Rose-Hulman Institute of TechnologyMechanical EngineeringVibrationsToday’s Objectives:Students will be able to:a) Understand the procedure for modelingb) Understand system elementsc) Find the EOM for a SDOF systemTrivia of the daymckLecture 2 – Modeling of SDOF SystemsGalileo was an excellent lute playerRose-Hulman Institute of TechnologyMechanical EngineeringVibrationsIdeal springs model a restoring forceFront suspensionAirplane wingLeaf SpringRose-Hulman Institute of TechnologyMechanical EngineeringVibrationsWhere’s the spring?Floating barrelPendulum clockRose-Hulman Institute of TechnologyMechanical EngineeringVibrationsIdeal springsForceRelative DisplacementAssumptions: _________________________________x1x2FFk⇒Translationθ1θ2TTk⇒RotationRose-Hulman Institute of TechnologyMechanical EngineeringVibrationsTable of equivalent springs is in the inside cover of the bookRose-Hulman Institute of TechnologyMechanical EngineeringVibrationskeq= ___________________________________2. Replace the three springs with an equivalent springReview examples of springs in series/parallelxkeqxk1k2k3keqxk1k2k3xEI, Lkeqmxreplace withmk11. Replace the three springs with an equivalent spring (no proof required).keq= ___________________________________Example in using the Tablekeq= ___________________________________Rose-Hulman Institute of TechnologyMechanical EngineeringVibrations1. Select system2. Model system elements (identify spring, mass and damping elements)3. Define inputs and outputs• Note: If measuring displacements from the static equilibrium point (SEP) you can leave gravity off the FBD in most cases.4. Displace mass in the positive direction5. Draw FBD (we usually do not need a kinetic diagram in this class)6. Apply the rate form of linear and angular momentum 7. Repeat steps 4-6 for each mass in the system8. Place final equations in standard form (if necessary)• SDOF system:• Second order matrix form:Example illustrating the note on step 3Procedure for finding equations of motion()tKfxxxnn=++&&&ωζω22[]{}[]{}[]{}{})t(fxKxCxM=++&&&δStatic equilibruim point (SEP)Point where spring is undeflectedxLo= free length of springf(t)mkcFBDxSEPk(x + δ)mgxc&Rose-Hulman Institute of TechnologyMechanical EngineeringVibrationsAll springs are non-linear springsForceRelative DisplacementUse linearization – only consider small displacements about an operating point. 1. Find the operating point2. Find the slope at the operating point: so kdxdFopx=x~kF~or =∆=∆ xkFRose-Hulman Institute of TechnologyMechanical EngineeringVibrationsExample problemnon-linear spring, FsxFsxFs= kx + γx3k = 3 lb/inγ = 0.5 lb/in310 lbFind the equation of motion for oscillations about the static equilibrium point (SEP). What is the natural frequency of this system?Rose-Hulman Institute of TechnologyMechanical EngineeringVibrationsConcept quizA spring has a mass of 0.2 kg and a spring constant of 50 N/m. Estimate the natural frequency of the spring. k, mk, mIf the spring is problem 1.2 originally has a length of L, what will its stiffness be if it is cut in half?Rose-Hulman Institute of TechnologyMechanical EngineeringVibrationsDamping models energy dissipationv1v2FFccDashpotViscous Friction (good for devices with lubrication)RotationcOther types of damping: Coulomb (dry friction), structural damping, hysteretic damping,