Problem 5-81: A model of a vehicle of m transporting two crates of masses m and

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 1,5-81

Explanation / 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

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