In this section, we use the techniques of Chapter 4 to simulate a rotational mechanical sys- tem. We specify a nonlinear load torque and show that this is easily added to the simulation. In Chapter 9 we will show how to treat such a nonlinearity analytically.
~,,, EXAMPLE 5.13
Draw a Simulink diagram to solve (42) and simulate the system in Example 5.5. As-sume that TL(t) is given by TL = alw2lw2. The parameter values are 11 = 10 kg-m/, h = 150 kg·m2, K = 800 Nrn, B = 50 N·m·s, anda= 50 N·m·s2. Plot w1 and w2 during the interval O~ t ~ 10 s when the input Ta(t) is a step of 100 N·m applied at t = 1 s.
SOLUTION We start the Simulink diagram with three integrators, whose inputs are the first derivatives of the three state variables. Each of these derivatives is the output of a gain block preceded by a summer. In order to avoid long crossing lines when obtaining the inputs for the summers, we use two new blocks: the Gato and From blocks, both of which are found in the Signals & Systems Iibrary. A signa! that is the input to a Gato block appears at ali From blocks that have the same tag. Using these two blocks can simplify the Simulink diagram by avoiding the need for long lines to carry signals created at one location to a remote site where they are needed. The complete diagram is shown in Figure 5.31.
We cannot show the variable TL(t) asan externa! input, because its value is completely determined by the variable w2. We create TL, which is aiw2lw2, in the lower left comer of the diagram by using two nonlinear blocks that are found in the Math library. The block named Abs is used to create the absolute value of its input, which is w2. The next block is the Product function, used to produce the product of its two inputs. Here these are
thetaR
The applied torque Ta(t) is obtained from the Step function in the Sources library.
We let the default option set the initial conditions on ali the integrators equal to zero. The following M-file will produce the desired responses, which are shown in Figure 5.32.
Jl 10; J2
=
150; % kg.mA2In this chapter we extended the techniques of Chapters 2 and 3 to include bodies that rotate about fixed axes. Moment of inertia, friction, and stiffness elements are characterized by an algebraic relationship between the torque and the angular acceleration, velocity, or displacement, respectively. In contrast, levers and gears are described by algebraic equa- tions that relate two variables of the same type (two displacements, for example, or two velocities). We showed how systems with levers, gears, or pulleys can have both rotational and translational motion.
1
-126 Rotational Mechanical Systems Problems 126
3
2.5
~~¡:;. 2
·c:;o
~(¡; 1.5
"S
Oel
<(
0.5
/ e- (02
»:
/ / / / / /
- -
- - --2 4 6 8 10 Time(s)
Figure 5.32 Angular velocities produced by a constant torque of 100 N·m applied at t = l s.
As in Chapter 2, we drew free-body diagrams to help us obtain the basic equations goveming the motion of the system. We then combined these equations into sets of state-variable equations or input-output equations. The same general modeling procedures are used for other types of systems in later chapters.
fJ> PROBLEMS
5.1. Write a differential equation for the system shown in Figure P5.1 and determine the equivalent stiffness constant.
K2 /
Figure PS.l
*5.2. The left side of the fluid drive element denoted by B in Figure P 5.2 moves with the angular velocity w0(t). Find the input-output differential equation relating w: and w0(t).
5.3.a. Choose a set of state variables for the system shown in Figure P 5.3, assuming that the values of the individual angular displacements 81 ,82,83, and 84 with respect
Figure PS.2
B
Figure PS.3
to a fixed reference are not of interest. Then write the state-variable equations describing the system.
b. Find the input-output differential equation relating w4 and Ta(t).
5.4. Repeat part (a) of Problem 5.3 if the torque input Ta(t) is replaced by the angular velocity input w1 (t).
5.5. a. Find a state-variable model for the system shown in Figure P5.5. The input is the applied torque Ta(t), and the output is the viscous torque on h, with the positive sense counterclockwise.
b. Write the input-output differential equation relating w2 and Ta(t) when K1
=
K2=
B
=
]¡=
Ji=
l in a consistent set of units.Negligible inertia
Figure PS.5
*5.6.
a. For the system shown in Figure P5.6, select state variables and write the model in state-variable form. The input is the applied torque T0(t), and the output is the viscous torque acting on J: through the fluid drive element, with the positive sense counterclockwise.
b. Write the input-output differential equation relating fh and T0(t) when K1
=
K2=
B
=
J¡=
J:=
1 in a consistent set of units.Figure PS.6
*5.7. Find the value of Keq in Figure P5.7(b) such that the relationship between Ta(t) and ()
is the same as for Figure P5.7(a).
