1. Consider the system with 7-m(²+²6² +2²) and U-mgz where z-cr²,6-wt, by using the Lagrangian method we can find the equation of motion A. F(1 +4c²r²) + +(4c²r) +r(2gc-w²) - 0 B. C. *(1-4c²r²)+²(4c²r) +r(2gc+w²) - 0 (1+4c²r²) + ²(4c²r) +r(2ge+w²) = 0 (1+4c²r²)-²(4c²r) +r(2gc-w²) = 0 D.

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Find the inverse Laplace transforms using the table. 10s s2+100 3s s2+7s

2. The equations of motion of this system are; 12Y+7Y+24Y-6Z"-122-0 6Z" +6Z"+12Z-6Y-12Y=f(t) 7W+7W+14W-7Y'-14Y=ft) Where Y,Z and W are deformations of the masses and springs. Put these equations into state variable form and express the model as matrix vector equation if output of the system is Y. Energy Storage Element m1 m₂ M3 k₁ k₂ k₂ State Variable X₁ = Y' X₂ = Z' X3 = W' X₁ = Y X5 = Z X6 = W Sea Suspension Wheel Road Datum leve

Decouple the following equation of motion into two decoupled equations of motion. That is, find r (t). 4 [ 11 ] * (t) + +[ 4 -2 -2 2 ] x (t) = 0

PROBLEM1: Determine the equation of motion of the following systems: 1080 al*k www 0.51 ghiv Vin & Masa Less, Rigid L J Figure 2-Problem 1-2 a3*m

Obtain the state model for the two-mass system whose equations of motion are m₁x₁ + k₁(x₁ − x₂) = f(t) m₂x₂ − k₁ (x₁ - x₂)+k₂x₂ = 0

Derive the governing differential equation for each system with the chosen generalized coordinate. SEE THE IMAGE BELOW Answers: 1. GDE: (5/2) mẍ + (5/4) kx = 0 2. GDE: (7/48) mL² ϴ [note: theta symbol has two dots above) + (3/8) cL² ϴ [ note: theta symbol has one dot above] + 5 kL² ϴ = 0

In this lab, we will observe the free response of a first order mechanical system: a rotating bicycle wheel. As shown in the figure, suppose that the bicycle wheel is suspended above the ground, and is free to rotate on its axle. Assume that the only source of damping is the axle bearing, and that after time t=0 no external torques act on the wheel. ✈0(1) a) Derive the differential equation describing the angu- lar velocity of the wheel (t), in terms of the moment of inertia J and the damping B. b) Assume that a time t= 0 the wheel has initial an- gular velocity 22o. Solve the differential equation for response (1). c) Normally, we plot the response as (t) versus t. In- stead, suppose we plot the response as log, (n(t)/Ng) versus t. What type of curve is this plot? If you mea- sured experimentally logg ((t)/no) as a function of t, could you determine the system time constant 7 from the curve? d) What is the time constant 7 of the system expressed in terms of the moment of inertia J and…