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engineering
mechanical vibration analysis
Mechanical Vibrations 6th Edition Singiresu S Rao - Solutions
Fill in the Blank.The phenomenon of ___________ can occur when the forcing frequency is close to the natural frequency of the system.
Fill in the Blank.When the base of system is subject to harmonic motion with amplitude \(Y\) resulting in a response amplitude \(X\), the ratio \(\frac{X}{Y}\) is called the displacement ___________.
Fill in the Blank.\(Z(i \omega)=-m \omega^{2}+i \omega c+k\) is called the mechanical ___________ .
Fill in the Blank.The difference between the frequencies associated with half-power points is called the ___________ of the system.
Fill in the Blank.The value of the amplitude ratio at resonance is called ___________ factor.
Fill in the Blank.The dry-friction damping is also known as ___________ damping.
Fill in the Blank.For ___________ values of dry-friction damping, the motion of the mass will be discontinuous.
Fill in the Blank.The quantity \(k(1+i \beta)\) in hysteresis damping is called ___________ stiffness.
Fill in the Blank.Quadratic or velocity-squared damping is present whenever a body moves in a(n) ___________ fluid flow.
Fill in the Blank.In self-excited systems, the ___________ itself produces the exciting force.
Fill in the Blank.The flutter of turbine blades is an example of ___________ vibration.
Fill in the Blank.The motion ___________ and the system becomes unstable during self-excitation.
Fill in the Blank.The transfer function approach is based on ___________transform.
Fill in the Blank.___________ identifies the input, system, and output clearly.
Fill in the Blank.The Laplace transform of \(f(t)\) is denoted as ___________.
Fill in the Blank.The Laplace transform converts a linear differential equation into a(n) ___________ expression.
The response of an undamped system under resonance will bea. very largeb. infinityc. zero
The reduction of the amplitude ratio in the presence of damping is very significanta. near \(\omega=\omega_{n}\)b. near \(\omega=0\)c. near \(\omega=\infty\)
The frequency of beating isa. \(\omega_{n}-\omega\)b. \(\omega_{n}\)c. \(\omega\)
The energy dissipated in a cycle by dry-friction damping is given bya. \(4 \mu N X\)b. \(4 \mu \mathrm{N}\)c. \(4 \mu N X^{2}\)
The complex frequency response, \(H(i \omega)\), is defined bya. \(\frac{k X}{F_{0}}\)b. \(\frac{X}{F_{0}}\)c. \(\left|\frac{k X}{F_{0}}\right|\)
The energy dissipated over the following duration is considered in finding the equivalent viscous-damping constant of a system with Coulomb damping:a. half cycleb. full cyclec. one second
The damping force depends on the frequency of the applied force in the case ofa. viscous-dampingb. Coulomb dampingc. hysteresis damping
The system governed by the equation \(m \ddot{x}+c \dot{x}+k x=0\) is dynamically stable ifa. \(k\) is positiveb. \(c\) and \(k\) are positivec. \(c\) is positive
Complex stiffness or complex damping is defined in the case ofa. hysteresis dampingb. Coulomb dampingc. viscous damping
The equation of motion of a machine (rotating at frequency \(\omega\) ) of mass \(M\), with an unbalanced mass \(m\), at radius \(e\), is given bya. \(m \ddot{x}+c \dot{x}+k x=m e \omega^{2} \sin \omega t\)b. \(M \ddot{x}+c \dot{x}+k x=m e \omega^{2} \sin \omega t\)c. \(M \ddot{x}+c \dot{x}+k x=M e
The force transmissibility of a system, subjected to base excitation (with amplitude \(Y\) ) resulting in a transmitted force \(F_{T}\), is defined asa. \(\frac{F_{T}}{k Y}\)b. \(\frac{X}{k Y}\)c. \(\frac{F_{T}}{k}\)