(a) (b) Figure PS.7
5.8. Use the p-operator technique to derive (43).
5.9. For the system shown in Figure P5.9, the angular motion of the ideal lever from the vertical posítion is small, so the motion of the top and midpoint can be regarded as horizontal. The input is the force .f~ (t) applied at the top of the lever, and the output is the support force on the lever, taking the positive sense to the right.
a. Select a suitable set of state variables and write the corresponding state-variable model.
b. Find the input-output dífferential equation relating x and .f~(t ).
f~(I)---Figure PS.9
*5.10. In the mechanical system shown in Figure P5.10, the input is the applied force .fa(t), and the output is the tensile force in the spring K2. The lever is ideal and is horizontal when the system is in static equilibrium with
f;
1(t) =O and M supported by the spring K¡. The displacements x1 ,x2,x3, and 8 are measured with respect to this equilibrium position. Thelever angle
e
remains small.a. Taking x¡, v1, and 8 as state variables, write the state-variable model.
b. Also write the algebraic output equation for the reaction force of the pivot on the lever, taking the positive sense upward.
Vz
Figure PS.10
5.11. For the system shown in Figure P 5.11, the input is the force f;,(t ), and the output is x¡. Assume that the lever is ideal and that
IBI
issmall.
a. Select a suitable set of state variables and write the corresponding state-variable model.
K¡ , fJt)
I~
FigurePS.11
b. Find the input-output differential equation when a= 2 and b
=
K1=
K2=
B=
M1
=
M2=
1 in a consistent set of units.5.12. Repeat part (a) of Problem 5.9 when the moment of inertia of the lever cannot be neglected and the output is the displacement x.
*5.13. Repeat part (a) of Problem 5.10 when the moment of inertia of the lever cannot be neglected.
5.14. Repeat part (a) of Problem 5.11 when the moment of inertia of the lever cannot be neglected.
5.15. Figure P 5.15 shows two pendulums suspended from frictionless pivots and connected at their midpoints by a spring. Each pendulum may be considered a point mass M at the end of a rigid, massless bar of length L. Assume that 1e11 and 1e2I are sufficiently small to allow use of the small-angle approximations sine'.::::'.
e
and cose'.::::'. l. The spring is unstretched whene1
=e2.
a. Draw a free-body diagram for each pendulum.
b. Define a set of state variables and write the state-variable equations.
c. Write an algebraic output equation for the spring force. Consider the force to be positive when the spring is in tension.
M
Figure PS.15 Figure PS.16
5.16. Find the equivalent stiffness constant Keq such that the algebraic model of the gears and shafts shown in Figure PS.16 can be written as B1 = (l/Keq)10(t).
5.17. In the system shown in Figure PS.17, a torque 10(t) is applied to the cylinder ]¡.
The gears are ideal with gear ratio N
=
R2/R1.a. Write the input-output differential equation when the inertia J, is reftected to h and w2 is the output.
b. Write the equation when J: is reftected to 11 and w1 is the output.
Figure PS.17
*5.18. A torque input 10(t) is applied to the lower gear shown in Figure PS.18. There are two outputs: the total angular momentum and the viscous torque acting on ]¡, with the positive sense clockwise.
a. Write a pair of simultaneous differential equations describing the system, where the contact force between the gears is included.
b. Select state variables and write the model in state-variable form.
c. Derive an input-output differential equation relating ()¡ and 111(t ).
*5.19. The input to the rotational system shown in Figure PS.19 is the applied torque 10(t) and the output is ()3. The gear ratio is N = R2/R3. Using B1,w1,B2, and w2 as the state variables, write the model in state-variable form.
5.20. Repeat Problem 5.19 when the state variables are ()R = ()¡ - B2, w1, ()3, and w3.
132 ''" Rotational Mechanical Systems Problems <i 132
FigurePS.18
FigurePS.19
5.21. The input for the drive system shown in Figure P5.21 is the applied torque ra(t), and a load attached to the moment of inertia J: produces the load torque TL = Alw2 lw2.
a. Taking as state variables W¡' w2, Wa, <l>a = B1 ea and <l>b = B2 eb, write the state-variable equations.
b. Write an algebraic output equation for the contact force on gear a, with the positive sense upward.
FigurePS.21
5.22. The input to the combined translational and rotational system shown in Figure P 5.22 is the force j;1(t) applied to the mass M. The elements K1 and K2 are undeflected when x= O ande =0.
a. Write a single input-output differential equation for x.
b. Write a single input-output differential equation for
e.
f,,(t)
Figure P5.22
5.23. Starting with (63), find the input-output differential equation for the system shown in
Figure 5.27, taking Ta(t) as the input and x as the output.
