0 -109 -52v-235q0 -47 10 -59.5t48 -17.5v-26h-211v26q50 5 62.5 17t12.5 60v207q0 48 -9 59.5
t-52 18.5v23q79 14 139 41v-81q3 2 29 20.5t44 29.5q45 28 81 28t66 -22t46 -61q25 20 83 57q39 26 76 26q59 0 93 -40t34 -113v-193q0 -49 10.5 -60.5t54.5 -16.5v-26z" id="x6D"></path></g><g transform="matrix(.017,-0,0,-.017,37.001,12.012)"><path d="M368 703l-264 -866h-60l265 866h59z" id="x2F"></path></g><g transform="matrix(.017,-0,0,-.017,44.004,12.012)"><path d="M319 325l-25 -7q-33 99 -103 99q-29 0 -47 -19.5t-18 -49.5t22 -49.5t62 -36.5q63 -26 95 -57t32 -79q0 -64 -50 -101t-115 -37q-35 0 -67.5 10.5t-46.5 23.5q-5 11 -11 51t-6 67l27 5q14 -53 46.5 -88.5t75.5 -35.5q29 0 50 19.5t21 50.5t-19.5 51.5t-59.5 39.5
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</svg>    </td></tr></table></td></tr><tr class="fig-image" id="b"><td><img alt="760939.fig.0010b" src="https://static.hindawi.com/articles/mpe/volume-2013/760939/figures/760939.fig.0010b.jpg"/></td></tr><tr class="fig-caption"><td align="center"><table><tr><td><b>(b) </b>Flow velocity <svg height="14.85" id="M159" style="vertical-align:-2.22495pt" version="1.1" viewbox="0 0 38.275002 14.85" width="38.275002" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,12.012)"><path d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z" id="x31"></path></g><g transform="matrix(.017,-0,0,-.017,11.044,12.012)"><path d="M797 0h-214v26q46 5 58 17.5t12 63.5v167q0 116 -88 116q-48 0 -112 -52q3 -18 3 -42v-195q0 -46 11 -58t53 -17v-26h-214v26q48 5 59.5 17t11.5 60v172q0 115 -88 115q-47 0 -109 -52v-235q0 -47 10 -59.5t48 -17.5v-26h-211v26q50 5 62.5 17t12.5 60v207q0 48 -9 59.5
t-52 18.5v23q79 14 139 41v-81q3 2 29 20.5t44 29.5q45 28 81 28t66 -22t46 -61q25 20 83 57q39 26 76 26q59 0 93 -40t34 -113v-193q0 -49 10.5 -60.5t54.5 -16.5v-26z" id="x6D"></path></g><g transform="matrix(.017,-0,0,-.017,24.966,12.012)"><path d="M368 703l-264 -866h-60l265 866h59z" id="x2F"></path></g><g transform="matrix(.017,-0,0,-.017,31.969,12.012)"><path d="M319 325l-25 -7q-33 99 -103 99q-29 0 -47 -19.5t-18 -49.5t22 -49.5t62 -36.5q63 -26 95 -57t32 -79q0 -64 -50 -101t-115 -37q-35 0 -67.5 10.5t-46.5 23.5q-5 11 -11 51t-6 67l27 5q14 -53 46.5 -88.5t75.5 -35.5q29 0 50 19.5t21 50.5t-19.5 51.5t-59.5 39.5
q-28 12 -46 22.5t-38.5 27t-30.5 38.5t-10 49q0 54 42.5 92t109.5 38q48 0 88 -18q6 -15 13 -50.5t9 -55.5z" id="x73"></path></g>
</svg></td></tr></table></td></tr></table>

<table>Comparison of the linearized and quadratic Drag.</table>

Mathematical Problems in Engineering

fig10

Figure 10

Figure 10: Vibrations due to Flow-Driven Repeated Impacts 