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2. (15 scores) Consider the mechanical system shown in Figure 1. A spring exerts a force...
Consider a mass-spring-damper system whose motion is described by the following system of differe...
Consider a mass-spring-damper system whose motion is described by the following system of differentiat equations [c1(f-k)+k,(f-х)-c2(x-9), f=f(t), y:' y(t) with x=x( t), where the function fit) is the input displacement function (known), while xit) and yt) are the two generalized coordinates (both unknown) of the mass-spring-damper systenm. 1. Identify the type of equations (e.g. H/NH, ODE/PDE, L/NL, order, type of coefficients, etc.J. 2. Express this system of differential equations in matrix form, assume f 0 and then determine its general...
PROBLEM 1 (35 %) The mechanical system in the figure below consists of a disk of radius r, a block of mass m, a spr...
PROBLEM 1 (35 %) The mechanical system in the figure below consists of a disk of radius r, a block of mass m, a spring of stiffness (spring constant) k, and a damper with damping ratio b. The disk has moment of inertia Jabout its center of mass (pivot point O), and the block is subjected to an external force t) as shown in the figure. The spring is unstressed when x 0= 0. Assume small 0. (a) (10 points)...
Question8 n the spring-mass-damper system in Figure 8, the force F, is applied to the mass and it...
Question8 n the spring-mass-damper system in Figure 8, the force F, is applied to the mass and its displacement is measured via r(t), whilst k and c are the spring and damper constants, respectively x(t) Figure 8: A spring-mass-damper system. a) Obtain the differential equation that relates the input force F, to the measured dis- (6 marks) placement x(t) for the system in Figure 8. b) Draw the block diagram representation of the system in Figure 8. c) Based on...
The mechanical system shown in the figure below is excited by a sinusoidal force f(t)-Fi cos(ut...
The mechanical system shown in the figure below is excited by a sinusoidal force f(t)-Fi cos(ut + ?) N. The differential equation of the displacement x(t) is Use phasor techniques to solve for the displacement phasor Xin terms of the excitation frequency ? , and the mechanical elements M = 0.1 kg, D = 8 N-s/m , and K = 2,000 N/m . If Fi-10 N and ?? = 30°, determine the excitation frequency w (in rad/s) at which the...
Problem 2: Transfer Functions of Mechanical Systems. (20 Points) A model sketch for a two-mass mechanical...
Problem 2: Transfer Functions of Mechanical Systems. (20 Points) A model sketch for a two-mass mechanical system subjected to fluctuations (t) at the wall is provided in figure 2. Spring k, is interconnected with both spring ka and damper Os at the nodal point. The independent displacement of mass m is denoted by 1, the independent displacement of mass m, is denoted by r2, and the independent displacement of the node is denoted by ra. Assume a linear force-displacement/velocity relationship...
Given the the mass-spring-damper system in Figure 3.10, assume that the contact forces are viscous friction....
Given the the mass-spring-damper system in Figure 3.10, assume
that the contact forces are viscous friction. 1. State the number
of degrees of freedom in the system.
2. Derive the equations of motion and state them in matrix
notation. 3. If f(t) = a (a constant), what is the long term state
of the system? 4. If the forcing is f(t) = A sin(ωt), and the
system parameters are given in Table 3.1, simulate the response
from rest. Plot all...
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...
A second order mechanical system of a mass connected to a spring and a damper is subjected to a s...
A second order mechanical system of a mass connected to a spring and a damper is subjected to a sinusoidal input force mx+cx + kx = A sin(at) The mass is m-5 kg, the damping constant is c = 1 N-sec/m, the spring stiffness is 2 N/m, and the amplitude of the input force is A- 3 N. For this system give explicit numerical values for the damping factor 5 and the un-damped natural frequency Using the given formulas for...
Consider the mass-spring-damper system shown below left. A spring-based force gage, with spring constant Kgage, is...
Consider the mass-spring-damper system shown below left. A
spring-based force gage, with spring constant Kgage, is to be
inserted between the spring K and mass M (as shown below right) to
measure the force in K in response to the applied position xin.
With ftrue representing the force in K without the gage present,
and fgage representing the force in K with the gage present, the
Laplace transforms of fgage and ftrue satisfy
For a mass-spring-damper mechanical systems shown below, x200) K1-1 N/m 0000 -X,(0) K-1 N/m 00004 =...
