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- Question : 1P - A study of the response of a human body subjected to vibration/shock is important in many applications. In a standing posture, the masses of head, upper torso, hips, and legs and the elasticity/damping of neck, spinal column, abdomen, and legs influence the response characteristics. Develop a sequence of three improved approximations for modeling the human body.
- Question : 2P - Figure 1.62 shows a human body and a restraint system at the time of an automobile collision [1.47]. Suggest a simple mathematical model by considering the elasticity, mass, and damping of the seat, human body, and restraints for a vibration analysis of the system.
- Question : 3P - A reciprocating engine is mounted on a foundation as shown in Fig. 1.63. The unbalanced forces and moments developed in the engine are transmitted to the frame and the foundation. An elastic pad is placed between the engine and the foundation block to reduce the transmission of vibration. Develop two mathematical models of the system using a gradual refinement of the modeling process.
- Question : 4P - A car moving over a rough road (Fig. 1.64) can be modeled considering (a) weight of the car body, passengers, seats, front wheels, and rear wheels; (b) elasticity of tires
- Question : 5P - The consequences of a head-on collision of two cars can be studied by considering the impact of the car on a barrier, as shown in Fig. 1.65. Construct a mathematical model by considering the masses of the car body, engine, transmission, and suspension and the elasticity of the bumpers, radiator, sheet metal body, driveline, and engine mounts.
- Question : 6P - Develop a mathematical model for the tractor and plow shown in Fig. 1.66 by considering the mass, elasticity, and damping of the tires, shock absorbers, and plows (blades).
- Question : 7P - Determine the equivalent spring constant of the system shown in Fig. 1.67.
- Question : 8P - Consider a system of two springs, with stiffnesses k1 and k2, arranged in parallel as shown in Fig. 1.68. The rigid bar to which the two springs are connected remains horizontal when the
- Question : 9P - In Fig. 1.69, find the equivalent spring constant of the system in the direction of u
- Question : 10P - Find the equivalent torsional spring constant of the system shown in Fig. 1.70. Assume that k1, k2, k3, and k4 are torsional and k5 and k6 are linear spring constants.
- Question : 11P - A machine of mass m = 500 kg is mounted on a simply supported steel beam of length l = 2 m having a rectangular cross section 1depth = 0.1 m, width = 1.2 m2 and Young
- Question : 12P - A bar of length L and Young
- Question : 13P - A cantilever beam of length L and Young
- Question : 14P - An electronic instrument, weighing 1000 N, is supported on a rubber mounting whose forcedeflection relationship is given by F1x2 = 157 x + 0.2 x3, where the force (F) and the deflection (x) are in newtons and millimeters, respectively. Determine the following: a. Equivalent linear spring constant of the mounting at its static equilibrium position. b. Deflection of the mounting corresponding to the equivalent linear spring constant.
- Question : 15P - The force-deflection relation of a steel helical spring used in an engine is found experimentally as F1x2 = 34.6 x + 0.34 x2 + 0.002 x3, where the force (F) and deflection (x) are measured in newtons and millimeters, respectively. If the spring undergoes a steady deflection of 12.7 mm during the operation of the engine, determine the equivalent linear spring constant of the spring at its steady deflection.
- Question : 16P - Four identical rigid bars
- Question : 17P - The tripod shown in Fig. 1.73 is used for mounting an electronic instrument that finds the distance between two points in space. The legs of the tripod are located symmetrically about the mid-vertical axis, each leg making an angle a with the vertical. If each leg has a length l and axial stiffness k, find the equivalent spring stiffness of the tripod in the vertical direction.
- Question : 18P - The static equilibrium position of a massless rigid bar, hinged at point O and connected with springs k1 and k2, is shown in Fig. 1.74. Assuming that the displacement (x) resulting from the application of a force F at point A is small, find the equivalent spring constant of the system, ke, that relates the applied force F to the displacement x as F = kex.
- Question : 19P - Figure 1.75 shows a system in which the mass m is directly connected to the springs with stiffnesses k1 and k2 while the spring with stiffness k3 or k4 comes into contact with the mass based on the value of the displacement of the mass. Determine the variation of the spring force exerted on the mass as the displacement of the mass (x) varies.
- Question : 20P - Figure 1.76 shows a uniform rigid bar of mass m that is pivoted at point O and connected by springs of stiffnesses k1 and k2. Considering a small angular displacement u of the rigid bar about the point O, determine the equivalent spring constant associated with the restoring moment.
