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Vibrations

Consider the system of the figure. For c1 = c2 = c3 = 0 derive the equation of motion and calculate the mass and stiffness matricies. Note that setting k3=0 in your solution should result in the stiffness matrix given by some equation.

To analyze the motions of an automobile on its tires, we approximate a car by a bar of uniform mass (1000 kg) with a moment of inertia equal to 1333.3 kg m^2. The bar is 4 m long. The tire/suspension is approximated by a pair of springs with spring constant equal to 11.697 N/m^2 for each one. If the road profile is sinusoidal, with an amplitude of 20 cm and a wavelength of 10 m, at what speed will the car experience resonance? with what mode is this resonance associated?

For initial conditions x(0)=[1 0]^7 and xdot(0)=[0 0]^7 calculate the free response of the system of the previous problem. Plot the response x1 and x2.

Calculate and plot the response of an undamped system to a step function with a finite rise time of t1 for the case m=1 kg, k=1 N/m, t1=4s, and F0=20 N. This function is described by: F(t)=F0t/t1 and F0

A lathe can be modeled as an electric motor mounted on a steel table. The table plus the motor have a mass of 50 kg. The rotating parts of the lathe have a mass of 5 kg at a distance 0.1 m from the center. The damping ratio of the system is measured to be 0.06 (viscous damping) and its natural frequency in 7.5 Hz. Calculate the amplitude of the steady-state displacement of the motor, assuming wr=30 Hz.

a machine weighing 2000 N rests on a support, as illustrated in the fiugre. The support deflects about 5 cm as a result of the weight of the machine. The floor under the support is somewhat flexible and moves, because of the motion of a nearby machine, harmonically near resonance (r=1) with an amplitude of 0.2 cm. Model the floor as base motion, assume a damping ratio of 0.01, and calculate the transmitted force and the amplitude of the transmitted displacement.

A 100 kg mass is suspended by a spring of stiffness 30 x 10^3 N/m with a viscous-damping constant of 1000 N s/m. The mass is initially at rest and in equilibrium. Calculate the steady-state displacement amplitude and phase if the mass is excited by a harmonic cosine force of 80 N at 3 Hz.

An air conditioner chiller unit which weighs 2000 lbf is supported by four air springs. Find the spring constant value so that the natural frequency of the unit lies between 5 and 10 rad/sec. List the spring constant of a single spring below.

To analyze the motions of an automobile on its tires, we approximate a car by a bar of uniform mass (1000 kg) with a moment of inertia equal to 1333.3 kg m^2. The bar is 4 m long. The tire/suspension is approximated by a pair of springs with spring constant equal to 11.697 N/m^2 for each one. If the road profile is sinusoidal, with an amplitude of 20 cm and a wavelength of 10 m, at what speed will the car experience resonance? with what mode is this resonance associated?

For initial conditions x(0)=[1 0]^7 and xdot(0)=[0 0]^7 calculate the free response of the system of the previous problem. Plot the response x1 and x2.

Calculate and plot the response of an undamped system to a step function with a finite rise time of t1 for the case m=1 kg, k=1 N/m, t1=4s, and F0=20 N. This function is described by: F(t)=F0t/t1 and F0

A lathe can be modeled as an electric motor mounted on a steel table. The table plus the motor have a mass of 50 kg. The rotating parts of the lathe have a mass of 5 kg at a distance 0.1 m from the center. The damping ratio of the system is measured to be 0.06 (viscous damping) and its natural frequency in 7.5 Hz. Calculate the amplitude of the steady-state displacement of the motor, assuming wr=30 Hz.

a machine weighing 2000 N rests on a support, as illustrated in the fiugre. The support deflects about 5 cm as a result of the weight of the machine. The floor under the support is somewhat flexible and moves, because of the motion of a nearby machine, harmonically near resonance (r=1) with an amplitude of 0.2 cm. Model the floor as base motion, assume a damping ratio of 0.01, and calculate the transmitted force and the amplitude of the transmitted displacement.

