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

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

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.

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Questions A and B

Please solve with step and explaining, thank u

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

Three springs are connected to a mass and to fixed supports. When the coordinate y is equal to zero, there is no spring force. Your tasks: Tw m k3 assume Staur C équilibrition Fload Figure 1: System schematic for Problem 1. A Draw the free body diagram for the mass m including all 3 spring forces and the load force Fload). B Draw the free body diagram for the mass m, but combine the springs into one equivalent spring with spring constant keq. Write the equivalent stiffness of the spring keq in terms of the other spring constants. (3 C Given that k₁ = 300 N/m, k₂ = 100 N/m, and k3 = 200 N/m, and Fload = 300 N, calculate the loaded deflection SA Yloaded- D Compute the amount of potential energy stored in each spring. E Compute the forces, in Newtons, in each spring and denote them as F₁, F2, and F3 when the load from Part D is applied. www