Page 1 Vibrations of Structures Module I: Vibrations of Strings and Bars Lesson 11: Forced Vibration Analysis Contents: 1. Introduction 2. Harmonic Forcing 3. General Forcing Keywords: Forced vibrations, Boundary value problem, Eigenfunction ex- pansion, Green's function, Traveling force Page 2 Vibrations of Structures Module I: Vibrations of Strings and Bars Lesson 11: Forced Vibration Analysis Contents: 1. Introduction 2. Harmonic Forcing 3. General Forcing Keywords: Forced vibrations, Boundary value problem, Eigenfunction ex- pansion, Green's function, Traveling force Forced Vibration Analysis 1 Introduction The dynamics of one-dimensional continuous systems discussed above, subjected to an arbitrary distributed forcing q(x; t), can be represented in a general form (x)w ;tt +K[w] = q(x; t): (1) In the following, we will discuss the solution of (1) under harmonic and gen- eral forcing. 2 Harmonic Forcing Consider a forcing q(x; t) in (1) that is separable in space and time, and has a harmonic time function such that (x)w ;tt +K[w] =R[Q(x)e i t ]; (2) where is the circular forcing frequency, Q(x) species the force distribution, andR[] represents the real part. Let us consider a solution of (2) in the form w(x; t) = w H (x; t) + w P (x; t) = 1 X k=1 [C k cos ! k t + S k sin ! k t]W k (x) +R[X(x)e i t ]; (3) where w H (x; t) represents the homogeneous solution, and w P (x; t) is the par- ticular solution. The amplitude function X(x) in (3) is an unknown (real or 2 Page 3 Vibrations of Structures Module I: Vibrations of Strings and Bars Lesson 11: Forced Vibration Analysis Contents: 1. Introduction 2. Harmonic Forcing 3. General Forcing Keywords: Forced vibrations, Boundary value problem, Eigenfunction ex- pansion, Green's function, Traveling force Forced Vibration Analysis 1 Introduction The dynamics of one-dimensional continuous systems discussed above, subjected to an arbitrary distributed forcing q(x; t), can be represented in a general form (x)w ;tt +K[w] = q(x; t): (1) In the following, we will discuss the solution of (1) under harmonic and gen- eral forcing. 2 Harmonic Forcing Consider a forcing q(x; t) in (1) that is separable in space and time, and has a harmonic time function such that (x)w ;tt +K[w] =R[Q(x)e i t ]; (2) where is the circular forcing frequency, Q(x) species the force distribution, andR[] represents the real part. Let us consider a solution of (2) in the form w(x; t) = w H (x; t) + w P (x; t) = 1 X k=1 [C k cos ! k t + S k sin ! k t]W k (x) +R[X(x)e i t ]; (3) where w H (x; t) represents the homogeneous solution, and w P (x; t) is the par- ticular solution. The amplitude function X(x) in (3) is an unknown (real or 2 complex), yet to be determined. Now, substituting the solution (3) in (2) yields on simplication 2 (x)X(x) +K[X(x)] = Q(x): (4) This equation along with the boundary conditions constitute a boundary value problem. In the following, we discuss two methods of solving (4), namely the eigenfunction expansion method, and Green's function method. Eigenfunction Expansion Method: Assume the solution of (4) as the eigenfunction expansion X(x) = 1 X k=1 k W k (x); (5) where k are unknown coecients, and W k (x) satises ! 2 k (x)W k (x) +K[W k (x)] = 0; k = 1; 2; : : :;1: (6) Substituting (5) in (4) yields 2 (x) 1 X k=1 k W k (x) +K " 1 X k=1 k W k (x) # = Q(x) ) 2 (x) 1 X k=1 k W k (x) + 1 X k=1 k K[W k (x)] = Q(x) ) 1 X k=1 (! 