Magnification factor of an undamped system Using the notation:\(r=\) frequency ratio \(=\frac{\omega}{\omega_{n}}\)\(\omega=\) forcing frequency \(\omega_{n}=\) natural frequency \(\zeta=\) damping ratioa. \(\frac{2 \pi}{\omega_{n}-\omega}\)b. \(\left[\frac{1+(2 \zeta
Period of beating Using the notation:\(r=\) frequency ratio \(=\frac{\omega}{\omega_{n}}\)\(\omega=\) forcing frequency \(\omega_{n}=\) natural frequency \(\zeta=\) damping ratioa. \(\frac{2 \pi}{\omega_{n}-\omega}\)b. \(\left[\frac{1+(2 \zeta r)^{2}}{\left(1-r^{2}\right)^{2}+(2 \zeta
Magnification factor of a damped system Using the notation:\(r=\) frequency ratio \(=\frac{\omega}{\omega_{n}}\)\(\omega=\) forcing frequency \(\omega_{n}=\) natural frequency \(\zeta=\) damping ratioa. \(\frac{2 \pi}{\omega_{n}-\omega}\)b. \(\left[\frac{1+(2 \zeta
Damped frequency Using the notation:\(r=\) frequency ratio \(=\frac{\omega}{\omega_{n}}\)\(\omega=\) forcing frequency \(\omega_{n}=\) natural frequency \(\zeta=\) damping ratioa. \(\frac{2 \pi}{\omega_{n}-\omega}\)b. \(\left[\frac{1+(2 \zeta r)^{2}}{\left(1-r^{2}\right)^{2}+(2 \zeta
Quality factor Using the notation:\(r=\) frequency ratio \(=\frac{\omega}{\omega_{n}}\)\(\omega=\) forcing frequency \(\omega_{n}=\) natural frequency \(\zeta=\) damping ratioa. \(\frac{2 \pi}{\omega_{n}-\omega}\)b. \(\left[\frac{1+(2 \zeta r)^{2}}{\left(1-r^{2}\right)^{2}+(2 \zeta
Displacement transmissibility Using the notation:\(r=\) frequency ratio \(=\frac{\omega}{\omega_{n}}\)\(\omega=\) forcing frequency \(\omega_{n}=\) natural frequency \(\zeta=\) damping ratio \(\omega_{1}, \omega_{2}=\) frequencies corresponding to half-power pointsa. \(\frac{2
\(m \ddot{z}+c \dot{z}+k z=-m \ddot{y}\)a. System with Coulomb dampingb. System with viscous dampingc. System subject to base excitationd. System with hysteresis dampinge. System with rotating unbalance
a. System with Coulomb dampingb. System with viscous dampingc. System subject to base excitationd. System with hysteresis dampinge. System with rotating unbalance\(M \ddot{x}+c \dot{x}+k x=m e \omega^{2} \sin \omega t\)
a. System with Coulomb dampingb. System with viscous dampingc. System subject to base excitationd. System with hysteresis dampinge. System with rotating unbalance\(m \ddot{x}+k x \pm \mu N=F(t)\)
a. System with Coulomb dampingb. System with viscous dampingc. System subject to base excitationd. System with hysteresis dampinge. System with rotating unbalance\(m \ddot{x}+k(1+i \beta) x=F_{0} \sin \omega t\)
a. System with Coulomb dampingb. System with viscous dampingc. System subject to base excitationd. System with hysteresis dampinge. System with rotating unbalance\(m \ddot{x}+c \dot{x}+k x=F_{0} \sin \omega t\)
Suggest a method for determining the damping constant of a highly damped vibrating system that uses viscous damping.
Can you apply the results of Section 2.2 to systems where the restoring force is not proportional to the displacement-that is, where \(k\) is not a constant?
State the parameters corresponding to \(m,c, k\), and \(x\) for a torsional system.
What effect does a decrease in mass have on the frequency of a system?
What effect does a decrease in the stiffness of the system have on the natural period?
Why does the amplitude of free vibration gradually diminish in practical systems?
Why is it important to find the natural frequency of a vibrating system?
How many arbitrary constants must a general solution to a second-order differential equation have? How are these constants determined?
Can the energy method be used to find the differential equation of motion of all singledegree-of-freedom systems?