- Question : 21P - Figure 1.77 shows a U-tube manometer open at both ends and containing a column of liquid mercury of length l and specific weight g. Considering a small displacement x of the manometer meniscus from its equilibrium position (or datum), determine the equivalent spring constant associated with the restoring force.
- Question : 22P - An oil drum of diameter d and mass m floats in a bath of sea water of density rw as shown in Fig. 1.78. Considering a small displacement x of the oil drum from its static equilibrium position, determine the equivalent spring constant associated with the restoring force.
- Question : 23P - Find the equivalent spring constant and equivalent mass of the system shown in Fig. 1.79 with references to u. Assume that the bars AOB and CD are rigid with negligible mass.
- Question : 24P - Find the length of the equivalent uniform hollow shaft of inner diameter d and thickness t that has the same axial spring constant as that of the solid conical shaft shown in Fig. 1.80.
- Question : 25P - Figure 1.81 shows a three-stepped bar fixed at one end and subjected to an axial force F at the other end. The length of step i is li and its cross sectional area is Ai, i = 1, 2, 3. All the steps are made of the same material with Young
- Question : 26P - Find the equivalent spring constant of the system shown in Fig. 1.82.
- Question : 27P - igure 1.83 shows a three-stepped shaft fixed at one end and subjected to a torsional moment T at the other end. The length of step i is li and its diameter is Di, i = 1, 2, 3. All the steps are made of the same material with shear modulus Gi = G, i = 1, 2, 3. a. Find the torsional spring constant (or stiffness) kti of step i 1i = 1, 2, 32
- Question : 28P - The force-deflection characteristic of a spring is described by F = 500x + 2x3, where the force (F) is in newton and the deflection (x) is in millimeters. Find (a) the linearized spring constant at x = 10 mm and (b) the spring forces at x = 9 mm and x = 11 mm using the linearized spring constant. Also find the error in the spring forces found in (b).
- Question : 29P - Figure 1.84 shows an air spring. This type of spring is generally used for obtaining very low natural frequencies while maintaining zero deflection under static loads. Find the spring constant of this air spring by assuming that the pressure p and volume v change adiabatically when the mass m moves.
- Question : 30P - Find the equivalent spring constant of the system shown in Fig. 1.85 in the direction of the load P.
- Question : 31P - Derive the expression for the equivalent spring constant that relates the applied force F to the resulting displacement x of the system shown in Fig. 1.86. Assume the displacement of the link to be small.
- Question : 32P - The spring constant of a helical spring under axial load is given by
- Question : 33P - Two helical springs, one made of steel and the other made of aluminum, have identical values of d and D. (a) If the number of turns in the steel spring is 10, determine the number of turns required in the aluminum spring whose weight will be same as that of the steel spring. (b) Find the spring constants of the two springs.
- Question : 34P - Figure 1.87 shows three parallel springs, one with stiffness k1 = k and each of the other two with stiffness k2 = k. The spring with stiffness k1 has a length l and each of the springs with stiffness k2 has a length of l - a. Find the force-deflection characteristic of the system. k2 k k1 k k2 k x F
- Question : 35P - Design an air spring using a cylindrical container and a piston to achieve a spring constant of 12 kN/m. Assume that the maximum air pressure available is 1.5 MPa.
- Question : 36P - The force (F)-deflection (x) relationship of a nonlinear spring is given by F = ax + bx3 where a and b are constants. Find the equivalent linear spring constant when the deflection is 0.01 m with a = 20,000 N>m and b = 40 * 106 N>m3.
- Question : 37P - Two nonlinear springs, S1 and S2, are connected in two different ways as indicated in Fig. 1.88. The force, Fi, in spring Si is related to its deflection 1xi2 as Fi = aixi + bixi 3, i = 1, 2 where ai and bi are constants. If an equivalent linear spring constant, keq, is defined by W = k eqx, where x is the total deflection of the system, find an expression for keq in each case.
- Question : 38P - Design a steel helical compression spring to satisfy the following requirements: Spring stiffness 1k2
- Question : 39P - Find the spring constant of the bimetallic bar shown in Fig. 1.89 in axial motion.
- Question : 40P - Consider a spring of stiffness k stretched by a distance x0 from its free length. One end of the spring is fixed at point O and the other end is connected to a roller as shown in Fig. 1.90. The
- Question : 41P - One end of a helical spring is fixed and the other end is subjected to five different tensile forces. The lengths of the spring measured at various values of the tensile forces are given below:
- Question : 42P - A tapered solid steel propeller shaft is shown in Fig. 1.91. Determine the torsional spring constant of the shaft.
