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Question 5 Consider the spring loaded system shown below. Assume that the springs are undeformed when...
4Consider the spring-loaded pendalum system shown in Figure 4. Assume that the spring force acting on...
4Consider the spring-loaded pendalum system shown in Figure 4. Assume that the spring force acting on the pendulum is zero when the pendulum is vertical, or θ "。. Assume also that the friction involved is negligible and the angle of oscillation θ is small. Obtain a mathematical model of the system. Lincarize the nonlincar equations and obtain the frequency of oscillation of the pendulum Mg Figure 4
(5 marks) Write the equation of motion for the double pendulum system shown below. Assume that...
(5 marks) Write the equation of motion for the double pendulum system shown below. Assume that the displacement angles of the pendulum are small enough to ensure that the spring is always horizontal. The pendulum rods are taken to be massless, of length I, and the springs are 75% of the way down the rods. 3. k, m2
Consider the system shown in the figure below. The mass moment of inertia of the bar...
Consider the system shown in the figure below. The mass moment
of inertia of the bar about the point O is JO, and the torsional
stiffness of the spring attached to the pivot point is kt . Assume
that there is gravity loading. The centre of gravity of the bar is
midways, as shown in the figure.
Question 2 Consider the system shown in the figure below. The mass moment of inertia of the bar about the point O is...
2. Consider the system shown in the figure below, comprised of the same motor, steel beam,...
2. Consider the system shown in the figure below, comprised of the same motor, steel beam, steel cable and crate All assumptions and properties are the same with one exception; the cable is no longer considered as rigid Cable properties: length = 4 m, diameter = 0.007 m, E = 207 GPa, Calculate the equivalent stiffness of the cable, in units of N/m. (See table 4.1.1 in your textbook) Draw an equivalent system diagram where the beam and cable each...
2. Consider the mass-spring system shown in the figure below. It can be shown that the...
2. Consider the mass-spring system shown in the figure below. It can be shown that the motion of the mass is governed by the equation a=-sw^2, where s and a are the position and acceleration of the mass, respectively, and w is a constant (which is referred to as the natural frequency of the system). Derive the equation describing the velocity of the mass in terms of the position. Assume that the velocity of the mass is v(subzero) when s=0...
1. Springs and a mass are attached to a rigid bar, as shown in Fig 1....
1. Springs and a mass are attached to a rigid bar, as shown in Fig 1. The hinges are free to rotate. 0 denotes the rotational angle of the rod, and 0-0 when all springs are not stretched. The mass of the bar and the size of the mass are negligible. Neglect gravitational force, and assume 0 is very small. 1) Derive the equation of motion for this system with Lagrange's method. (20pt) 2) Find the natural frequency of the...
IV. Spring-Mass System Application - Consider the system of two masses and three springs as shown...
IV. Spring-Mass System Application - Consider the system of two masses and three springs as shown in the figure below. Let z(t) be the position mass m, and y(t) be the position of mass m. Let m, = 1, m, = 1, k, = 4, k, = 6, and k, = 4. ksi-ik, a.) Model the system with two second order differential equations. 6a.) System: b) Find the general solution to the system using the constants.) head of your choice....
3. Consider the spring - mass system shown below, consisting of two masses mi and ma sus- pended ...
3. Consider the spring - mass system shown below, consisting of two masses mi and ma sus- pended from springs with spring constants ki and k, respectively. Assume that there is no damping in the system. a) Show that the displacements z1 and 2 of the masses from their respective equilibrium positions satisfy the differential equations b) Use the above resuit to show that the spring-mass system satisfies the following fourth order differential equation. and ) Find the general solution...
Consider the inverted pendulum system shown below. The inverted pendulum is mounted on top of a...
Consider the inverted pendulum system shown below. The inverted pendulum is mounted on top of a motor driven cart. The pendulum and cart have two degrees of freedom in plane together, i.e x and θ. It is desired to keep the pendulum upright in the presence of disturbances, such as unexpected force applied on the cart. The slanted pendulum can be brought back to the vertical position by applying a control force u applied to the cart. Once the pendulum...
