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engineering
introduction mechanical engineering
Questions and Answers of
Introduction Mechanical Engineering
A load of \(2 \mathrm{kN}\) is suspended from two rods as shown is Fig. 13.36. Rod AB is of steel of \(15 \mathrm{~mm}\) diameter and rod CD of copper having \(10 \mathrm{~mm}\) diameter.Calculate
\(A\) rigid bar \(A B C\) is hinged at \(A\) and suspended at two points \(B\) and \(C\) by two bars BD and CE made of aluminium and steel respectively as shown in Fig. 13.37. The aluminium bar is of
A bar of \(20 \mathrm{~mm}\) diameter is subjected to a pull of \(30 \mathrm{kN}\). The measured extensions over a guage length of \(200 \mathrm{~mm}\) is \(0.1 \mathrm{~mm}\) and the change in
A steel tube \(0.75 \mathrm{~m}\) long, \(25 \mathrm{~mm}\) external diameter and \(20 \mathrm{~mm}\) internal diamameter encloses a copper rod of the same length and \(15 \mathrm{~mm}\) in diameter.
A steel rod \(1 \mathrm{~m}\) long and \(15 \mathrm{~mm}\) diameter is held between rigid supports as shown in Fig. 13.40 The temperature of the rod is increased by \(40^{\circ} \mathrm{C}\). The
Three identical vertical wires of \(1 \mathrm{~m}\) length each and \(4 \mathrm{~mm}\) diameter are suspended from a horizontal support as shown in Fig.13.41. a load if \(5 \mathrm{kN}\) is applied
A compound bar is shown in Fig. 13.42. Its temperature is raised by \(120^{\circ} \mathrm{F}\). Calculate the stresses in each metal and the change in length. \(E_{s}=210 \mathrm{GPa} E_{c u} 105\)
A composite rod of steel and copper of lengths \(1.5 \mathrm{~m}\) and \(1 \mathrm{~m}\) respectively is clamped at the ends, as shown in Fig. 13.43. Its temperature is then raised by \(50^{\circ}
Determine the stresses in the bar shown in Fig. 13.44 if its temperature is increased by \(30^{\circ} \mathrm{C} . E_{s}=2 E_{c}=200 \mathrm{GPa} \alpha_{s}=12.5 \times 10^{-6} \mathrm{per}{
A load of \(45 \mathrm{kN}\) is transmitted through a slab to a composite solid steel cylinder \(\left(A_{s}=15 \mathrm{~cm}^{2}\right)\) and hollow copper cylinder \(\left(A_{c}=20
A tapered circular bar is rigidly fixed at both the ends as shown in Fig. 13.45. If the temperature is raised by \(50^{\circ} \mathrm{C}\), calculate the stress in the bar. Take \(E=202
A trapezoidal flat steel plate of thickness \(10 \mathrm{~mm}\) tapers uniformly from a width \(150 \mathrm{~mm}\) to \(100 \mathrm{~mm}\) in a length of \(500 \mathrm{~mm}\). Calculate the extension
In a uniaxial state of stress, the maximum normal stress occurs on a plane which is inclined to the load line at(a) \(0^{\circ}\)(b) \(45^{\circ}\)(c) \(60^{\circ}\)(d) \(90^{\circ}\)
In a uniaxial state of stress, the maximum shear stress occurs on a plane which is inclined to the load line at(a) \(0^{\circ}\)(b) \(45^{\circ}\)(c) \(60^{\circ}\)(d) \(90^{\circ}\)
In a uniaxial state of stress, the ratio of maximum shear stress to normal stress is :(a) \(1: 2\)(b) \(2: 1\)(d) \(1: 4\)(d) \(4: 1\)
On the principal planes, the maximum stress is the(a) normal stress(b) shear stress(c) both normal and shear stress(d) either normal or shear stress
In a uniaxial state of stress, the resultant stress on a plane inclined at an angle \(\theta\) is :(a) \(\sigma \sin \theta\)(b) \(\sigma \cos \theta\)(c) \(0.5 \sigma \sin 2 \theta\)(d) \(0.5 \sigma
In a biaxial state of stress, the shear stress is maximum on a plane making an angle with the stresses, equal to:(a) \(22.5^{\circ}\)(b) \(30^{\circ}\)(c) \(45^{\circ}\)(d) \(67.5^{\circ}\)
In a biaxial state of stress, the maximum shear stress is equal to:(a) sum of the stresses(b) difference of the stresses(c) average of the stresses(d) half of the difference of stresses
In a body subjected to a complex state of stress, the planes of maximum shear stress are inclined to the principal planes at:(a) \(22.5^{\circ}\)(b) \(45^{\circ}\)(c) \(67.5^{\circ}\)(d)
For a complex state of stress, the radius of the Mohr's circle is:(a) \(0.5\left[\left(\sigma_{x}-\sigma_{y}\right)^{2}+4 \tau_{x y}{ }^{2}\right]^{1 / 2}\)(b)
Pure shear is equal to(a) simple shear(b) simple shear plus rotation(c) simple shear minus rotation(d) simple shear minus twice the rotation
Maximum shear stress is equal to the(a) sum of the principal stress(b) difference of the principle stresses(c) average of the principal stresses(d) half of the difference of the principal stresses