- Question : 43P - A composite propeller shaft, made of steel and aluminum, is shown in Fig. 1.92. a. Determine the torsional spring constant of the shaft. b. Determine the torsional spring constant of the composite shaft when the inner diameter of the aluminum tube is 5 cm instead of 10 cm.
- Question : 44P - Consider two helical springs with the following characteristics: Spring 1: material
- Question : 45P - Solve Problem 1.44 by assuming the wire diameters of springs 1 and 2 to be 0.125 m and 0.0125 m instead of 0.05 m and 0.025 m, respectively.
- Question : 46P - The arm AD of the excavator shown in Fig. 1.93 can be approximated as a steel tube of outer diameter 0.25 m, inner diameter 0.24 m, and length 2.5 m with a viscous damping coefficient of 70 N-s/m. The arm DE can be approximated as a steel tube of outer diameter 0.18 m, inner diameter 0.168 m, and length 1.9 m with a viscous damping coefficient of 52 N-s/m. Estimate the equivalent spring constant and equivalent damping coefficient of the excavator, assuming that the base AC is fixed.
- Question : 47P - A heat exchanger consists of six identical stainless steel tubes connected in parallel as shown in Fig. 1.94. If each tube has an outer diameter 0.00825 m, inner diameter 0.008 m, and length 1.3 m, determine the axial stiffness and the torsional stiffness about the longitudinal axis of the heat exchanger.
- Question : 48P - Two sector gears, located at the ends of links 1 and 2, are engaged together and rotate about O1 and O2, as shown in Fig. 1.95. If links 1 and 2 are connected to springs k1 to k4 and kt1 and kt2 as shown, find the equivalent torsional spring stiffness and equivalent mass moment of inertia of the system with reference to u1. Assume (a) the mass moment of inertia of link 1 (including the sector gear) about O1 is J1 and that of link 2 (including the sector gear) about O2 is J2, and (b) the angles u1 and u2 are small.
- Question : 49P - In Fig. 1.96 find the equivalent mass of the rocker arm assembly with respect to the x coordinate.
- Question : 50P - Find the equivalent mass moment of inertia of the gear train shown in Fig. 1.97 with reference to the driving shaft. In Fig. 1.97, Ji and ni denote the mass moment of inertia and the number of teeth, respectively, of gear i, i = 1, 2, c, 2N.
- Question : 51P - Two masses, having mass moments of inertia J1 and J2, are placed on rotating rigid shafts that are connected by gears, as shown in Fig. 1.98. If the numbers of teeth on gears 1 and 2 are n1 and n2, respectively, find the equivalent mass moment of inertia corresponding to u1.
- Question : 52P - A simplified model of a petroleum pump is shown in Fig. 1.99, where the rotary motion of the crank is converted to the reciprocating motion of the piston. Find the equivalent mass, meq, of the system at location A.
- Question : 53P - Find the equivalent mass of the system shown in Fig. 1.100.
- Question : 54P - Figure 1.101 shows an offset slider-crank mechanism with a crank length r, connecting rod length l, and offset d. If the crank has a mass and mass moment of inertia of mr and Jr, respectively, at its center of mass A, the connecting rod has a mass and mass moment of inertia of m c and Jc, respectively, at its center of mass C, and the piston has a mass mp, determine the equivalent rotational inertia of the system about the center of rotation of the crank, point O
- Question : 55P - Find a single equivalent damping constant for the following cases: a. When three dampers are parallel. b. When three dampers are in series. c. When three dampers are connected to a rigid bar (Fig. 1.102) and the equivalent damper is at site c1.
- Question : 56P - Consider a system of two dampers, with damping constants c1 and c2, arranged in parallel as shown in Fig. 1.104. The rigid bar to which the two dampers are connected remains horizontal when the force F is zero. Determine the equivalent damping constant of the system 1ce2 that relates the force applied (F) to the resulting velocity (v) as F = cev. Hint: Because the damping constants of the two dampers are different and the distances l1 and l2 are not the same, the rigid bar will not remain horizontal when the force F is applied.
- Question : 57P - Design a piston-cylinder-type viscous damper to achieve a damping constant of 175 N-s/m using a fluid of viscosity 35 * 10-3 N@s/m2.