Exercises 1. (introduction) Sketch or plot the displacement of the mass in a mass-spring system for at least two per...
Exercises 1. (introduction) Sketch or plot the displacement of the mass in a mass-spring system for at least two periods for the case when Wn-2rad/s, 괴,-1mm, and eto =-v/5mm/s. 2. (introduction) The approximation sin θ ะ θ is reasonable for θ < 10°. If a pendulum of length 0.5m, has an initial position of 0()0, what is the maximum value of the initial angular velocity that can be given to the pendulum without violating this smll angle approximation? 3. (harmonic...
4Consider the spring-loaded pendalum system shown in Figure 4. Assume that the spring force acting on the pendulum is zero when the pendulum is vertical, or θ "。. Assume also that the friction involved is negligible and the angle of oscillation θ is small. Obtain a mathematical model of the system. Lincarize the nonlincar equations and obtain the frequency of oscillation of the pendulum Mg Figure 4
(5 marks) Write the equation of motion for the double pendulum system shown below. Assume that the displacement angles of the pendulum are small enough to ensure that the spring is always horizontal. The pendulum rods are taken to be massless, of length I, and the springs are 75% of the way down the rods. 3. k, m2
Consider the system shown in the figure below. The mass moment
of inertia of the bar about the point O is JO, and the torsional
stiffness of the spring attached to the pivot point is kt . Assume
that there is gravity loading. The centre of gravity of the bar is
midways, as shown in the figure.
Question 2 Consider the system shown in the figure below. The mass moment of inertia of the bar about the point O is...
2. Consider the system shown in the figure below, comprised of the same motor, steel beam, steel cable and crate All assumptions and properties are the same with one exception; the cable is no longer considered as rigid Cable properties: length = 4 m, diameter = 0.007 m, E = 207 GPa, Calculate the equivalent stiffness of the cable, in units of N/m. (See table 4.1.1 in your textbook) Draw an equivalent system diagram where the beam and cable each...
1. Springs and a mass are attached to a rigid bar, as shown in Fig 1. The hinges are free to rotate. 0 denotes the rotational angle of the rod, and 0-0 when all springs are not stretched. The mass of the bar and the size of the mass are negligible. Neglect gravitational force, and assume 0 is very small. 1) Derive the equation of motion for this system with Lagrange's method. (20pt) 2) Find the natural frequency of the...
IV. Spring-Mass System Application - Consider the system of two masses and three springs as shown in the figure below. Let z(t) be the position mass m, and y(t) be the position of mass m. Let m, = 1, m, = 1, k, = 4, k, = 6, and k, = 4. ksi-ik, a.) Model the system with two second order differential equations. 6a.) System: b) Find the general solution to the system using the constants.) head of your choice....
3. Consider the spring - mass system shown below, consisting of two masses mi and ma sus- pended from springs with spring constants ki and k, respectively. Assume that there is no damping in the system. a) Show that the displacements z1 and 2 of the masses from their respective equilibrium positions satisfy the differential equations b) Use the above resuit to show that the spring-mass system satisfies the following fourth order differential equation. and ) Find the general solution...
Consider the inverted pendulum system shown below. The inverted pendulum is mounted on top of a motor driven cart. The pendulum and cart have two degrees of freedom in plane together, i.e x and θ. It is desired to keep the pendulum upright in the presence of disturbances, such as unexpected force applied on the cart. The slanted pendulum can be brought back to the vertical position by applying a control force u applied to the cart. Once the pendulum...
Exercises 1. (introduction) Sketch or plot the displacement of the mass in a mass-spring system for at least two periods for the case when Wn-2rad/s, 괴,-1mm, and eto =-v/5mm/s. 2. (introduction) The approximation sin θ ะ θ is reasonable for θ < 10°. If a pendulum of length 0.5m, has an initial position of 0()0, what is the maximum value of the initial angular velocity that can be given to the pendulum without violating this smll angle approximation? 3. (harmonic...
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