We refer to thesystem of Fig. 4.12, representing the two-dof iconic model of a terrestrial vehicle,under the conditions described in Exercise 4.7. If we assume that the bump does notaffect the horizontal, uniform motion of the vehicle, determine its time response,namely x₁(t) and x₂(t), for t ≥ 0, if time is set equal to zero at the instant in whichthe vehicle hits the bump.

Figure 4.12

Data From in Exercise 4.7

Obtain the mathematical model of the system of Fig. 4.1, Example 4.2.1, witha different set of generalized coordinates, q1 = θ1, q2 =θ2−θ1.

Figure 4.1

Example 4.2.1

A two-linkrobotic arm suspended from the ceiling is shown in the iconic model of Fig. 4.1.In this model we have assumed that the two links are rigid bodies pinned at O₁ andO₂, the moment of inertia of body 1 with respect to O₁ being denoted by J₁, thatof body 2 with respect to its center of mass being denoted by J₂, while the massesof these bodies are labelled m₁ and m₂, respectively. Moreover, we assume that thejoints are lubricated with a fluid that provides linearly viscous damping, while themotors at the joints provide torques τ₁ and τ₂. As indicated in the figure, the effect ofgravity is considered and the first link is assumed to be inertially symmetric. Whatwe mean by this is that its center of mass C₁ is aligned with the two joint centers.Derive the Lagrange equations of motion of this system.

It will be helpful to recall the total time response of a single-dof undampedsystem to the function sin(ωt)u(t), as given in Eq. 2.143, which is reproduced belowin a form suitable for the problem at hand:

Also note that

Transcribed Image Text:

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