- Question : 58P - Design a shock absorber (piston-cylinder-type dashpot) to obtain a damping constant of 1.8 * 107 N-s/m using SAE 30 oil at 21
- Question : 59P - Develop an expression for the damping constant of the rotational damper shown in Fig. 1.105 in terms of D, d, l, h, v, and m, where v denotes the constant angular velocity of the inner cylinder, and d and h represent the radial and axial clearances between the inner and outer cylinders.
- Question : 60P - Consider two nonlinear dampers with the same force-velocity relationship given by F = 1000v + 400v2 + 20v3 with F in newton and v in meters/second. Find the linearized damping constant of the dampers at an operating velocity of 10 m/s.
- Question : 61P - If the linearized dampers of Problem 1.60 are connected in parallel, determine the resulting equivalent damping constant.
- Question : 62P - If the linearized dampers of Problem 1.60 are connected in series, determine the resulting equivalent damping constant.
- Question : 63P - The force-velocity relationship of a nonlinear damper is given by F = 500v + 100v2 + 50v3, where F is in newton and v is in meters/second. Find the linearized damping constant of the damper at an operating velocity of 5 m/s. If the resulting linearized damping constant is used at an operating velocity of 10 m/s, determine the error involved.
- Question : 64P - The experimental determination of damping force corresponding to several values of the velocity of the damper yielded the following results:
- Question : 65P - A flat plate with a surface area of 0.25 m2 moves above a parallel flat surface with a lubricant film of thickness 1.5 mm in between the two parallel surfaces. If the viscosity of the lubricant is 0.5 Pa-s, determine the following: a. Damping constant. b. Damping force developed when the plate moves with a velocity of 2 m/s.
- Question : 66P - Find the torsional damping constant of a journal bearing for the following data: Viscosity of the lubricant 1m2: 0.35 Pa-s, Diameter of the journal or shaft (2 R): 0.05 m, Length of the bearing (l): 0.075 m, Bearing clearance (d): 0.005 m. If the journal rotates at a speed (N) of 3000 rpm, determine the damping torque developed.
- Question : 67P - If each of the parameters (m, R, l, d, and N) of the journal bearing described in Problem 1.66 is subjected to a {5% variation about the corresponding value given, determine the percentage fluctuation in the values of the torsional damping constant and the damping torque developed. Note: The variations in the parameters may have several causes, such as measurement error, manufacturing tolerances on dimensions, and fluctuations in the operating temperature of the bearing.
- Question : 68P - Consider a piston with an orifice in a cylinder filled with a fluid of viscosity m as shown in Fig. 1.106. As the piston moves in the cylinder, the fluid flows through the orifice, giving rise to a friction or damping force. Derive an expression for the force needed to move the piston with a velocity v and indicate the type of damping involved. Hint: The mass flow rate of the fluid (q) passing through an orifice is given by q = a 2?p, where a is a constant for a given fluid, area of cross section of the cylinder (or area of piston), and area of the orifice [1.52].
- Question : 69P - The force (F)-velocity 1x # 2 relationship of a nonlinear damper is given by F = ax # + bx # 2 where a and b are constants. Find the equivalent linear damping constant when the relative velocity is 5 m/s with a = 5 N-s/m and b = 0.2 N@s2>m2.
- Question : 70P - The damping constant (c) due to skin-friction drag of a rectangular plate moving in a fluid of viscosity m is given by (see Fig. 1.107): c = 100ml2d Design a plate-type damper (shown in Fig. 1.42) that provides an identical damping constant for the same fluid.
- Question : 71P - The damping constant (c) of the dashpot shown in Fig. 1.108 is given by [1.27]: c = 6pml h3 J
- Question : 72P - In Problem 1.71, using the given data as reference, find the variation of the damping constant c when a. r is varied from 0.5 cm to 1.0 cm. b. h is varied from 0.05 cm to 0.10 cm. c. a is varied from 2 cm to 4 cm.
- Question : 73P - A massless bar of length 1 m is pivoted at one end and subjected to a force F at the other end. Two translational dampers, with damping constants c1 = 10 N@s>m and c2 = 15 N@s>m are connected to the bar as shown in Fig. 1.109. Determine the equivalent damping constant, ceq, of the system so that the force F at point A can be expressed as F = ceqv, where v is the linear velocity of point A.
- Question : 74P - Find an expression for the equivalent translational damping constant of the system shown in Fig. 1.110 so that the force F can be expressed as F = ceqv, where v is the velocity of the rigid bar A.
- Question : 75P - Express the complex number 5 + 2i in the exponential form Aeiu.