5.24. In the mechanical system shown in Figure P 5.24, the cable is wrapped around the disk and does not slip or stretch. The input is the force
f;,
(t), and the output is the displacement x. The springs are undeflected when ()=
x=
O.a. Write the system modelas a pair of differential equations involving only the vari-ables x, (), [a (z), and their derivatives.
b. Select state variables and write the model in state-variable form.
c. Find an input-output differential equation relating x and .f~(t) when K1
=
K2=
K3
=
B1=
B2=
M=
J=
R=
l in a consistent set of units.Figure P5.24
5.25. A mass and a translational spring are suspended by cables wrapped around two sections of a drum as shown in Figure P,5.25. The cables are assumed not to stretch, and the moment of inertia of the drum is J. The viscous-friction coefficient between the drum anda fixed surface is denoted by B. The spring is neither stretched nor compressed when ()=O.
a. Write a differential equation describing the system in terms of the variable é.
b. For what value of () will the rotational element remain motionless?
c. Rewrite the system's differential equation in terms of ef¡, the relative angular dis-placement with respect to the static-equilibrium position you found in part (b ).
134 Rotational Mechanical Systems Problems 134
Figure PS.25
*5.26. In the system shown in Figure P5.26, the cable around the cylinder J does not stretch and does not slip. The input is the applied force j~(t) and the outputs are () and z, where z is the additional elongation of K from its static-equilibrium position dueto j~(t).
The spring is undeftected when x¡
=
x2= () =
O. Select a set of state variables and write themodel in state-variable form.
f~(t)
Figure PS.26
5.27. In the system shown in Figure P5.27, the two masses are equal and the cable neither stretches nor slips. The springs are undeftected when ()
=
x1=
x2=
O. The input is the force fa (t) applied to the right mass, and the output is .rj • Selecta set of state variables andwrite the model in state-variable form,5.28. The input to the system in Figure P5.28 is the applied torque Ta(t). The shaft K1 and the spring K2 are undeflected when () 1
= ()
2=
x=
O.a. Write a set of differential equations describing the system in terms of ()¡, e2,x, and Ta(t ).
R J
M
M
Figure P5.27
b. Select a set of state variables and write the state-variable equations.
c. Find the constant value of Ta(t) for which the system will reach an equilibrium position with the mass remaining motionless. Determine the corresponding de-flections of K¡ and K2.
Figure P5.28
*5.29. The input to the combined translational and rotational system shown in Figure P5.29
is the displacement x1 (t ). The output is the torque applied to the gear by the shaft, with the positive sense clockwise. The springs are undeflected when Select an
e =
x1=
x2=
O.appropriate set of state variables and write the model in state-variable form.
5.30. Starting with (66), find the input-output differential equation for the system shown in Figure 5.29, taking
f;
1(t) and the gravitational force as inputs ande as the output.Negligible
Figure PS.29
5.31.
a. For the rotational mechanical system discussed in Example 5.3, write the state- variable equations (5.31) in matrix form. Show why we cannot obtain either 81 or
82 from this model.
b. Add 81 as the fourth state variable and write the state-variable equations in matrix form.
*5.32. For the combined translational and rotational system modeled in Example 5.12, write (66) in matrix form. Note that the system has two inputs, the gravitational constant
g and the applied force fa(t).
5.33. For the combined translational and rotational system modeled in Example 5.12, write (70) in matrix form. Also write a matrix output equation for the displacements 8 and X defined in (69).
5.34. Draw a block diagram for the rotational mechanical system described by the follow-ing two equations, where 80(t) is the input.
2(}1
+
e1+
3(ei - e2)+
81 = e0+
8a(t) 2(}2+
e2=
3(e1 - e2)+
82Hint: Define a new variable z such that
z
= 2(}1 -e
0 and substitute the first modeling equation into this new equation.Problems <i 137 5.35. Draw a block diagram for the system described by Equation (5.27) that was modeled in Example 5.2.
5.36. Draw a block diagram for the system described by Equations (5.64) and (5.66) that was modeled in Example 5.12.
5.37. When the input is a constant, then we expect that ali variables in the modeling equa-tions will eventually become constants, in which case their derivatives will eventually be-come zero. Use this fact as a partía! check on the curves in Figure 5.32. For Example 5.13, calculate what the final values of w¡ and w2 should be, and compare the results with the plots obtained by MATLAB.
5.38. Draw a Simulink diagram for the system shown in Figure 5.27, and described by (63). Use the following parameter values: M
=
50 kg, K=
20 N·m, R=
0.1 m, J=
30
kg·m2, B1 = 50 N·m·s, and B2 = 100 N·s/m. Simulate the first lO seconds of the
response
when the applied torque is a step of l N·m starting at t = l s. Plot the angular velocity w and the linear velocity
v.