A 100 kg mass is suspended by a spring of stiffness 30 x 10^3 N/m with a viscous-damping constant of 1000 N s/m. The mass is initially at rest and in equilibrium. Calculate the steady-state displacement amplitude and phase if the mass is excited by a harmonic cosine force of 80 N at 3 Hz.

An air conditioner chiller unit which weighs 2000 lbf is supported by four air springs. Find the spring constant value so that the natural frequency of the unit lies between 5 and 10 rad/sec. List the spring constant of a single spring below.

Compute the total response of a spring-mass system with the following values: k=1000 N/m, m=10 kg, subject to a harmonic force of magnitude F0=100 N and frequency of 8.162 rad/s, and initial conditions given by x0=0.01 m and v0=0.01 m/s. Plot the response.

Solve the following system for the response x(t) using Laplace transforms: 100xdoubdot(t) + 2000x(t) = 50S(t), where the units are in Newtons and the initial conditions are both zero.

Calculate the characteristic equation from the previous problem for the case m1=9 kg, m2=1 kg, k1=24 N/m, k2=3 N/m, k3=3 N/m and solve for the system's natural frequencies.

A slightly more sophistticated model of a vehicle suspension system is given in the figure. Write the equations of motion in matrix form. Calculate the natural frequencies for k1=10^3 N/m, k2=10^4 N/m, m2=50 kg, and m1=2000 kg.

Given: k=27 N/m; M=3 kg; C=9 Nsec/m, Find: A) Undamped Natural Frequency B) Damped Natural Frequency C) Damping Ratio D) Critical Damping

Describe in words the difference between viscous damping and Coulomb damping. Give a real world example of each.

The free response of an electric motor of weight 500 N mounted on a foundation is shown in the figure below. a) What is the damping ratio of the system? b) What is the undamped and damped natural frequency? c) What is the spring constant, K, and the damping constant, C, of the system?

Consider a simple model of an airplane wing given in the figure. The wing is approximated as vibrating back and forth in its plane, massless compared to the missile carriage system (of mass m). The modulus and moment of inertia of the wing are approximated by E and I, respectively, and L is the length of the wing. The wing is modeled as a simple cantilever for the purpose of estimating the vibration resulting from the release of the missile, which is apprroximated by the impulse function FS(t). Calculate the response and plot your results for the case of an aluminum wing 4 m long with m=1000 kg, z=0.01, and I=0.5 m^4. Model F as 1000 N lasting over 10^-2 s.

Solve the following system for the response x(t) using Laplace transforms: 100xdoubdot(t) + 2000x(t) = 50S(t), where the units are in Newtons and the initial conditions are both zero.

Calculate the characteristic equation from the previous problem for the case m1=9 kg, m2=1 kg, k1=24 N/m, k2=3 N/m, k3=3 N/m and solve for the system's natural frequencies.

A slightly more sophistticated model of a vehicle suspension system is given in the figure. Write the equations of motion in matrix form. Calculate the natural frequencies for k1=10^3 N/m, k2=10^4 N/m, m2=50 kg, and m1=2000 kg.

Given: k=27 N/m; M=3 kg; C=9 Nsec/m, Find: A) Undamped Natural Frequency B) Damped Natural Frequency C) Damping Ratio D) Critical Damping

Describe in words the difference between viscous damping and Coulomb damping. Give a real world example of each.

The free response of an electric motor of weight 500 N mounted on a foundation is shown in the figure below. a) What is the damping ratio of the system? b) What is the undamped and damped natural frequency? c) What is the spring constant, K, and the damping constant, C, of the system?

Consider a simple model of an airplane wing given in the figure. The wing is approximated as vibrating back and forth in its plane, massless compared to the missile carriage system (of mass m). The modulus and moment of inertia of the wing are approximated by E and I, respectively, and L is the length of the wing. The wing is modeled as a simple cantilever for the purpose of estimating the vibration resulting from the release of the missile, which is apprroximated by the impulse function FS(t). Calculate the response and plot your results for the case of an aluminum wing 4 m long with m=1000 kg, z=0.01, and I=0.5 m^4. Model F as 1000 N lasting over 10^-2 s.