For a mass-spring-damper mechanical systems shown below, x200) K1-1 N/m 0000 -X,(0) K-1 N/m 00004 = 1 N-s/m fr2 M1=1 kg = 2 N-s/m M2 -1 kg 13 = 1 N-s/m 1. Find the differential equations relating input force f(t) and output displacement xi(t) and x2(C) in the system. (40 marks) (Hint: K, fy and M are spring constant, friction coefficient and mass respectively) 2. Determine the transfer function G(s)= X1(s)/F(s) (20 marks)
Consider a mass-spring-damper system whose motion is described by the following system of differentiat equations [c1(f-k)+k,(f-х)-c2(x-9), f=f(t), y:' y(t) with x=x( t), where the function fit) is the input displacement function (known), while xit) and yt) are the two generalized coordinates (both unknown) of the mass-spring-damper systenm. 1. Identify the type of equations (e.g. H/NH, ODE/PDE, L/NL, order, type of coefficients, etc.J. 2. Express this system of differential equations in matrix form, assume f 0 and then determine its general...
PROBLEM 1 (35 %) The mechanical system in the figure below consists of a disk of radius r, a block of mass m, a spring of stiffness (spring constant) k, and a damper with damping ratio b. The disk has moment of inertia Jabout its center of mass (pivot point O), and the block is subjected to an external force t) as shown in the figure. The spring is unstressed when x 0= 0. Assume small 0. (a) (10 points)...
Question8 n the spring-mass-damper system in Figure 8, the force F, is applied to the mass and its displacement is measured via r(t), whilst k and c are the spring and damper constants, respectively x(t) Figure 8: A spring-mass-damper system. a) Obtain the differential equation that relates the input force F, to the measured dis- (6 marks) placement x(t) for the system in Figure 8. b) Draw the block diagram representation of the system in Figure 8. c) Based on...
The mechanical system shown in the figure below is excited by a sinusoidal force f(t)-Fi cos(ut + ?) N. The differential equation of the displacement x(t) is Use phasor techniques to solve for the displacement phasor Xin terms of the excitation frequency ? , and the mechanical elements M = 0.1 kg, D = 8 N-s/m , and K = 2,000 N/m . If Fi-10 N and ?? = 30°, determine the excitation frequency w (in rad/s) at which the...
Problem 2: Transfer Functions of Mechanical Systems. (20 Points) A model sketch for a two-mass mechanical system subjected to fluctuations (t) at the wall is provided in figure 2. Spring k, is interconnected with both spring ka and damper Os at the nodal point. The independent displacement of mass m is denoted by 1, the independent displacement of mass m, is denoted by r2, and the independent displacement of the node is denoted by ra. Assume a linear force-displacement/velocity relationship...
Given the the mass-spring-damper system in Figure 3.10, assume
that the contact forces are viscous friction. 1. State the number
of degrees of freedom in the system.
2. Derive the equations of motion and state them in matrix
notation. 3. If f(t) = a (a constant), what is the long term state
of the system? 4. If the forcing is f(t) = A sin(ωt), and the
system parameters are given in Table 3.1, simulate the response
from rest. Plot all...
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...
A second order mechanical system of a mass connected to a spring and a damper is subjected to a sinusoidal input force mx+cx + kx = A sin(at) The mass is m-5 kg, the damping constant is c = 1 N-sec/m, the spring stiffness is 2 N/m, and the amplitude of the input force is A- 3 N. For this system give explicit numerical values for the damping factor 5 and the un-damped natural frequency Using the given formulas for...
Consider the mass-spring-damper system shown below left. A
spring-based force gage, with spring constant Kgage, is to be
inserted between the spring K and mass M (as shown below right) to
measure the force in K in response to the applied position xin.
With ftrue representing the force in K without the gage present,
and fgage representing the force in K with the gage present, the
Laplace transforms of fgage and ftrue satisfy
For a mass-spring-damper mechanical systems shown below, x200) K1-1 N/m 0000 -X,(0) K-1 N/m 00004 = 1 N-s/m fr2 M1=1 kg = 2 N-s/m M2 -1 kg 13 = 1 N-s/m 1. Find the differential equations relating input force f(t) and output displacement xi(t) and x2(C) in the system. (40 marks) (Hint: K, fy and M are spring constant, friction coefficient and mass respectively) 2. Determine the transfer function G(s)= X1(s)/F(s) (20 marks)
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