B$.8$ GRADUATE WORK: CALCULATE

HORIZONTAL FLEX$-$BEAM RESPONSE TO IMPACT

Figure $10$: impact excitation of a horizontal flex$-$beam vibration system

Consider a cantilever beam of length $L=420mm$ with a mass

$M=50kg$ attached at its tip as shown in Figure $10.$ The beam material

is steel with $E=200$GPa. The beam cross$-$section dimensions are

$h=3mm,b=30mm.$ The beam is initially at rest.

A small projectile of mass $m=8$ grams and velocity $v=930\frac{m}{s}$

hits the mass and remains embedded in it$.$ This impact imparts an

initial velocity to the system that starts to oscillate with motion $u(t)$

contained in the horizontal plane. Do the following:

$(1)$ Display input data

$(2)$ Calculate the second moment of area I for flexing consistent

with the direction of motion $u(t)$

$(3)$ Calculate the cross$-$sectional stiffness $EI$ of the beam

$(4)$ Calculate the flex$-$beam spring constant $k$

$(5)$ Calculate the initial velocity of the system $u_0^{}$ as a result of the

interaction between the projectile and the mass

$(6)$ Calculate the $ra\frac{d}{sec}$ natural frequency ${}_n$ and period of

oscillation $$

$(7)$ Calculate the position of the mass $u(t_1)$ after $t_1=10sec$

$(8)$ Plot the response of the system for $N_{\text{o}\text{s}\text{c}}=10$ oscillations and find

on the plot:

$($a$)$ the largest response

$($b$)$ the response calculated at item $(7)$

$(9)$ Discuss your results