A control pedal of an aircraft can be modeled as the single degree of freedom system of the figure. Consider the lever as a massless shaft and the pedal as a lumped mass at the end of the shaft. Use the energy method to determine the equation of motion in theta and calculate the natural frequency of the system. Assume the spring to be unstretched at theta=0.

Numerically integrate and plot the response of an underdamped system determined by m=100 kg, k=20000 N/m, and c=200 kg/s, subject to the initial conditions of x0=0.01 m and v0=0.1 m/s and the applied force F(t)=150cos(5t). Then plot the exact response as computed by the equation. Compare the plot of the exact solution to the numerical simulation.

A machine oscillates in simple harmonic motion and appears to be well modeled by an undamped single degree of freedom oscillation. Its acceleration is measured to have an amplitude of 10000 mm/s^2 at 8 Hz. What is the machine's maximum displacement? Briefly describe a real scenario where a system's displacement due to vibration should not go beyond a given limit for maximum displacement and explain why.

An electric motor has an eccentric mass of 10 kg (10% of the total mass) 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 vertical vibration.

Determine the equation of motion in matrix form, then calculate the natural frequencies and mode shapes of the torsional system of the Figure. Assume that the torsional stiffness values provided by the shaft are equal (k1=k2) and that disk 1 has three times the inertia as that of disk 2 (J1=3J2).

Compute the natural frequency and plot the soution of a sping-mass system with mass of 1 kg and stiffness of 4 N/m, and initial conditions of x0=1mm and v0=0 mm/s, for at least two periods.

An undamped system vibrates with a frequency of 10 Hz and amplitude 1 mm. Calculate the maximum amplitude of the system's velocity and acceleration. What is the vibrational frequency in radians/sec? Why do we care about units?

Determine the natural frequency of the two systems illustrated. Which system (a or b) has the higher natural frequency? Note that K1 and K2 are the same in both a and b and K3 is nonzero. Explain your answer. Which system (a or b) has the higher natural frequency if K3=0 (K1 and K2 are the same for both a and b)? Explain your answer.

Numerically integrate and plot the response of an underdamped system determined by m=100 kg, k=20000 N/m, and c=200 kg/s, subject to the initial conditions of x0=0.01 m and v0=0.1 m/s and the applied force F(t)=150cos(5t). Then plot the exact response as computed by the equation. Compare the plot of the exact solution to the numerical simulation.

A machine oscillates in simple harmonic motion and appears to be well modeled by an undamped single degree of freedom oscillation. Its acceleration is measured to have an amplitude of 10000 mm/s^2 at 8 Hz. What is the machine's maximum displacement? Briefly describe a real scenario where a system's displacement due to vibration should not go beyond a given limit for maximum displacement and explain why.

An electric motor has an eccentric mass of 10 kg (10% of the total mass) 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 vertical vibration.

Determine the equation of motion in matrix form, then calculate the natural frequencies and mode shapes of the torsional system of the Figure. Assume that the torsional stiffness values provided by the shaft are equal (k1=k2) and that disk 1 has three times the inertia as that of disk 2 (J1=3J2).

Compute the natural frequency and plot the soution of a sping-mass system with mass of 1 kg and stiffness of 4 N/m, and initial conditions of x0=1mm and v0=0 mm/s, for at least two periods.

An undamped system vibrates with a frequency of 10 Hz and amplitude 1 mm. Calculate the maximum amplitude of the system's velocity and acceleration. What is the vibrational frequency in radians/sec? Why do we care about units?

Determine the natural frequency of the two systems illustrated. Which system (a or b) has the higher natural frequency? Note that K1 and K2 are the same in both a and b and K3 is nonzero. Explain your answer. Which system (a or b) has the higher natural frequency if K3=0 (K1 and K2 are the same for both a and b)? Explain your answer.

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