• Author / Uploaded
• kyzhang12

Citation preview

220

ME  360  Section  1   W15  1/22/2015  

1.

CHAPTER 4

Spring and Damper Elements in Mechanical Systems

Figure   P4.36

 

I = 0.8 kg · m2 , and cT = 0.1 N · m · s. Find the speed v Due   1/29/15   initially at rest and the torque T is a step function of ma cT 4.37 Derive the equation of motion for the lever system show Problem Set 3 the force f as the input and the angle θ as the output. T R corresponds to the equilibrium position when f = 0. T ! about the pivot. Assume small displacements. A heavy disk is fastened to a wall with aT cylindrical shaft. Both shaft and I ! ! !" ! In the system shown in Figure P4.38, the input is the di 4.38 disk are made of steel (𝜌 = 7.88 ⋅ 10  kg/m , 𝐺 = 7.6 ⋅ 10  N/m ). A output is theTdisplacement x of the mass m. The equilib ! motor torque actspalm-38591 on the disk. disk has radius 25 cm, thickness 5 cm. book The December 17, 2008 12:4 v corresponds to x = y = 0. Neglect any friction between The shaft has radius 1.5 cm, length 8 cm. m surface. Derive the equation of motion and find the tran a. Draw a lumped parameter model for the system. Hint: Use one of the handout X (s)/Y (s). tables. g

 

b. Calculate parameter on the dimensions 220 values for the model, C H A Pbased T E R 4 Spring and Damper Elements inabove. Mechanical Systems c. Find an input-output differential equation for the system, T asFigure input,P4.37 and angular displacement 𝜃 2 Figure P4.36 I = 0.8 kg · m , and c = 0.1 N · m · s. Find the speed v(t) if the system is of the disk as output. (Leave the parameters as symbols, notT numbers.) initially at rest and the torque T is a step cfunction of magnitude 300 N · m. d. Suppose the system is found to chave response time 10 ms, meaning the time for an impulse at the T 4.37 Derive the equation of motion for the lever Lsystem shown in Figure P4.37, with 2 motor to be felt at the wall. Now draw a multi-segment parameter model thatposition would the force f as thelumped input and the angle θ as the output. The θ =0 " R corresponds to the equilibriumchosen. position when f = 0. The lever has an inertia I capture that behavior. Be sure to justify the number of elements k ! T

I

4.38

x k2

m

about the pivot. Assume small displacements. f In the system shown in Figure P4.38, the input is the displacement y and the I output is theinput displacement t , with u t x=of1thetL1mass andm. The x 0equilibrium = 4 . position corresponds to x = y = 0. Neglect any friction between the mass and the surface. the equation and find the Ltransfersolving function the and tryDerive to sketch without explicitly x tof motion 3 X (s)/Y (s).

2. Consider the differential equation 2 x +1x = 2u ( v

Figure P4.38

)

() () ()

( )

m a. Sketch the input and output time response, g differential equation, as discussed in class.. Figure P4.37 Figure P4.38 y b. Sketch the input and output for u (t ) = 1(t ) −1(t −10) , with zero initial conditions. Alsoa Houdaille express 4.39 Figure P4.39a shows damper, which is a de x

k1 c crankshaft to reduce vibrations. The damper has mathematically either using multiple time intervals or using the unitengine step function. k2 L2 to rotate within an m enclosure filled with viscous fluid. T " cthose c. In class we described two methods to find an impulse response to uinertia t ) . fan-belt What are (t ) = δof(the pulley. Modeling the crankshaft a k f the damper system canand be modeled as shown in part (b) methods? Use both methods to find the impulse response with zero initial conditions, sketch I L1 the equation of motion with the angular displacements θ the input and output time response. and the crankshaft angular displacement φ as the input. d. Find the impulse response with initial condition 𝑥 0 L= 1. 3

Figure P4.39

3. From textbook (4.39): The figure shows a Houdaille damper, Ip 4.39 Figure P4.39a showswhich a Houdaille damper, which is a device attached to an engine crankshaft to reduce vibrations. The damper has an inertia Id that is free attaches to an engine crankshaft to reduce vibrations. The damper to rotate within an enclosure filled with viscous fluid. The inertia I p is the Viscous fluid has an inertia 𝐼! that is free to rotate within an enclosure filled kT 12:4 as aIdtorsional Crankshaft inertia of the fan-belt Modeling crankshaft spring k T , palm-38591pulley. book Decemberthe 17, 2008 the damper system can be modeled as shown in part (b) of the figure. Derive with viscous fluid. The inertia 𝐼! is of the fan-belt pulley, # the equation of motion with the angular displacements θ p and θd as the outputs P4.40 which rotates with the crankshaft, itself Figure modeled torsional Pulley andastheacrankshaft angular displacement φ as the input.

