What to bring and what to studyMaterial (sections) CoveredAlso bringExpect 4 to 5 problemsFree-body diagram and equations of motion2nd Order Ordinary Differential Equation with Constant CoefficientsPeriodic MotionFrequencyAmplitude & Phase from the ICsOther forms of the solution:Peak ValuesSpring-mass-damper systemsSolution: Given m, c, k, x0, v0 find x(t)PowerPoint PresentationThree possibilities:Slide 16Slide 17Underdamped 0 < z < 1Potential and Kinetic EnergyConservation of EnergyDeriving equation of motionNatural frequencyStatic DeflectionCombining SpringsHarmonically Excited Systems Equations of motion (c =0):Linear nonhomogenous ode:Substitute into the equation of motion:Add particular and homogeneous solutions to get general solution:Slide 292.2 Harmonic excitation of damped systemsSlide 31Slide 32Slide 33Slide 34Feb 18, 20021/34Mechanical Engineering at Virginia Tech What to bring and what to study •One 8.5 X 11 formula sheet, one side only, no examples. Save the other side for test 2. •Put your name on it and turn it in with the test. •If you number the formulas. Suggest you use the same numbers as the text, you will be free to refer to them on the test. That is: “ from equation (1.37):d20 1 (0.01)2 19.9999 20 rad/s "Feb 18, 20022/34Mechanical Engineering at Virginia Tech Material (sections) Covered•1.1,1.2, 1.3, 1.4, 1.5, 1.7•2.1, 2.2•Log decrement from 1.6Feb 18, 20023/34Mechanical Engineering at Virginia Tech Also bring•Paper, pencil, calculator•No other resources allowed•Your honor, but no anxiety•Knowledge of all examples worked in class or presented in the text•All assigned homeworkFeb 18, 20024/34Mechanical Engineering at Virginia Tech Expect 4 to 5 problems1. An example covered in class 2. A homework problem 3. An example from the book, not covered in class4. A problem involving combining parts of any of the above in “two steps” and/or5. A derivation•25% (or 20%) each•the last problem(4 and/or 5) intended to sort out the A’s and B’sFeb 18, 20025/34Mechanical Engineering at Virginia Tech Free-body diagram and equations of motion•Newton’s Law:00)0(,)0(0)()()()(vxxxtkxtxmtkxtxmFeb 18, 20026/34Mechanical Engineering at Virginia Tech 2nd Order Ordinary Differential Equation with Constant Coefficients)sin()(rad/s infrequency natural 0)()(:by Divide2tAtxmktxtxmnnnFeb 18, 20027/34Mechanical Engineering at Virginia Tech Periodic Motion2nx(0)Time usually secDisplacement amplitudePhaseMaximumVelocityFeb 18, 20028/34Mechanical Engineering at Virginia Tech Frequencyn is in rad/s is the natural frequencyfnn rad/s2 rad/cyclen cycles2 sn2HzT 2n s is the periodWe often speak of frequency in Hertz, but we need rad/s in the arguments of the trigonometric functions.Feb 18, 20029/34Mechanical Engineering at Virginia Tech Amplitude & Phase from the ICs x0Asin(n0 ) Asinv0nA cos(n0 ) nA cosSolving yieldsA 1nn2x02 v02 Amplitude1 2 4 4 4 3 4 4 4 , tan 1nx0v0Phase1 2 4 4 3 4 4Feb 18, 200210/34Mechanical Engineering at Virginia Tech Other forms of the solution:x(t) Asin(nt )x(t) A1sinnt  A2cosntx(t) a1ejnt a2e jntSee window 1.4, page 12 for relationships among these.Feb 18, 200211/34Mechanical Engineering at Virginia Tech Peak ValuesAxAxAxnn2maxmaxmax :onaccelerati :velocity :ntdisplaceme:of peak valueor max Feb 18, 200212/34Mechanical Engineering at Virginia Tech Spring-mass-damper systems•From Newton’s law:00)0( ,)0( 0)()()()()( )(vxxxtkxtxctxmtkxtxcfftxmkcFeb 18, 200213/34Mechanical Engineering at Virginia Tech Solution: Given m, c, k, x0, v0 find x(t)less)(dimension ratio damping 2= and where 0)()(2)(by motion of equation Divide2kmcmktxtxtxmnnnFeb 18, 200214/34Mechanical Engineering at Virginia Tech Let x(t) aet & subsitute into eq. of motion a2et a2netn2aet0which is now an algebraic equation in : 1,2nn2 1from the roots of a quadratic equationHere the discriminant 2 1, determinesthe nature of the roots 1001Feb 18, 200215/34Mechanical Engineering at Virginia Tech Three possibilities:0020121 , :conditions initial theUsing)( 221=damped critically calledrepeated & equal are roots1 )1xvaxateaeatxmkmccnttncrnnFeb 18, 200216/34Mechanical Engineering at Virginia Tech 12)1( 12)1( where)()(1 :roots realdistinct two-goverdampin called ,1 )22020220201121122,122nnnntttnnxvaxvaeaeaetxnnnFeb 18, 200217/34Mechanical Engineering at Virginia Tech 1 re whe 1 :as formcomplex in roots writepairs conjugate as rootscomplex Twocommonmost -motion dunderdampe called ,1 )322,1jjnnFeb 18, 200218/34Mechanical Engineering at Virginia Tech Underdamped 0 <  < 100012020021211tan)()(1frequency natural damped ,1)]cos()sin([=)sin( )()(22xvxxxvAtBtAetAeeaeaetxnddndndddtdttjtjtnnnnnReduces to undamped formulasfor  = 0Feb 18, 200219/34Mechanical Engineering at Virginia Tech Potential and Kinetic EnergyThe potential energy of mechanical systems U is often stored in “springs” (remember that for a spring F=kx)20002100kxxkxxFUxxspringd d The kinetic energy of mechanical systems T is due to the motion of the “mass” in the system222121JTxmTrottransMkx0MassSpringx=0The potential energy of mechanical systems U is also gravitational: mghmgdxUhgrav0Feb 18, 200220/34Mechanical Engineering at Virginia Tech Conservation of EnergyT U constantor ddt(T U) 0 For a simply, conservative (i.e. no damper), mass spring system the energy must be conserved: At two different times t1 and t2 the increase in potential energy must be equal to a decrease in kinetic energy (or visa-versa). U1