MODELING MECHANICAL AND ELECTRICAL
2.7 CHAPTER SUMMARY
∂VT
∂u1 =ku1+αk(u2−u1)(−1)
∂VT
∂u2 =αk(u2−u1)+ku2. Taking the time derivative of the kinetic energy terms yields
d dt
∂T
∂u˙1 =mu¨1
d dt
∂T
∂u˙2
=mu¨2.
The generalized mechanical forces are F1= f1 F2= f2.
Combining the terms according to equation (2.205) produces the governing equations f1 =ku1−αk(u2−u1)+mu¨1
f2 =αk(u2−u1)+ku2+mu¨2.
The two equations can be rewritten with the forcing terms on the right-hand side as mu¨1+k(1+α)u1−αku2= f1
mu¨2−αku1+k(1+α)u2= f2
and the two equations can also be rewritten in matrix form as
m 1 0
0 1
¨ u1
¨ u2
+k
1+α −α
−α 1+α u1 u2
= f1
f2
,
which is a standard second-order form for vibrating systems.
properties of materials. Similarly, a review of electrostatics was based on the defi- nitions of charge, electric potential, and electric field. These relationships were then used to define insulating and conducting materials. The review of work and energy methods presented for mechanical and electrical systems will serve as a precursor to discussions later when we develop equations of motion derived from variational principles of mechanics. One of the central features of work and energy methods is that the terms associated with the analysis are scalar quantities. This aspect of work and energy analysis often simplifies the procedures associated with finding equations of motion for smart material systems.
Defining the fundamental elements of mechanical and electrical analysis allowed us to study several representative problems in mechanical and electrostatics. The axial deformation of a bar was studied to highlight one-dimensional mechanics analysis.
Beam analysis was also presented to demonstrate how the equations of mechanics could be used to derive expressions for the static displacement of beams for various boundary conditions. Common electrical elements such as a capacitor were studied using definitions from electrostatics. All of these basic elements are used later to analyze and design smart material systems.
In the final section of the chapter we reviewed variational methods for deriving equations of motion based on the work and energy concepts introduced earlier in the chapter. Variational approaches for static and dynamic systems were presented.
PROBLEMS
2.1. A solid has the displacement field
u1 =6x1
u2 =8x2
u3 =3x32. Determine the strain field in the material.
2.2. A solid has the displacement field
u1=x12+x22 u2=2x2x1
u3=0.
Determine the strain field in the material.
2.3. Determine if the stress field
T11=4x12x3 T33 =4
3x32 T13= −4x1x32
is in equilibrium when the body forces are assumed to be equal to zero.
2.4. A material is said to be inplane strainif S3 =S4=S5=0.
(a) Write the stress–strain relationships for a linear elastic, isotropic material assumed to be in a state of plane strain.
(b) Compute the state of stress for an isotropic material with a modulus of 62 GPa and a Poisson’s ratio of 0.3 if the strain state is
S1 =150µstrain S2=50µstrain S6 = −35µstrain.
2.5. Compute the electrostatic force vector between a charge of 200µC located at (0,2) and a second charge of−50µC located at (−1,−6) in a two-dimensional plane in free space. Draw a schematic of this problem, identifying the locations of the charged particles and the electrostatic force vector.
2.6. A linear spring of stiffnessk has a charged particle ofq coulombs fixed at each end. Determine the expression for the deflection of the spring at static equilibrium if the spring is constained to move in only the linear direction.
2.7. A set of fixed charges is located in free space as shown in Figure 2.21.
(a) Compute the electric field at the origin of the coordinate system.
(b) Compute the electric field at (0,c/2) and (0,−c/2).
2.8. The charge density profile at the interface between two materials is modeled as ρv(x)=αxe−β|x|.
A representative plot of the charge density is shown in Figure 2.22.
(a) Compute the function for the electric displacement. Assume that the electric displacement is continuous atx→0.
–q
–q
–q q
q
q
b c
Figure 2.21 Fixed charges located in free space.
ρv(x)
x
Figure 2.22 Charge density profile at the interface between two materials.