The ratio of direct stress to volumetric strain in case of a body subjected to three mutually perpendicular stresses of equal intensity, is equal to(a) Young's modulus(b) bulk modulus(c) shear
If the radius of a wire stretched by a load is doubled, then its Young's modulus will be(a) doubled(b) four times(c) one-fourth(d) unaffected
The relationship between modulus of elasticity \(E\) and modulus of rigidity \(G\) is :(a) \(E=G(1+v)\)(b) \(G=E(2-v)\)(c) \(G=0.5 E /(1+v)\)(d) \(G=E(1+2 v)\)wher \(v=\) Poisson's ratio
The ratio of shear modulus to the modulus of elasticity, if Poisson's ratio is 0.25 , will be(a) 0.40(b) 0.25(c) 4.00(d) 0.50
The value of Poisson's ratio for steel lies in the range(a) 0.10 to 0.20(b) 0.21 to 0.25(c) 0.26 to 0.30(d) 0.30 to 0.35
If the modulus of elasticity of a material is \(200 \mathrm{GPa}\) and Poisson's ratio is 0.25 , then the modulus of rigidity will be(a) \(80 \mathrm{GPa}\)(b) \(125 \mathrm{GPa}\)(c) \(133.3
Choose the correct relationship between \(\mathrm{E}, \mathrm{G}\) and \(\mathrm{K}\).(a) \(E=(\mathrm{G}+3 \mathrm{~K}) / 9 \mathrm{GK}\)(b) \(E=(3 \mathrm{G}+\mathrm{K}) / 9 \mathrm{GK}\)(c) \(E=9
Choose the correct relationship between \(E, K\), and \(v\).(a) \(E=3 K(1-2 \mathrm{v})\)(b) \(E=3 K(1-v)\)(c) \(E=K(1-2 v)\)(d) \(E=K(1-v)\)
Maximum shear stress in Mohr's circle is equal to:(a) radius of the circle(b) diameter of the circle(c) centre of circle from \(y\)-axis(d) chord of circle
The convolution integral can be applied to solve nonlinear problems.Indicate whether the statement presented is true or false. If true, state why. If false, rewrite the statement to make it true.
A mass attached to a linear spring sliding on a surface with Coulomb damping is an example of a nonlinear system.Indicate whether the statement presented is true or false. If true, state why. If
The swinging spring is an example of a two degree-of-freedom system with a cubic nonlinearity.Indicate whether the statement presented is true or false. If true, state why. If false, rewrite the
The period of free vibrations of a nonlinear system depends upon initial conditions.Indicate whether the statement presented is true or false. If true, state why. If false, rewrite the statement to
The free response of a system with a cubic nonlinearity occurs only at the linear natural frequency of the system.Indicate whether the statement presented is true or false. If true, state why. If
A focus is always unstable.Indicate whether the statement presented is true or false. If true, state why. If false, rewrite the statement to make it true.
A saddle point is always unstable.Indicate whether the statement presented is true or false. If true, state why. If false, rewrite the statement to make it true.
Secular terms must be removed from the response of a system.Indicate whether the statement presented is true or false. If true, state why. If false, rewrite the statement to make it true.
When a superharmonic resonance occurs, the free oscillation term does not decay exponentially but combines with the forced response.Indicate whether the statement presented is true or false. If true,
A SDOF system with viscous damping subject to a single frequency excitation always has a free response which decays exponentially.Indicate whether the statement presented is true or false. If true,
A SDOF system with a cubic nonlinearity is excited by a harmonic force at a frequency of \(100 \mathrm{rad} / \mathrm{s}\). The forced response occurs only at \(300 \mathrm{rad} /
A MDOF system has a combination resonance when the parameters are such that one of the system's linear natural frequencies is in a certain combination with another of the system's natural
A bifurcation is a split in natural frequencies for one value of a parameter.Indicate whether the statement presented is true or false. If true, state why. If false, rewrite the statement to make it
Period doubling is a route to chaos.Indicate whether the statement presented is true or false. If true, state why. If false, rewrite the statement to make it true.
Why can’t the Laplace transform method be applied to nonlinear systems?
A spring has a cubic nonlinearity which is an example of a (geometric, material) _______________________________ nonlinearity.