- Question : 76P - Add the two complex numbers 11 + 2i2 and 13 - 4i2 and express the result in the form Aeiu
- Question : 77P - Subtract the complex number 11 + 2i2 from 13 - 4i2 and express the result in the form Aeiu.
- Question : 78P - Find the product of the complex numbers z1 = 11 + 2i2 and z2 = 13 - 4i2 and express the result in the form Aeiu.
- Question : 79P - Find the quotient, z1>z2, of the complex numbers z1 = 11 + 2i2 and z2 = 13 - 4i2 and express the result in the form Aeiu.
- Question : 80P - The foundation of a reciprocating engine is subjected to harmonic motions in x and y directions: x1t2 = X cos vt y1t2 = Y cos1vt + f2 where X and Y are the amplitudes, v is the angular velocity, and f is the phase difference. a. Verify that the resultant of the two motions satisfies the equation of the ellipse given by (see Fig. 1.111):
- Question : 81P - The foundation of an air compressor is subjected to harmonic motions (with the same frequency) in two perpendicular directions. The resultant motion, displayed on an oscilloscope, appears as shown in Fig. 1.112. Find the amplitudes of vibration in the two directions and the phase difference between them.
- Question : 82P - A machine is subjected to the motion x1t2 = A cos150t + a2 mm. The initial conditions are given by x102 = 3 mm and x # 102 = 1.0 m>s. a. Find the constants A and a. b. Express the motion in the form x1t2 = A1 cos vt + A2 sin vt, and identify the constants A1 and A2.
- Question : 83P - Show that any linear combination of sin vt and cos vt such that x1t2 = A1 cos vt + A2 sin vt 1A1, A2 = constants2 represents a simple harmonic motion.
- Question : 84P - Find the sum of the two harmonic motions x11t2 = 5 cos13t + 12 and x21t2 = 10 cos13t + 22. Use: a. Trigonometric relations b. Vector addition c. Complex-number representation
- Question : 85P - If one of the components of the harmonic motion x1t2 = 10 sin1vt + 60
- Question : 86P - Consider the two harmonic motions x11t2 = 1 2 cos p 2 t and x21t2 = sin pt. Is the sum x11t2 + x21t2 a periodic motion? If so, what is its period?
- Question : 87P - Consider two harmonic motions of different frequencies: x11t2 = 2 cos 2t and x21t2 = cos 3t. Is the sum x11t2 + x21t2 a harmonic motion? If so, what is its period?
- Question : 88P - Consider the two harmonic motions x11t2 = 1 2 cos p 2 t and x21t2 = cos pt. Is the difference x1t2 = x11t2 - x21t2 a harmonic motion? If so, what is its period?
- Question : 89P - Find the maximum and minimum amplitudes of the combined motion x1t2 = x11t2 + x21t2 when x11t2 = 3 sin 30t and x21t2 = 3 sin 29t. Also find the frequency of beats corresponding to x(t).
- Question : 90P - A machine is subjected to two harmonic motions, and the resultant motion, as displayed by an oscilloscope, is shown in Fig. 1.113. Find the amplitudes and frequencies of the two motions.
- Question : 91P - A harmonic motion has an amplitude of 0.05 m and a frequency of 10 Hz. Find its period, maximum velocity, and maximum acceleration.
- Question : 92P - An accelerometer mounted on a building frame indicates that the frame is vibrating harmonically at 15 cps, with a maximum acceleration of 0.5g. Determine the amplitude and the maximum velocity of the building frame.
- Question : 93P - The maximum amplitude and the maximum acceleration of the foundation of a centrifugal pump were found to be xmax = 0.25 mm and x $max = 0.4g, respectively. Find the operating speed of the pump.
- Question : 94P - An exponential function is expressed as x1t2 = Ae-at with the values of x1t2 known at t = 1 and t = 2 as x112 = 0.752985 and x122 = 0.226795, respectively. Determine the values of A and a.
- Question : 95P - When the displacement of a machine is given by x1t2 = 18 cos 8t, where x is measured in millimeters and t in seconds, find (a) the period of the machine in s, and (b) the frequency of oscillation of the machine in rad/s as well as in Hz.
- Question : 96P - If the motion of a machine is described as 8 sin15t + 12 = A sin 5t + B cos 5t, determine the values of A and B.
- Question : 97P - Express the vibration of a machine given by x1t2 = -3.0 sin 5t - 2.0 cos 5t in the form x1t2 = A cos15t + f2.