(b) Plot the charge density and electric displacement forα=10 andβ=3.
2.9. The charge density within the material shown in Figure 2.22 has the profile ρv(x)=αxe−β|x|(1−e−λt).
Compute the expression for flux in thexdirection.
2.10. An electric field of 10 mV/m is applied to a conductive wire with a circular cross section. The wire has a diameter of 2 mm and a conductivity of 50 (·µm)−1. Compute the current in the wire.
2.11. An isotropic dielectric material with the permittivity matrix ε=diag(500,500,1500)×8.54×10−12F/m has an applied electric field of
E=100 ˆx1+500 ˆx2V/mm. Compute the electric displacement in the material.
2.12. Compute the work required to lift a 5-kg box from the ground to a height of 1.3 m.
2.13. A model for a nonlinear softening spring is f(u)= −ktan−1 u
us
,
wherek/us represents the small displacement spring constant andus is the saturation displacement.
(a) Compute the energy function U and the potential energy function V for this spring.
(b) Plot the force versus displacement over the range−10 to 10 for the values k=100 N/mm and us =3 mm. Compute the work required to stretch the spring from 0 to 5 mm and illustrate this graphically on a plot of force versus displacement.
2.14. The potential energy function for a spring is found to be
V= 1
2k1u21+k2u1u2+1
2k3u22+1 2k4u23. Determine the force vector for this spring.
2.15. Determine the displacement function for a cantilevered bending beam with a load applied atx1=Lf, whereLf <L. Note that the solution will be in the form of a piecewise continuous function.
2.16. Determine the displacement function for a cantilevered bending beam with mo- mentM1applied atx1=L1and moment−M1applied atx1 =L2, whereL2>
L1. Note that the solution will be in the form of a piecewise continuous function.
2.17. (a) Determine the expression for the stored energy of a cantilevered bending beam with a load applied at the free end.
(b) Repeat part (a) for a pinned–pinned beam with a load applied at the center.
2.18. (a) Compute the electric field in a capacitor of thickness 250 µm with an applied voltage difference of 100 V. The dielectric material in the capacitor has a relative dielectric constant of 850 and a surface area of 10 mm2. (b) Compute the charge stored in the capacitor with the properties given in
part (a).
(c) Compute the stored energy in the capacitor with the properties given in part (a).
k
ψ u
gravity m2
m1
Figure 2.23 Two-mass mechanical system.
ψ –q
pivot with torsional spring kt –q
q
Figure 2.24 Rigid link.
2.19. Use the variational approach for static systems to derive the equations of motion for the mechanical system shown in Figure 2.23. Assume that the masses are both zero for this analysis. Use u and ψ as the generalized coordinates for the analysis.
2.20. Determine the governing equations for a system that has the potential energy and work expressions
VT = 1
2ku2+duq+ 1 2Cq2 WM+WE = f u+vq,
where the generalized coordinates areuandq.
2.21. A rigid link of length 2ahas two charges attached to its end (Figure 2.24). It is placed in a free space with two fixed charges. The fixed charge−q is located at (0,b) and the fixed charge+q is located at (0,−b), whereb>a. At the center of the rigid link is a pivot that contains a linear torsional spring of spring constantkt.
(a) Determine an expression for the potential energy of this system (ignoring gravity).
(b) Use the variational principle to determine the governing equations of static equilibrium.
2.22. Use the variational approach for dynamic systems to derive the equations of motion for the mechanical system shown in Figure 2.23. Useuandψas the generalized coordinates for the analysis.
2.23. Repeat Problem 2.20 including a kinetic energy term of the form T=12mu˙2.
NOTES
The material in this chapter was drawn from several textbooks on the subjects of mechanics, electrostatics, and work and energy methods. The book by Gere and
Timoshenko [12] was used as a reference for the sections on mechanics of materials, as was to the text of Allen and Haisler [13]. References on work and energy methods included Pilkey and Wunderlich [14] and Reddy [15] for mechanical systems and the excellent text by Crandall et al. [16] for electromechanical systems. The latter text includes a thorough discussion of systems that incorporate both mechanical and electrical energy.