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/termsSimple Elastic Continuatransient situations. At the same time the frequency-wavenumber picture ofthe dynamics, represented by the dispersion equation introduced in Section9.2, provides the unifying theme for Chapter 10.PROBLEMS9.1. A long thin steel cable of unstressed length I is hanging from a fixed support, asillustrated in Fig. 9P.1. Assume that the origin of coordinates is at the support and that xmeasures positive as shown. Assume that the steel cable has the following constants:Cross-sectional area A = 10- 4 m2Young's modulus E = 2.0 x 1011 N/m2Mass density p = 7.8 x 10i kg/msMaximum allowable stress Tmax = 2 x 109 N/m2xSteel cable -Fig. 9P.1(a) Find the length of cable Ifor which the maximum stress in the cable just equals themaximum 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 ofelasticity El, density Pl, and cross-sectional area A1.It is attached at one end to a rigidwall and at the other to a very thin rigid plate of mass m and area Am. On the other side ofV0Fig. 9P.21Problemsthis plate is attached a second thin elastic rod with elastic modulus E2, density P2, and cross-sectional area A2.The other end of the second rod is fixed to a perfectly conducting thinplate with mass M and area AM. This plate is held at a potential Vo with respect to a secondcapacitor plate a distance d away. In the absence of gravity and with Vo = 0, the lengthof the first rod is L1 and the length of the second is L2.Assuming now that the system isimmersed 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 rod6(2)(z).9.3. In Fig. 9P.3 a thin elastic rod of cross-sectional area A, equilibrium length 1, elasticmodulus E, and mass density p is fixed at one end (x = 0) and attached to a rigid mass Mx=OI u Equilibrium length _.0 f(t)uirbliuqem le Melastic modulus Emass density pFig. 9P.3at the other (x = 1). The mass is driven by a force source f(t). The system constants aresuch that the mass M is much greater than the mass of the elastic rod; that is,M > pAl.The force source is constrained to bef(t) = Re (foei't),where fo and ao are positive real constants. The system is operating in the sinusoidal steadystate. 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 thesystem in Fig. 9P.3 for low frequencies (from wO= 0 up to and includingthe lowest resonance frequency). Evaluate the equivalent elements in terms of thegiven parameters.9.4. A long thin elastic rod with cross-sectional area A, unstressed length 1, modulus ofelasticity E,and mass density p is constrained at one end by the three ideal, lumped elementsElastic rod.--.. ..- : -... .. Ax=0Fig. 9P.4Simple Elastic ContinuaB, K, and M and at the other by a stress source To(t), as shown in Fig. 9P.4. Determine thevalues of B, K, and M that are necessary for the response of the rod to To(t) to be rep-resentable purely as a wave traveling in the negative x-direction. (Zero is an acceptablevalue for an element.)9.5. A long thin rod of elastic modulus E, mass density p, and cross-sectional area A isfixed at one end (x = 0) and constrained at the other (x = 1)by a force source f(t) and aLong thin rodelastic modulus Emass density pcross-sectional area A f(txO 6(1, t)Fig. 9P.5lumped linear damper of coefficient B, as illustrated in Fig. 9P.5. The applied force issinusoidal f(t) = Re (Foeiwt), where F0 and w are positive constants. The system is oper-ating in the sinusoidal steady state.(a) Write the boundary condition at x = I in terms of the stress T(1, t), the displace-ment 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 thecomplex amplitude 6(x) in terms of given data.(c) If the damper coefficient B is positive and the frequency w is real, can the systemexhibit a resonance? That is, can the displacement 6 be infinite with a zero appliedforce? Give justification for your answer.9.6. A thin, circular magnetic rod is fixed at one end and constrained at the other end by atransducer (Fig. 9P.6). In the absence of an excitation, the transducer is simply biased byEquilibrium gapsThin rodmass density pSla•ctire modullus E/ cross-sectional area AIgnorable-x gapsIFig. 9P.6jProblemsthe constant current source L When the rod is in static equilibrium, its length is I and thegap spacing is d. Compute the natural frequencies of the system under the assumption thatthe magnetization force on the rod acts on the end surface. A graphical representation ofthe eigenfrequencies is an adequate solution.9.7. In Fig. 9P.7 two identical thin elastic rods are connected by a thin plate of mass Mand area AM.The plate is positioned between four springs, each having constant K. All.M AKK /e KrE, p,A6(-L, t)= 6osifwtE, p, A_W4 1IK KL >i _L >Fig. 9P.7springs are relaxed when the plate is at x = 0. The system is driven on the left with a dis-placement 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) everywherein 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 the delay line shown in Fig. 9.1.14 to a transientinput signal. In this example design approximations were made concerning the effect of theself-inductance in the output circuit [see approximation following (t)]. You wish to computethe sinusoidal steady-state response of this system without making this approximation.Confine your attention to the sinusoidal
View Full Document
Unlocking...