Homework 12: IDP. Due Tuesday November 25 at 5:00 PM

Complete problems 5.1, 5.4, 5.7, and 5.10.

Instructions for these problems

1. Please make sure to follow all homework guidelines.

2. Please do not staple problems together, we will once again turn problems in, in four

separate stacks. If more then one page is used on a given problem, please staple these

together.

3. For all problems you must print out at least the sheet with the links sepa-rated and do your FBDs on this.

4. For all problems, you may neglect the forces due to gravity.

5. For problems 5.1, 5.4, and 5.7, make sure to follow all instructions stated in the book.

If there are any discrepancies between the book and the PDF posted on Sakai, follow

what the book says.

6. For problem 5.10, please follow the instruction in the PDF posted on Sakai, not

the book. I believe that these instructions are more straight forward and make the

problem easier.

1

Chapter 5

Machine Dynamics Part I: The

Inverse Dynamics Problem -

Homework Problems

Problem 5.1

The input to the machine below is θ2. The values of θ2, θ2 and θ2 are known, i.e. the “state ofmotion” is known. The load torque, T5, is known. This is a two loop mechanism.The figure shows appropriate vector loops. There are two vector loop equations,

r2 + r3 + r4 − r5 + r1 = 0

r8 − r7 − r6 − r5 + r1 = 0

and three geometric constraints,

θ8 + γ − θ2 = 0

θ6 + (π − ψ)− θ4 = 0

and θ6 +π

2− θ7 = 0

Withscalar knowns: r1, θ1 = π, r2, r3, r4, r5, r6, r8, φ, ψ.andscalar unknowns: θ2, θ3, θ4, θ5, θ6, r7, θ7, θ8.With seven equations in eight unknowns there is 1 degree-of-freedom. Take θ2 as Si.

The figure below defines the locations of the mass centers. vectors r9, r10, r11, r12 and angles φ2and φ4 are known.The inertias, m2, Ig2 , m3, Ig3 , m4, Ig4 and m5, Ig5 are known.

Assume that steps 1-4 in Example ?? are completed. You should complete Steps 5 and 6 hereand derive a system of equations in matrix form which could be solved for the magnitude of thedriving torque, T2 and all the bearing forces. The following figures are for your free body diagrams.

For link 2 sum moments about point A.For link 3 sum moments about point B.

77

fig501

T2

T5

r1

r2

r3 r4

r5

r6

r7

r8

γ

ψ

1

1

2

3

4

5

X

Y

A

B

C

D

E

F

Figure 5.1: The machine to be analyzed and its vector loop

For link 4 sum moments about point E.For link 5 sum moments about point F .For each moment equation specify which unknowns were eliminated by summing moments aboutthe designated point versus summing moments about the mass center.

78

fig501a

G2

G3

G4

G5

r2

r3r4

r5

r9

r10

r11

r12

φ2

φ4

A

B

C

D

E

F

Figure 5.2: Definition of inertia properties

79

fig501b

G2

G3

G4

G5

2

3

4

5

A

B

B

C

C

D

D

E

E

F

Figure 5.3: Free body diagrams

80

Problem 5.4

The machine below is a scissor jack. It is commonly used to lift an automobile in order to changea flat tire. W is a known weight of the automobile. F is the driving force. This force is usuallydeveloped through a screw which is turned by the operator of the jack.

This is a two loop mechanism. The right side shows two appropriate vector loops.

fig205d

W

F

r1

r2

r3

r4

r5

r6 r7

X

Y

11

2

3

4

AB

C

Figure 5.10: A scissor jack

The mechanism’s vector loop equations and geometric constraints are,

r1 + r4 + r3 − r2 = 0 and r1 + r7 − r6 = 0

θ2 − θ6 = 0 and θ4 − θ7 = 0.

Together these comprise 6 scalar equations withscalar knowns: θ1, r2, r3, r4, r6, r7 andscalar unknowns: r1, θ2, r3, θ3, θ4, θ6 and θ7.

You may assume that r2 = r4 = R and r6 = r7 = r (realize that this would only be true to a certaintolerance) in which case unknown θ3 = 0 and is constant.

The system has one dof. Suppose that r1 is a given input and that r1 and r1 are known (i.e.the state of motion is known). The weight being lifted is large and it is being lifted very slowly.

87

This means that it is reasonable to neglect inertia effects. Likewise, the weights of the links aresignificantly less than weight W and they can also be neglected.

Suppose that steps 1-4 in example ?? have been completed. Complete steps 5 and 6 to deriveall equations necessary in order to compute F and the bearing forces in terms of the prescribed W ,link lengths and state of motion. Present all systems of equations in matrix form.The following figure is for your free body diagrams. For your moment equations please sum momentsas stated:for link 2 sum moments about the point B,for link 3 sum moments about the point C andfor link 4 sum moments about the point A.

fig205e

2

3

4

AB

C

Figure 5.11: Free body diagrams

88

Problem 5.7

The load P applied to link 4 is known. The driving torque Q applied to link 2 is unknown. Thefigure shows locations of mass centers. Inertia properties of all links are known.

fig505

1

2

2

3

3

4

4

X

YQ

P

G2

G3

G4

A

C

r1

r2

r3

r4

r5

r6

r7

Figure 5.17: A machine

The mechanism’s vector loops and geometric constraint are,

r2 − r3 − r4 − r1 = 0 , r6 − r5 − r4 − r1 = 0 , θ3 − θ5 = 0,

so there are 5 scalar position equations with

94

scalar knowns: r1, θ1, θ4, r6, θ6, r2 andscalar unknowns: r4, r5, θ5, r3, θ3 and θ2.

and there is one degree of freedom. Suppose the state of motion, θ2, θ2 and θ2 is known and steps1-4 in Example ?? are completed. Complete Steps 5 and 6 here and derive a system of equations inmatrix form which could be solved for the magnitude of the driving torque, Q and all the bearingforces. For link 2 sum moments about G2, for link 3 sum moments about C and for link 4 summoments about G4

fig505a

C

2

3

4

G2

G3

G4

Figure 5.18: Free body diagrams

95

Problem 5.10

The machine below moves slowly. The load F3 is known and is an order of magnitude larger thanthe weight of any link. An appropriate vector loop is shown on the following page. The vector loopequation, r1 + r2 − r3 = 0 has 2 scalar components andscalar knowns: θ1, θ2, r3 andscalar unknowns: r1, r2 and θ3,

so it has one degree of freedom. Suppose the angular position, velocity and acceleration of link 2are known and steps 1-4 in example ?? have been completed. Complete steps 5 and 6 in Example?? and derive and system of equations in matrix form which could be used to compute T2.

In your free body diagrams, model the force between 1 and 3 as a force with unknown line ofaction, like in example ??, where the force between 1 and 4, F14, has an unknown line of actiondetermined by the magnitude of r9.

Do you notice anything odd when you do this? Please discuss.

fig524

+

+1

1

1

2

3

F3

T2

r4

Figure 5.24: A machine

101

+

fig524a

+

1

r3

r2

r1

X

Y

Figure 5.25: The vector loop

102

fig524b

+

+

+

+

2

3

F3

T2

r1

r4

X

Y

A

B

Figure 5.26: Free body diagrams

103