Figure 2 shows a translating mass on a friction-less surface, acted on by an input force fa(t). The mass m is connected to ground via a linear damper b and spring k, arranged in series. The coordinate z2 tracks the relative position of the mass-less node between the damper and mass, shown with a small dark circle. The relative position of the mass (from rest/equilibrium) is z1. This lumped-element model could serve, for example, as a basic model of linear creep behavior of a polymer rope under load.
(a) Draw a free-body diagram for mass m clearly showing your sign conventions for the coordinates and external forces.
(b) Find the equation of motion for mass m. [Hint: You might find it easiest to consider the mass-less node to have a finite mass, say m0, and write its equation of motion, and then take the limit m0 → 0. The resulting expression can be used to complete the equation of motion for m, such that it does not depend on z2. Think about why coordinate z2 is not independent of coordinate z1.]
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Figure 2 shows a translating mass on a friction-less surface,
acted on by an input force...
An ideal mass m is sitting on a plane,attached to a rigid surface via a spring. The spring consta...
An ideal mass m is sitting on a plane,attached to a rigid surface via a spring. The spring constant is k, damping coefficient is c, and r(t) is the displacement of the mass with respect to the equilibrium position at time t. damper r 丑 spring Whent 0, we start to measure this of mass v(0)0 system and displacement o )1, velocity a) How many times will the mass pass through the equilibrium position in one b) Please find the...
Please show work 3. Given a mass-spring-damper system, the 2kg mass is connected to two linear...
Please show work
3. Given a mass-spring-damper system, the 2kg mass is connected to two linear springs with stiffness coefficients ki- 100 N/m and ki 150 N/m and a viscous damper with b 50 Ns/m. A constant force of SN is applied as shown. The effect of friction is negligible. ki m b 3.1 [2pts] Determine the equivalent stiffness of the springs. 3.2 [3pts] Draw the free-body diagram of the system. Define the generalized coordinate and label your forces and...
Figure 1 shows a mechanical system suspended between a fixed ceiling and a moving base. Taking...
Figure 1 shows a mechanical system suspended between a fixed ceiling and a moving base. Taking the base motion zin(t) as the input to this system, please answer the following: Fixed Zin() Moving base Figure 1: Problem 1: One-mass moving base system. (a) (2 points) Draw a free-body diagram for mass m clearly showing your sign conventions for the co- ordinates and external forces. (Assume the spring is pre-compressed under the weight mg, e.g. the z and Zin coordinates are...
Question 6.3 6.3 Consider a double mass-spring system with two masses of M and m on...
Question 6.3
6.3 Consider a double mass-spring system with two masses of M and m on a frictionless surface, as shown in Figure 6.30. Mass m is connected to M by a spring of constant k and rest length lo. Mass M is connected to a fixed wall by a spring of constant k and rest length lo and a damper with constant b. Find the equations of motion of each mass. (HINT: See Tutorial 2.1.) risto M wa ww...
Consider a mass m suspended from a massless spring that obeys Hooke's Law (i.e. the force...
Consider a mass m suspended from a massless spring that obeys Hooke's Law (i.e. the force required to stretch or compress it is proportional to the distance stretched/compressed). The kinetic energy T of the system is mv2/2, where v is the velocity of the mass, and the potential energy V of the system is kr-/2, where k is the spring constant and x is the displacement of the mass from its gravitational equilibrium position. Using Lagrange's equations for mechanics (with...
Consider the mass-spring-damper system depicted in the figure below, where the input of the system is...
Consider the mass-spring-damper system depicted in the figure below, where the input of the system is the applied force F(t) and the output of the system is xít) that is the displacement of the mass according to the coordinate system defined in that figure. Assume that force F(t) is applied for t> 0 and the system is in static equilibrium before t=0 and z(t) is measured from the static equilibrium. b m F Also, the mass of the block, the...
1 Q2. Figure 2 shows a system in which mass m is connected with a cylinder of mass m2 and moment of inertia Jo through...
1 Q2. Figure 2 shows a system in which mass m is connected with a cylinder of mass m2 and moment of inertia Jo through a horizontal spring k. The cylinder is m1 rolling on the rough surface without slipping. (1) Find its total kinetic energy, total potential energy TN and Lagrangian, Figure 2 (2) Derive the equations of motion using Lagrangian equation method, and (3) Calculate its natural frequencies
1 Q2. Figure 2 shows a system in which mass...
The figure below shows a uniform slender bar supported by cantilevers at A and C. At B a linear spring with stiffness K' is connected to an additional point mass 'm'. Note the physical pr...
The figure below shows a uniform slender bar supported by cantilevers at A and C. At B a linear spring with stiffness K' is connected to an additional point mass 'm'. Note the physical properties of the bar include cross sectional area A, Young's modulus E, second moment of area I, and, density ρ, and length AB-BC-L. 1. 2. Develop the matrix equation of motion for the FEM system in the model How many natural frequencies are in the system?...