A spring with a cubic nonlinearity equal to \(-3 x^{3}\) is an example of a (hardening, softening) _______________________________ nonlinearity.
Trajectories near an equilibrium point in the state space are shown in Figure SP12.18. Identify the equilibrium point that is(a) an unstable saddle point,(b) a stable focus,(c) a center, and(d) an
The eigenvalues of the differential equation are \(\beta_{1}\) and \(\beta_{2}\) when the equation is linearized about an equilibrium point. Determine the type of the equilibrium point and its
The eigenvalues of the differential equation are \(\beta_{1}\) and \(\beta_{2}\) when the equation is linearized about an equilibrium point. Determine the type of the equilibrium point and its
The eigenvalues of the differential equation are \(\beta_{1}\) and \(\beta_{2}\) when the equation is linearized about an equilibrium point. Determine the type of the equilibrium point and its
The eigenvalues of the differential equation are \(\beta_{1}\) and \(\beta_{2}\) when the equation is linearized about an equilibrium point. Determine the type of the equilibrium point and its
The eigenvalues of the differential equation are \(\beta_{1}\) and \(\beta_{2}\) when the equation is linearized about an equilibrium point. Determine the type of the equilibrium point and its
Explain the use of the detuning parameter.
The frequency-response curve shown in Figure SP12.25 is for the primary resonance of a SDOF system with a cubic nonlinearity.(a) Is the curve drawn for a hardening spring or a softening spring?(b)
A SDOF system with a cubic nonlinearity has a linear natural frequency of \(30 \mathrm{rad} / \mathrm{s}\). At what excitation frequency does the system have(a) A primary resonance?(b) A
\(\omega_{1}\) is near \(30 \mathrm{rad} / \mathrm{s}\) and \(\omega_{2}\) is near \(60 \mathrm{rad} / \mathrm{s}\)A DOF system with a cubic nonlinearity has a linear natural frequency of \(120
\(\omega_{1}\) is near \(90 \mathrm{rad} / \mathrm{s}\) and \(\omega_{2}\) is near \(60 \mathrm{rad} / \mathrm{s}\)A DOF system with a cubic nonlinearity has a linear natural frequency of \(120
\(\omega_{1}\) is near \(20 \mathrm{rad} / \mathrm{s}\) and \(\omega_{2}\) is near \(260 \mathrm{rad} / \mathrm{s}\)A DOF system with a cubic nonlinearity has a linear natural frequency of \(120
\(\omega_{1}\) is near \(50 \mathrm{rad} / \mathrm{s}\) and \(\omega_{2}\) is near \(180 \mathrm{rad} / \mathrm{s}\)A DOF system with a cubic nonlinearity has a linear natural frequency of \(120
\(\omega_{1}\) is near \(40 \mathrm{rad} / \mathrm{s}\) and \(\omega_{2}\) is near \(200 \mathrm{rad} / \mathrm{s}\)A DOF system with a cubic nonlinearity has a linear natural frequency of \(120
\(\omega_{1}\) is near \(120 \mathrm{rad} / \mathrm{s}\) and \(\omega_{2}\) is near \(40 \mathrm{rad} / \mathrm{s}\)A DOF system with a cubic nonlinearity has a linear natural frequency of \(120
\(\quad \omega_{1}\) is near \(240 \mathrm{rad} / \mathrm{s}\) and \(\omega_{2}\) is near \(360 \mathrm{rad} / \mathrm{s}\)A DOF system with a cubic nonlinearity has a linear natural frequency of
Explain why a superharmonic resonance occurs.A DOF system with a cubic nonlinearity has a linear natural frequency of \(120 \mathrm{rad} / \mathrm{s}\). The system is forced by harmonic excitations
What is an internal resonance in a MDOF system?A DOF system with a cubic nonlinearity has a linear natural frequency of \(120 \mathrm{rad} / \mathrm{s}\). The system is forced by harmonic excitations
Describe the Poincaré section corresponding to a periodic function when the sampling interval is one-third of the period.A DOF system with a cubic nonlinearity has a linear natural frequency of
What is the signature of the Fourier transform?A DOF system with a cubic nonlinearity has a linear natural frequency of \(120 \mathrm{rad} / \mathrm{s}\). The system is forced by harmonic excitations
What is Feigenbaum's constant?A DOF system with a cubic nonlinearity has a linear natural frequency of \(120 \mathrm{rad} / \mathrm{s}\). The system is forced by harmonic excitations at different
The linearized differential equation around an equilibrium point is\[\Delta \ddot{x}+2 \Delta \dot{x}+3 \Delta x=0\]Classify the equilibrium point and determine its stability.
