Homework 1

Flexible links in robot arms are encountered in many industrial applications. Figures 1 and 2 show sketches of a laboratory model of such a link. The manipulator positions a load of mass M and length I at an angle q1. The link's flexible joint is modeled as a spring of stiffness k, and is driven by a de motor with a rotor that has an inertia J and friction B. The link's inertia is I. Flexible joint Mgl Motor and encoder Encoder Arm Fig. 1. Sketch of experimental hardware. Fig. 2. Model of single-link flexible-joint arm. (a) Find the equations of motions in time domain and Laplace domain. u(t) is the torque generated by the motor and g is the acceleration due to gravity. (assuming zero initial conditions)

The EOM for the following 1-DOF gyroscope is: JÖ + BO + KO = Hw w is the angular velocity of the gyroscope about the input axis, is the angular position of the spin axis, H is the angular momentum stored in the spinning wheel, J is the mass moment of inertia of the wheel about the output axis, B is the viscous friction coefficient about the output axis, and K is the spring constant of the restraining spring attached to the spin axis. Determine the transfer function relating output to the input w. Case K spin axis output axis BOD Go Wheel spins at constant velocity input axis

dynamic of machinery question photo

Figure below shows a mass-spring model of the type used to design packaging systems and vehicle suspensions, for example. The springs exert a force that is proportional to their compression, and the proportionality constant is the spring constant k. The two side springs provide additional resistance if the weight W is too heavy for the center spring. When the weight W is gently placed, it moves through a distance x before coming to rest. From statics, the weight force must balance the spring forces at this new position. (if x=d) These relations can be used to generate the plot of x versus W. (a) The following values are given: k1=104 N/m; k2 = 1.5 *104 N/m; d =0.1 m. Create a function that computes the distance x, using the input parameter W; Then, test your function for the following two cases, using the values: W=500 N W=2000 N. (b) Create another script and use your function to plot x versus W for 0 < W < 3000 N for the values of k1, k2, and d given in part a.

* F2 # 3 E Problem 4. EOM Derivation Derive the equation of motion for the following system: We want to measure the spin rate w of the rotating shaft of a turbine. To do it we push a small mass m against the end of the shaft. The shaft makes friction contact with the drag cup mounted on the mass. The frictional torque is b (w). Assume that the rotational spring torque is ke and that the moment of inertia of mass m is J. J Rotating Shaft F3 @ $ 4 R b J, m Drag Cup F4 Q Search % 5 F5 T 0 Wall * A 6 F6 Y 2 * F7 & 7 PrtScn U F8 4 8 Home 6. F9 9 End F10 Po 0

For the problem related to Fundamental.6 sketches of the system showing: • the respective velocity and acceleration and the frame of reference considered • the forces acting on the system of considered, in other words, a free body diagram (FBD) are mandatory. Their absences will automatically make the problem false. The tractor is used to lift the 50-kg load B with the 27-m long rope, boom, and pulley system. The height h=4-m, SA=15m and sg=1.3m. SB B hm -SA If the tractor travels to the right with an acceleration of 3 m.s2. The maximum of tension of the allowable tension force in the rope is 3-kN. Determine the maximum of the allowable velocity of the tractor.

Theory of machine

Problem 4: Use the principle of virtual work to determine the equation of motion for the system shown below. The rod is rigid and has a total mass of 1.5m. Assume small rotations. Rigid 1.5m L/2 L/2

4

Please don't provide handwritten solution ....

Determine an equivalent lumped mass model for the crankpin.

igure 2. Assume that the rod is massless, perfectly rigid, and pivoted at point P. When the rod is perfectly horizontal, the angle 0 = 0, the displacement y 0, and the springs are in neither tension nor compression. Gravity acts on the system (e.g. on mass M). We assume that y is a small displacement. A mass M is attached at the end of the rod. DE only 225 Your tasks: X k 0 3k a F a M y A Derive an equation of motion for the system in terms of the angular displacement 0, and its derivatives (you should not have y or its derivatives in this equation.) B Derive an equation of motion for the system in terms of the displacement y, and its derivatives (you should not have or its derivatives in this equation. C Assuming there is no external actuator force F acting on the system, write down the total energy H of the system in terms of 0,0 and element constants. Derive an expression for the time derivative H of the total energy. D Transform the equation from part B, which is in y, to another…

Questions A and B

Question 10: Figure 4 shows a mechanical system. Mass m is attached at the end of the rigid rod, which has an inertia I about the pivot. Assume negligible pivot friction. The input is the displacement z. When z = 0 0 degrees, the spring is undeflected. Assuming a small 0, identify the equation of motion for with z as an input. m Figure 4: Mechanical System