Thanks in advance 4. (8 points) We consider the mass-spring-dnmper systen in Figure &: A block...
Thanks in advance
4. (8 points) We consider the mass-spring-dnmper systen in Figure &: A block af mes is saspended fron the eeiling by n spring with eonistant k nnd unstretched length lo and a damper with damping eoeficient d The block ean anly move vertienlly nnd its poesition is theresore fually deseribed by the z-courdinnte. An nlternative poeition is mensued from the statie rest position Furthermore, ngravitational neeclerstioa with mngnitude g is present. SYSTEM Fgure & A rigid body...
A block having mass m and charge +Q is connected to an insulating spring having a force constant k.
A block having mass m and charge +Q is connected to an insulating spring having a force constant k. The block lies on a frictionless, insulating, horizontal track, and the system is immersed In a uniform electric field of magnitude E directed as shown in the figure below. The block Is released from rest when the spring Is unstretched (at x = 0). We wish to show that the ensuing motion of the block is simple harmonic. (a) Consider the system...
An ideal mass m is sitting on a plane,attached to a rigid surface via a spring. The spring constant is k, damping coefficient is c, and r(t) is the displacement of the mass with respect to the equilibrium position at time t. damper r 丑 spring Whent 0, we start to measure this of mass v(0)0 system and displacement o )1, velocity a) How many times will the mass pass through the equilibrium position in one b) Please find the...
Please show work
3. Given a mass-spring-damper system, the 2kg mass is connected to two linear springs with stiffness coefficients ki- 100 N/m and ki 150 N/m and a viscous damper with b 50 Ns/m. A constant force of SN is applied as shown. The effect of friction is negligible. ki m b 3.1 [2pts] Determine the equivalent stiffness of the springs. 3.2 [3pts] Draw the free-body diagram of the system. Define the generalized coordinate and label your forces and...
Figure 1 shows a mechanical system suspended between a fixed ceiling and a moving base. Taking the base motion zin(t) as the input to this system, please answer the following: Fixed Zin() Moving base Figure 1: Problem 1: One-mass moving base system. (a) (2 points) Draw a free-body diagram for mass m clearly showing your sign conventions for the co- ordinates and external forces. (Assume the spring is pre-compressed under the weight mg, e.g. the z and Zin coordinates are...
Question 6.3
6.3 Consider a double mass-spring system with two masses of M and m on a frictionless surface, as shown in Figure 6.30. Mass m is connected to M by a spring of constant k and rest length lo. Mass M is connected to a fixed wall by a spring of constant k and rest length lo and a damper with constant b. Find the equations of motion of each mass. (HINT: See Tutorial 2.1.) risto M wa ww...
Consider a mass m suspended from a massless spring that obeys Hooke's Law (i.e. the force required to stretch or compress it is proportional to the distance stretched/compressed). The kinetic energy T of the system is mv2/2, where v is the velocity of the mass, and the potential energy V of the system is kr-/2, where k is the spring constant and x is the displacement of the mass from its gravitational equilibrium position. Using Lagrange's equations for mechanics (with...
Consider the mass-spring-damper system depicted in the figure below, where the input of the system is the applied force F(t) and the output of the system is xít) that is the displacement of the mass according to the coordinate system defined in that figure. Assume that force F(t) is applied for t> 0 and the system is in static equilibrium before t=0 and z(t) is measured from the static equilibrium. b m F Also, the mass of the block, the...
1 Q2. Figure 2 shows a system in which mass m is connected with a cylinder of mass m2 and moment of inertia Jo through a horizontal spring k. The cylinder is m1 rolling on the rough surface without slipping. (1) Find its total kinetic energy, total potential energy TN and Lagrangian, Figure 2 (2) Derive the equations of motion using Lagrangian equation method, and (3) Calculate its natural frequencies
1 Q2. Figure 2 shows a system in which mass...
The figure below shows a uniform slender bar supported by cantilevers at A and C. At B a linear spring with stiffness K' is connected to an additional point mass 'm'. Note the physical properties of the bar include cross sectional area A, Young's modulus E, second moment of area I, and, density ρ, and length AB-BC-L. 1. 2. Develop the matrix equation of motion for the FEM system in the model How many natural frequencies are in the system?...
Thanks in advance
4. (8 points) We consider the mass-spring-dnmper systen in Figure &: A block af mes is saspended fron the eeiling by n spring with eonistant k nnd unstretched length lo and a damper with damping eoeficient d The block ean anly move vertienlly nnd its poesition is theresore fually deseribed by the z-courdinnte. An nlternative poeition is mensued from the statie rest position Furthermore, ngravitational neeclerstioa with mngnitude g is present. SYSTEM Fgure & A rigid body...
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