- Question : 98P - If the displacement of a machine is given by x1t2 = 0.2 sin15t + 32, where x is in meters and t is in seconds, find the variations of the velocity and acceleration of the machine. Also find the amplitudes of displacement, velocity, and acceleration of the machine.
- Question : 99P - If the displacement of a machine is described as x1t2 = 0.4 sin 4t + 5.0 cos 4t, where x is in centimetres and t is in seconds, find the expressions for the velocity and acceleration of the machine. Also find the amplitudes of displacement, velocity, and acceleration of the machine.
- Question : 100P - The displacement of a machine is expressed as x1t2 = 0.05 sin16t + f2, where x is in meters and t is in seconds. If the displacement of the machine at t = 0 is known to be 0.04 m, determine the value of the phase angle f.
- Question : 101P - The displacement of a machine is expressed as x1t2 = A sin16t + f2, where x is in meters and t is in seconds. If the displacement and the velocity of the machine at t = 0 are known to be 0.05 m and 0.005 m/s, respectively, determine the values of A and f.
- Question : 102P - A machine is found to vibrate with simple harmonic motion at a frequency of 20 Hz and an amplitude of acceleration of 0.5g. Determine the displacement and velocity of the machine. Use the value of g as 9.81 m>s2.
- Question : 103P - The amplitudes of displacement and acceleration of an unbalanced turbine rotor are found to be 0.5 mm and 0.5g, respectively. Find the rotational speed of the rotor using the value of g as 9.81 m>s2.
- Question : 104P - The root mean square (rms) value of a function, x(t), is defined as the square root of the average of the squared value of x(t) over a time period t: x rms = A 1 t L 0t[x1t2]2 dt Using this definition, find the rms value of the function
- Question : 105P - Using the definition given in Problem 1.104, find the rms value of the function shown in Fig. 1.54(a).
- Question : 106P - Prove that the sine Fourier components 1bn2 are zero for even functions
- Question : 107P - Find the Fourier series expansions of the functions shown in Figs. 1.58(ii) and (iii). Also, find their Fourier series expansions when the time axis is shifted down by a distance A.
- Question : 108P - The impact force created by a forging hammer can be modeled as shown in Fig. 1.114. Determine the Fourier series expansion of the impact force
- Question : 109P - Find the Fourier series expansion of the periodic function shown in Fig. 1.115. Also plot the corresponding frequency spectrum.
- Question : 110P - Find the Fourier series expansion of the periodic function shown in Fig. 1.116. Also plot the corresponding frequency spectrum.
- Question : 111P - Find the Fourier series expansion of the periodic function shown in Fig. 1.117. Also plot the corresponding frequency spectrum.
- Question : 112P - The Fourier series of a periodic function, x(t), is an infinite series given by
- Question : 113P - Conduct a harmonic analysis, including the first three harmonics, of the function given below:
- Question : 114P - In a centrifugal fan (Fig. 1.118(a)), the air at any point is subjected to an impulse each time a blade passes the point, as shown in Fig. 1.118(b). The frequency of these impulses is determined by the speed of rotation of the impeller n and the number of blades, N, in the impeller. For n = 100 rpm and N = 4, determine the first three harmonics of the pressure fluctuation shown in Fig. 1.118(b).
- Question : 115P - Solve Problem 1.114 by using the values of n and N as 200 rpm and 6 instead of 100 rpm and 4, respectively
- Question : 116P - The torque 1Mt2 variation with time, of an internal combustion engine, is given in Table 1.3. Make a harmonic analysis of the torque. Find the amplitudes of the first three harmonics.
- Question : 117P - Make a harmonic analysis of the function shown in Fig. 1.119 including the first three harmonics.
- Question : 118P - Plot the Fourier series expansion of the function x(t) given in Problem 1.113 using MATLAB.
- Question : 119P - Use MATLAB to plot the variation of the force with time using the Fourier series expansion determined in Problem 1.117.
- Question : 120P - Use MATLAB to plot the variations of the damping constant c with respect to r, h, and a as determined in Problem 1.72.
- Question : 121P - Use MATLAB to plot the variation of spring stiffness (k) with deformation (x) given by the relations: a. k = 1000x - 100x2; 0
- Question : 122P - A mass is subjected to two harmonic motions given by x11t2 = 3 sin 30t and x21t2 = 3 sin 29t. Plot the resultant motion of the mass using MATLAB and identify the beat frequency and the beat period.

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