The equation of motion of a simple pendulum of length \(\ell\) is\[\ddot{\theta}+\frac{g}{\ell} \sin \theta=0\](a) Determine the pendulum's equilibrium points.(b) Classify the equilibrium points and
The differential equation governing the motion of a nonlinear system is\[\ddot{x}-0.5 \dot{x}+x-0.1 x^{3}=0\](a) Determine the equilibrium points.(b) Classify the equilibrium points and determine
The differential equation governing the motion of a nonlinear system is\[\ddot{x}-x+0.1 x^{3}=0\](a) Determine the equilibrium points.(b) Classify the equilibrium points and determine their
The equation of motion for a particle moving on a rotating circular frame (Figure SP12.45) is\[\ddot{\theta}+\frac{g}{R}\left(\sin \theta-\frac{\omega^{2}}{g} \cos \theta\right)=0\](a) Determine the
Determine the free response to the nondimensional undamped Duffing's equation for \(\varepsilon=0.01, x(0)=1\), and \(\dot{x}(0)=0\).
Determine the free response to the nondimensional damped Duffing's equation for \(\varepsilon=0.01, \zeta=0.05, x(0)=1\), and \(\dot{x}(0)=0\).
Determine the steady-state amplitudes for the equation\[\ddot{x}+0.05 \dot{x}+x+0.01 x^{3}=0.03 \sin 1.01 t\]
Determine the steady-state amplitudes for the equation\[\ddot{x}+0.05 \dot{x}+x+0.01 x^{3}=0.03 \sin 0.33 t\]
Determine the steady-state amplitudes for the equation\[\ddot{x}+0.05 \dot{x}+x+0.01 x^{3}=0.03 \sin 3.06 t\]
Suggest any internal resonances for a fixed-free beam.
Suggest any internal resonances for a beam fixed at one end with a mass of \(0.25 ho A L\) attached at its other end.
Determine the Fourier transform of\[F(t)=2 \sin 3 t+4 \sin 4.5 t\]
What are the dimensions of the following quantities?(a) Coefficient multiplying \(x^{3}\) in nonlinear spring stiffness, \(k_{3}\)(b) The perturbation parameter, \(\varepsilon\)(c) A detuning
The free-vibration response of a block hanging from a linear spring is the same as that of the block attached to the same spring, but sliding on a frictionless surface. Is the response the same if
The system of Figure P12.2 is one of the few for which an exact solution is available. Its solution is obtained in a manner analogous to that of free vibrations with Coulomb damping. The block is
The block in Figure P12.3 is not attached to the springs. Determine the period of the resulting oscillations if the block is displaced a distance \(x_{0}\) to the right from equilibrium and released.
Without making linearizing assumptions, use Lagrange's equations to derive the nonlinear differential equation(s) governing the motion of the systems shown. Use the generalized coordinates indicated
Without making linearizing assumptions, use Lagrange's equations to derive the nonlinear differential equation(s) governing the motion of the systems shown. Use the generalized coordinates indicated
Without making linearizing assumptions, use Lagrange's equations to derive the nonlinear differential equation(s) governing the motion of the systems shown. Use the generalized coordinates indicated
Without making linearizing assumptions, use Lagrange's equations to derive the nonlinear differential equation(s) governing the motion of the systems shown. Use the generalized coordinates indicated
A wedge of specific weight \(\gamma\) floats stably on the free surface of a fluid of specific weight \(\gamma_{w}\) (Figure P12.8). The wedge is given a vertical displacement \(\delta\) from this
Repeat Chapter Problem 12.8 for the inverted cone of Figure P12.9.Data From Chapter Problem 12.8:A wedge of specific weight \(\gamma\) floats stably on the free surface of a fluid of specific weight
Determine the equation defining the state plane for the system of Figure P12.6. Sketch trajectories in the phase plane when the following are given.(a) \(p=1.5 \mathrm{~m}^{-1}, \omega=5 \mathrm{rad}
Plot the trajectory in the state plane corresponding to the motion of a mass attached to a linear spring free to slide on a surface with Coulomb damping when the mass is displaced from equilibrium
Determine the equilibrium points and their type for the differential equation\[ \ddot{x}+2 \zeta \dot{x}-x+\epsilon x^{3}=0 \]
Determine the equilibrium points and their type for the differential equation\[ \ddot{x}+2 \zeta \dot{x}-x-\epsilon x^{3}=0 \]
Determine the equilibrium points and their type for the differential equation\[ \ddot{x}+2 \zeta \dot{x}+x+\epsilon x^{2}=0 \]
Determine the equilibrium points and their type for the differential equation\[ \ddot{x}+2 \zeta \dot{x}+x-\epsilon x^{2}=0 \]
The equation of motion for the free oscillations of a pendulum subject to quadratic damping is\[ \ddot{\theta}+2 \zeta \dot{\theta}^{2}+\sin \theta=0 \](a) Determine an exact equation defining the
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