Appendix A: Conservaion of Mechanical Energy = Conservaion of Linear Momenum Consider he moion of a nd order mechanical sysem comprised of he fundamenal mechanical elemens: ineria or mass (M), siffness (K), and viscous damping coefficien, (D). The Principle of Conservaion of Linear Momenum (Newon s nd Law of Moion) leads o he following nd order differenial equaion: M D K F where he coordinae () describes he sysem moion. has is origin a he sysem saic equilibrium posiion (SEP). () () W In he free body diagram above, F () =F ex is he exernal force acing on he sysem, Fk KY K s is he reacion force from he spring. s =W/K represens he saic deflecion Y s is he oal deflecion of he spring from is unsreched posiion. FD D is he reacion force from he dashpo elemen. MEEN 67 noes: Appendix A Luis San Andrés, 03 A-
M D K( ) F W () is recas as s () () Now, inegrae his Eq. () beween wo arbirary displacemens ( ), ( ) occurring a imes and, respecively A hese imes he sysem velociies are ( ), ( ), respecively. The process gives: s () M d Dd K d F W d (a) Since Y s hen dy=d, hen wrie Eq. (a) as Y M d Dd KYdY F W d Y () (b) d The acceleraion and velociy are d Using hese definiions, wrie Eq. (b) as:, d d, respecively. d d d M d D d K d Y F d W d d d d Y () Y or, Y d M d D d Kd Y Wd F() d d Y MEEN 67 noes: Appendix A Luis San Andrés, 03 A-
Y Y M d D d K Y W F d ( ) (3) and since (M, K, D) are consan parameers, express Eq. (3) as: ( ) ( ) ( ) ( ) M D d K Y Y W F d (4) Le s recognize several of he erms in he equaion above. These are known as Change in kineic energy, T T M M (5.a) Change in poenial energy (elasic srain and graviaional) V V KY KY W W (5.b) Toal work from exernal force inpu ino he sysem, W F d (5.c) ( ) Se v D as he viscous power dissipaion, Then, he dissipaed viscous energy (removed from sysem) is, E v D d v d (5.d) Wih hese definiions, wrie Eq. (4) as T T V V E W (6) v MEEN 67 noes: Appendix A Luis San Andrés, 03 A-3
Tha is, he change in (kineic energy + poenial energy) + he viscous dissipaed energy = Exernal work. This is also known as he Principle of Conservaion of Mechanical Energy (PCME). Noe ha Eq. () and Eq. (6) are NOT independen. They acually represen he same physical law. Noe also ha Eq. (6) is no o be misaken wih he firs-law of hermodynamics since i does no accoun for hea flows and/or changes in emperaure. One can paricularize Eqn. (6) for he iniial ime 0 wih iniial displacemen and velociies given as (, 0 0 ), and a an arbirary ime () wih displacemens and velociies equal o (, ), respecively, i.e., Thus, he PCME saes () () T V W E T V (7) ( ) () (0 ) v(0 ) 0 0 where T0 V0 is he iniial sae of energy for he sysem a ime =0 s. Eqn. (7a) is also wrien as ( ) M () KY() W F() d D d M 0 KY0 W 0 0 0 (8) Taking he ime derivaive of Eq. (7) gives d T d d d V ( ) () dw de v ex v (9) MEEN 67 noes: Appendix A Luis San Andrés, 03 A-4
where ex, v are he mechanical power from exernal forces acing on he sysem and he power dissipaed by a viscous-ype forces, respecively. Work wih Eq. (8) o obain d d () M () K Y() W F() d D d M 0 K Y0 W 0 0 0 d dy d d M KY W F D d d d d () () () () () () () (0) To obain Recall ha he derivaive of an inegral funcion is jus he inegrand. M KY Y W F D Since Y s () () () () () () and Y, Eq. () becomes () M () K() s W F() D (a) Canceling he saic load balance erms, W=K s, and facoring ou he velociy, obain M K D F () () () () () () Since for mos imes he sysem velociy is differen from zero, i.e., ; ha is, he sysem is moving; hen () 0 M D K F () i.e., he original equaion derived from Newon s Law (conservaion of linear momenum). () MEEN 67 noes: Appendix A Luis San Andrés, 03 A-5
Suggesion/recommended work: Rework he problem for a roaional (orsional) mechanical sysem and show he equivalence of conservaion of mechanical energy o he principle of angular momenum, i.e. sar wih he following Eqn. I D K T () where (I, D θ, K θ ) are he equivalen mass momen of ineria, roaional viscous damping and siffness coefficiens, T () =T ex is an applied exernal momen or orque, and θ() is he angular displacemen of he roaional sysem. MEEN 67 noes: Appendix A Luis San Andrés, 03 A-6