kT

Probl z P4.39 # spring 𝑘 ! . The lumpedFigure parameter model of thisk system is shown at right. c T I a. Derive equations of motion with outputs 𝜃! and 𝜃! and inputIp𝜙.4.41 In the system shown(a) in Figure P4.41, thedinput is the force f and the output of point A. When x = "xdA the spring is at its free length the displacement x A b. Eliminate 𝜃! to find a single equation Lwith output 𝜃!kT and Iinput 𝜙. fluid 1 Viscous Derive the equation Crankshaft d kT of motion. L2 L Ip c. Find the steady-state 𝜙! for a unit step in 𝜙. Is this # 3 related to the Figure P4.41 FigureExamples P4.42 Section 4.5 Additional Modeling Figure P4.40 "p Pulley I xA y x x homogenous or particular solution? c z 4.40 The mass#m in Figure P4.40 is attached to a rigid rod ha k the steady-state plus a transient. How d. The full response includes m m m the pivot and negligible pivot friction. f c The input A c (b) k1 k2 is the d k many exponential terms do you expect in the transient,(a)and what basic requirement do you have of z = θ = 0, the spring is at its free length. Assuming tha L1 these terms if you expect the L 2 Lcrankshaft will not the systemof shown in Figure the input theinput. displacement y and th " destroy itself? 4.42 In equation motion forP4.42, θ with z as isthe 3 output is the displacement x. When x = y = 0 the springs are at their free Section 4.5what Additional Examples e. If a sinusoidal input 𝜙c 𝑡 = Icos 𝜔𝑡 is applied, formModeling will the steady-state response take? lengths. Derive the equation of motion. 4.40 The mass m in Figure P4.404.43 is attached to asolve rigid rod having an inertia I about Answer briefly with a short phrase or general math expression, but do not for coefficients. Figure P4.43 shows a rack-and-pinion gear in which a damping force and a m the pivot and negligible pivot friction. input is the displacement z. When springThe force act are against the rack. Develop the equivalent rotational model of f. In order to find the total solution for a unit step input, what initial conditions needed? (E.g., z = θ = 0, the spring is at its freethe length. Assuming that θ torque is small, derive the variable and the angular system with the applied T as the input " equation of motion for θ with z asdisplacement the input. θ is the output variable. Neglect any twist in the shaft. for a first-order system, only one initial condition x(0) is needed.) g. Suppose a unit impulse is applied to 𝜙. What is the new initial Figure condition immediately after? P4.43 Figure P4.44

4. From textbook (4.43): This is a rack and pinion, with a rigid shaft transmitting input torque T and producing output motion x. a. Derive the differential equation for the system. b. Develop an equivalent rotational system with 𝜃 as output, and find its natural frequency.

R Ip

x

c mr

Im T

!

I1 k

T1

!1

4.44 Figure P4.44 shows a drive train with a spur-gear pair. The first shaft turns N times faster than the second shaft. Develop a model of the system includi the elasticity of the second shaft. Assume the first shaft is rigid, and neglect gear and shaft masses. The input is the applied torque T1 . The outputs are th angles θ1 and θ3 . 4.45 Assuming that θ is small, derive the equations of motion of the systems

ME  360  Section  1   W15  1/22/2015  

   

  Due  1/29/15  

5. Consider the differential equation 𝑥 + 5𝑥 + 4𝑥 = 2𝑢 𝑡 . a. Find the unit step response. b. Find the impulse response. c. Find the free response with initial conditions 𝑥 0 = 1, 𝑥 0 = 0. d. Find the free response with initial conditions 𝑥 0 = 0, 𝑥 0 = 2. Compare with your answer from (b), and comment on the similarity. e. What is the response to a unit step input, with initial conditions 𝑥 0 = 2, 𝑥 0 = 0? Try to solve this without much effort, using information already known. f. What is the response to input 𝑢 𝑡 = 2 ⋅ 1 𝑡 − 1 𝑡 − 4 ? Sketch both the input and output, no need to express mathematically. g. Suppose a unit impulse is applied 𝑢 𝑡 = 𝛿 𝑡 . What is the new initial condition just after t = 0? Find the values for x and its time-derivatives as appropriate. 6. Second-order responses are often described in terms of the damping ratio ζ, which determines the shape of a time response, as shown in textbook Fig. 9.3.2, and in the handout on second-order responses. Use that information to find the value of ζ for each of the response (A – D) below left. One of the responses is first-order, in which case find the time constant instead of ζ . 5 4

ω1

3

T

2

J B

1

2

4

6

8

10

t

7. Shown above right, a rotor with moment of inertia J (50 kg-m2) is mounted in a bearing with unknown rotational damping B (N-s/m). The rotor at time t = 0 is at rest. A step torque T is then applied and the angular velocity is shown. Find the numerical values of rotational damping B and torque T.