1 Equations of Motion 3: Equivalent System Method

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1 8 Mechanica Vibrations Equations of Motion : Equivaent System Method In systems in which masses are joined by rigid ins, evers, or gears and in some distributed systems, various springs, dampers, and masses can be epressed in terms of one coordinate at a specific point and the system is simpy transformed into a singe DOF system This method is caed the equivaent system method used to simpify higher DOF probems We aready discussed this method for systems of rigid bodies in Lesson In this esson, we present two more eampes of distributed systems Eampe : [] Consider a spring-mass system in equiibrium beow Assuming that the spring has mass m per unit ength z, use the equivaent system method to find the equivaent mass of the system and determine its effect on the natura frequency s Soution For the equivaent system, the inetic energy is y T eq = meq The inetic energy of the actua system is given by dy z z y ms z z T = dy+ m ms m = + m ms Therefore, the equivaent mass is meq = + m, and the natura frequency of the system is ω n = ms + m Copyright 7 by Withit Chatatanaguchai

2 Eampe : [] A simpy supported beam has tota mass m b Supposed that the equivaent spring stiffness of the beam is 48 EI / and the defection under the oad due to a concentrated force appied at midspan is y= yma 4, determine the effective mass of the system at midspan and find its fundamenta frequency 8 Mechanica Vibrations / M m b Soution The effective system has inetic energy T eq = meq y ma where y ma is the maimum defection at midspan The actua system has inetic energy m b ma 4 T = y d+ My = ( 4857 mb + M) y ma Copyright 7 by Withit Chatatanaguchai, ma Therefore, the equivaent mass is m = 4857 m + M, and the natura frequency is ω = n 48EI eq ( ) mb M b

3 Equations of Motion 4: Virtua Wor Method The virtua wor method is another scaar method besides the wor and energy method It is usefu especiay for systems of interconnected bodies of higher DOF The principe of virtua wor states that If a system in equiibrium under the action of a set of forces is given a virtua dispacement, the virtua wor done by the forces wi be zero 8 Mechanica Vibrations Eampe : [] Use the virtua wor method, determine the equation of motion for the system beow m Equiibrium position δ Soution Draw the system in the dispaced position and pace the forces acting on it, incuding inertia and gravity forces Give the system a sma virtua dispacement δ and determine the wor done by each force Using the fact that virtua wor done by eterna forces equas virtua wor done by inertia forces, we then obtain the equation of motion for the system The virtua wor done by inertia forces is δw = m δ The virtua wor done by eterna forces is δw = δ Equating the two quantities above and canceing δ, we have the equation of motion m + = Copyright 7 by Withit Chatatanaguchai

4 Eampe 4: [] ( ) p f t 8 Mechanica Vibrations / / c M θ ( ) p f t d Soution Draw the beam in the dispaced position θ and pace the forces acting on it, incuding the inertia and damping forces Give the beam a virtua dispacement δθ and determine the wor done The virtua wor done by inertia forces is θ δθ θ cθ δw M = θ δθ The virtua wor done by eterna forces is δw = ( c θ) δθ θ δθ + ( p f ( t) d) δθ = ( c θ) δθ θ δθ + p f ( t) δθ Therefore, we have the equation of motion M θ + θ + θ = 4 ( c ) p f ( t) 4 Copyright 7 by Withit Chatatanaguchai

5 Eampe 5: [] Two simpe penduums are connected together with the bottom mass restricted to vertica motion in a frictioness guide as shown beow Using the virtua wor method, determine the equation of motion 8 Mechanica Vibrations δθ θ m θ δθ sinθ m δθ sinθ ma m g m mg θ ( θ ) θ a ( θ ) B θ A 5 Copyright 7 by Withit Chatatanaguchai

6 Soution First, we find acceeration of mass m by using the reative motion anaysis with transating aes using the formua a a a B = A+ B / A Pacing point A on the transating aes, the formua above in the vertica direction becomes ( a ) = ( a ) + ( a ) B A B / A θ cosθ + θ sinθ = a θ cosθ θ sinθ ( θ θ θ θ) a = cos + sin Homewor Probems None 8 Mechanica Vibrations Homewor probems are from the required tetboo (Mechanica Vibrations, by Singiresu S Rao, Prentice Ha, 4) References [] Theory of Vibration with Appications, by Wiiam T Thomson and Marie Dion Daheh, Prentice Ha, 998 The virtua wor done by inertia forces become ( )( ) ( )( ) ( ) δw = mθ δθ + m θ cosθ + θ sinθ δθ sin θ The virtua wor done by eterna forces become ( ) ( ) δw = m g δθ sinθ m g δθ sin θ Therefore, the equation of motion becomes ( ( ))( ) = m θ + m θ cosθ + θ sinθ sinθ ( θ) ( θ) + m g sin + m g sin If θ is sma, sin θ θ zero The equation of motion then becomes a is sma and can be assumed to be m g θ + + = m θ 6 Copyright 7 by Withit Chatatanaguchai

Lobontiu: System Dynamics for Engineering Students Website Chapter 3 1. z b z

Lobontiu: System Dynamics for Engineering Students Website Chapter 3 1. z b z Chapter W3 Mechanica Systems II Introduction This companion website chapter anayzes the foowing topics in connection to the printed-book Chapter 3: Lumped-parameter inertia fractions of basic compiant