Question

4. (8 points) We consider the mass-spring-dnmper systen in Figure &: A block af mes is saspended fron the eeiling by n spring with eonistant k nnd unstretched length lo and a damper with damping eoeficient d The block ean anly move vertienlly nnd its poesition is theresore fually deseribed by the z-courdinnte. An nlternative poeition is mensued from the statie rest position Furthermore, ngravitational neeclerstioa with mngnitude g is present. SYSTEM Fgure & A rigid body ofmi pended frm the ailing by pringand damper. a. ( pts) Dr afree-body dingram and use Newton's second lua to find the seeand-arder differentinl equntion that deseribes the motion of the system in the nhsolute coordinate r. b. (3 pts) Check the pluasibility of the equation found in part n in three different ways 1. check units, 2. check the rest position,-0,-0 does the expreession still make sense? 3 check the efeet of parameters; how does seing ny parneter to uro influence the rest position? c. (3 pts) Find the poeition for statie equilibrium and use this to show thnt the movenent of the blok in terms of with respeet to the static equilibrium ean be described by the second order differential equation d. ( pts) Trunsfoem (2) to aset of frst-order ordinary differestinl equations(ODE) e. (δ pts) Insert your custion of motion to oomplete the ODE in the supplied MATLAB' ile nassspringdampar-om. Then run the file sinulata,nassspringdanpar.n, which uses MATLABT* o integrate the ODE for speeified time und thea nimates the systems motion by enlling the funetion mansspringdampar anination. Exemplary mlues of parsme ters m, d nnd k, nnd initinl pasitiond initinl veloeity t ure speciied, but ceck if your reesults mnke sense by comparing resalts for different values,eg, by exeluding damping, d-0 f. (3 pts) The motion of the maes-spring system enn be described by with damping rutio ζ-d/(2mw), oscillation frequency ω, phnse angle φ, sund tunplitude A Whnt sre the units of al constants and varinbles in (3)? Give your answer in theform: -, ete. Do the units correspond with the units required in (2)? g. (4 pts) For d-0, show that indeed (3) tisies the equntion of motion (2). Do this by finding the value for w. h. ( pts) Compre the namerienl solutioa of part e and the ninlytie solution of part f of the spring-mnes system to e ch other (by plotting positions ณะเ funetion of time in the sure igure). Are they similar?

Thanks in advance

Answer 1

As per Chegg guidelines first 4 sections have been answered.