2 k 2 ) k (x)W k (x) = Q(x); (using (6)): (7) 3 Page 4 Vibrations of Structures Module I: Vibrations of Strings and Bars Lesson 11: Forced Vibration Analysis Contents: 1. Introduction 2. Harmonic Forcing 3. General Forcing Keywords: Forced vibrations, Boundary value problem, Eigenfunction ex- pansion, Green's function, Traveling force Forced Vibration Analysis 1 Introduction The dynamics of one-dimensional continuous systems discussed above, subjected to an arbitrary distributed forcing q(x; t), can be represented in a general form (x)w ;tt +K[w] = q(x; t): (1) In the following, we will discuss the solution of (1) under harmonic and gen- eral forcing. 2 Harmonic Forcing Consider a forcing q(x; t) in (1) that is separable in space and time, and has a harmonic time function such that (x)w ;tt +K[w] =R[Q(x)e i t ]; (2) where is the circular forcing frequency, Q(x) species the force distribution, andR[] represents the real part. Let us consider a solution of (2) in the form w(x; t) = w H (x; t) + w P (x; t) = 1 X k=1 [C k cos ! k t + S k sin ! k t]W k (x) +R[X(x)e i t ]; (3) where w H (x; t) represents the homogeneous solution, and w P (x; t) is the par- ticular solution. The amplitude function X(x) in (3) is an unknown (real or 2 complex), yet to be determined. Now, substituting the solution (3) in (2) yields on simplication 2 (x)X(x) +K[X(x)] = Q(x): (4) This equation along with the boundary conditions constitute a boundary value problem. In the following, we discuss two methods of solving (4), namely the eigenfunction expansion method, and Green's function method. Eigenfunction Expansion Method: Assume the solution of (4) as the eigenfunction expansion X(x) = 1 X k=1 k W k (x); (5) where k are unknown coecients, and W k (x) satises ! 2 k (x)W k (x) +K[W k (x)] = 0; k = 1; 2; : : :;1: (6) Substituting (5) in (4) yields 2 (x) 1 X k=1 k W k (x) +K " 1 X k=1 k W k (x) # = Q(x) ) 2 (x) 1 X k=1 k W k (x) + 1 X k=1 k K[W k (x)] = Q(x) ) 1 X k=1 (! 2 k 2 ) k (x)W k (x) = Q(x); (using (6)): (7) 3 Taking the inner product on both sides of (7) with W j (x), j = 1; 2; : : :;1, and using the orthogonality property, we get (! 2 j 2 ) j h(x)W j (x); W j (x)i =hQ(x); W j (x)i; j = 1; 2; : : :;1 ) j = R l 0 Q(x)W j (x) dx (! 2 j 2 ) R l 0 (x)W 2 j (x) dx ; j = 1; 2; : : :;1; (8) where it has been assumed that the forcing is non-resonant, i.e., 6= ! j for all j. This completes the solution (5) of (4) for a non-resonant harmonic forcing. In case = ! j for some j, we have resonance, which is characterized by a very high response amplitude for the j th mode (innite as far as the linear theory is concerned). To determine the response of the system at resonance, we use the method of variation of parameters in which the particular solution is assumed in the form w P (x; t) =R 2 6 4 0 B @ j (t)W j (x) + 1 X k=1 k 6=j k W k (x) 1 C A e i! j t 3 7 5 : (9) It may be noted that the j th modal coordinate j (t) has been taken as a function of time. Substituting this solution form in (2) and proceeding as discussed above, one can easily obtain the equation of modal dynamics of the j th mode as j + 2i! j _ j = R l 0 Q(x)W j (x) dx R l 0 (x)W 2 j (x) dx : 4 Page 5 Vibrations of Structures Module I: Vibrations of Strings and Bars Lesson 11: Forced Vibration Analysis Contents: 1. Introduction 2. Harmonic Forcing 3. General Forcing Keywords: Forced vibrations, Boundary value problem, Eigenfunction ex- pansion, Green's function, Traveling force Forced Vibration Analysis 1 Introduction The dynamics of one-dimensional continuous systems discussed above, subjected to an arbitrary distributed forcing q(x; t), can be represented in a general form (x)w ;tt +K[w] = q(x; t): (1) In the following, we will discuss the solution of (1) under harmonic and gen- eral forcing. 