Problem 5-81: A model of a vehicle of m transporting two crates of masses m and
ID: 3281189 • Letter: P
Question
Problem 5-81: A model of a vehicle of m transporting two crates of masses m and m3 is sketched in Figure P5-81. The crates are connected to each other by a spring of spring constant ki. The crate m is attached to the vehicle by a spring of spring con stant k. The dissipation due to the relative motion be- tween the crates and the vehicle can be modeled as viscous damping having an equivalent dashpot con- stant c. The vehicle's power plant generates a force F(t) that moves the system. Derive the equation(s) of motion for the system F(O) ko 3 ,Viscous damping, c 1GI RE 1,5-81Explanation / Answer
When the vehicle moves an acceleration will be produced
F(t)=(m1+m2+m3) a(t)
a(t)= F(t)/(m1+m2+m3)
a(t)= F(t)/M
due to this acceleration a pseudo force will act on the two spring mass system and there will be forced oscillatons. So we will have two seperate equation of motion
for mass m2 considering all the forces acting
F2= -k2x2-k1x2-cdx2/dt+m2*a(t)
k2x is the restoring force due to spring
cdx/dt is the damping force whre c is damping coefficient and dx/dt is velocity
m2*a(t) is the pseudo force
if a2 is the nt acceleration then
m2 a2 = -(k1+k2)x2-cdx2/dt+m2 F(t)/M
m2 d2x2/dt2+(k1+k2)x2+cdx2/dt= m2 F(t)/M
this is equation of motion for mass m2
considering all the forces on m3
F3= -k1x3-cdx3/dt+m3 a(t)
m3 d2x3/dt2+k1x3+cdx3/dt= m3 F(t)/M
this is equation of motion for mass m3