What assumptions are made in finding the natural frequency of a single-degree-offreedom system using the energy method?
Is the frequency of a damped free vibration smaller or greater than the natural frequency of the system?
What is the use of the logarithmic decrement?
Is hysteresis damping a function of the maximum stress?
What is critical damping, and what is its importance?
What happens to the energy dissipated by damping?
What is equivalent viscous damping? Is the equivalent viscous-damping factor a constant?
What is the reason for studying the vibration of a single-degree-of-freedom system?
How can you find the natural frequency of a system by measuring its static deflection?
Give two practical applications of a torsional pendulum.
Define these terms: damping ratio, logarithmic decrement, loss coefficient, and specific damping capacity.
In what ways is the response of a system with Coulomb damping different from that of systems with other types of damping?
What is complex stiffness?
Define the hysteresis damping constant.
Give three practical applications of the concept of center of percussion.
What is the order of the equation of motion given by \(m \dot{v}+c v=0\) ?
Define the time constant.
What is a root locus plot?
What is the significance of \(c
What is a time-invariant system?
True or False.The amplitude of an undamped system will not change with time.
True or False.A system vibrating in air can be considered a damped system.
True or False.The equation of motion of a single-degree-of-freedom system will be the same whether the mass moves in a horizontal plane or an inclined plane.
True or False.When a mass vibrates in a vertical direction, its weight can always be ignored in deriving the equation of motion.
True or False.The principle of conservation of energy can be used to derive the equation of motion of both damped and undamped systems.
True or False.The damped frequency can in some cases be larger than the undamped natural frequency of the system.
True or False.The damped frequency can be zero in some cases.
True or False.The natural frequency of vibration of a torsional system is given by \(\sqrt{\frac{k}{m}}\), where \(k\) and \(m\) denote the torsional spring constant and the polar mass moment of inertia, respectively.
True or False.Rayleigh's method is based on the principle of conservation of energy.
True or False.The final position of the mass is always the equilibrium position in the case of Coulomb damping.
True or False.The undamped natural frequency of a system is given by \(\sqrt{g / \delta_{\mathrm{st}}}\), where \(\delta_{\mathrm{st}}\) is the static deflection of the mass.
True or False.For an undamped system, the velocity leads the displacement by \(\pi / 2\).
True or False.For an undamped system, the velocity leads the acceleration by \(\pi / 2\).
True or False.Coulomb damping can be called constant damping.
True or False.The loss coefficient denotes the energy dissipated per radian per unit strain energy.
True or False.The motion diminishes to zero in both underdamped and overdamped cases.
True or False.The logarithmic decrement can be used to find the damping ratio.
True or False.The hysteresis loop of the stress-strain curve of a material causes damping.
True or False.The complex stiffness can be used to find the damping force in a system with hysteresis damping.
True or False.Motion in the case of hysteresis damping can be considered harmonic.
True or False.In the \(s\)-plane, the locus corresponding to constant natural frequency will be a circle.
True or False. The characteristic equation of a single-degree-of-freedom system can have one real root and one complex root.
Fill in the Blanks.The free vibration of an undamped system represents interchange of__________ and energies __________.
Fill in the Blanks.A system undergoing simple harmonic motion is called \(a(n)\) __________ oscillator.
Fill in the Blanks.The mechanical clock represents a(n) __________ pendulum.
Fill in the Blanks.The center of __________ can be used advantageously in a bat.
Fill in the Blanks.With viscous and hysteresis damping, the motion __________ forever, theoretically.
Fill in the Blanks.The damping force in Coulomb damping is given by __________ .
Fill in the Blanks.The __________ coefficient can be used to compare the damping capacity of different engineering materials.
Fill in the Blanks.Torsional vibration occurs when a(n) __________ body oscillates about an axis.
Fill in the Blanks.The property of __________ damping is used in many practical applications, such as large guns.
Fill in the Blanks.The logarithmic decrement denotes the rate at which the __________ of a free damped vibration decreases.
Fill in the Blanks.Rayleigh's method can be used to find the __________ frequency of a system directly.
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