2 Harmonic Forcing Consider a forcing q(x; t) in (1) that is separable in space and time, and has a harmonic time function such that (x)w ;tt +K[w] =R[Q(x)e i t ]; (2) where is the circular forcing frequency, Q(x) species the force distribution, andR[] represents the real part. Let us consider a solution of (2) in the form w(x; t) = w H (x; t) + w P (x; t) = 1 X k=1 [C k cos ! k t + S k sin ! k t]W k (x) +R[X(x)e i t ]; (3) where w H (x; t) represents the homogeneous solution, and w P (x; t) is the par- ticular solution. The amplitude function X(x) in (3) is an unknown (real or 2 complex), yet to be determined. Now, substituting the solution (3) in (2) yields on simplication 2 (x)X(x) +K[X(x)] = Q(x): (4) This equation along with the boundary conditions constitute a boundary value problem. In the following, we discuss two methods of solving (4), namely the eigenfunction expansion method, and Green's function method. Eigenfunction Expansion Method: Assume the solution of (4) as the eigenfunction expansion X(x) = 1 X k=1 k W k (x); (5) where k are unknown coecients, and W k (x) satises ! 2 k (x)W k (x) +K[W k (x)] = 0; k = 1; 2; : : :;1: (6) Substituting (5) in (4) yields 2 (x) 1 X k=1 k W k (x) +K " 1 X k=1 k W k (x) # = Q(x) ) 2 (x) 1 X k=1 k W k (x) + 1 X k=1 k K[W k (x)] = Q(x) ) 1 X k=1 (! 2 k 2 ) k (x)W k (x) = Q(x); (using (6)): (7) 3 Taking the inner product on both sides of (7) with W j (x), j = 1; 2; : : :;1, and using the orthogonality property, we get (! 2 j 2 ) j h(x)W j (x); W j (x)i =hQ(x); W j (x)i; j = 1; 2; : : :;1 ) j = R l 0 Q(x)W j (x) dx (! 2 j 2 ) R l 0 (x)W 2 j (x) dx ; j = 1; 2; : : :;1; (8) where it has been assumed that the forcing is non-resonant, i.e., 6= ! j for all j. This completes the solution (5) of (4) for a non-resonant harmonic forcing. In case = ! j for some j, we have resonance, which is characterized by a very high response amplitude for the j th mode (innite as far as the linear theory is concerned). To determine the response of the system at resonance, we use the method of variation of parameters in which the particular solution is assumed in the form w P (x; t) =R 2 6 4 0 B @ j (t)W j (x) + 1 X k=1 k 6=j k W k (x) 1 C A e i! j t 3 7 5 : (9) It may be noted that the j th modal coordinate j (t) has been taken as a function of time. Substituting this solution form in (2) and proceeding as discussed above, one can easily obtain the equation of modal dynamics of the j th mode as j + 2i! j _ j = R l 0 Q(x)W j (x) dx R l 0 (x)W 2 j (x) dx : 4 Solving this and substituting in (9), the particular solution is nally obtained as w P (x; t) = t 2! j R l 0 Q(x)W j (x) dx R l 0 (x)W 2 j (x) dx W j (x) sin! j t + 1 X k=1 k 6=j k W k (x) cos! j t; where the constants k are obtained from (8). For = ! j if Q(x) is such that Z l 0 Q(x)W j (x) dx = 0; (i.e., Q(x) is orthogonal to W j (x)), the solution is still nite since the j th mode cannot be excited by the force. This situation is referred to as apparent- resonance. Green's Function Method: Let G(x; ; ) be the solution of (4) excited by a concentrated unit force at x = x2 [0; l], i.e., 2 (x)G(x; x; ) +K[G(x; x; )] = (x x); (10) with all the boundary conditions of (4), which are assumed homogeneous. Consider the function X(x) = Z l 0 Q( x)G(x; x; ) d x: (11) 5Read More

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