Vibrations

3. A mass m connected to an ideal spring slides frictionlessly on a surface.

A cam is used to provide the following profile. Determine the following: MATLAB or Excel output will not be accepted. Carry calculations to 4th decimal point. a. Time for a full cycle: T = b. Omega cam = c. Beta1 in degrees = d. Beta2 in degrees = e. Beta3 in degrees = f. Beta4 in degrees = g. Height of the cam profile at Theta= 100°. h. Height of the cam profile at 0 = 270º. end

A variable speed compressor operates between 400 and 600 rpm. The compressor has a mass of 450 kg and is mounted on springs having stiffness of 2000 kN/m. a.) Determine the maximum unbalance (me) allowed if the maximum allowable displacement within the operating speed range is 0.5 mm assuming no damping. b.) Repeat part a, if damping is added with c=2000 N*s/m.

Task 3 : Analysis of the singe spring-mass system (as defined in Task 2)using state space equations. 1. Develop a state space equation model for the system, considering u(t) the input and y(t) the output. 2. Find the eigenvalues of the system. Compare them to the poles obtained in Task 2. 3. Compute the state transition matrix O= e^-4t. Express the complete solution of the state space equation, in time-domain, with a general input u(t). 5. Examine to see if the system has complete reachability and observability.

Two masses are connected via three springs to each other and to the wall at left and right. The equations of motion are -k_{1} x_{1}-k\left(x_{1}-x_{2}\right)=m_{1} \ddot{x}_{1} -k_{2} x_{2}-k\left(x_{2}-x_{1}\right)=m_{2} \ddot{x}_{2} Your job is to calculate expressions giving the position of each mass as a function of time using the diagonalization method discussed in class. This classic problem can be solved in a number of different ways, but I will not give you credit unless you solve using the diagonalization method from the course. Suppose the force constants of the left and right springs are k1= k2 =2 and that of the center spring is k=1 let the masses be m1=m2=1 the two masses are given initial displacements at t =0 of x1 ==1 and x2=0 the initial velocity of each mass at t=0 is zero

3. An electric motor (Figure 2) has an eccentric mass of 10 kg (10% of the total mass of 100 kg) and is set on two identical springs (k = 3200 N/m). The motor runs at 1750 rpm, and the mass eccentricity is 100 mm from the center. The springs are mounted 250 mm apart with the motor shaft in the center. Neglect damping and determine the amplitude of the vertical vibration.

1.91 Calculate the natural frequency for the system in Figure P1.91 given the values m=10kg, k1=4000 N/m, k2=200 N/m and k3=1000 N/m. Assume that no friction acts on the rollers.

We choose mechanical systems as the second group of dynamic systems to study. We first consider a simple case of the motion of a spring attached with a rigid body of mass. Newton's law of motion governs the motion. With assumptions about the spring and the mass, we derive a differential equation model for the motion. A system which connects multiple springs and masses expands the number of dynamic variables. A thorough understanding of the modeling of a singe mass-on-spring will allow is to derive system equations for complex cases. They give us useful examples for studying system issues and the effect of feedback. Many mechanical systems produce oscillations. A simple case is one mass on one spring. The motion follows a linear differential equation with constant coefficients. Since there is oscillation, it cannot be modeled with a first-order equation. We know that Newton's law of motion governs the system. So, it is a second-order differential equation.A figure is shown below to describe the spring-mass systems Begin with a simple case of a single spring attached with a mass. The variable y(t) is the position of the mass m. It is measured from where the spring-mass is at rest,where y = 0. That position is an equilibrium. Away from that point, the spring is stretched and it has a restoring force to pulls the spring back toward the equilibrium. We assume that the restoring force is linear proportional to the stretch, that is it is -ky. The negative sign is important, for the spring force is against the stretch. Now suppose the mass is not at y = 0, the spring force will cause the mass to move. The motion follows Newton's law of motion: m\frac{d^2y}{\differentialD t^2}=-ky We follow the same approach of system analysis as we did with water-tank systems. We first investigate a system, reasoning about the nature of the system and applicable laws or rules. From that we establish a mathematical model which the dynamic behavior of the system follows. Then we analyze the system using the mathematical models to gain definitive and quantitative understanding of the system. The quantitative analysis is aimed for system design ideas or compensation strategies.The model of the spring-mass system, Equation (1), is a second-order linear differential equation. From the knowledge of differential equations, the solution of equation (1) describes pure oscillations. Task 1 : 1. Suppose at t= 0, y(0) = C1 and y(0) = C2. Find the solution of Equation(1). 2. Suppose the motion of m induces a friction force. The friction is against the motion and is assumed linearly proportional to the velocity: -c(dy/dt).Include this term in the equation and find the solution.

5. A fan of 45 kg has an unbalance that creates a harmonic force. A spring-damper system is designed to minimize the force transmitted to the base of the fan. A damper is used having a damping ratio of = 0.2. Calculate the required spring stiffness so that only 10% of the force is transmitted to the ground when the fan is running at 10,000 rpm.

Using kinetic energy equivalence, find the equivalent inertia of the system shown below with respect to è of the sphere, where strings are inextensible and mass moment of inertia \text { of the entire bell crank is } \mathrm{J}_{0} \text {. Small vibrations and no friction. Spherical inertia is } \frac{2}{5} m_{s} r_{s}^{2} \text {. } Other parameters are given in figure.