MIT OpenCourseWare http ocw mit edu Electromechanical Dynamics For any use or distribution of this textbook please cite as follows Woodson Herbert H and James R Melcher Electromechanical Dynamics 3 vols Massachusetts Institute of Technology MIT OpenCourseWare http ocw mit edu accessed MM DD YYYY License Creative Commons Attribution NonCommercial Share Alike For more information about citing these materials or our Terms of Use visit http ocw mit edu terms Simple Elastic Continua transient situations At the same time the frequency wavenumber picture of the dynamics represented by the dispersion equation introduced in Section 9 2 provides the unifying theme for Chapter 10 PROBLEMS 9 1 A long thin steel cable of unstressed length I is hanging from a fixed support as illustrated in Fig 9P 1 Assume that the origin of coordinates is at the support and that x measures positive as shown Assume that the steel cable has the following constants A 10 4 m2 Cross sectional area E 2 0 x 1011 N m 2 Young s modulus Mass density p 7 8 x 10i kg m s Maximum allowable stress Tmax 2 x 109 N m2 1 x Steel cable Fig 9P 1 a Find the length of cable Ifor which the maximum stress in the cable just equals the maximum allowable stress b Find the displacement 6 and stress T in the cable as functions of z c Find the total elongation of the cable 9 2 Two thin elastic rods are arranged as shown in Fig 9P 2 The first rod has modulus of elasticity El density Pl and cross sectional area A1 It is attached at one end to a rigid wall and at the other to a very thin rigid plate of mass m and area Am On the other side of V0 Fig 9P 2 Problems this plate is attached a second thin elastic rod with elastic modulus E2 density P2 and crosssectional area A2 The other end of the second rod is fixed to a perfectly conducting thin plate with mass M and area AM This plate is held at a potential Vo with respect to a second capacitor plate a distance d away In the absence of gravity and with Vo 0 the length of the first rod is L1 and the length of the second is L 2 Assuming now that the system is immersed in a gravitational field g and that Vo 0 find the following a The stress in the first rod TM x and the displacement in the first rod 6V b The stress in the second rod T 2 x and the displacement in the second rod 6 2 z 9 3 In Fig 9P 3 a thin elastic rod of cross sectional area A equilibrium length 1 elastic modulus E and mass density p is fixed at one end x 0 and attached to a rigid mass M x O I u Equilibrium length 0 f t M le m uirbliuqe elastic modulus E mass density p Fig 9P 3 at the other x 1 The mass is driven by a force source f t The system constants are such that the mass M is much greater than the mass of the elastic rod that is M pAl The force source is constrained to be f t Re foei t where fo and ao are positive real constants The system is operating in the sinusoidal steady state Neglect gravity a Find the displacement 6 x t and stress T x t in the elastic rod b Show a lumped parameter mechanical system that represents the behavior of the system in Fig 9P 3 for low frequencies from wO 0 up to and including the lowest resonance frequency Evaluate the equivalent elements in terms of the given parameters 9 4 A long thin elastic rod with cross sectional area A unstressed length 1 modulus of elasticity E and mass density p is constrained at one end by the three ideal lumped elements Elastic rod x 0 Fig 9P 4 A Simple Elastic Continua B K and M and at the other by a stress source To t as shown in Fig 9P 4 Determine the values of B K and M that are necessary for the response of the rod to To t to be representable purely as a wave traveling in the negative x direction Zero is an acceptable value for an element 9 5 A long thin rod of elastic modulus E mass density p and cross sectional area A is fixed at one end x 0 and constrained at the other x 1 by a force source f t and a Long thin rod elastic modulus E mass density p cross sectional area A f t 6 1 t xO Fig 9P 5 lumped linear damper of coefficient B as illustrated in Fig 9P 5 The applied force is sinusoidal f t Re Foeiwt where F0 and w are positive constants The system is operating in the sinusoidal steady state a Write the boundary condition at x I in terms of the stress T 1 t the displacement 6 1 t and the applied forcef t b Assume that the displacement has the form 6 x t Re 6 x ej t Find the complex amplitude 6 x in terms of given data c If the damper coefficient B is positive and the frequency w is real can the system exhibit a resonance That is can the displacement 6 be infinite with a zero applied force Give justification for your answer 9 6 A thin circular magnetic rod is fixed at one end and constrained at the other end by a transducer Fig 9P 6 In the absence of an excitation the transducer is simply biased by Equilibrium gap s Thin rod mass density p Sla ctire modullus E cross sectional area A Ignorable j x gaps I Fig 9P 6 Problems the constant current source L When the rod is in static equilibrium its length is I and the gap spacing is d Compute the natural frequencies of the system under the assumption that the magnetization force on the rod acts on the end surface A graphical representation of the eigenfrequencies is an adequate solution 9 7 In Fig 9P 7 two identical thin elastic rods are connected by a thin plate of mass M and area AM The plate is positioned between four springs each having constant K All M K e V c r K Wall N E p A E p A 6 L t 6osifwt K L AK I1 W4 K i L Fig 9P 7 springs are relaxed when the plate is at x 0 The system is driven on the left with a displacement S L t So sin wt Assume that steady state has been established a Write the general solution for the stress T x and the displacement S x everywhere in both rods in terms of arbitrary constants b What are the boundary conditions that determine the constants in a c Find the stress T x t everywhere in both rods 9 8 Example 9 1 5 considers the response of …
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